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  "year": 1899,
  "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
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   "url": "https://www.gutenberg.org/ebooks/39041",
   "released": "March 3, 2012",
   "licence_terms": "This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever.  You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.net"
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   "exercise_sets": []
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  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "number": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "title": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "name": "De Morgan 1899, On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
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   "number": "The Drawing of a Tangent to a Curve",
   "title": "The Drawing of a Tangent to a Curve",
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   "number": "Rational Explanation of the Language of Leibnitz",
   "title": "Rational Explanation of the Language of Leibnitz",
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   "number": "Orders of Infinity",
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   "number": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "title": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
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   "id": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "number": "The Same Problem Solved by the Principles of Leibnitz",
   "title": "The Same Problem Solved by the Principles of Leibnitz",
   "name": "De Morgan 1899, The Same Problem Solved by the Principles of Leibnitz",
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   "number": "An Illustration from Dynamics: Velocity, Acceleration, etc",
   "title": "An Illustration from Dynamics: Velocity, Acceleration, etc",
   "name": "De Morgan 1899, An Illustration from Dynamics: Velocity, Acceleration, etc",
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   "number": "The Method of Fluxions",
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   "title": "Accelerated Motion",
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   "number": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "title": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "name": "De Morgan 1899, Limiting Ratios of Magnitudes that Increase Without Limit",
   "pages": [
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   "title": "Recapitulation of Results Reached in the Theory of Functions",
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   "number": "Differential Coefficients of Differential Coefficients",
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   "name": "De Morgan 1899, Differential Coefficients of Differential Coefficients",
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   "number": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "title": "Total and Partial Differential Coefficients. Implicit Differentiation",
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   "number": "Applications of the Theorem for Implicit Differentiation",
   "title": "Applications of the Theorem for Implicit Differentiation",
   "name": "De Morgan 1899, Applications of the Theorem for Implicit Differentiation",
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   "title": "Inverse Functions",
   "name": "De Morgan 1899, Inverse Functions",
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    "106"
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   "name": "De Morgan 1899, Determination of Curvilinear Areas. The Parabola",
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   "number": "Concluding Remarks on the Study of the Calculus",
   "title": "Concluding Remarks on the Study of the Calculus",
   "name": "De Morgan 1899, Concluding Remarks on the Study of the Calculus",
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  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-d46087aa6d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-and-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "1",
   "location": "Differential and Integral Calculus",
   "latex": "Differential and Integral Calculus, or, as it was formerly called in this country [England], the Doctrine of Fluxions, has always been supposed to present remarkable obstacles to the beginner.",
   "markdown": "Differential and Integral Calculus, or, as it was formerly called in this country [England], the Doctrine of Fluxions, has always been supposed to present remarkable obstacles to the beginner.",
   "why": "It names the subject and its older English name, and it frames the calculus as a subject with a reputation for difficulty.",
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   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-and-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "1",
   "location": "Differential and Integral Calculus",
   "latex": "\\First{The} Differential and Integral Calculus, or, as it was formerly called in this country [England], the Doctrine of Fluxions, has always been supposed to present remarkable obstacles to the beginner.",
   "markdown": "The Differential and Integral Calculus, or, as it was formerly called in this country [England], the Doctrine of Fluxions, has always been supposed to present remarkable obstacles to the beginner.",
   "why": "It names the subject, gives its older English name, and admits up front that beginners find it hard.",
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   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "1",
   "location": "Differential and Integral Calculus",
   "latex": "It is matter of common observation, that any one who commences this study, even with the best elementary works, finds himself in the dark as to the real meaning of the processes which he learns, until, at a certain stage of his progress, depending upon his capacity, some accidental combination of his own ideas throws light upon the subject.",
   "markdown": "It is matter of common observation, that any one who commences this study, even with the best elementary works, finds himself in the dark as to the real meaning of the processes which he learns, until, at a certain stage of his progress, depending upon his capacity, some accidental combination of his own ideas throws light upon the subject.",
   "why": "It tells a struggling learner that confusion at the start is common and that understanding often arrives suddenly.",
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  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-3f72ea146b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-and-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "1",
   "location": "Differential and Integral Calculus",
   "latex": "The reason of this may be, that it is usual to introduce him at the same time to new principles, processes, and symbols, thus preventing his attention from being exclusively directed to one new thing at a time.",
   "markdown": "The reason of this may be, that it is usual to introduce him at the same time to new principles, processes, and symbols, thus preventing his attention from being exclusively directed to one new thing at a time.",
   "why": "It gives a teaching diagnosis: too many new things at once block understanding.",
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  {
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   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-and-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "2",
   "location": "Differential and Integral Calculus",
   "latex": "We would not, nevertheless, that the student should imagine we can remove all obstacles; we must introduce notions, the consideration of which has not hitherto occupied his mind; and shall therefore consider our object as gained, if we can succeed in so placing the subject before him, that two independent difficulties shall never occupy his mind at once.",
   "markdown": "We would not, nevertheless, that the student should imagine we can remove all obstacles; we must introduce notions, the consideration of which has not hitherto occupied his mind; and shall therefore consider our object as gained, if we can succeed in so placing the subject before him, that two independent difficulties shall never occupy his mind at once.",
   "why": "It states the book's honest aim: not to remove every difficulty, but never to present two new difficulties together.",
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  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c5eaa1c118",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "2",
   "location": "On the Ratio or Proportion of Two Magnitudes",
   "latex": "Let the given ratio be that of the numbers $m$~and~$n$. Then, $P$~being a line, $mP$~and~$nP$ are in the proportion of $m$ to~$n$; and it is evident, that let $m$,~$n$, and~$A$ be what they may, $P$~can be so taken that $mP$~shall be less than~$A$.",
   "markdown": "Let the given ratio be that of the numbers $m$ and $n$. Then, $P$ being a line, $mP$ and $nP$ are in the proportion of $m$ to $n$; and it is evident, that let $m$, $n$, and $A$ be what they may, $P$ can be so taken that $mP$ shall be less than $A$.",
   "why": "Shows how a ratio of numbers is realised by two magnitudes, and how one can always be made smaller than any given size.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/number",
    "concept/ratio",
    "quantity/magnitude"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a982f55088",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "2",
   "location": "On the Ratio or Proportion of Two Magnitudes",
   "latex": "Thus, the ratio of the diagonal of a square to its side is that of $\\sqrt{2}$ to~$1$, which is very nearly that of $14142$ to~$10000$, and is certainly between this and that of $14143$ to~$10000$.",
   "markdown": "Thus, the ratio of the diagonal of a square to its side is that of $\\sqrt{2}$ to $1$, which is very nearly that of $14142$ to $10000$, and is certainly between this and that of $14143$ to $10000$.",
   "why": "Shows concretely how an incommensurable ratio is bracketed as closely as we please by whole-number ratios.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/approximation",
    "concept/diagonal",
    "concept/incommensurable-magnitudes",
    "concept/ratio",
    "concept/root",
    "concept/square-root-of-2"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-d350e40051",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "2",
   "location": "On the Ratio or Proportion of Two Magnitudes",
   "latex": "We are not, therefore, entitled to say that because two magnitudes are diminished, their ratio is diminished; it is possible that~$B$, which we will suppose to be at first a hundredth part of~$C$, may, after a diminution of both, be its tenth or thousandth, or may still remain its hundredth, as the following example will show:",
   "markdown": "We are not, therefore, entitled to say that because two magnitudes are diminished, their ratio is diminished; it is possible that $B$, which we will suppose to be at first a hundredth part of $C$, may, after a diminution of both, be its tenth or thousandth, or may still remain its hundredth, as the following example will show:",
   "why": "Warns against the common mistake that shrinking two quantities shrinks their ratio.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/ratio",
    "quantity/magnitude"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c06a89fe72",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "3",
   "location": "On the Ratio or Proportion of Two Magnitudes",
   "latex": "In estimating the approach to, or departure from equality, which two magnitudes undergo in consequence of a change in their values, we must not look at their differences, but at the proportions which those differences bear to the whole magnitudes.",
   "markdown": "In estimating the approach to, or departure from equality, which two magnitudes undergo in consequence of a change in their values, we must not look at their differences, but at the proportions which those differences bear to the whole magnitudes.",
   "why": "States the key principle that nearness to equality is judged by ratio rather than by difference.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/difference",
    "concept/equality",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-7d99b88a66",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "3",
   "location": "On the Ratio or Proportion of Two Magnitudes",
   "latex": "For example, if a geometrical figure, two of whose sides are $3$~and $4$~inches now, be altered in dimensions, so that the corresponding sides are $100$~and $101$~inches, they are nearer to equality in the second case than in the first; because, though the difference is the same in both, namely one inch, it is one third of the least side in the first case, and only one hundredth in the second.",
   "markdown": "For example, if a geometrical figure, two of whose sides are $3$ and $4$ inches now, be altered in dimensions, so that the corresponding sides are $100$ and $101$ inches, they are nearer to equality in the second case than in the first; because, though the difference is the same in both, namely one inch, it is one third of the least side in the first case, and only one hundredth in the second.",
   "why": "A simple numerical example in which the same difference means very different closeness to equality.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/difference",
    "concept/equality",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-35f5f1eb56",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "3",
   "location": "On the Ratio or Proportion of Two Magnitudes",
   "latex": "Thus, twenty miles would be a material error in talking of a day's journey, but would not be considered worth mentioning in one of three months, and would be called totally insensible in stating the distance between the earth and sun.",
   "markdown": "Thus, twenty miles would be a material error in talking of a day’s journey, but would not be considered worth mentioning in one of three months, and would be called totally insensible in stating the distance between the earth and sun.",
   "why": "A vivid everyday analogy for why size must be judged relative to the whole.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/difference",
    "concept/error-of-measurement",
    "concept/ratio",
    "quantity/magnitude"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ccb6b238ba",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "4",
   "location": "On the Ratio or Proportion of Two Magnitudes",
   "latex": "In future, when we talk of an approach towards equality, we mean that the ratio is made more nearly equal to unity, not that the difference is more nearly equal to nothing. The second may follow from the first, but not necessarily; still less does the first follow from the second.",
   "markdown": "In future, when we talk of an approach towards equality, we mean that the ratio is made more nearly equal to unity, not that the difference is more nearly equal to nothing. The second may follow from the first, but not necessarily; still less does the first follow from the second.",
   "why": "Fixes the meaning of 'approach to equality' that later limit arguments will rely on.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/difference",
    "concept/equality",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fbded606c7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "2",
   "location": "On the Ratio or Proportion of Two Magnitudes",
   "latex": "This is only saying that $P$~can be taken less than the $m$\\th~part of~$A$, which is obvious, since~$A$, however small it may be, has its tenth, its hundredth, its thousandth part,~etc., as certainly as if it were larger.",
   "markdown": "This is only saying that $P$ can be taken less than the $m$th part of $A$, which is obvious, since $A$, however small it may be, has its tenth, its hundredth, its thousandth part, etc., as certainly as if it were larger.",
   "why": "Explains why any ratio can be had between magnitudes as small as we please.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/proportion",
    "concept/ratio",
    "quantity/magnitude"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-87df7086eb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "4",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "while the magnitudes diminish, we may not assume either that their ratio increases, diminishes, or remains the same, for we have shown that a diminution of two magnitudes is consistent with either of these.",
   "markdown": "while the magnitudes diminish, we may not assume either that their ratio increases, diminishes, or remains the same, for we have shown that a diminution of two magnitudes is consistent with either of these.",
   "why": "It warns the learner that shrinking both terms of a fraction says nothing by itself about whether the fraction grows or shrinks.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/ratio",
    "quantity/magnitude"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-84795adbf1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "5",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "Here both $M$~and~$N$ decrease at every step, but $M$~loses at each step a larger fraction of itself than~$N$, and their ratio continually diminishes.",
   "markdown": "Here both $M$ and $N$ decrease at every step, but $M$ loses at each step a larger fraction of itself than $N$, and their ratio continually diminishes.",
   "why": "It explains in plain words why the ratio of two shrinking quantities can fall steadily toward zero.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c4fcbb2886",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "7",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "This is what we mean by saying that $\\dfrac{M}{N}$~is an increasing ratio, the limit of which is~$2$.",
   "markdown": "This is what we mean by saying that $\\dfrac{M}{N}$ is an increasing ratio, the limit of which is $2$.",
   "why": "It gives the precise meaning of 'limit' as used for a ratio, so learners know what the term asserts.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/limit",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-16792ebada",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "4",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "For example, let a point~$A$ move on a circle towards a fixed point~$B$. The arc~$AB$ will then diminish, as also the chord~$AB$, and by bringing the point~$A$ sufficiently near to~$B$, we may obtain an arc and its chord, both of which shall be smaller than a given line, however small this last may be.",
   "markdown": "For example, let a point $A$ move on a circle towards a fixed point $B$. The arc $AB$ will then diminish, as also the chord $AB$, and by bringing the point $A$ sufficiently near to $B$, we may obtain an arc and its chord, both of which shall be smaller than a given line, however small this last may be.",
   "why": "A concrete picture of two magnitudes, an arc and its chord, vanishing together.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/arc-of-a-circle",
    "concept/chord",
    "concept/magnitudes-vanishing-together",
    "concept/point",
    "quantity/magnitude"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-f596991fe5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "4",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "But while the magnitudes diminish, we may not assume either that their ratio increases, diminishes, or remains the same, for we have shown that a diminution of two magnitudes is consistent with either of these.",
   "markdown": "But while the magnitudes diminish, we may not assume either that their ratio increases, diminishes, or remains the same, for we have shown that a diminution of two magnitudes is consistent with either of these.",
   "why": "Warns learners not to guess how a ratio behaves just because both terms shrink.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/magnitudes-vanishing-together",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-0e1b247626",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "5",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "The first possible case is that the ratio of $M$ to~$N$ may decrease without limit, that is, $M$~may be a smaller fraction of~$N$ after a decrease than it was before, and a still smaller after a further decrease, and so on; in such a way, that there is no fraction so small, to which $\\dfrac{M}{N}$~shall not be equal or inferior, if the decrease of $M$~and~$N$ be carried sufficiently far.",
   "markdown": "The first possible case is that the ratio of $M$ to $N$ may decrease without limit, that is, $M$ may be a smaller fraction of $N$ after a decrease than it was before, and a still smaller after a further decrease, and so on; in such a way, that there is no fraction so small, to which $\\dfrac{M}{N}$ shall not be equal or inferior, if the decrease of $M$ and $N$ be carried sufficiently far.",
   "why": "States precisely what it means for a ratio to shrink past every fraction you can name.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/ratio",
    "concept/ratio-decreasing-without-limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a48a0caa11",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "6",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "The second possible case is that in which the ratio of $M$ to~$N$, though it increases or decreases, does not increase or decrease without limit, that is, continually approaches to some ratio, which it never will exactly reach, however far the diminution of $M$ and~$N$ may be carried.",
   "markdown": "The second possible case is that in which the ratio of $M$ to $N$, though it increases or decreases, does not increase or decrease without limit, that is, continually approaches to some ratio, which it never will exactly reach, however far the diminution of $M$ and $N$ may be carried.",
   "why": "Introduces the idea of a ratio approaching a value it never reaches, which is the seed of a limit.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/approximation",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c489b9d7d2",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "6",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "The difference between this case and the last is, that the ratio of $M$ to~$N$, though perpetually increasing, does not increase without limit; it is never so great as~$2$, though it may be brought as near to~$2$ as we please.",
   "markdown": "The difference between this case and the last is, that the ratio of $M$ to $N$, though perpetually increasing, does not increase without limit; it is never so great as $2$, though it may be brought as near to $2$ as we please.",
   "why": "Sets an unbounded ratio against a ratio that grows forever but stays below a ceiling.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/limit",
    "concept/ratio-decreasing-without-limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-07fb32e5bf",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "7",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "Therefore (1)~$\\dfrac{M}{N}$~continually increases; (2)~may be brought as near to~$2$ as we please; (3)~can never be greater than~$2$. This is what we mean by saying that $\\dfrac{M}{N}$~is an increasing ratio, the limit of which is~$2$.",
   "markdown": "Therefore (1) $\\dfrac{M}{N}$ continually increases; (2) may be brought as near to $2$ as we please; (3) can never be greater than $2$. This is what we mean by saying that $\\dfrac{M}{N}$ is an increasing ratio, the limit of which is $2$.",
   "why": "Spells out the three conditions that make 2 the limit of an increasing ratio.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/approximation",
    "concept/limit",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-601d97c476",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "4",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "In introducing the notion of time, we consult only simplicity. It would do equally well to write any number of successive values of the two quantities, and place them in two columns.",
   "markdown": "In introducing the notion of time, we consult only simplicity. It would do equally well to write any number of successive values of the two quantities, and place them in two columns.",
   "why": "Shows that time in these examples is only a convenience and that a table of paired values would serve as well.",
   "use": [
    "history",
    "lesson"
   ],
   "concepts": [
    "concept/magnitudes-vanishing-together",
    "quantity/magnitude"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-99de5ad42a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "8",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "assign a line and an angle, however small, $B$~can be placed so near to~$A$ that the lines and angles above alluded to shall be severally less than the assigned line and angle.",
   "markdown": "assign a line and an angle, however small, $B$ can be placed so near to $A$ that the lines and angles above alluded to shall be severally less than the assigned line and angle.",
   "why": "It states the meaning of 'without limit' in plain terms: any assigned smallness can be reached by moving the point close enough.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-26173e591e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "9",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "To illustrate this result from the trigonometrical tables, observe that if the radius~$OA$ be the linear unit, and $\\angle BOA = \\theta$, $BM$~and~$BA$ are respectively $\\sin\\theta$ and $2\\sin\\frac{1}{2}\\theta$.",
   "markdown": "To illustrate this result from the trigonometrical tables, observe that if the radius $OA$ be the linear unit, and $\\angle BOA = \\theta$, $BM$ and $BA$ are respectively $\\sin\\theta$ and $2\\sin\\frac{1}{2}\\theta$.",
   "why": "It links the geometric argument to the sine tables with a concrete numerical check the learner can repeat.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/sine",
    "unit/degree-of-angle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ad03db25e0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "7",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "In geometry and mechanics,\nit is necessary to consider quantities as\nincreasing or decreasing \\emph{continuously}; that is, a magnitude\ndoes not pass from one value to another without\npassing through every intermediate value. Thus\nif one point move towards another on a circle, both\nthe arc and its chord decrease continuously.",
   "markdown": "In geometry and mechanics, it is necessary to consider quantities as increasing or decreasing *continuously*; that is, a magnitude does not pass from one value to another without passing through every intermediate value. Thus if one point move towards another on a circle, both the arc and its chord decrease continuously.",
   "why": "It defines continuous change in plain words and gives a concrete picture of an arc and chord shrinking together.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/arc-of-a-circle",
    "concept/chord",
    "concept/continuous-quantity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2cf19b5ddb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "8",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "Again, $OT$~diminishes\nand $OM$~increases, but neither without limit,\nfor the first is never less, nor the second greater, than\nthe radius.",
   "markdown": "Again, $OT$ diminishes and $OM$ increases, but neither without limit, for the first is never less, nor the second greater, than the radius.",
   "why": "It shows that a quantity can change continuously and still be bounded, a distinction a learner needs before meeting limits.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/decreasing-without-limit",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-3e04ffee72",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "9",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "Let $\\theta = 1°$; then $\\sin\\theta = .0174524$\nand $2\\sin\\frac{1}{2}\\theta = .0174530$; whence $2\\sin\\frac{1}{2}\\theta ÷ \\sin\\theta = 1.00003$ very nearly, so that $BM$~differs from~$BA$ by\nless than four of its own hundred-thousandth parts.",
   "markdown": "Let $\\theta = 1°$; then $\\sin\\theta = .0174524$ and $2\\sin\\frac{1}{2}\\theta = .0174530$; whence $2\\sin\\frac{1}{2}\\theta ÷ \\sin\\theta = 1.00003$ very nearly, so that $BM$ differs from $BA$ by less than four of its own hundred-thousandth parts.",
   "why": "It checks the claim that the ratio tends to 1 against trigonometrical tables, with actual numbers.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/limit",
    "concept/ratio",
    "concept/sine",
    "unit/degree-of-angle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-243749ab47",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "9",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "Thus if $\\angle BOA = 1°$,\n$BM ÷ MA = 114.589$ and $BA ÷ MA = 114.593$ very\nnearly; that is, $BM$ and $BA$ both contain~$MA$\nmore than $114$~times.",
   "markdown": "Thus if $\\angle BOA = 1°$, $BM ÷ MA = 114.589$ and $BA ÷ MA = 114.593$ very nearly; that is, $BM$ and $BA$ both contain $MA$ more than $114$ times.",
   "why": "It makes a ratio that grows without limit concrete with a computed value.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/ratio",
    "concept/tends-to-infinity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-af0e988ca6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "10",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "The arc~$BA$ always lies between $BA$ and $BN + NA$,\nor $BM + MA$; hence $\\dfrac{\\arc BA}{\\chord BA}$ lies between $1$ and\n$\\dfrac{BM}{BA} + \\dfrac{MA}{BA}$. But $\\dfrac{BM}{BA}$~has been shown to approach\ncontinually towards~$1$, and $\\dfrac{MA}{BA}$~to decrease without\nlimit; hence $\\dfrac{\\arc BA}{\\chord BA}$ continually approaches towards~$1$.",
   "markdown": "The arc $BA$ always lies between $BA$ and $BN + NA$, or $BM + MA$; hence $\\dfrac{\\arc BA}{\\chord BA}$ lies between $1$ and $\\dfrac{BM}{BA} + \\dfrac{MA}{BA}$. But $\\dfrac{BM}{BA}$ has been shown to approach continually towards $1$, and $\\dfrac{MA}{BA}$ to decrease without limit; hence $\\dfrac{\\arc BA}{\\chord BA}$ continually approaches towards $1$.",
   "why": "It is a short squeeze argument showing that the arc-to-chord ratio approaches 1.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/arc-of-a-circle",
    "concept/chord",
    "concept/limit",
    "concept/ratio",
    "quantity/arc-of-a-circle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2ba6b11533",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "11",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "Since $(a  +  h)^{2} = a^{2} + 2ah + h^{2}$, it appears that if $a$~be increased by~$h$, $a^{2}$~is increased by~$2ah + h^{2}$.",
   "markdown": "Since $(a + h)^{2} = a^{2} + 2ah + h^{2}$, it appears that if $a$ be increased by $h$, $a^{2}$ is increased by $2ah + h^{2}$.",
   "why": "It gives a worked start to the square-increment example that a learner can follow line by line.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/increment",
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b7b542f990",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "13",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "The smaller $h$~is made, the more near does this proportion diminish towards that of $2a$ to~$1$, to which it may be made to approach within any quantity, if it be allowable to take $h$ as small as we please.",
   "markdown": "The smaller $h$ is made, the more near does this proportion diminish towards that of $2a$ to $1$, to which it may be made to approach within any quantity, if it be allowable to take $h$ as small as we please.",
   "why": "It states the limit idea in place of infinitely small quantities, showing how the ratio settles on a definite value.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/limit",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a999a7b983",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "13",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "The proposition, therefore, that $h$~can be taken so small that $2ah + h^{2}$ and~$2ah$ are rigorously equal, though not true, and therefore entailing error upon all its subsequent consequences, yet is of this character, that, by taking $h$ sufficiently small, all errors may be made as small as we please.",
   "markdown": "The proposition, therefore, that $h$ can be taken so small that $2ah + h^{2}$ and $2ah$ are rigorously equal, though not true, and therefore entailing error upon all its subsequent consequences, yet is of this character, that, by taking $h$ sufficiently small, all errors may be made as small as we please.",
   "why": "It explains why an error that can be made as small as we like is still a true approximation, which is a key idea for students.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c838ba3ea6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "14",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "Again, it would be said (\\Fig{1}) that if $AB$~be infinitely small, $MA$~is infinitely less than~$BM$.",
   "markdown": "Again, it would be said (1) that if $AB$ be infinitely small, $MA$ is infinitely less than $BM$.",
   "why": "It shows the circle-and-arc picture in the author's words, letting a learner link infinitely small arcs to chords.",
   "use": [
    "history",
    "website"
   ],
   "concepts": [
    "concept/arc-of-a-circle",
    "concept/chord",
    "concept/circle",
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c07e5cd04a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "11",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "In this case he used the phrase that $n$~is \\emph{infinitely} small with respect to~$m$.",
   "markdown": "In this case he used the phrase that $n$ is *infinitely* small with respect to $m$.",
   "why": "Gives the origin of the phrase 'infinitely small' in Leibnitz's own usage.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/infinitesimal",
    "person/gottfried-wilhelm-leibniz"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-04971a73cc",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "12",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "In this reasoning there is evidently an absolute error; for it is impossible that $h$~can be so small, that $2ah + h^{2}$ and $2ah$ shall be the same.",
   "markdown": "In this reasoning there is evidently an absolute error; for it is impossible that $h$ can be so small, that $2ah + h^{2}$ and $2ah$ shall be the same.",
   "why": "Honestly flags that the infinitely small shortcut is strictly wrong, a useful caution for learners.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/infinitesimal",
    "method/neglecting-higher-order-small-quantities"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-309884c9fd",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "12",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "Nothing is either small or great in itself, these terms only implying a relation to some other magnitude of the same kind, and even then varying their meaning with the subject in talking of which the magnitude occurs, so that both terms may be applied to the same magnitude: thus a large field is a very small part of the earth.",
   "markdown": "Nothing is either small or great in itself, these terms only implying a relation to some other magnitude of the same kind, and even then varying their meaning with the subject in talking of which the magnitude occurs, so that both terms may be applied to the same magnitude: thus a large field is a very small part of the earth.",
   "why": "Shows with a vivid example why 'small' needs a comparison before it can be used mathematically.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/infinitesimal",
    "concept/standard-of-approximation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-892fd7cd5f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "13",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "Here all dispute about a standard of smallness is avoided, because, be the standard whatever it may, the proportion of~$h^{2}$ to~$h$ may be brought under it.",
   "markdown": "Here all dispute about a standard of smallness is avoided, because, be the standard whatever it may, the proportion of $h^{2}$ to $h$ may be brought under it.",
   "why": "Explains why the argument works however strictly 'small' is judged.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/proportion",
    "concept/standard-of-approximation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-e029409be3",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "13",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "The desire of combining simplicity with the appearance of rigorous demonstration, probably introduced the notion of infinitely small quantities; which was further established by observing that their careful use never led to any error.",
   "markdown": "The desire of combining simplicity with the appearance of rigorous demonstration, probably introduced the notion of infinitely small quantities; which was further established by observing that their careful use never led to any error.",
   "why": "A historical remark on why infinitely small quantities became accepted.",
   "use": [
    "history",
    "website"
   ],
   "concepts": [
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ebd6cb0f5e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "13",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "If $a$~be increased by~$h$, $a^{2}$~is increased by $2ah + h^{2}$, which, whatever may be the value of~$h$, is to~$h$ in the proportion of $2a + h$ to~$1$. The smaller $h$~is made, the more near does this proportion diminish towards that of $2a$ to~$1$, to which it may be made to approach within any quantity, if it be allowable to take $h$ as small as we please. Hence the ratio, $\\emph{increment of } a^{2} ÷ \\emph{increment of } a$, is a decreasing ratio, whose limit is~$2a$.",
   "markdown": "If $a$ be increased by $h$, $a^{2}$ is increased by $2ah + h^{2}$, which, whatever may be the value of $h$, is to $h$ in the proportion of $2a + h$ to $1$. The smaller $h$ is made, the more near does this proportion diminish towards that of $2a$ to $1$, to which it may be made to approach within any quantity, if it be allowable to take $h$ as small as we please. Hence the ratio, $\\emph{increment of } a^{2} ÷ \\emph{increment of } a$, is a decreasing ratio, whose limit is $2a$.",
   "why": "Shows the rigorous, limit-based replacement for the infinitely small argument, worked on a^2.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/increment",
    "concept/limit",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-738869f2e0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "14",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "This should be considered as an abbreviation of the proposition proved (\\PageRef{10}), and of the following: If a polygon be inscribed in a circle, the greater the number of its sides, and the smaller their lengths, the more nearly will the perimeters of the polygon and circle be equal to one another; and further, if any straight line be given, however small, the difference between the perimeters of the polygon and circle may be made less than that line, by sufficient increase of the number of sides and diminution of their lengths.",
   "markdown": "This should be considered as an abbreviation of the proposition proved (10), and of the following: If a polygon be inscribed in a circle, the greater the number of its sides, and the smaller their lengths, the more nearly will the perimeters of the polygon and circle be equal to one another; and further, if any straight line be given, however small, the difference between the perimeters of the polygon and circle may be made less than that line, by sufficient increase of the number of sides and diminution of their lengths.",
   "why": "Translates the loose claim that a circle is a polygon of infinitely many sides into a precise statement.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/infinitesimal",
    "concept/perimeter",
    "concept/polygon"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-0ef727e97c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "14",
   "location": "On Functions",
   "latex": "Such are $x^{2} + a^{2}$, $\\dfrac{a + x}{a - x}$, $\\log(x + y)$, $\\sin 2x$.",
   "markdown": "Such are $x^{2} + a^{2}$, $\\dfrac{a + x}{a - x}$, $\\log(x + y)$, $\\sin 2x$.",
   "why": "Four concrete examples let a learner see that a function can be a sum, a fraction, a logarithm or a trigonometric expression.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/algebraic-expression",
    "concept/function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5383d86947",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "14",
   "location": "On Functions",
   "latex": "Thus if in $x^{2} + a^{2}$ $x$~only is considered as changing its value, this is called a function of~$x$; if $x$~and~$a$ both change, it is called a function of $x$~and~$a$.",
   "markdown": "Thus if in $x^{2} + a^{2}$ $x$ only is considered as changing its value, this is called a function of $x$; if $x$ and $a$ both change, it is called a function of $x$ and $a$.",
   "why": "It shows that what counts as a function depends on which quantities are allowed to change, which is the key point for functions of several variables.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/function",
    "concept/function-of-several-variables",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-65b25baf5b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "15",
   "location": "On Functions",
   "latex": "Here it must be borne in mind that $\\phi$~and~$\\psi$ do not represent numbers which multiply~$x$, but are \\emph{the abbreviated directions to perform certain operations with $x$ and constant quantities}.",
   "markdown": "Here it must be borne in mind that $\\phi$ and $\\psi$ do not represent numbers which multiply $x$, but are *the abbreviated directions to perform certain operations with $x$ and constant quantities*.",
   "why": "Warns against the common mistake of reading phi(x) as phi times x.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/function",
    "concept/function-notation",
    "concept/operation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-9c37135fd4",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "15",
   "location": "On Functions",
   "latex": "Thus, if $\\phi x = x + x^{2}$, $\\phi$~is equivalent to a direction to add~$x$ to its square, and the whole~$\\phi x$ stands for the result of this operation. Thus, in this case, $\\phi(1) = 2$; $\\phi(2) = 6$; $\\phi a = a +  a^{2}$; $\\phi(x +  h) = x + h + (x +  h)^{2}$; $\\phi \\sin x = \\sin x + (\\sin x)^{2}$.",
   "markdown": "Thus, if $\\phi x = x + x^{2}$, $\\phi$ is equivalent to a direction to add $x$ to its square, and the whole $\\phi x$ stands for the result of this operation. Thus, in this case, $\\phi(1) = 2$; $\\phi(2) = 6$; $\\phi a = a + a^{2}$; $\\phi(x + h) = x + h + (x + h)^{2}$; $\\phi \\sin x = \\sin x + (\\sin x)^{2}$.",
   "why": "A worked example showing how one rule phi is applied to numbers, letters, sums and other functions.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/function",
    "concept/function-notation",
    "concept/operation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fc14f3b25e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "15",
   "location": "On Functions",
   "latex": "It may be easily conceived that this notion is useless, unless there are propositions which are generally true of all functions, and which may be made the foundation of general reasoning.",
   "markdown": "It may be easily conceived that this notion is useless, unless there are propositions which are generally true of all functions, and which may be made the foundation of general reasoning.",
   "why": "Explains why general function symbols are worth having: they allow results that hold for every function.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/function",
    "concept/function-notation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-07d0c83f06",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "14",
   "location": "On Functions",
   "latex": "An expression may be a function of more quantities than one, but it is usual only to name those quantities of which it is necessary to consider a change in the value. Thus if in $x^{2} + a^{2}$ $x$~only is considered as changing its value, this is called a function of~$x$; if $x$~and~$a$ both change, it is called a function of $x$~and~$a$.",
   "markdown": "An expression may be a function of more quantities than one, but it is usual only to name those quantities of which it is necessary to consider a change in the value. Thus if in $x^{2} + a^{2}$ $x$ only is considered as changing its value, this is called a function of $x$; if $x$ and $a$ both change, it is called a function of $x$ and $a$.",
   "why": "Shows that what counts as a function of what depends on which quantities we choose to let change.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant",
    "concept/function",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-9d0eb71507",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "15",
   "location": "On Functions",
   "latex": "Thus in \\Fig{1}, the length of the radius~$OB$ is a constant, the arc~$AB$ is the independent variable, while $BM$,~$MA$, the chord~$AB$,~etc., are dependent.",
   "markdown": "Thus in 1, the length of the radius $OB$ is a constant, the arc $AB$ is the independent variable, while $BM$, $MA$, the chord $AB$, etc., are dependent.",
   "why": "Gives a concrete geometric picture of constant, independent and dependent quantities.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/constant",
    "concept/dependent",
    "concept/independent-variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-6d33ed0759",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "15",
   "location": "On Functions",
   "latex": "And, as in algebra we reason on numbers by means of general symbols, each of which may afterwards be particularised as standing for any number we please, unless specially prevented by the conditions of the problem, so, in treating of functions, we use general symbols, which may, under the restrictions of the problem, stand for any function whatever.",
   "markdown": "And, as in algebra we reason on numbers by means of general symbols, each of which may afterwards be particularised as standing for any number we please, unless specially prevented by the conditions of the problem, so, in treating of functions, we use general symbols, which may, under the restrictions of the problem, stand for any function whatever.",
   "why": "Presents general function symbols as the same step of abstraction that letters for numbers made in algebra.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/function",
    "concept/function-notation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c1b9c1ff0e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "15",
   "location": "Infinite Series",
   "latex": "If in~$\\phi x$, any function of~$x$, the\nvalue of~$x$ be increased by~$h$, or $x +  h$~be substituted\ninstead of~$x$, the result is denoted by~$\\phi(x + h)$.",
   "markdown": "If in $\\phi x$, any function of $x$, the value of $x$ be increased by $h$, or $x + h$ be substituted instead of $x$, the result is denoted by $\\phi(x + h)$.",
   "why": "Introduces the notation phi(x + h) that underlies Taylor's theorem and the idea of an increment.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/function",
    "concept/increment",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2dbf6596cc",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "16",
   "location": "Infinite Series",
   "latex": "It will happen, however, in many functions,\nthat one or more values can be given to~$x$ for which\nit is impossible to expand $f(x + h)$ without introducing\nnegative or fractional powers.",
   "markdown": "It will happen, however, in many functions, that one or more values can be given to $x$ for which it is impossible to expand $f(x + h)$ without introducing negative or fractional powers.",
   "why": "Warns learners that the expansion has exceptions, which motivates singular values.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/singular-value",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-3fef7f36fe",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "16",
   "location": "Infinite Series",
   "latex": "As the notion of a series which has no end of its\nterms, may be new to the student, we will now proceed\nto show that there may be series so constructed,\nthat the addition of any number of their terms, however\ngreat, will always give a result less than some\ndeterminate quantity.",
   "markdown": "As the notion of a series which has no end of its terms, may be new to the student, we will now proceed to show that there may be series so constructed, that the addition of any number of their terms, however great, will always give a result less than some determinate quantity.",
   "why": "Frames the surprising idea that endlessly many terms can add up to a bounded amount.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/infinite-sequence",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-201ce37498",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "16",
   "location": "Infinite Series",
   "latex": "The\nfirst two terms of this series may be obtained by dividing\n$1 - x^{2}$ by $1 - x$; the first three by dividing\n$1 - x^{3}$ by $1 - x$; and the first $n$~terms by dividing\n$1 - x^{n}$ by $1 - x$.",
   "markdown": "The first two terms of this series may be obtained by dividing $1 - x^{2}$ by $1 - x$; the first three by dividing $1 - x^{3}$ by $1 - x$; and the first $n$ terms by dividing $1 - x^{n}$ by $1 - x$.",
   "why": "Shows concretely how partial sums of the geometric series come from simple division.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/geometric-series",
    "concept/geometrical-progression",
    "concept/infinite-sequence"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-f8e08f0b9c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "17",
   "location": "Infinite Series",
   "latex": "Hence by taking $n$~sufficiently\ngreat, $\\dfrac{1 - x^{n}}{1 - x}$ or $\\dfrac{1}{1 - x} - \\dfrac{x^{n}}{1 - x}$ may be\nbrought as near to~$\\dfrac{1}{1 - x}$ as we please, than which,\nhowever, it must always be less, since $\\dfrac{x^{n}}{1 - x}$ can never\nentirely vanish, whatever value $n$~may have, and therefore\nthere is always something subtracted from $\\dfrac{1}{1 - x}$.",
   "markdown": "Hence by taking $n$ sufficiently great, $\\dfrac{1 - x^{n}}{1 - x}$ or $\\dfrac{1}{1 - x} - \\dfrac{x^{n}}{1 - x}$ may be brought as near to $\\dfrac{1}{1 - x}$ as we please, than which, however, it must always be less, since $\\dfrac{x^{n}}{1 - x}$ can never entirely vanish, whatever value $n$ may have, and therefore there is always something subtracted from $\\dfrac{1}{1 - x}$.",
   "why": "Gives the core argument for why the partial sums approach but never reach 1/(1 - x).",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/approximation",
    "concept/geometric-series",
    "concept/infinite-sequence",
    "concept/limit",
    "concept/power"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a19e4a097a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "16",
   "location": "Infinite Series",
   "latex": "Thus it generally happens that $x^{2} - 10x + 40$ is greater than~$15$, with\n  the exception only of the case where $x = 5$. It is generally true that a line\n  which meets a circle in a given point meets it again, with the exception only\n  of the tangent.",
   "markdown": "Thus it generally happens that $x^{2} - 10x + 40$ is greater than $15$, with the exception only of the case where $x = 5$. It is generally true that a line which meets a circle in a given point meets it again, with the exception only of the tangent.",
   "why": "Makes the word 'generally' concrete with two small examples that have a single exception each.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/singular-value"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ed1e74bb52",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "17",
   "location": "Convergent and Divergent Series",
   "latex": "On the other\nhand, a series is said to be \\emph{divergent} when the sum of\na number of terms may be made to surpass any quantity,\nhowever great.",
   "markdown": "On the other hand, a series is said to be *divergent* when the sum of a number of terms may be made to surpass any quantity, however great.",
   "why": "It sets divergence beside convergence so the learner can see the two outcomes as opposites.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/convergent-series",
    "concept/divergent-series"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-f17ab51c11",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "18",
   "location": "Convergent and Divergent Series",
   "latex": "We have introduced the new terms, $\\dfrac{b}{a}$,~$\\dfrac{c}{b}$,~etc., or the\nratios which the several terms of the original series\nbear to those immediately preceding.",
   "markdown": "We have introduced the new terms, $\\dfrac{b}{a}$, $\\dfrac{c}{b}$, etc., or the ratios which the several terms of the original series bear to those immediately preceding.",
   "why": "It explains the idea of rewriting a series in terms of the ratios of its successive terms, the key device for testing convergence.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/ratio",
    "theorem/ratio-test-for-convergence"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-cc1e79c25f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "17",
   "location": "Convergent and Divergent Series",
   "latex": "A series is said to be \\emph{convergent} when the sum of its terms tends towards some limit; that is, when, by taking any number of terms, however great, we shall never exceed some certain quantity. On the other hand, a series is said to be \\emph{divergent} when the sum of a number of terms may be made to surpass any quantity, however great.",
   "markdown": "A series is said to be *convergent* when the sum of its terms tends towards some limit; that is, when, by taking any number of terms, however great, we shall never exceed some certain quantity. On the other hand, a series is said to be *divergent* when the sum of a number of terms may be made to surpass any quantity, however great.",
   "why": "Gives both definitions side by side in plain words, so a learner sees convergence and divergence as opposites.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/convergent-series",
    "concept/divergent-series",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-615126b778",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "17",
   "location": "Convergent and Divergent Series",
   "latex": "A series cannot be convergent, unless its separate terms decrease, so as, at last, to become less than any given quantity.",
   "markdown": "A series cannot be convergent, unless its separate terms decrease, so as, at last, to become less than any given quantity.",
   "why": "States the necessary condition that terms must shrink, a first check a learner can apply to any series.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/convergent-series",
    "concept/infinite-sequence",
    "theorem/necessary-condition-for-convergence"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-0280f41905",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "18",
   "location": "Convergent and Divergent Series",
   "latex": "And the terms of a series may at first increase and afterwards decrease, being apparently divergent for a finite number of terms, and convergent afterwards. It will only be necessary to consider the latter part of the series.",
   "markdown": "And the terms of a series may at first increase and afterwards decrease, being apparently divergent for a finite number of terms, and convergent afterwards. It will only be necessary to consider the latter part of the series.",
   "why": "Warns that early terms can mislead and that only the tail decides convergence.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/convergent-series",
    "concept/divergent-series"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-8fa413d97d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "18",
   "location": "Convergent and Divergent Series",
   "latex": "It may be shown (1)~that if the terms of the series $\\dfrac{b}{a}$,~$\\dfrac{c}{b}$,~$\\dfrac{d}{c}$, etc., come at last to be less than unity, and afterwards either continue to approximate to a limit which is less than unity, or decrease without limit, the series $a + b + c + \\etc.$, is convergent; (2)~if the limit of the terms $\\dfrac{b}{a}$,~$\\dfrac{c}{b}$,~etc., is either greater than unity, or if they increase without limit, the series is divergent.",
   "markdown": "It may be shown (1) that if the terms of the series $\\dfrac{b}{a}$, $\\dfrac{c}{b}$, $\\dfrac{d}{c}$, etc., come at last to be less than unity, and afterwards either continue to approximate to a limit which is less than unity, or decrease without limit, the series $a + b + c + \\etc.$, is convergent; (2) if the limit of the terms $\\dfrac{b}{a}$, $\\dfrac{c}{b}$, etc., is either greater than unity, or if they increase without limit, the series is divergent.",
   "why": "The chapter's central result: a test for convergence and divergence using the ratios of successive terms.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/limit",
    "concept/ratio",
    "concept/ratio-of-successive-terms",
    "theorem/ratio-test",
    "theorem/ratio-test-for-convergence"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5cd6f38aa4",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "19",
   "location": "Convergent and Divergent Series",
   "latex": "But since $\\dfrac{l}{k}$~is less than unity, the first can never surpass $k × \\dfrac{1}{1 - \\dfrac{l}{k}}$, or~$\\dfrac{k^{2}}{k - l}$, and is convergent; the second is therefore convergent.",
   "markdown": "But since $\\dfrac{l}{k}$ is less than unity, the first can never surpass $k × \\dfrac{1}{1 - \\dfrac{l}{k}}$, or $\\dfrac{k^{2}}{k - l}$, and is convergent; the second is therefore convergent.",
   "why": "Shows the key step of the proof: a smaller series is convergent because a larger one is bounded.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/comparison-of-series",
    "theorem/ratio-test"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-82dbee3fec",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "19",
   "location": "Convergent and Divergent Series",
   "latex": "(2) The second theorem on the divergence of series we leave to the student's consideration, as it is not immediately connected with our object.",
   "markdown": "(2) The second theorem on the divergence of series we leave to the student’s consideration, as it is not immediately connected with our object.",
   "why": "An honest note that the divergence half of the test is left unproved, and an invitation to the student to supply it.",
   "use": [
    "history",
    "website"
   ],
   "concepts": [
    "concept/divergent-series",
    "theorem/ratio-test"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ef442b718b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "Thus, when $\\phi x = x^{n}$, $\\phi' x = nx^{n-1}$, $\\phi'' x = n(n - 1)x^{n-2}$, etc.; when $\\phi x = \\sin x$, $\\phi' x = \\cos x$, $\\phi'' x = -\\sin x$,~etc.",
   "markdown": "Thus, when $\\phi x = x^{n}$, $\\phi' x = nx^{n-1}$, $\\phi'' x = n(n - 1)x^{n-2}$, etc.; when $\\phi x = \\sin x$, $\\phi' x = \\cos x$, $\\phi'' x = -\\sin x$, etc.",
   "why": "Two worked cases let a learner check the pattern of derivatives against familiar power and trigonometric functions.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cosine",
    "concept/derivative",
    "concept/higher-order-derivative",
    "concept/sine"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-09a9cd96b8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "22",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "Here the coefficient of~$h$ is~$-\\dfrac{1}{x^{2}}$, which is the same as $\\phi'' x$~in the third example.",
   "markdown": "Here the coefficient of $h$ is $-\\dfrac{1}{x^{2}}$, which is the same as $\\phi'' x$ in the third example.",
   "why": "A short worked check that a second derivative equals the h-coefficient of the first derivative's expansion, useful for practising the idea.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2c23f169af",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "The proof of this is equivalent to \\emph{Taylor's Theorem} already alluded to (\\PageRef{15}); and the fact may be verified in the examples already given.",
   "markdown": "The proof of this is equivalent to *Taylor’s Theorem* already alluded to (15); and the fact may be verified in the examples already given.",
   "why": "It ties the general pattern of derivatives to Taylor's theorem and tells the learner the examples are a check on it.",
   "use": [
    "history",
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-0d27742c57",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "20",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "This never happens in the developments which we shall be required to consider in the Differential Calculus.",
   "markdown": "This never happens in the developments which we shall be required to consider in the Differential Calculus.",
   "why": "It states plainly when the series converges for small h, so a learner knows the developments used in calculus are safe.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/calculus",
    "concept/convergent-series"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-0ffd6d930f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "It appears, then, that the development of~$\\phi(x + h)$ consists of certain functions of~$x$, the first of which is $\\phi x$~itself, and the remainder of which are multiplied by $h$,~$\\dfrac{h^{2}}{2}$, $\\dfrac{h^{3}}{2·3}$, $\\dfrac{h^{4}}{2·3·4}$, and so on.",
   "markdown": "It appears, then, that the development of $\\phi(x + h)$ consists of certain functions of $x$, the first of which is $\\phi x$ itself, and the remainder of which are multiplied by $h$, $\\dfrac{h^{2}}{2}$, $\\dfrac{h^{3}}{2·3}$, $\\dfrac{h^{4}}{2·3·4}$, and so on.",
   "why": "It states the shape of the expansion of phi(x + h) and sets up the derived functions as its coefficients.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/function-notation",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-0fd8000cae",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "The following relation exists between $\\phi x$,~$\\phi' x$, $\\phi'' x$,~etc. In the same manner as $\\phi' x$~is the coefficient of~$h$ in the development of~$\\phi(x + h)$, so $\\phi'' x$~is the coefficient of~$h$ in the development of~$\\phi'(x + h)$, and $\\phi''' x$~is the coefficient of~$h$ in the development of~$\\phi''(x + h)$; $\\phi^{\\text{iv}} x$~is the coefficient of~$h$ in the development of $\\phi'''(x + h)$, and so on.",
   "markdown": "The following relation exists between $\\phi x$, $\\phi' x$, $\\phi'' x$, etc. In the same manner as $\\phi' x$ is the coefficient of $h$ in the development of $\\phi(x + h)$, so $\\phi'' x$ is the coefficient of $h$ in the development of $\\phi'(x + h)$, and $\\phi''' x$ is the coefficient of $h$ in the development of $\\phi''(x + h)$; $\\phi^{\\text{iv}} x$ is the coefficient of $h$ in the development of $\\phi'''(x + h)$, and so on.",
   "why": "It shows that each derived function comes from the previous one by the same expansion step, which is the core idea of successive derivatives.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-92f138648e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "20",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "In this case, by theorem~(1\\ib), the series is convergent; it follows, therefore, that a value of~$h$ can always be found so small that $ph + qh^{2} + rh^{3} + \\etc.$, shall be convergent, at least unless the coefficients $p$,~$q$,~$r$,~etc., be such that the ratio of any one to the preceding increases without limit, as we take more distant terms of the series. This never happens in the developments which we shall be required to consider in the Differential Calculus.",
   "markdown": "In this case, by theorem (1*b*), the series is convergent; it follows, therefore, that a value of $h$ can always be found so small that $ph + qh^{2} + rh^{3} + \\etc.$, shall be convergent, at least unless the coefficients $p$, $q$, $r$, etc., be such that the ratio of any one to the preceding increases without limit, as we take more distant terms of the series. This never happens in the developments which we shall be required to consider in the Differential Calculus.",
   "why": "It shows why a small enough h makes the series converge, and states the one exceptional case plainly.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/convergent-series",
    "concept/power-series",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ce9b504082",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "When $\\phi x = a^{x}$, $\\phi' x = ka^{x}$, and $\\phi'(x + h) = ka^{x+h} = k(a^{x} + ka^{x}\\, h + \\etc.)$. The coefficient of~$h$ is here~$k^{2} a^{x}$, which is the same as~$\\phi'' x$.",
   "markdown": "When $\\phi x = a^{x}$, $\\phi' x = ka^{x}$, and $\\phi'(x + h) = ka^{x+h} = k(a^{x} + ka^{x}\\, h + \\etc.)$. The coefficient of $h$ is here $k^{2} a^{x}$, which is the same as $\\phi'' x$.",
   "why": "A short worked check that the coefficient of h in the expansion of phi'(x + h) really is phi'' x.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/exponential-function",
    "concept/general-power",
    "concept/higher-order-derivative",
    "method/verification"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a8d74ead19",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "22",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "Again, if $\\phi x = \\log x$, $\\phi' x = \\dfrac{1}{x}$, and $\\phi'(x + h) = \\dfrac{1}{x + h} = \\dfrac{1}{x} - \\dfrac{h}{x^{2}} + \\etc.$, as appears by common division. Here the coefficient of~$h$ is~$-\\dfrac{1}{x^{2}}$, which is the same as $\\phi'' x$~in the third example.",
   "markdown": "Again, if $\\phi x = \\log x$, $\\phi' x = \\dfrac{1}{x}$, and $\\phi'(x + h) = \\dfrac{1}{x + h} = \\dfrac{1}{x} - \\dfrac{h}{x^{2}} + \\etc.$, as appears by common division. Here the coefficient of $h$ is $-\\dfrac{1}{x^{2}}$, which is the same as $\\phi'' x$ in the third example.",
   "why": "A second worked check, using ordinary division to expand 1/(x + h) and read off phi'' x for the logarithm.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/logarithm",
    "method/verification"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fd7a039cc2",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "22",
   "location": "Differential Coefficients",
   "latex": "Let there be any function of~$x$, which we call~$\\phi x$, in which $x$~is increased by an increment~$h$; the function then becomes",
   "markdown": "Let there be any function of $x$, which we call $\\phi x$, in which $x$ is increased by an increment $h$; the function then becomes",
   "why": "It sets up the whole idea by letting a function's variable grow by a small amount h and asking what happens to the function.",
   "use": [
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-d955f0ee8c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "24",
   "location": "Differential Coefficients",
   "latex": "Therefore to find the coefficient of~$h$ in the development of~$\\phi(x + h)$, find $\\phi(x + h) - \\phi x$, divide it by~$h$, and find the limit towards which it tends as $h$~is diminished.",
   "markdown": "Therefore to find the coefficient of $h$ in the development of $\\phi(x + h)$, find $\\phi(x + h) - \\phi x$, divide it by $h$, and find the limit towards which it tends as $h$ is diminished.",
   "why": "It states the working procedure for a derivative in one step a learner can follow and practise.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-426a08cf16",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "24",
   "location": "Differential Coefficients",
   "latex": "Hence the ratio of the increments of $\\phi x$ and~$x$, produced by changing $x$ into~$x + h$, though never equal to~$\\phi' x$, approaches towards it as $h$~is diminished, and may be brought as near as we please to it, by sufficiently diminishing~$h$.",
   "markdown": "Hence the ratio of the increments of $\\phi x$ and $x$, produced by changing $x$ into $x + h$, though never equal to $\\phi' x$, approaches towards it as $h$ is diminished, and may be brought as near as we please to it, by sufficiently diminishing $h$.",
   "why": "It warns that the difference quotient is never exactly the derivative, yet can be brought as close to it as wanted, which is the key idea behind a limit.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b160c237b0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "24",
   "location": "Differential Coefficients",
   "latex": "It follows, therefore, that if, instead of the full development of~$\\phi(x + h)$, we use only its two first terms $\\phi x +  \\phi' x\\, h$, the error thereby introduced may, by taking $h$ sufficiently small, be made as small a portion as we please of the small term~$\\phi' x\\, h$.",
   "markdown": "It follows, therefore, that if, instead of the full development of $\\phi(x + h)$, we use only its two first terms $\\phi x + \\phi' x\\, h$, the error thereby introduced may, by taking $h$ sufficiently small, be made as small a portion as we please of the small term $\\phi' x\\, h$.",
   "why": "It shows why a linear approximation is good for small changes, the reason first-order approximation is used in practice.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b76ee1fa7a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "25",
   "location": "The Notation of the Differential Calculus",
   "latex": "When any quantity is increased by an increment, which, consistently with the conditions of the problem, may be supposed as small as we please, this increment is denoted, not by a separate letter, but by prefixing the letter~$d$, either followed by a full stop or not, to that already used to signify the quantity.",
   "markdown": "When any quantity is increased by an increment, which, consistently with the conditions of the problem, may be supposed as small as we please, this increment is denoted, not by a separate letter, but by prefixing the letter $d$, either followed by a full stop or not, to that already used to signify the quantity.",
   "why": "It gives the learner the basic rule for reading the d-notation, stated in the book's own words.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/differential",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fe5b8decb8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "25",
   "location": "The Notation of the Differential Calculus",
   "latex": "Thus, if $x$~becomes $x + dx$, $x^{2}$~becomes $x^{2} + d.x^{2}$. But this is also $(x + dx)^{2}$ or $x^{2} + 2x\\, dx + (dx)^{2}$; whence $d.x^{2} = 2x\\, dx + (dx)^{2}$.",
   "markdown": "Thus, if $x$ becomes $x + dx$, $x^{2}$ becomes $x^{2} + d.x^{2}$. But this is also $(x + dx)^{2}$ or $x^{2} + 2x\\, dx + (dx)^{2}$; whence $d.x^{2} = 2x\\, dx + (dx)^{2}$.",
   "why": "It works a small example of the increment of x squared, so the learner can see where the extra term comes from.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-23d4ec6aea",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "25",
   "location": "The Notation of the Differential Calculus",
   "latex": "Care must be taken not to confound $d.x^{2}$, the increment of~$x^{2}$, with~$(dx)^{2}$, or, as it is often written,~$dx^{2}$, the square of the increment of~$x$.",
   "markdown": "Care must be taken not to confound $d.x^{2}$, the increment of $x^{2}$, with $(dx)^{2}$, or, as it is often written, $dx^{2}$, the square of the increment of $x$.",
   "why": "It warns the learner about a common confusion between the increment of x squared and the square of the increment.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-9ed1161a79",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "26",
   "location": "The Notation of the Differential Calculus",
   "latex": "It must not be imagined that because $x$~occurs in the symbol~$dx$, the value of the latter in any way depends upon that of the former: both the first value of~$x$, and the quantity by which it is made to differ from its first value, are at our pleasure, and the letter~$d$ must merely be regarded as an abbreviation of the words ``\\emph{difference of}.''",
   "markdown": "It must not be imagined that because $x$ occurs in the symbol $dx$, the value of the latter in any way depends upon that of the former: both the first value of $x$, and the quantity by which it is made to differ from its first value, are at our pleasure, and the letter $d$ must merely be regarded as an abbreviation of the words “*difference of*.”",
   "why": "It tells the learner that dx does not depend on the value of x, which is a point beginners often miss.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/difference",
    "concept/differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-47c9095102",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "28",
   "location": "The Notation of the Differential Calculus",
   "latex": "If $x = 4$ and $dx = .01$, then $dy = .0801$ and $\\dfrac{dy}{dx} = 8.01$. If $dx = .0001$, $dy = .00080001$ and $\\dfrac{dy}{dx} = 8.0001$.",
   "markdown": "If $x = 4$ and $dx = .01$, then $dy = .0801$ and $\\dfrac{dy}{dx} = 8.01$. If $dx = .0001$, $dy = .00080001$ and $\\dfrac{dy}{dx} = 8.0001$.",
   "why": "These numbers show concretely how the ratio dy/dx approaches its limit as dx shrinks.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ae58aafec0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "29",
   "location": "The Notation of the Differential Calculus",
   "latex": "We must adopt the first of these explanations when $dy$ and $dx$ appear in a fraction, and the second when they are on opposite sides of an equation.",
   "markdown": "We must adopt the first of these explanations when $dy$ and $dx$ appear in a fraction, and the second when they are on opposite sides of an equation.",
   "why": "It tells the learner which reading of the notation applies in each form of expression.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c04c26be53",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "29",
   "location": "The Notation of the Differential Calculus",
   "latex": "\\emph{The equations which we thus use are not absolutely true in any case, but may be brought as near as we please to the truth}, by making $dy$~and~$dx$ sufficiently small.",
   "markdown": "*The equations which we thus use are not absolutely true in any case, but may be brought as near as we please to the truth*, by making $dy$ and $dx$ sufficiently small.",
   "why": "It states plainly that the working equations are approximations that become true in the limit, which keeps the learner honest about what is being claimed.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-64d8e94343",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "26",
   "location": "The Notation of the Differential Calculus",
   "latex": "In the Differential Calculus, the limit of the ratio only is retained, to the exclusion of the rest, which may be explained in either of the two following ways:",
   "markdown": "In the Differential Calculus, the limit of the ratio only is retained, to the exclusion of the rest, which may be explained in either of the two following ways:",
   "why": "It names the central idea of the chapter, that the calculus keeps only the limit of a ratio, before the two explanations begin.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/calculus",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-4c22d3bc92",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-algebraical-geometry",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "29",
   "location": "Algebraical Geometry",
   "latex": "If two straight lines be drawn at right angles to each other, dividing the whole of their plane into four parts, one lying in each right angle, the situation of any point is determined when we know, (1)~in which angle it lies, and (2)~its perpendicular distances from the two right lines.",
   "markdown": "If two straight lines be drawn at right angles to each other, dividing the whole of their plane into four parts, one lying in each right angle, the situation of any point is determined when we know, (1) in which angle it lies, and (2) its perpendicular distances from the two right lines.",
   "why": "It gives the learner the basic idea of a point being located by two perpendicular distances, before any symbols are introduced.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cartesian-coordinates"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2d4c94dc26",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-algebraical-geometry",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "29",
   "location": "Algebraical Geometry",
   "latex": "for, though there is an infinite number of points whose distance from~$OA$ only is the same as that of~$P$, and an infinite number of others, whose distance from~$OB$ is the same as that of~$P$, there is no other point whose distances from both lines are the same as those of~$P$.",
   "markdown": "for, though there is an infinite number of points whose distance from $OA$ only is the same as that of $P$, and an infinite number of others, whose distance from $OB$ is the same as that of $P$, there is no other point whose distances from both lines are the same as those of $P$.",
   "why": "It explains why two distances are needed: one distance alone never picks out a single point, but both together do.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cartesian-coordinates"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-4cc06a163f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-algebraical-geometry",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "29",
   "location": "Algebraical Geometry",
   "latex": "The line~$OA$ is called the axis of~$x$, because it is usual to denote any variable distance measured on or parallel to~$OA$ by the letter~$x$.",
   "markdown": "The line $OA$ is called the axis of $x$, because it is usual to denote any variable distance measured on or parallel to $OA$ by the letter $x$.",
   "why": "It shows how the letters x and y come from the two axes, so the learner can see where the notation begins.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cartesian-coordinates"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-e3e4d89b00",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-algebraical-geometry",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "30",
   "location": "Algebraical Geometry",
   "latex": "It is moreover usual to call the co-ordinate~$OM$, the \\emph{abscissa}, and $MP$, the \\emph{ordinate}, of the point~$P$.",
   "markdown": "It is moreover usual to call the co-ordinate $OM$, the *abscissa*, and $MP$, the *ordinate*, of the point $P$.",
   "why": "It gives the two standard names for the coordinates along x and along y.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/abscissa",
    "concept/ordinate"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c2d5d96512",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-algebraical-geometry",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "30",
   "location": "Algebraical Geometry",
   "latex": "As $O$~moves towards~$A$, the point~$P$ will, by its motion on~$MP$, compounded with the motion of the line $MP$ itself, describe a curve~$OP$, in which $PM$~is less than, equal to, or greater than,~$OM$, according as $OM$~is less than, equal to, or greater than the linear unit.",
   "markdown": "As $O$ moves towards $A$, the point $P$ will, by its motion on $MP$, compounded with the motion of the line $MP$ itself, describe a curve $OP$, in which $PM$ is less than, equal to, or greater than, $OM$, according as $OM$ is less than, equal to, or greater than the linear unit.",
   "why": "It describes how a rule relating y to x traces a curve, a vivid picture a learner can follow with a sketch.",
   "use": [
    "website",
    "lesson"
   ],
   "concepts": [
    "concept/cartesian-coordinates"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-4d829ef47c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "31",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "And conversely it may be proved by any number of examples, that when an equation in which $a$~occurs has been deduced strictly on the supposition that $a$~is a line measured in one direction, a change of sign in~$a$ will turn the equation into that which would have been deduced by the same reasoning, had we begun by measuring the line~$a$ in the contrary direction.",
   "markdown": "And conversely it may be proved by any number of examples, that when an equation in which $a$ occurs has been deduced strictly on the supposition that $a$ is a line measured in one direction, a change of sign in $a$ will turn the equation into that which would have been deduced by the same reasoning, had we begun by measuring the line $a$ in the contrary direction.",
   "why": "It states the central rule plainly: flipping the sign of a symbol is the same as reversing the direction of its line, so a learner can see why signs carry geometric meaning.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/negative-direction",
    "concept/sign"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-84547d81c6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "32",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "Hence the equation $ay + bx = ab$ belongs to all parts of the straight line~$EF$, if we agree to consider $M''P''$ as negative, when $MP$~is positive, and $OM'$~as negative when $OM$~is positive.",
   "markdown": "Hence the equation $ay + bx = ab$ belongs to all parts of the straight line $EF$, if we agree to consider $M''P''$ as negative, when $MP$ is positive, and $OM'$ as negative when $OM$ is positive.",
   "why": "It shows a learner the agreement that makes one equation cover every part of a line, which is the heart of using signed coordinates.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/line",
    "concept/sign"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-f99c8a41e0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "35",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "In this manner, if four points be taken similarly situated in the four angles, the numerical values of whose co-ordinates are $x = 4$ and  $y = 6$, and if the co-ordinates of that point which lies in the angle~$AOB$, are called $+4$ and~$+6$; those of the points lying in the angle~$BOC$ will be $-4$~and~$+6$; in the angle~$COD$ $-4$~and~$-6$; and in the angle~$DOE$ $+4$~and~$-6$.",
   "markdown": "In this manner, if four points be taken similarly situated in the four angles, the numerical values of whose co-ordinates are $x = 4$ and $y = 6$, and if the co-ordinates of that point which lies in the angle $AOB$, are called $+4$ and $+6$; those of the points lying in the angle $BOC$ will be $-4$ and $+6$; in the angle $COD$ $-4$ and $-6$; and in the angle $DOE$ $+4$ and $-6$.",
   "why": "It gives a concrete table of signs for the four angles, so a learner can check the sign pattern of coordinates quadrant by quadrant.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/sign"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-58078c3273",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "33",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "The latter method is preferable, inasmuch as it enables us to contain, in one investigation, all the different cases of a problem.",
   "markdown": "The latter method is preferable, inasmuch as it enables us to contain, in one investigation, all the different cases of a problem.",
   "why": "It explains why a signed general equation is better than separate calculations for each case, a useful point about method choice.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/sign"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c35fd79311",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "33",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "Thus, if $OE = 4$, and $OF = 5$, and $OM = 1$, we can determine~$MP$ from the equation $ay + bx = ab$, or $4y + 5 = 20$, which gives $y$~or $MP = 3\\frac{3}{4}$.",
   "markdown": "Thus, if $OE = 4$, and $OF = 5$, and $OM = 1$, we can determine $MP$ from the equation $ay + bx = ab$, or $4y + 5 = 20$, which gives $y$ or $MP = 3\\frac{3}{4}$.",
   "why": "It is a short numerical example in which a learner can follow every step from the general equation to a single value.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/line",
    "concept/negative-direction"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5f0f367225",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "36",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "And thus, if $y$ be any function of~$x$, we can obtain a geometrical representation of the same, by making $y$ the ordinate, and $x$~the abscissa of a curve, every ordinate of which shall be the linear representation of the numerical value of the given function corresponding to the numerical value of the abscissa, the linear unit being a given line.",
   "markdown": "And thus, if $y$ be any function of $x$, we can obtain a geometrical representation of the same, by making $y$ the ordinate, and $x$ the abscissa of a curve, every ordinate of which shall be the linear representation of the numerical value of the given function corresponding to the numerical value of the abscissa, the linear unit being a given line.",
   "why": "It states in one sentence the idea of drawing a function as a curve, which is the origin of the graph for a general reader.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/abscissa",
    "concept/function",
    "concept/graph-of-a-function",
    "concept/ordinate"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fae104f011",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "37",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "The line~$TPV$ indicates the direction in which the point~$P$ is proceeding, and is called the \\emph{tangent} of the curve at the point~$P$.",
   "markdown": "The line $TPV$ indicates the direction in which the point $P$ is proceeding, and is called the *tangent* of the curve at the point $P$.",
   "why": "It gives the plain definition of the tangent as the line showing the direction of motion at a point.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/tangent"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-d6648d2d48",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "37",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "If, therefore, a line~$PV$ be drawn through~$P$, making with~$PQ$ an angle whose tangent is~$2x$, the chord~$PP'$ will, as $P'$~approaches towards~$P$, or as $dx$~is diminished, continually approximate towards~$PV$, so that the angle~$P'PV$ may be made smaller than any given angle, by sufficiently diminishing~$dx$.",
   "markdown": "If, therefore, a line $PV$ be drawn through $P$, making with $PQ$ an angle whose tangent is $2x$, the chord $PP'$ will, as $P'$ approaches towards $P$, or as $dx$ is diminished, continually approximate towards $PV$, so that the angle $P'PV$ may be made smaller than any given angle, by sufficiently diminishing $dx$.",
   "why": "It shows how the chord through two nearby points closes onto the tangent as the small increment shrinks, which is the idea behind the limit.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/chord",
    "concept/limit",
    "concept/tangent"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-9d809bb568",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "36",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "Since the relation $y = x^{2}$ is true for the co-ordinates of every point in the curve, we have $y + dy = (x +  dx)^{2}$, the subtraction of the former equation from which gives $dy = 2x\\, dx + (dx)^{2}$, or $\\dfrac{dy}{dx} = 2x + dx$.",
   "markdown": "Since the relation $y = x^{2}$ is true for the co-ordinates of every point in the curve, we have $y + dy = (x + dx)^{2}$, the subtraction of the former equation from which gives $dy = 2x\\, dx + (dx)^{2}$, or $\\dfrac{dy}{dx} = 2x + dx$.",
   "why": "It works a complete worked example, showing step by step how the ratio dy/dx is obtained for a parabola.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ecca7e3ba8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "38",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "There is some confusion between these different uses of the word tangent.",
   "markdown": "There is some confusion between these different uses of the word tangent.",
   "why": "It warns the learner that the geometrical tangent and the trigonometrical tangent are different things that share one word.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/tangent",
    "concept/tangent-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-020e3c0b22",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "37",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "If the curve were the interior of a small solid tube, in which an atom of matter were made to move, being projected into it at~$O$, and if all the tube above~$P$ were removed, the line~$PV$ is in the direction which the atom would take on emerging at~$P$, and is the line which it would describe.",
   "markdown": "If the curve were the interior of a small solid tube, in which an atom of matter were made to move, being projected into it at $O$, and if all the tube above $P$ were removed, the line $PV$ is in the direction which the atom would take on emerging at $P$, and is the line which it would describe.",
   "why": "Its vivid tube-and-atom image lets a learner picture the tangent as the path a moving body takes when it leaves the curve.",
   "use": [
    "website",
    "lesson"
   ],
   "concepts": [
    "concept/curve",
    "concept/tangent"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a8b3bf9814",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "38",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "This problem, of drawing a tangent to any curve, was one, the consideration of which gave rise to the methods of the Differential Calculus.",
   "markdown": "This problem, of drawing a tangent to any curve, was one, the consideration of which gave rise to the methods of the Differential Calculus.",
   "why": "It places the tangent problem as the historical origin of the differential calculus.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/derivative",
    "concept/tangent"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5ab95e6a55",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "39",
   "location": "Rational Explanation of the Language of Leibnitz",
   "latex": "We shall proceed to a strict proof of this; but in the meanwhile, as a familiar illustration, imagine a small arc to be cut off from a curve, and its extremities joined by a chord, thus forming an arch, of which the chord is the base.",
   "markdown": "We shall proceed to a strict proof of this; but in the meanwhile, as a familiar illustration, imagine a small arc to be cut off from a curve, and its extremities joined by a chord, thus forming an arch, of which the chord is the base.",
   "why": "It gives learners a concrete picture of a small arc and its chord before the formal argument begins.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/arc-of-a-curve",
    "concept/chord",
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2fef6a54a3",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "39",
   "location": "Rational Explanation of the Language of Leibnitz",
   "latex": "However the original arc may be diminished, let the magnified base continue of a given length. This is possible, since on any line a figure may be constructed similar to a given figure.",
   "markdown": "However the original arc may be diminished, let the magnified base continue of a given length. This is possible, since on any line a figure may be constructed similar to a given figure.",
   "why": "It shows learners how magnifying a shrinking arch keeps the picture fixed, which makes the idea of an infinitely small arc concrete.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/arc-of-a-curve",
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2f0539dd99",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "39",
   "location": "Rational Explanation of the Language of Leibnitz",
   "latex": "All which must be interpreted to mean that, the chord and arc being diminished, approach more and more nearly to a ratio of equality as to their lengths; and also that the greatest separation between an arc and its chord may be made as small a part as we please of the whole chord or arc, by sufficiently diminishing the chord.",
   "markdown": "All which must be interpreted to mean that, the chord and arc being diminished, approach more and more nearly to a ratio of equality as to their lengths; and also that the greatest separation between an arc and its chord may be made as small a part as we please of the whole chord or arc, by sufficiently diminishing the chord.",
   "why": "It states plainly what the language of infinitely small quantities is taken to mean, which keeps learners from reading it as literal magnitude.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/arc-of-a-curve",
    "concept/chord",
    "theorem/infinitesimal-arc-coincides-with-its-chord"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-4ebfca4919",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "41",
   "location": "Rational Explanation of the Language of Leibnitz",
   "latex": "Hence $PQ$~can be taken so small that $VQ$~shall contain~$VP'$ as often as we please, or the ratio of $VQ$ to~$VP'$ shall be as great as we please.",
   "markdown": "Hence $PQ$ can be taken so small that $VQ$ shall contain $VP'$ as often as we please, or the ratio of $VQ$ to $VP'$ shall be as great as we please.",
   "why": "It shows the tangent-line term dominating the remainder as the step shrinks, which is the core of the argument.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/infinitesimal",
    "concept/tangent"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-f217b8fbac",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "40",
   "location": "Rational Explanation of the Language of Leibnitz",
   "latex": "Let $PP'$ (\\Fig{4}) be a part of a curve, whose equation is $y = \\phi(x)$, that is, $PM$~may always be found by substituting the numerical value of~$OM$ in a given function of~$x$.",
   "markdown": "Let $PP'$ (4) be a part of a curve, whose equation is $y = \\phi(x)$, that is, $PM$ may always be found by substituting the numerical value of $OM$ in a given function of $x$.",
   "why": "It shows learners how a curve's equation is read as a function, which sets up the increment and tangent work that follows.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/curve",
    "concept/function-notation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5993730d2d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-orders-of-infinity",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "43",
   "location": "Orders of Infinity",
   "latex": "In the language of Leibnitz if $h$~be an infinitely small quantity, \\Eq{(1)}~is an infinitely small quantity of the first order, \\Eq{(2)}~is an infinitely small quantity of the second order, and so on.",
   "markdown": "In the language of Leibnitz if $h$ be an infinitely small quantity, (1) is an infinitely small quantity of the first order, (2) is an infinitely small quantity of the second order, and so on.",
   "why": "It links the chapter's order language to the older infinitesimal vocabulary a learner will meet in other books.",
   "use": [
    "history",
    "lesson"
   ],
   "concepts": [
    "concept/infinitesimal",
    "concept/order-of-smallness",
    "person/gottfried-wilhelm-leibniz"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-768d8cff7c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-orders-of-infinity",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "43",
   "location": "Orders of Infinity",
   "latex": "Hence \\Eq{(1)}~is said to be \\emph{comparable} to the first power of~$h$, or \\emph{of the first order}, since this is the only power of~$h$ whose ratio to~\\Eq{(1)} tends towards a finite limit.",
   "markdown": "Hence (1) is said to be *comparable* to the first power of $h$, or *of the first order*, since this is the only power of $h$ whose ratio to (1) tends towards a finite limit.",
   "why": "It gives the exact test for the order of a quantity: the one power whose ratio to it has a finite limit.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "concept/order-of-smallness"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-895da47c64",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-orders-of-infinity",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "42",
   "location": "Orders of Infinity",
   "latex": "As $h$~is diminished, all these expressions decrease without limit; but the first \\emph{increases} with respect to the second, that is, contains it more times after a decrease of~$h$ than it did before.",
   "markdown": "As $h$ is diminished, all these expressions decrease without limit; but the first *increases* with respect to the second, that is, contains it more times after a decrease of $h$ than it did before.",
   "why": "It shows why a ratio of two vanishing quantities can grow, which is the core intuition for comparing their orders.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/infinitesimal",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a537f429e6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-orders-of-infinity",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "43",
   "location": "Orders of Infinity",
   "latex": "Nevertheless this decrease increases the ratio of the first to the second, of the second to the third, and so on, and the increase is without limit.",
   "markdown": "Nevertheless this decrease increases the ratio of the first to the second, of the second to the third, and so on, and the increase is without limit.",
   "why": "It warns learners that a quantity going to zero does not mean the ratio of two such quantities goes to one.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "concept/order-of-smallness"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-f16acbbf3a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-orders-of-infinity",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "44",
   "location": "Orders of Infinity",
   "latex": "The converse proposition is readily shown, that if the ratio of two series arranged in powers of~$h$ continually approaches to some finite limit as $h$~is diminished, the two series are of the same order, or the exponent of the lowest power of~$h$ is the same in both.",
   "markdown": "The converse proposition is readily shown, that if the ratio of two series arranged in powers of $h$ continually approaches to some finite limit as $h$ is diminished, the two series are of the same order, or the exponent of the lowest power of $h$ is the same in both.",
   "why": "It states the criterion for when two vanishing quantities are of the same order, which a learner can apply directly.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/order-of-smallness",
    "concept/power-series"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-cae4f31855",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "46",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "As $A'B'$ moves towards~$AB$, $da$~and~$db$ are diminished without limit, $a$~and~$b$ remaining the same; hence the limit of the ratio~$\\dfrac{db}{da}$ is $\\dfrac{2a}{2b}$ or~$\\dfrac{a}{b}$.",
   "markdown": "As $A'B'$ moves towards $AB$, $da$ and $db$ are diminished without limit, $a$ and $b$ remaining the same; hence the limit of the ratio $\\dfrac{db}{da}$ is $\\dfrac{2a}{2b}$ or $\\dfrac{a}{b}$.",
   "why": "It shows a learner how a ratio of two vanishing increments takes a definite limit when both increments are made to diminish while the other quantities stay fixed.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "concept/limit-of-intersections"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c12ca703d6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "45",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "But here it is necessary to remark that $AB$~is itself one of the positions intermediate between $A'B'$ and~$A''B''$, and when two lines are, by the motion of one of them, brought into one and the same straight line, they intersect one another (if this phrase can be here applied at all) in every point, and all idea of one distinct point of intersection is lost.",
   "markdown": "But here it is necessary to remark that $AB$ is itself one of the positions intermediate between $A'B'$ and $A''B''$, and when two lines are, by the motion of one of them, brought into one and the same straight line, they intersect one another (if this phrase can be here applied at all) in every point, and all idea of one distinct point of intersection is lost.",
   "why": "It warns that when two lines coincide, the notion of a single intersection point breaks down, which is why a limit is needed to define it.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/intersection-of-two-curves",
    "concept/limit-of-intersections"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-6639f7b054",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "46",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "Let $P$~be the point of separation; then every point of~$P'P''$, except~$P$, is a real point of intersection of~$AB$, with one of the positions of~$A''B''$, and when $A''B''$~has moved very near to~$AB$, the point~$P''$ will be very near to~$P$; and there is no point so near to~$P$, that it may not be made the intersection of $A''B''$ and~$AB$, by bringing the former sufficiently near to the latter.",
   "markdown": "Let $P$ be the point of separation; then every point of $P'P''$, except $P$, is a real point of intersection of $AB$, with one of the positions of $A''B''$, and when $A''B''$ has moved very near to $AB$, the point $P''$ will be very near to $P$; and there is no point so near to $P$, that it may not be made the intersection of $A''B''$ and $AB$, by bringing the former sufficiently near to the latter.",
   "why": "It gives a clear verbal argument for why the point P is the limit of the intersections and not an intersection itself.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit-of-intersections"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-044cc972a7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "47",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "The same result may be more simply obtained, by diminishing $da$~and~$db$ in equation~\\Eq{(5)}, before obtaining the values of $y$~and~$x$.",
   "markdown": "The same result may be more simply obtained, by diminishing $da$ and $db$ in equation (5), before obtaining the values of $y$ and $x$.",
   "why": "It teaches a learner that a limit can be taken earlier in a calculation, which simplifies the algebra without changing the result.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "concept/limit-of-intersections"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5281fe27c4",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "48",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "This limit of the intersections is different for every different position of the line~$AB$, but may be determined, in every case, by the following simple construction.",
   "markdown": "This limit of the intersections is different for every different position of the line $AB$, but may be determined, in every case, by the following simple construction.",
   "why": "It tells a learner that the limit point depends on the position of the line and can be found for each case by a construction.",
   "use": [
    "website",
    "lesson"
   ],
   "concepts": [
    "concept/limit-of-intersections",
    "concept/line"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-6b23f9dfda",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "48",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "Hence $BP = AQ$ and $AP = BQ$, or the point~$P$ is as far from either extremity of~$AB$ as $Q$~is from the other.",
   "markdown": "Hence $BP = AQ$ and $AP = BQ$, or the point $P$ is as far from either extremity of $AB$ as $Q$ is from the other.",
   "why": "It states the geometric result of the construction in a form a learner can check with a drawing.",
   "use": [
    "website"
   ],
   "concepts": [
    "concept/limit-of-intersections",
    "concept/line"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-43dbb23a83",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "48",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "The inaccuracy of this supposition has been already pointed out; yet it must be confessed that this once got over, the results are deduced with a degree of simplicity and consequent clearness, not to be found in any other method.",
   "markdown": "The inaccuracy of this supposition has been already pointed out; yet it must be confessed that this once got over, the results are deduced with a degree of simplicity and consequent clearness, not to be found in any other method.",
   "why": "It tells the learner plainly that the method rests on an admitted inaccuracy while still giving clear results, which is the honest frame for the whole chapter.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-8d8a7b2402",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "49",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "The following cannot be regarded as a demonstration, except by a mind so accustomed to the subject that it can readily convert the various inaccuracies into their corresponding truths, and see, at one glance, how far any proposition will affect the final result.",
   "markdown": "The following cannot be regarded as a demonstration, except by a mind so accustomed to the subject that it can readily convert the various inaccuracies into their corresponding truths, and see, at one glance, how far any proposition will affect the final result.",
   "why": "It warns the beginner that this reasoning is not yet a proof and that the inaccuracies must be consciously corrected.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-026ab20d0f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "49",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "The beginner will be struck with the extraordinary assertions which follow, given in their most naked form, without any attempt at a less startling mode of expression.",
   "markdown": "The beginner will be struck with the extraordinary assertions which follow, given in their most naked form, without any attempt at a less startling mode of expression.",
   "why": "It prepares a learner for startling statements and signals that they are deliberately stated bluntly.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-3274a6bc8f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "50",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "But at the same time we observe that every one of these assumptions approaches the truth, as we diminish the angle~$A'PA$, so that there is no magnitude, line or angle, so small that the linear or angular errors, arising from the above-mentioned suppositions, may not be made smaller.",
   "markdown": "But at the same time we observe that every one of these assumptions approaches the truth, as we diminish the angle $A'PA$, so that there is no magnitude, line or angle, so small that the linear or angular errors, arising from the above-mentioned suppositions, may not be made smaller.",
   "why": "It explains that each approximation improves as the angle shrinks, so the error can always be made smaller, which is the core idea behind taking limits.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/infinitesimal",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-6e17544c12",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "Hence the ratio $A\\alpha$ to~$\\beta B'$ or $dp + \\mu$ to $dq + \\nu$ will continually approximate to that of $dp$ to~$dq$, or a ratio of equality.",
   "markdown": "Hence the ratio $A\\alpha$ to $\\beta B'$ or $dp + \\mu$ to $dq + \\nu$ will continually approximate to that of $dp$ to $dq$, or a ratio of equality.",
   "why": "It shows the stricter argument in which the neglected terms vanish, so that a ratio tends to equality, a clear model of a limit of a ratio.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "concept/order-of-smallness",
    "concept/proportion"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-beb97445d0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "49",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "An infinitely small arc of a circle is a straight line perpendicular to its radius; hence $A'aA$~and~$BbB'$ are right-angled triangles, the first similar to~$BOA$, the two having the angle~$A$ in common, and the second similar to~$B'OA'$.",
   "markdown": "An infinitely small arc of a circle is a straight line perpendicular to its radius; hence $A'aA$ and $BbB'$ are right-angled triangles, the first similar to $BOA$, the two having the angle $A$ in common, and the second similar to $B'OA'$.",
   "why": "It shows how a small arc is replaced by a perpendicular to its radius, producing right triangles that are similar to the figure's triangles.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/arc-of-a-circle",
    "concept/perpendicular",
    "concept/right-triangle",
    "concept/similarity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-dd41dee829",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-an-illustration-from-dynamics-velocity-acceleration-etc",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "52",
   "location": "An Illustration from Dynamics: Velocity, Acceleration, etc",
   "latex": "The number of units of length described in a unit of time is called the \\emph{velocity}; thus",
   "markdown": "The number of units of length described in a unit of time is called the *velocity*; thus",
   "why": "Gives the learner the plain definition of velocity as length per unit time before any calculus is introduced.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b27aa8fd63",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-an-illustration-from-dynamics-velocity-acceleration-etc",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "52",
   "location": "An Illustration from Dynamics: Velocity, Acceleration, etc",
   "latex": "Suppose a point moving along a straight line uniformly; that is, if the whole length described be divided into any number of equal parts, however great, each of those parts is described in the same time.",
   "markdown": "Suppose a point moving along a straight line uniformly; that is, if the whole length described be divided into any number of equal parts, however great, each of those parts is described in the same time.",
   "why": "Defines uniform motion in terms a learner can test by dividing a path into equal parts.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/uniform-motion"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-322ef48dbb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-an-illustration-from-dynamics-velocity-acceleration-etc",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "54",
   "location": "An Illustration from Dynamics: Velocity, Acceleration, etc",
   "latex": "we can, at the end of every time, assign a uniform velocity, which shall represent, more nearly than any other, the rate at which the point is moving.",
   "markdown": "we can, at the end of every time, assign a uniform velocity, which shall represent, more nearly than any other, the rate at which the point is moving.",
   "why": "Shows how an instantaneous velocity is assigned even when no uniform velocity holds over any interval.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/limit",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-227c7d7705",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-an-illustration-from-dynamics-velocity-acceleration-etc",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "57",
   "location": "An Illustration from Dynamics: Velocity, Acceleration, etc",
   "latex": "And since, when $x$~is the space described, $\\phi' t$~is the limit of~$\\dfrac{dx}{dt}$, the velocity is also this limit; that is, when a point does not move uniformly, the velocity is not represented by any increment of length divided by its increment of time, but by the limit to which that ratio continually tends, as the increment of time is diminished.",
   "markdown": "And since, when $x$ is the space described, $\\phi' t$ is the limit of $\\dfrac{dx}{dt}$, the velocity is also this limit; that is, when a point does not move uniformly, the velocity is not represented by any increment of length divided by its increment of time, but by the limit to which that ratio continually tends, as the increment of time is diminished.",
   "why": "States the limit-of-ratio definition of velocity that a learner needs before differentiating motion.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/limit",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-42179164f4",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-an-illustration-from-dynamics-velocity-acceleration-etc",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "57",
   "location": "An Illustration from Dynamics: Velocity, Acceleration, etc",
   "latex": "That is, at the end of four seconds a falling body moves at the rate of $128\\frac{2}{3}$~feet per~second.",
   "markdown": "That is, at the end of four seconds a falling body moves at the rate of $128\\frac{2}{3}$ feet per second.",
   "why": "A worked numerical result that shows the derivative of distance giving velocity for falling bodies.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/falling-bodies",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2c44cc7b3f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "58",
   "location": "Simple Harmonic Motion",
   "latex": "But if $d\\theta$ be taken sufficiently small, $\\sin d\\theta$, and~$d\\theta$, may be made as nearly in a ratio of equality as we please, and $1 - \\cos d\\theta$ may be made as small a part as we please, either of $d\\theta$ or $\\sin d\\theta$.",
   "markdown": "But if $d\\theta$ be taken sufficiently small, $\\sin d\\theta$, and $d\\theta$, may be made as nearly in a ratio of equality as we please, and $1 - \\cos d\\theta$ may be made as small a part as we please, either of $d\\theta$ or $\\sin d\\theta$.",
   "why": "It shows the learner why a small angle lets sin dθ be replaced by dθ, which is the idea behind the whole derivation.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/infinitesimal",
    "concept/sine"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-192413e739",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "58",
   "location": "Simple Harmonic Motion",
   "latex": "we have equations, which, though never exactly true, are such that by making $d\\theta$ sufficiently small, the errors may be made as small parts of~$d\\theta$ as we please.",
   "markdown": "we have equations, which, though never exactly true, are such that by making $d\\theta$ sufficiently small, the errors may be made as small parts of $d\\theta$ as we please.",
   "why": "It teaches that an equation can be approximately true and still give exact velocities once the limit is taken.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/differential",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-f226a23356",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "59",
   "location": "Simple Harmonic Motion",
   "latex": "[The motion of the point~$M$ or the point~$N$ is called in physics a \\emph{simple harmonic motion}.]",
   "markdown": "[The motion of the point $M$ or the point $N$ is called in physics a *simple harmonic motion*.]",
   "why": "It gives the physical name for the motion, linking the circle picture to the vibrating-body case a learner meets in physics.",
   "use": [
    "website",
    "lesson"
   ],
   "concepts": [
    "concept/simple-harmonic-motion"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-50dc29dafd",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "59",
   "location": "Simple Harmonic Motion",
   "latex": "If the angle so described be always increased by equal angles in equal portions of time, the angular velocity is said to be uniform, and is measured by the number of angular units described in a unit of time.",
   "markdown": "If the angle so described be always increased by equal angles in equal portions of time, the angular velocity is said to be uniform, and is measured by the number of angular units described in a unit of time.",
   "why": "It defines uniform angular velocity in terms a learner can picture, as equal angles in equal times.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/uniform-motion",
    "quantity/angle",
    "quantity/angular-velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-99a02d809f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "59",
   "location": "Simple Harmonic Motion",
   "latex": "The same considerations of velocity which have been applied to the motion of a point along a line may also be applied to the motion of a line round a point.",
   "markdown": "The same considerations of velocity which have been applied to the motion of a point along a line may also be applied to the motion of a line round a point.",
   "why": "It shows that the ideas of velocity learned for straight motion carry over unchanged to rotation.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "quantity/angular-velocity",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-005f3361f0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-method-of-fluxions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "60",
   "location": "The Method of Fluxions",
   "latex": "If we suppose $y$ to be any function of~$x$, and that $x$~increases with a given velocity, $y$~will also increase or decrease with a velocity depending: (1)~upon the velocity of~$x$; (2)~upon the function which $y$ is of~$x$.",
   "markdown": "If we suppose $y$ to be any function of $x$, and that $x$ increases with a given velocity, $y$ will also increase or decrease with a velocity depending: (1) upon the velocity of $x$; (2) upon the function which $y$ is of $x$.",
   "why": "It gives the learner the core idea of fluxions: a function's rate of change depends on both the rate of x and the form of the function.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/fluxional-notation",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-3e6b912648",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-method-of-fluxions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "60",
   "location": "The Method of Fluxions",
   "latex": "If we diminish~$dt$, the term $\\dfrac{dx}{dt}\\, dx$ will diminish without limit, since one factor continually approaches to a given quantity, viz., the velocity of~$x$, and the other diminishes without limit.",
   "markdown": "If we diminish $dt$, the term $\\dfrac{dx}{dt}\\, dx$ will diminish without limit, since one factor continually approaches to a given quantity, viz., the velocity of $x$, and the other diminishes without limit.",
   "why": "It shows the limiting argument in action: a term drops away because one factor tends to a finite velocity while the other vanishes.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/fluxional-notation",
    "concept/infinitesimal",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-814afcca55",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-method-of-fluxions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "60",
   "location": "The Method of Fluxions",
   "latex": "The processes are the same in both methods, since the ratio of the velocities is the limiting ratio of the corresponding increments, or, according to Leibnitz, the ratio of the infinitely small increments.",
   "markdown": "The processes are the same in both methods, since the ratio of the velocities is the limiting ratio of the corresponding increments, or, according to Leibnitz, the ratio of the infinitely small increments.",
   "why": "It tells the learner that fluxions and differentials are two notations for one process, a useful historical and conceptual point.",
   "use": [
    "history",
    "lesson"
   ],
   "concepts": [
    "concept/differential",
    "concept/fluxional-notation",
    "concept/infinitesimal",
    "concept/limit",
    "person/gottfried-wilhelm-leibniz"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-cd3d38155a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "61",
   "location": "Accelerated Motion",
   "latex": "Thus, an impulse which changes the velocity from $50$ to $70$~feet per~second, is twice as great as one which changes it from $50$ to $60$~feet.",
   "markdown": "Thus, an impulse which changes the velocity from $50$ to $70$ feet per second, is twice as great as one which changes it from $50$ to $60$ feet.",
   "why": "A concrete numerical example that makes the idea of an impulse proportional to the velocity change easy to grasp.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/impulse",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-979d409a70",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "61",
   "location": "Accelerated Motion",
   "latex": "It is said to act uniformly, when the velocity acquired by the point in any one interval of time is the same as that acquired in any other interval of equal duration.",
   "markdown": "It is said to act uniformly, when the velocity acquired by the point in any one interval of time is the same as that acquired in any other interval of equal duration.",
   "why": "Defines uniform action of a force by equal velocity gains in equal times, which is the condition for uniformly accelerated motion.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/uniformly-accelerated-motion",
    "quantity/force"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2e20ace9d0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "62",
   "location": "Accelerated Motion",
   "latex": "Hence the limit to which we approximate by diminishing~$t'$ without limit, is the length described in the time~$t$ by a uniformly accelerated velocity, which shall increase from~$0$ to~$v$ in that time.",
   "markdown": "Hence the limit to which we approximate by diminishing $t'$ without limit, is the length described in the time $t$ by a uniformly accelerated velocity, which shall increase from $0$ to $v$ in that time.",
   "why": "Shows how a limit of finer and finer subdivisions gives the exact distance travelled, the step that introduces the calculus idea.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "concept/uniformly-accelerated-motion",
    "quantity/length"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fe68745400",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "61",
   "location": "Accelerated Motion",
   "latex": "\\emph{Force} is a name given to that which causes a change in the velocity of a body.",
   "markdown": "*Force* is a name given to that which causes a change in the velocity of a body.",
   "why": "Gives learners the plain meaning of force as whatever changes a body's velocity, before any formula appears.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "quantity/force",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a480a8a8a5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "61",
   "location": "Accelerated Motion",
   "latex": "It is plain that we cannot, by supposing any succession of impulses, however small, and however quickly repeated, arrive at a uniformly accelerated motion; because the length described between any two impulses will be uniformly described, which is inconsistent with the idea of continually accelerated velocity.",
   "markdown": "It is plain that we cannot, by supposing any succession of impulses, however small, and however quickly repeated, arrive at a uniformly accelerated motion; because the length described between any two impulses will be uniformly described, which is inconsistent with the idea of continually accelerated velocity.",
   "why": "Shows why a string of sudden impulses can never give smooth acceleration, which is the reason the chapter needs a limiting argument.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/impulse",
    "concept/uniformly-accelerated-motion"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-65135efea0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "62",
   "location": "Accelerated Motion",
   "latex": "In this substitute $v$ for~$nv'$, and $t$ for~$nt'$, which gives for the space described $\\frac{1}{2}v(t + t')$. The smaller we suppose~$t'$, the more nearly will this approach to~$\\frac{1}{2}vt$.",
   "markdown": "In this substitute $v$ for $nv'$, and $t$ for $nt'$, which gives for the space described $\\frac{1}{2}v(t + t')$. The smaller we suppose $t'$, the more nearly will this approach to $\\frac{1}{2}vt$.",
   "why": "Shows a learner how a finite sum is pushed toward a limit by making the subdivision finer, which is the core idea of the chapter.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/approximation",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-1d77e70501",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "65",
   "location": "Accelerated Motion",
   "latex": "And it must be observed that $6t$~is the differential coefficient of~$3t^{2}$, or the coefficient of~$dt$, in the development of~$3(t + dt)^{2}$.",
   "markdown": "And it must be observed that $6t$ is the differential coefficient of $3t^{2}$, or the coefficient of $dt$, in the development of $3(t + dt)^{2}$.",
   "why": "Shows that the accelerating force is the coefficient of dt in the expansion, which is how a derivative is read off from an expansion; source typos in the velocity table (the C-line term 3(t + 2dt)^3 should read ^2, and the velocity at A printed as 3t^{3} should read 3t^2) are left as printed, not corrected here.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-4fabc915b5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "64",
   "location": "Accelerated Motion",
   "latex": "But as the terms involving $(dt)^{2}$ in the velocities, etc., cannot be rejected without error, the above supposition of a uniform force cannot be made.",
   "markdown": "But as the terms involving $(dt)^{2}$ in the velocities, etc., cannot be rejected without error, the above supposition of a uniform force cannot be made.",
   "why": "Warns learners that dropping the small terms is only an approximation, so the uniform-force model is valid only up to an error that shrinks with dt.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/uniformly-accelerated-motion"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-480a0d560f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "67",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "It has been shown that by taking $\\dfrac{1}{x}$ sufficiently small, that is, by taking $x$ sufficiently great, any term of this series may be made to contain the aggregate of the succeeding terms, as often as we please; which relation is not altered if we multiply every term by~$x^{m}$, and so restore the original series.",
   "markdown": "It has been shown that by taking $\\dfrac{1}{x}$ sufficiently small, that is, by taking $x$ sufficiently great, any term of this series may be made to contain the aggregate of the succeeding terms, as often as we please; which relation is not altered if we multiply every term by $x^{m}$, and so restore the original series.",
   "why": "It explains why dividing a series by its leading power does not change its meaning, which a learner often finds hard to see.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/arrangement",
    "concept/infinity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b8d6e0b2d5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "67",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "Hence $\\dfrac{(x + 1)^{m}}{x^{m}} = 1 + \\dfrac{mx^{m-1} + \\etc.}{x^{m}}$, the numerator of which last fraction decreases indefinitely as compared with its denominator.",
   "markdown": "Hence $\\dfrac{(x + 1)^{m}}{x^{m}} = 1 + \\dfrac{mx^{m-1} + \\etc.}{x^{m}}$, the numerator of which last fraction decreases indefinitely as compared with its denominator.",
   "why": "It gives the reason that (x+1)^m over x^m tends to one, in a form a learner can check line by line.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/limit",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-6ad0cd8e6f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "66",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "The limit of any ratio may be found by rejecting any terms or aggregate of terms~($Q$) which are connected with another term~($P$) by the sign of addition or subtraction, provided that by increasing~$x$, $Q$~may be made as small a part of~$P$ as we please.",
   "markdown": "The limit of any ratio may be found by rejecting any terms or aggregate of terms ($Q$) which are connected with another term ($P$) by the sign of addition or subtraction, provided that by increasing $x$, $Q$ may be made as small a part of $P$ as we please.",
   "why": "It gives the learner the rule for dropping terms when finding a limit, with the condition that makes dropping them legitimate.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/dominant-term",
    "concept/limit",
    "method/finding-a-limit-by-rejecting-negligible-terms"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-bc7a92ef4c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "66",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "Divide both numerator and denominator by~$x^{2}$, which gives $1 + \\dfrac{2}{x} + \\dfrac{3}{x^{2}}$, and $2 + \\dfrac{5}{x}$, for the numerator and denominator of a fraction equal in value to the one proposed.",
   "markdown": "Divide both numerator and denominator by $x^{2}$, which gives $1 + \\dfrac{2}{x} + \\dfrac{3}{x^{2}}$, and $2 + \\dfrac{5}{x}$, for the numerator and denominator of a fraction equal in value to the one proposed.",
   "why": "It shows the standard step of dividing by the highest power of x so that the small terms can be seen to vanish.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "method/dividing-numerator-and-denominator-by-the-highest-power"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-05de4723b7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "66",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "It is easy to show that the increase of two magnitudes may cause a decrease of their ratio; so that, as the two increase without limit, their ratio may diminish without limit.",
   "markdown": "It is easy to show that the increase of two magnitudes may cause a decrease of their ratio; so that, as the two increase without limit, their ratio may diminish without limit.",
   "why": "It warns that a ratio of two growing quantities can shrink, so growth alone does not settle a limit.",
   "use": [
    "website",
    "lesson"
   ],
   "concepts": [
    "concept/infinity",
    "concept/limit",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-d259aa955c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "66",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "We will now prove the following: That in any series of decreasing powers of~$x$, any one term will, if $x$~be taken sufficiently great, contain the aggregate of all which follow, as many times as we please.",
   "markdown": "We will now prove the following: That in any series of decreasing powers of $x$, any one term will, if $x$ be taken sufficiently great, contain the aggregate of all which follow, as many times as we please.",
   "why": "It states the key lemma that lets a learner discard the tail of a power series when x is large.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/arrangement",
    "concept/dominant-term",
    "concept/infinity",
    "concept/power"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fc62278ae9",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "73",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "This result will be of use when we come to the first principles of the integral calculus.",
   "markdown": "This result will be of use when we come to the first principles of the integral calculus.",
   "why": "It tells the learner why these limit results matter, by linking them forward to the integral calculus.",
   "use": [
    "history",
    "lesson"
   ],
   "concepts": [
    "concept/calculus",
    "concept/integral",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ab8b2a35fe",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-recapitulation-of-results-reached-in-the-theory-of-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "74",
   "location": "Recapitulation of Results Reached in the Theory of Functions",
   "latex": "The following is a recapitulation of the principal results which have hitherto been noticed in the general theory of functions:",
   "markdown": "The following is a recapitulation of the principal results which have hitherto been noticed in the general theory of functions:",
   "why": "It tells the learner that this chapter gathers the results already established in the theory of functions, so the list that follows can be read as a summary.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-371bf9eadc",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-recapitulation-of-results-reached-in-the-theory-of-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "74",
   "location": "Recapitulation of Results Reached in the Theory of Functions",
   "latex": "That if in the equation $y = \\phi(x)$, the variable~$x$ receives an increment~$dx$, $y$~is increased by the series \\[ \\phi' x\\, dx + \\phi'' x\\, \\frac{(dx)^{2}}{2} + \\phi''' x\\, \\frac{(dx)^{3}}{2·3} + \\etc. \\]",
   "markdown": "That if in the equation $y = \\phi(x)$, the variable $x$ receives an increment $dx$, $y$ is increased by the series ’ x  dx + ” x  (dx)^22 + ”’ x  (dx)^32·3 + etc.",
   "why": "It states the expansion of a function under an increment, showing how the successive derivatives supply the coefficients of the powers of dx.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/function",
    "concept/higher-order-derivative",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-64fef63503",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-recapitulation-of-results-reached-in-the-theory-of-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "74",
   "location": "Recapitulation of Results Reached in the Theory of Functions",
   "latex": "$\\phi' x$ is the limit of~$\\dfrac{dy}{dx}$, or the quantity to which the latter will approach, and to which it may be brought as near as we please, when $dx$~is diminished.",
   "markdown": "$\\phi' x$ is the limit of $\\dfrac{dy}{dx}$, or the quantity to which the latter will approach, and to which it may be brought as near as we please, when $dx$ is diminished.",
   "why": "It gives the chapter's definition of the derivative as a limit of the ratio dy/dx, with the idea of approaching as near as we please.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/approximation",
    "concept/derivative",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-6673b8db03",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-recapitulation-of-results-reached-in-the-theory-of-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "74",
   "location": "Recapitulation of Results Reached in the Theory of Functions",
   "latex": "That $\\phi'' x$ is derived in the same manner from~$\\phi' x$, that $\\phi' x$~is from~$\\phi x$; viz., that in like manner as $\\phi' x$~is the coefficient of~$dx$ in the development of $\\phi(x + dx)$, so $\\phi'' x$~is the coefficient of~$dx$ in the development of $\\phi'(x + dx)$; similarly $\\phi''' x$~is the coefficient of~$dx$ in the development of~$\\phi''(x + dx)$, and so on.",
   "markdown": "That $\\phi'' x$ is derived in the same manner from $\\phi' x$, that $\\phi' x$ is from $\\phi x$; viz., that in like manner as $\\phi' x$ is the coefficient of $dx$ in the development of $\\phi(x + dx)$, so $\\phi'' x$ is the coefficient of $dx$ in the development of $\\phi'(x + dx)$; similarly $\\phi''' x$ is the coefficient of $dx$ in the development of $\\phi''(x + dx)$, and so on.",
   "why": "It shows how each higher derivative is found by repeating the same coefficient procedure on the previous one.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5d0244ec86",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-recapitulation-of-results-reached-in-the-theory-of-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "74",
   "location": "Recapitulation of Results Reached in the Theory of Functions",
   "latex": "That in every case which occurs in practice, $dx$~may be taken so small, that any term of the series above written may be made to contain the aggregate of those which follow, as often as we please; whence, though $\\phi' x\\, dx$~is not the actual increment produced by changing~$x$ into~$x + dx$ in the function~$\\phi x$, yet, by taking $dx$ sufficiently small, it may be brought as near as we please to a ratio of equality with the actual increment.",
   "markdown": "That in every case which occurs in practice, $dx$ may be taken so small, that any term of the series above written may be made to contain the aggregate of those which follow, as often as we please; whence, though $\\phi' x\\, dx$ is not the actual increment produced by changing $x$ into $x + dx$ in the function $\\phi x$, yet, by taking $dx$ sufficiently small, it may be brought as near as we please to a ratio of equality with the actual increment.",
   "why": "It explains that the differential is not the true increment but can be made to agree with it as closely as wanted, which is the core warning for using differentials.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/approximation",
    "concept/differential",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ee49dc297a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-recapitulation-of-results-reached-in-the-theory-of-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "74",
   "location": "Recapitulation of Results Reached in the Theory of Functions",
   "latex": "It is called the differential coefficient of~$y$.",
   "markdown": "It is called the differential coefficient of $y$.",
   "why": "It shows the older name for the derivative, which learners will meet in older texts.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-6f0cc4ce85",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "75",
   "location": "Approximations by the Differential Calculus",
   "latex": "These last are in the proportion of $h$ to~$k$, and hence results a proposition of the utmost importance in every practical application of mathematics, viz., that if two different, but small, errors be committed in the valuation of any quantity, the errors arising therefrom at the end of any process, in which both the supposed values of~$x$ are successively adopted, are very nearly in the proportion of the errors committed at the beginning.",
   "markdown": "These last are in the proportion of $h$ to $k$, and hence results a proposition of the utmost importance in every practical application of mathematics, viz., that if two different, but small, errors be committed in the valuation of any quantity, the errors arising therefrom at the end of any process, in which both the supposed values of $x$ are successively adopted, are very nearly in the proportion of the errors committed at the beginning.",
   "why": "It states the principle that small errors carry through a calculation in nearly the same proportion, which a learner can check on the worked example that follows.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/error-of-measurement",
    "theorem/propagation-of-small-errors"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ae2eea8377",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "75",
   "location": "Approximations by the Differential Calculus",
   "latex": "For example, let there be a right-angled triangle, whose base is~$3$, and whose other side should be~$4$, so that the hypothenuse should be $\\sqrt{3^{2} + 4^{2}}$ or~$5$.",
   "markdown": "For example, let there be a right-angled triangle, whose base is $3$, and whose other side should be $4$, so that the hypothenuse should be $\\sqrt{3^{2} + 4^{2}}$ or $5$.",
   "why": "It sets up a concrete right triangle with a known hypotenuse so the learner can follow how a measurement error moves through the calculation.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/error-of-measurement",
    "concept/hypotenuse",
    "concept/right-triangle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-9c7ed61625",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "75",
   "location": "Approximations by the Differential Calculus",
   "latex": "The errors of the hypothenuse are then $.0008$ and $.0016$ nearly; and these last are in the proportion of $.001$ and~$.002$.",
   "markdown": "The errors of the hypothenuse are then $.0008$ and $.0016$ nearly; and these last are in the proportion of $.001$ and $.002$.",
   "why": "It gives the numerical result of the triangle example, showing the proportional-error principle in action.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/hypotenuse",
    "theorem/propagation-of-small-errors"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c7974cf3ad",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "76",
   "location": "Approximations by the Differential Calculus",
   "latex": "It also follows, that if $x$~increase by successive equal steps, any function of~$x$ will, for a few steps, increase so nearly in the same manner, that the supposition of such an increase will not be materially wrong.",
   "markdown": "It also follows, that if $x$ increase by successive equal steps, any function of $x$ will, for a few steps, increase so nearly in the same manner, that the supposition of such an increase will not be materially wrong.",
   "why": "It explains why a function changing in equal small steps grows almost linearly over a short range, which gives the learner an intuition for linear approximation.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/function",
    "concept/increment",
    "method/linear-approximation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-437fc7e689",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "76",
   "location": "Approximations by the Differential Calculus",
   "latex": "The sun's longitude is a function of the time; that is, the number of years and days from a given epoch being given, and called~$x$, the sun's longitude can be found by an algebraical expression which may be called~$\\phi x$.",
   "markdown": "The sun’s longitude is a function of the time; that is, the number of years and days from a given epoch being given, and called $x$, the sun’s longitude can be found by an algebraical expression which may be called $\\phi x$.",
   "why": "It shows a physical quantity treated as a function of time, which makes the abstract function notation concrete for a learner.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/function",
    "quantity/solar-longitude",
    "quantity/time"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-96304af24a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "76",
   "location": "Approximations by the Differential Calculus",
   "latex": "And even for this interval, though it can hardly be called \\emph{small} in an astronomical point of view, the increments or decrements will be found so nearly the same for four or five days together, as to enable the student to form an idea how much more near they would be to equality, if the interval had been less, say one hour instead of twenty-four.",
   "markdown": "And even for this interval, though it can hardly be called *small* in an astronomical point of view, the increments or decrements will be found so nearly the same for four or five days together, as to enable the student to form an idea how much more near they would be to equality, if the interval had been less, say one hour instead of twenty-four.",
   "why": "It is a vivid, period-flavoured remark that invites the learner to imagine shrinking the step size and see the increments become more nearly equal.",
   "use": [
    "history",
    "website"
   ],
   "concepts": [
    "concept/approximation",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2d73479cca",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "Let $a + h$ be the real value, in which $h$~will be a small quantity. It follows that $\\phi(a + h) = 0$, or, which is nearly true, $\\phi a + \\phi' a\\, h = 0$.",
   "markdown": "Let $a + h$ be the real value, in which $h$ will be a small quantity. It follows that $\\phi(a + h) = 0$, or, which is nearly true, $\\phi a + \\phi' a\\, h = 0$.",
   "why": "It shows the reasoning behind the correction step: expanding phi(a + h) to first order gives the linear equation that determines h.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "method/newton-s-method"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-6434bd59fe",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "For example, let $x^{2} + x - 4 = 0$ be the equation. Here $\\phi x = x^{2} + x - 4$, and $\\phi(x + h) = (x + h)^{2} + x + h - 4 = x^{2} + x - 4 + (2x + 1)h + h^{2}$; so that $\\phi' x = 2x + 1$.",
   "markdown": "For example, let $x^{2} + x - 4 = 0$ be the equation. Here $\\phi x = x^{2} + x - 4$, and $\\phi(x + h) = (x + h)^{2} + x + h - 4 = x^{2} + x - 4 + (2x + 1)h + h^{2}$; so that $\\phi' x = 2x + 1$.",
   "why": "It works a concrete example, letting a learner see where the derivative 2x + 1 comes from by expanding and collecting terms in h.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/function",
    "concept/function-notation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-e4f1657e50",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "A near value of~$x$ is~$1.57$; let this be~$a$. Then $\\phi a = .0349$, and $\\phi' a = 4.14$. Hence $-\\dfrac{\\phi a}{\\phi' a} = -.00843$.",
   "markdown": "A near value of $x$ is $1.57$; let this be $a$. Then $\\phi a = .0349$, and $\\phi' a = 4.14$. Hence $-\\dfrac{\\phi a}{\\phi' a} = -.00843$.",
   "why": "It runs one full correction numerically, so a learner can follow each number through to the improved estimate.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "method/newton-s-method"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-6e57f96a72",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "If we proceed in the same way with~$1.5616$, we shall find a still nearer value of~$x$, viz., $1.561553$.",
   "markdown": "If we proceed in the same way with $1.5616$, we shall find a still nearer value of $x$, viz., $1.561553$.",
   "why": "It shows that the method can be repeated, with each pass giving a closer value of the root.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/approximation",
    "method/newton-s-method"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-517d1d9413",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "78",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "We have here chosen an equation of the second degree, in order that the student may be able to verify the result in the common way; it is, however, obvious that the same method may be applied to equations of higher degrees, and even to those which are not to be treated by common algebraical method, such as $\\tan x = ax$.",
   "markdown": "We have here chosen an equation of the second degree, in order that the student may be able to verify the result in the common way; it is, however, obvious that the same method may be applied to equations of higher degrees, and even to those which are not to be treated by common algebraical method, such as $\\tan x = ax$.",
   "why": "It tells the learner why the example is quadratic (so the answer can be checked by ordinary algebra) and that the method also reaches equations algebra cannot solve, such as tan x = ax.",
   "use": [
    "history",
    "lesson"
   ],
   "concepts": [
    "concept/degree",
    "concept/equation",
    "method/newton-s-method"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-86684b0135",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "80",
   "location": "Partial and Total Differentials",
   "latex": "Therefore the numerator of each of the fractions $\\dfrac{p}{a}$,~$\\dfrac{p}{b}$, and~$\\dfrac{p}{c}$, must never be separated from its denominator, because the value of the former depends, in part, upon the latter; and one~$p$ cannot be distinguished from another without its denominator.",
   "markdown": "Therefore the numerator of each of the fractions $\\dfrac{p}{a}$, $\\dfrac{p}{b}$, and $\\dfrac{p}{c}$, must never be separated from its denominator, because the value of the former depends, in part, upon the latter; and one $p$ cannot be distinguished from another without its denominator.",
   "why": "It warns the learner that a derivative symbol is one inseparable unit, so the numerator must never be cancelled away from its denominator.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/denominator",
    "concept/mathematical-notation",
    "concept/numerator",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5a01cc731b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "81",
   "location": "Partial and Total Differentials",
   "latex": "The last equation gives a striking illustration of the method of notation. Treated according to the common rules of algebra, it is $du = du + du$, which is absurd, but which appears rational when we recollect that the second~$du$ arises from a change in $x$~only, the third from a change in $y$~only, and the first from a change in both.",
   "markdown": "The last equation gives a striking illustration of the method of notation. Treated according to the common rules of algebra, it is $du = du + du$, which is absurd, but which appears rational when we recollect that the second $du$ arises from a change in $x$ only, the third from a change in $y$ only, and the first from a change in both.",
   "why": "It gives a vivid picture of why the same symbol du can mean three different increments without contradiction.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/differential",
    "concept/mathematical-notation",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-4cbe7b1bfb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "82",
   "location": "Partial and Total Differentials",
   "latex": "The symbol $\\phi(x, y)$ must not be confounded with~$\\phi(xy)$. The former represents any function of $x$~and~$y$; the latter a function in which $x$~and~$y$ only enter so far as they are contained in their product.",
   "markdown": "The symbol $\\phi(x, y)$ must not be confounded with $\\phi(xy)$. The former represents any function of $x$ and $y$; the latter a function in which $x$ and $y$ only enter so far as they are contained in their product.",
   "why": "It teaches the learner that a function written with a special form is only a particular case of a general function of the same variables.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/function",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-4cf9d381fc",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "80",
   "location": "Partial and Total Differentials",
   "latex": "Neither are we allowed to say that $\\dfrac{p}{a}$ divided by~$\\dfrac{p}{b}$ is~$\\dfrac{b}{a}$; for this supposes that $p$~means the same thing in both quantities.",
   "markdown": "Neither are we allowed to say that $\\dfrac{p}{a}$ divided by $\\dfrac{p}{b}$ is $\\dfrac{b}{a}$; for this supposes that $p$ means the same thing in both quantities.",
   "why": "It warns against cancelling a symbol whose meaning depends on its denominator, a common error with differential notation.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/mathematical-notation",
    "concept/partial-derivative",
    "method/division"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-380152ec4a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "83",
   "location": "Partial and Total Differentials",
   "latex": "The \\emph{etc.}\\ is the representative of an infinite series of terms, the aggregate of which diminishes continually with respect to $dp$,~$dq$,~etc., as the latter are diminished, and which, therefore, has no effect on the \\emph{limit} of the ratio of~$d.z$ to any other quantity.",
   "markdown": "The *etc.* is the representative of an infinite series of terms, the aggregate of which diminishes continually with respect to $dp$, $dq$, etc., as the latter are diminished, and which, therefore, has no effect on the *limit* of the ratio of $d.z$ to any other quantity.",
   "why": "It explains in plain terms why the omitted higher-order terms do not affect the limit that the calculus is seeking.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/infinitesimal",
    "concept/limit",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-722801497c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "79",
   "location": "Partial and Total Differentials",
   "latex": "in which, however, it must be remembered, that $du$~does not stand for the same thing in any two of the three equations: it is true that it always represents an increment of~$u$, but as far as we have yet gone, we have used it indifferently, whether the increment of~$u$ was the result of a change in $x$~only, or $y$~only, or both together.",
   "markdown": "in which, however, it must be remembered, that $du$ does not stand for the same thing in any two of the three equations: it is true that it always represents an increment of $u$, but as far as we have yet gone, we have used it indifferently, whether the increment of $u$ was the result of a change in $x$ only, or $y$ only, or both together.",
   "why": "It shows the learner that a single symbol can be used loosely until a clearer notation is needed to tell its cases apart.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/differential",
    "concept/increment",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-1b5cf29bd6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-application-of-the-theorem-for-total-differentials-to-the-determination-of-total-resultant-errors",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "84",
   "location": "Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors",
   "latex": "For if any result be obtained from a set of \\textit{data}, no one of which is exactly correct, the error in the result would be a very complicated function of the errors in the \\textit{data}, if the latter were considerable.",
   "markdown": "For if any result be obtained from a set of *data*, no one of which is exactly correct, the error in the result would be a very complicated function of the errors in the *data*, if the latter were considerable.",
   "why": "It explains why errors in several data are hard to handle together, which is the motivation for the method that follows.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/error-of-measurement",
    "concept/function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-cfa0964183",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-application-of-the-theorem-for-total-differentials-to-the-determination-of-total-resultant-errors",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "84",
   "location": "Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors",
   "latex": "When they are small, the error in the results is very nearly the sum of the errors which would arise from the error in each \\textit{datum}, if all the others were correct.",
   "markdown": "When they are small, the error in the results is very nearly the sum of the errors which would arise from the error in each *datum*, if all the others were correct.",
   "why": "It states the central idea in one sentence: small errors can be treated one at a time and then added.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/error-of-measurement",
    "theorem/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b81e8b4c80",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-application-of-the-theorem-for-total-differentials-to-the-determination-of-total-resultant-errors",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "85",
   "location": "Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors",
   "latex": "Next suppose only the second error, and then only the third to exist, and calculate the effect of each separately, all which may be done by simple formulæ.",
   "markdown": "Next suppose only the second error, and then only the third to exist, and calculate the effect of each separately, all which may be done by simple formulæ.",
   "why": "It shows the practical procedure of isolating each error in turn, a method a learner can copy in any measurement problem.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/error-of-measurement",
    "instrument/transit-instrument"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b788dc34d7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-application-of-the-theorem-for-total-differentials-to-the-determination-of-total-resultant-errors",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "85",
   "location": "Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors",
   "latex": "The effect of all the errors will then be the sum of the effects of each separate error, at least with sufficient accuracy for practical purposes.",
   "markdown": "The effect of all the errors will then be the sum of the effects of each separate error, at least with sufficient accuracy for practical purposes.",
   "why": "It makes clear that the summed result is an approximation, not an exact value, which a learner should keep in mind.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/error-of-measurement"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-d6ff53c393",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-application-of-the-theorem-for-total-differentials-to-the-determination-of-total-resultant-errors",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "85",
   "location": "Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors",
   "latex": "The formulæ employed, like the equations in \\PageRef{28}, are not actually true in any case, but approach more near to the truth as the errors are diminished.",
   "markdown": "The formulæ employed, like the equations in 28, are not actually true in any case, but approach more near to the truth as the errors are diminished.",
   "why": "It is an honest statement that the method is approximate and improves as the errors shrink, a point worth teaching alongside the method.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/error-of-measurement",
    "theorem/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a2f813b0aa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "This term, in theory, is the only one on which the \\emph{limit} of the ratio of the increments depends; in practice, it is sufficiently near to the real increment of~$y$, if the increment of~$x$ be small.",
   "markdown": "This term, in theory, is the only one on which the *limit* of the ratio of the increments depends; in practice, it is sufficiently near to the real increment of $y$, if the increment of $x$ be small.",
   "why": "It tells the learner that the derivative is the part of the increment that matters as the increment of x shrinks, and that the rest is negligible in practice.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/increment",
    "concept/limit",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-8764efbbaa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "When the exponent is negative, or when $y = \\dfrac{1}{x^{m}}$, $dy = -\\dfrac{m\\, dx}{x^{m+1}}$, or when $y = x^{-m}$, $dy = -mx^{-m-1}\\, dx$, which is according to the rule.",
   "markdown": "When the exponent is negative, or when $y = \\dfrac{1}{x^{m}}$, $dy = -\\dfrac{m\\, dx}{x^{m+1}}$, or when $y = x^{-m}$, $dy = -mx^{-m-1}\\, dx$, which is according to the rule.",
   "why": "It shows the power rule carrying over to negative exponents, so the learner can check the sign and the exponent change by hand.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/exponent",
    "theorem/power-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-3d0b68f365",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "The negative sign indicates that an increase in~$x$ decreases the value of~$y$; which, in this case, is evident.",
   "markdown": "The negative sign indicates that an increase in $x$ decreases the value of $y$; which, in this case, is evident.",
   "why": "It explains in plain words what the minus sign in a derivative means, so the learner can read it as a direction of change.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential",
    "theorem/power-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b16d51ec5b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "(2) $y = a^{x}$. Here $dy = a^{x}\\log a\\, dx$ where the logarithm (as is always the case in analysis, except where the contrary is specially mentioned) is the Naperian or hyperbolic logarithm.",
   "markdown": "(2) $y = a^{x}$. Here $dy = a^{x}\\log a\\, dx$ where the logarithm (as is always the case in analysis, except where the contrary is specially mentioned) is the Naperian or hyperbolic logarithm.",
   "why": "It states the convention that log means the natural logarithm in analysis, which a learner needs before reading any exponential derivative.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/exponential-function",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-0ddc2aa34e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "$y = \\log x$ (the Naperian logarithm). Here $dy = \\dfrac{dx}{x}$.",
   "markdown": "$y = \\log x$ (the Naperian logarithm). Here $dy = \\dfrac{dx}{x}$.",
   "why": "It gives the single most-used logarithm derivative in its simplest form, so the learner can recall it at a glance.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/natural-logarithm",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-89de2a30cd",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-illustration-of-the-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Illustration of the Rules for Differentiation",
   "latex": "At the risk of being tedious to some readers, we will proceed to illustrate these formulæ by examples from the tables of logarithms and sines",
   "markdown": "At the risk of being tedious to some readers, we will proceed to illustrate these formulæ by examples from the tables of logarithms and sines",
   "why": "It tells the learner plainly why the chapter stops to check the general rules against numerical tables before going on.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/common-logarithm",
    "concept/sine",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-eae3e7b2b7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-illustration-of-the-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "87",
   "location": "Illustration of the Rules for Differentiation",
   "latex": "Let $x = 1000$, whence $y = \\text{common log}~ 1000 =3$; and let $dx = 1$, or let it be required to find the common logarithm of $1000 + 1$, or~$1001$.",
   "markdown": "Let $x = 1000$, whence $y = \\text{common log}~ 1000 =3$; and let $dx = 1$, or let it be required to find the common logarithm of $1000 + 1$, or $1001$.",
   "why": "It gives a concrete numerical case in which the increment of a logarithm can be checked against a table.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/common-logarithm",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b036f20ea7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-illustration-of-the-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "87",
   "location": "Illustration of the Rules for Differentiation",
   "latex": "The tables give $3.0004341$, differing from the former only in the $7$\\Chg{th}{\\th}~place of decimals.",
   "markdown": "The tables give $3.0004341$, differing from the former only in the $7$thth place of decimals.",
   "why": "It shows the learner how close a first-term approximation is, and how far it can be trusted, when checked against a table.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/common-logarithm",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-7c23a73046",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-illustration-of-the-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "87",
   "location": "Illustration of the Rules for Differentiation",
   "latex": "Let $x = 16°$, in which case $\\sin x = .2756374$, and $\\cos x = .9612617$.",
   "markdown": "Let $x = 16°$, in which case $\\sin x = .2756374$, and $\\cos x = .9612617$.",
   "why": "It supplies the sine and cosine values needed to carry out the sine example by hand.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cosine",
    "concept/sine",
    "quantity/angle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fd43c4f2f8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-illustration-of-the-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "87",
   "location": "Illustration of the Rules for Differentiation",
   "latex": "These examples may serve to show how nearly the real ratio of two increments approaches to their limit, when the increments themselves are small.",
   "markdown": "These examples may serve to show how nearly the real ratio of two increments approaches to their limit, when the increments themselves are small.",
   "why": "It states the central idea of the chapter: a ratio of increments approaches its limit as the increments shrink.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/increment",
    "concept/limit",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-452dd44f98",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients-of-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "88",
   "location": "Differential Coefficients of Differential Coefficients",
   "latex": "has been found, the result, being a function of~$x$, may be also differentiated, which gives the differential coefficient of the differential coefficient, or, as it is called, the \\emph{second} differential coefficient.",
   "markdown": "has been found, the result, being a function of $x$, may be also differentiated, which gives the differential coefficient of the differential coefficient, or, as it is called, the *second* differential coefficient.",
   "why": "It shows in plain words that a derivative is itself a function, so it can be differentiated again to give the second derivative.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-531305048b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients-of-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "88",
   "location": "Differential Coefficients of Differential Coefficients",
   "latex": "Similarly the differential coefficient of the second differential coefficient is called the third differential coefficient, and so on.",
   "markdown": "Similarly the differential coefficient of the second differential coefficient is called the third differential coefficient, and so on.",
   "why": "It gives the naming pattern for successive orders, which learners can extend to any order.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-195fa2b09d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients-of-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "88",
   "location": "Differential Coefficients of Differential Coefficients",
   "latex": "We have already had occasion to notice these successive differential coefficients in \\PageRef{22}, where it appears that $\\phi' x$~being the first differential coefficient of~$\\phi x$, $\\phi'' x$~is the coefficient of~$h$ in the development $\\phi'(x + h)$, and is therefore the differential coefficient of~$\\phi' x$, or what we have called the second differential coefficient of~$\\phi x$.",
   "markdown": "We have already had occasion to notice these successive differential coefficients in 22, where it appears that $\\phi' x$ being the first differential coefficient of $\\phi x$, $\\phi'' x$ is the coefficient of $h$ in the development $\\phi'(x + h)$, and is therefore the differential coefficient of $\\phi' x$, or what we have called the second differential coefficient of $\\phi x$.",
   "why": "It links the second derivative to the coefficient of h in the expansion of phi'(x + h), giving a concrete meaning to the notation.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-22ba1963fd",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients-of-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "88",
   "location": "Differential Coefficients of Differential Coefficients",
   "latex": "In order to avoid so cumbrous a system of notation, the following symbols are usually preferred,",
   "markdown": "In order to avoid so cumbrous a system of notation, the following symbols are usually preferred,",
   "why": "It explains why the compact d^2y/dx^2 form replaced the nested fractions, a useful point about notation for learners to remember.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/mathematical-notation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-97622ff023",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "89",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "The symbol~$\\Delta x$ is called the \\emph{difference} of~$x$, being the difference between the value of the variable~$x$, before and after its increase.",
   "markdown": "The symbol $\\Delta x$ is called the *difference* of $x$, being the difference between the value of the variable $x$, before and after its increase.",
   "why": "It gives the learner a plain definition of the difference Delta x as the change in the variable across one step.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/difference"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-1a32f30e91",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "90",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "And the student must recollect, that in like manner as $\\Delta$~is not the symbol of a number, but of an operation, so $\\Delta^{2}$~does not denote a number multiplied by itself, but an operation repeated upon its own result; just as the logarithm of the logarithm of~$x$ might be written $\\log^{2} x$; $(\\log x)^{2}$~being reserved to signify the square of the logarithm of~$x$.",
   "markdown": "And the student must recollect, that in like manner as $\\Delta$ is not the symbol of a number, but of an operation, so $\\Delta^{2}$ does not denote a number multiplied by itself, but an operation repeated upon its own result; just as the logarithm of the logarithm of $x$ might be written $\\log^{2} x$; $(\\log x)^{2}$ being reserved to signify the square of the logarithm of $x$.",
   "why": "It warns the learner that Delta squared names a repeated operation, not a squared number, and compares it with log squared.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/higher-order-difference",
    "concept/mathematical-notation",
    "concept/operation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-cea321d8f1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "89",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "If $y$~be a function which decreases when $x$~is increased, $y_{1} - y$, or $\\Delta y$ is negative.",
   "markdown": "If $y$ be a function which decreases when $x$ is increased, $y_{1} - y$, or $\\Delta y$ is negative.",
   "why": "It shows that a decreasing function gives a negative difference, which makes the sign of a difference meaningful.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/decrement",
    "concept/difference"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-8c38b46153",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "90",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "And as we have denoted the operation which deduces the second column from the first by~$\\Delta$, so that which deduces the third from the second may be denoted by~$\\Delta\\Delta$, which is abbreviated into~$\\Delta^{2}$.",
   "markdown": "And as we have denoted the operation which deduces the second column from the first by $\\Delta$, so that which deduces the third from the second may be denoted by $\\Delta\\Delta$, which is abbreviated into $\\Delta^{2}$.",
   "why": "It explains in one sentence how the repeated difference columns of a table acquire their notation.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/calculus-of-finite-differences",
    "concept/higher-order-difference"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-948378a4a7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "92",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "Hence we have a succession of ratios $\\dfrac{dy}{dx}$, $\\dfrac{d^{2} y}{dx^{2}}$, $\\dfrac{d^{3} y}{dx^{3}}$, etc., which tend towards finite limits when $dx$~is diminished.",
   "markdown": "Hence we have a succession of ratios $\\dfrac{dy}{dx}$, $\\dfrac{d^{2} y}{dx^{2}}$, $\\dfrac{d^{3} y}{dx^{3}}$, etc., which tend towards finite limits when $dx$ is diminished.",
   "why": "It links the successive ratios dy/dx, d2y/dx2 and so on to the idea of a limit, which is the core of higher-order derivatives.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-dd489a7eaa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "93",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "Write $dx$ for~$\\Delta x$, etc., and recollect that $h - dx$, $h - 2\\, dx$, etc., continually approximate to~$h$.",
   "markdown": "Write $dx$ for $\\Delta x$, etc., and recollect that $h - dx$, $h - 2\\, dx$, etc., continually approximate to $h$.",
   "why": "It explains in plain terms why the steps can be made smaller without changing the result, which is the reasoning behind Taylor's series.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/limit",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-cce5c84c3b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "96",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "Generally, the complete differential coefficient of~$z$ with respect to~$x$, will contain as many terms as there are different ways in which $z$ contains~$x$. From looking at a complete differential coefficient, we may see in what manner the function contained its variable.",
   "markdown": "Generally, the complete differential coefficient of $z$ with respect to $x$, will contain as many terms as there are different ways in which $z$ contains $x$. From looking at a complete differential coefficient, we may see in what manner the function contained its variable.",
   "why": "It tells the learner how to read a total derivative: each term corresponds to one route by which the function depends on the variable.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/function",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a0f72cc48f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "100",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "Find the differential coefficient belonging to each of the ways in which $z$ will contain~$x$, as if it were the only way; the sum of these results (with their proper signs) will be the total differential coefficient.",
   "markdown": "Find the differential coefficient belonging to each of the ways in which $z$ will contain $x$, as if it were the only way; the sum of these results (with their proper signs) will be the total differential coefficient.",
   "why": "It states the procedure in one sentence: differentiate along each route separately, then add the results.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/partial-differential",
    "concept/total-differential",
    "method/implicit-differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-4d168900a3",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "99",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "In saying that $z$~is a function of $x$~and~$y$, and that $y$~is a function of~$x$, we have first supposed~$x$ to vary, $y$~remaining the same. The student must not imagine that $y$~is \\emph{then} a function of~$x$; for if so, it would vary when $x$~varied.",
   "markdown": "In saying that $z$ is a function of $x$ and $y$, and that $y$ is a function of $x$, we have first supposed $x$ to vary, $y$ remaining the same. The student must not imagine that $y$ is *then* a function of $x$; for if so, it would vary when $x$ varied.",
   "why": "It warns the learner against the common mistake of letting y change with x during the partial step, which would double-count the indirect route.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/indirect-function",
    "concept/partial-differential",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-f6b0394639",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "100",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "let $z = \\log(x^{2} + a^{2})$. If we make $y = x^{2} + a^{2}$, we have $z = \\log y$, and $\\dfrac{dz}{dy} = \\dfrac{1}{y}$; while from the first equation $\\dfrac{dy}{dx} = 2x$.",
   "markdown": "let $z = \\log(x^{2} + a^{2})$. If we make $y = x^{2} + a^{2}$, we have $z = \\log y$, and $\\dfrac{dz}{dy} = \\dfrac{1}{y}$; while from the first equation $\\dfrac{dy}{dx} = 2x$.",
   "why": "It shows the method on a short example: introduce an intermediate variable, differentiate each link, and multiply the results.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/logarithm",
    "method/implicit-differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-6c0fe33a3d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "100",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "If $z = \\log\\log\\sin x$, or the logarithm of the logarithm of~$\\sin x$, let $\\sin x = y$ and $\\log y = a$; whence $z= \\log a$, and contains~$x$, because $a$ contains~$y$, which contains~$x$.",
   "markdown": "If $z = \\log\\log\\sin x$, or the logarithm of the logarithm of $\\sin x$, let $\\sin x = y$ and $\\log y = a$; whence $z= \\log a$, and contains $x$, because $a$ contains $y$, which contains $x$.",
   "why": "It shows a chain of three intermediate quantities, so the learner can see how each link adds one factor to the total derivative.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/indirect-function",
    "concept/logarithm",
    "method/implicit-differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-746a6d3021",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "97",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "When $x$~is contained in~$y$, and $y$~is contained in~$z$, we shall say that $z$~is an indirect function of~$x$ \\emph{through}~$y$.",
   "markdown": "When $x$ is contained in $y$, and $y$ is contained in $z$, we shall say that $z$ is an indirect function of $x$ *through* $y$.",
   "why": "It gives the chapter's precise meaning of an indirect function, which the rest of the argument depends on.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/indirect-function",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-d633470383",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "101",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "We must leave $\\dfrac{da}{dx}$ and $\\dfrac{db}{dx}$ as we find them, until we\nknow \\emph{what} functions $a$~and~$b$ are of~$x$; but as we\nknow what function $z$~is of $a$~and~$b$, we substitute for\n$\\dfrac{dz}{da}$ and~$\\dfrac{dz}{db}$.",
   "markdown": "We must leave $\\dfrac{da}{dx}$ and $\\dfrac{db}{dx}$ as we find them, until we know *what* functions $a$ and $b$ are of $x$; but as we know what function $z$ is of $a$ and $b$, we substitute for $\\dfrac{dz}{da}$ and $\\dfrac{dz}{db}$.",
   "why": "It shows the learner the logic of the method: leave the inner derivatives unevaluated until the functions are known, and substitute the partial derivatives of the outer function first.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/partial-derivative",
    "theorem/chain-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-e8c5bb9df0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "Let $z = \\dfrac{a}{b}$. If $a$~become $a + da$, $z$~becomes\n$\\dfrac{a + da}{b}$ or $\\dfrac{a}{b} + \\dfrac{da}{b}$, and $\\dfrac{dz}{da}$ is~$\\dfrac{1}{b}$.",
   "markdown": "Let $z = \\dfrac{a}{b}$. If $a$ become $a + da$, $z$ becomes $\\dfrac{a + da}{b}$ or $\\dfrac{a}{b} + \\dfrac{da}{b}$, and $\\dfrac{dz}{da}$ is $\\dfrac{1}{b}$.",
   "why": "It walks through a quotient case step by step, showing how a small change in the numerator produces the partial derivative.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/partial-derivative",
    "theorem/quotient-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2dd78d8657",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "Again,\n$a^{b+db} = a^{b}\\, a^{db} = a^{b}(1 + \\log a\\, db + \\etc.)$ whence $\\dfrac{dz}{db} = a^{b} \\log a$.",
   "markdown": "Again, $a^{b+db} = a^{b}\\, a^{db} = a^{b}(1 + \\log a\\, db + \\etc.)$ whence $\\dfrac{dz}{db} = a^{b} \\log a$.",
   "why": "It shows how the logarithm of the base enters the derivative of an exponential when the exponent is varied.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/logarithm",
    "theorem/chain-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-189ffb2d36",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "101",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "In this case, and part\nof the following, the limiting ratio of the increments\nis the same as that of the increments themselves.",
   "markdown": "In this case, and part of the following, the limiting ratio of the increments is the same as that of the increments themselves.",
   "why": "It warns that the equality of a ratio of increments and its limit holds only in these particular cases, not in general.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/increment",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-f92a3e12c8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Inverse Functions",
   "latex": "It is not necessary that we should be able to solve the equation $y = \\phi x$ in finite terms, that is, so as to give a value of~$x$ without infinite series; it is sufficient that $x$~can be so expressed that the value of~$x$ corresponding to any value of~$y$ may be found as near as we please from $x = \\psi y$, in the same manner as the value of~$y$ corresponding to any value of~$x$ is found from $y = \\phi x$.",
   "markdown": "It is not necessary that we should be able to solve the equation $y = \\phi x$ in finite terms, that is, so as to give a value of $x$ without infinite series; it is sufficient that $x$ can be so expressed that the value of $x$ corresponding to any value of $y$ may be found as near as we please from $x = \\psi y$, in the same manner as the value of $y$ corresponding to any value of $x$ is found from $y = \\phi x$.",
   "why": "It explains that an inverse need not be solvable in finite terms, only computable to any wanted accuracy, which a learner needs before trusting inverse functions.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/approximation",
    "concept/function",
    "concept/inverse-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fdc09fa29e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "103",
   "location": "Inverse Functions",
   "latex": "That is, the effect of the operation or set of operations denoted by~$\\psi$ is destroyed by the effect of those denoted by~$\\phi$; as in the instances $(x^{2})^{\\efrac{1}{2}}$, $(x^{3})^{\\efrac{1}{3}}$, $e^{\\log x}$, angle whose sine is~$(\\sin x)$, etc., each of which is equal to~$x$.",
   "markdown": "That is, the effect of the operation or set of operations denoted by $\\psi$ is destroyed by the effect of those denoted by $\\phi$; as in the instances $(x^{2})^{\\efrac{1}{2}}$, $(x^{3})^{\\efrac{1}{3}}$, $e^{\\log x}$, angle whose sine is $(\\sin x)$, etc., each of which is equal to $x$.",
   "why": "It gives concrete cases where undoing one operation after another returns x, making the idea of an inverse easy to check.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/composition-of-functions",
    "concept/inverse-function",
    "concept/logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-7f7251efdf",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "103",
   "location": "Inverse Functions",
   "latex": "Hence $\\dfrac{dx}{dy}$ as deduced from the second, and $\\dfrac{dy}{dx}$ as deduced from the first, are reciprocals for every value of~$dx$.",
   "markdown": "Hence $\\dfrac{dx}{dy}$ as deduced from the second, and $\\dfrac{dy}{dx}$ as deduced from the first, are reciprocals for every value of $dx$.",
   "why": "It shows that the derivatives of a function and its inverse are reciprocals, a result a learner can test on examples.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/inverse-function",
    "concept/reciprocal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5b87b971e4",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "104",
   "location": "Inverse Functions",
   "latex": "There is no very obvious analogy between $\\dfrac{d^{2} y}{dx^{2}}$ and $\\dfrac{d^{2} x}{dy^{2}}$; indeed no such appears from the method in which these coefficients were first formed.",
   "markdown": "There is no very obvious analogy between $\\dfrac{d^{2} y}{dx^{2}}$ and $\\dfrac{d^{2} x}{dy^{2}}$; indeed no such appears from the method in which these coefficients were first formed.",
   "why": "A candid remark from the author that the two second derivatives look unrelated at first, which invites the reader to look for the connection.",
   "use": [
    "history",
    "website"
   ],
   "concepts": [
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-03b4a33dd5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "104",
   "location": "Inverse Functions",
   "latex": "Therefore $dy$,~$dy_{1}$, etc., are not equal; whence arises the next column of second differences, or $d^{2} y$, $d^{2} y_{1}$, etc.",
   "markdown": "Therefore $dy$, $dy_{1}$, etc., are not equal; whence arises the next column of second differences, or $d^{2} y$, $d^{2} y_{1}$, etc.",
   "why": "It shows how second differences arise from unequal first differences, the starting point of the limiting ratio definition.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential",
    "concept/higher-order-difference"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-7ed41c3747",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "104",
   "location": "Inverse Functions",
   "latex": "The resulting values of~$y$, or $y$,~$y_{1}$, etc., are not equidistant, except in one function only, when $y = ax + b$, where $a$~and~$b$ are constant.",
   "markdown": "The resulting values of $y$, or $y$, $y_{1}$, etc., are not equidistant, except in one function only, when $y = ax + b$, where $a$ and $b$ are constant.",
   "why": "It identifies the single case, a straight line, in which equal steps in x give equal steps in y, a useful check for learners.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant",
    "concept/function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-db7b930150",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "104",
   "location": "Inverse Functions",
   "latex": "The limiting ratio of $d^{2} y$ to~$(dx)^{2}$, expressed by~$\\dfrac{d^{2} y}{dx^{2}}$, is the second differential coefficient of~$y$ with respect to~$x$.",
   "markdown": "The limiting ratio of $d^{2} y$ to $(dx)^{2}$, expressed by $\\dfrac{d^{2} y}{dx^{2}}$, is the second differential coefficient of $y$ with respect to $x$.",
   "why": "It defines the second derivative as a limiting ratio of differences, tying the notation dy/dx to a concrete limit.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/higher-order-difference"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b272c3af95",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "107",
   "location": "Implicit Functions",
   "latex": "In this case $y$~is said to be \\emph{implicitly} a function of~$x$, or an implicit function.",
   "markdown": "In this case $y$ is said to be *implicitly* a function of $x$, or an implicit function.",
   "why": "It gives the defining sentence for an implicit function, the idea the whole section is built on.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/implicit-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-7c6e08f887",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "107",
   "location": "Implicit Functions",
   "latex": "For example, in $x^{2} - xy + y^{2} = a$, when $x$~is known, $y$~must be determined by the solution of an equation of the second degree.",
   "markdown": "For example, in $x^{2} - xy + y^{2} = a$, when $x$ is known, $y$ must be determined by the solution of an equation of the second degree.",
   "why": "A concrete quadratic shows the learner why y is hidden inside an equation and must be solved for.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/implicit-function",
    "method/solving-an-equation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-038f54f7d7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "107",
   "location": "Implicit Functions",
   "latex": "Here, though we know that $y$~must be a function of~$x$, we do not know, without further investigation, what function it is.",
   "markdown": "Here, though we know that $y$ must be a function of $x$, we do not know, without further investigation, what function it is.",
   "why": "It names the key difficulty: a function can exist without its formula being known.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/explicit-function",
    "concept/implicit-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-d6b8ff72bd",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "108",
   "location": "Implicit Functions",
   "latex": "$y$~and~$x$ are no longer independent; for, one of them being given, the other must be so taken that the equation $\\phi(x, y) = 0$ may be satisfied.",
   "markdown": "$y$ and $x$ are no longer independent; for, one of them being given, the other must be so taken that the equation $\\phi(x, y) = 0$ may be satisfied.",
   "why": "It explains why the differentials of x and y are tied together once the constraint holds.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/relation-between-variables",
    "concept/variable",
    "method/implicit-differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b9cb535ed4",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "110",
   "location": "Implicit Functions",
   "latex": "Hence $\\dfrac{dy}{dx}$ (meaning the limit) is~$-\\dfrac{1}{x^{2}}$, which will also be the result of~\\Eq{(3)} if $1 + \\dfrac{1}{x}$ be substituted for~$y$.",
   "markdown": "Hence $\\dfrac{dy}{dx}$ (meaning the limit) is $-\\dfrac{1}{x^{2}}$, which will also be the result of (3) if $1 + \\dfrac{1}{x}$ be substituted for $y$.",
   "why": "It checks the implicit-differentiation result against direct differentiation of the solved function, a good habit for learners to copy.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/derivative",
    "concept/limit",
    "method/implicit-differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-2e7a78e6e3",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-fluxions-and-the-idea-of-time",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "111",
   "location": "Fluxions, and the Idea of Time",
   "latex": "Here we want a new word, which has not been invented for the world at large, since none but mathematicians consider the subject; which word, if the change considered were change of place, depending upon change of time, would be \\emph{velocity}.",
   "markdown": "Here we want a new word, which has not been invented for the world at large, since none but mathematicians consider the subject; which word, if the change considered were change of place, depending upon change of time, would be *velocity*.",
   "why": "It shows how the word velocity arises from a change of place over time, giving learners a physical way into the idea.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "quantity/time",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-04d04dcc26",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-fluxions-and-the-idea-of-time",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "111",
   "location": "Fluxions, and the Idea of Time",
   "latex": "Imagine the diameter of a circle divided into a million of equal parts, from each of which a perpendicular is drawn meeting the circle.",
   "markdown": "Imagine the diameter of a circle divided into a million of equal parts, from each of which a perpendicular is drawn meeting the circle.",
   "why": "It gives learners a concrete picture of many changing quantities that are considered together.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/circle",
    "concept/diameter",
    "concept/perpendicular"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-f7b20f15d0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-fluxions-and-the-idea-of-time",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "111",
   "location": "Fluxions, and the Idea of Time",
   "latex": "We may answer that the notion of time is only necessary, inasmuch as we are not able to consider more than one thing at a time.",
   "markdown": "We may answer that the notion of time is only necessary, inasmuch as we are not able to consider more than one thing at a time.",
   "why": "It explains why time enters the study of changing quantities: the mind considers one thing at a time.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "quantity/time"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-93fb8b751b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-fluxions-and-the-idea-of-time",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "112",
   "location": "Fluxions, and the Idea of Time",
   "latex": "But we, who cannot consider all these perpendiculars at once, are obliged to take one after another.",
   "markdown": "But we, who cannot consider all these perpendiculars at once, are obliged to take one after another.",
   "why": "It gives a plain reason for taking successive states of a function one after another.",
   "use": [
    "website"
   ],
   "concepts": [
    "quantity/time"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-8141791aec",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-fluxions-and-the-idea-of-time",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "112",
   "location": "Fluxions, and the Idea of Time",
   "latex": "and succession, disguise it as we may, is the identical idea of time introduced in Newton's Method of Fluxions.",
   "markdown": "and succession, disguise it as we may, is the identical idea of time introduced in Newton’s Method of Fluxions.",
   "why": "It places Newton's fluxions in the historical debate about whether time belongs in mathematics.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/fluxional-notation",
    "quantity/time"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fc2d97d5af",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-fluxions-and-the-idea-of-time",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "110",
   "location": "Fluxions, and the Idea of Time",
   "latex": "who should never take it for granted that because he has made some progress in a science, he understands the first principles, which are often, if not always, the last to be learned well.",
   "markdown": "who should never take it for granted that because he has made some progress in a science, he understands the first principles, which are often, if not always, the last to be learned well.",
   "why": "It warns learners not to assume that progress in a subject means they understand its basic principles.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/differentiating-from-first-principles"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-27158e53e6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-fluxions-and-the-idea-of-time",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "111",
   "location": "Fluxions, and the Idea of Time",
   "latex": "The number which represents this line (reference being made to a given linear unit) is in a corresponding state of increase or decrease, and so is every function of this number, or every algebraical expression in the formation of which it is required.",
   "markdown": "The number which represents this line (reference being made to a given linear unit) is in a corresponding state of increase or decrease, and so is every function of this number, or every algebraical expression in the formation of which it is required.",
   "why": "It shows how a changing distance, measured in units, makes every function built from it change too.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/algebraic-expression",
    "concept/function",
    "concept/number",
    "unit/unit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-30a6c07b6f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-fluxions-and-the-idea-of-time",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "110",
   "location": "Fluxions, and the Idea of Time",
   "latex": "This not being the case, it is a cause of embarrassment to the student, that he is introduced at once to a definition so refined as that of the limiting ratio which the increment of a function bears to the increment of its variable.",
   "markdown": "This not being the case, it is a cause of embarrassment to the student, that he is introduced at once to a definition so refined as that of the limiting ratio which the increment of a function bears to the increment of its variable.",
   "why": "It names the difficulty of the limiting-ratio definition and so prepares learners to take it slowly.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/increment",
    "concept/limit",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-721f28bb80",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-differential-coefficient-considered-with-respect-to-its-magnitude",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "112",
   "location": "The Differential Coefficient Considered with Respect to its Magnitude",
   "latex": "value of the variable, is, if we may use the phrase, the \\emph{index} of the change which the function would receive if the value of the variable were increased.",
   "markdown": "value of the variable, is, if we may use the phrase, the *index* of the change which the function would receive if the value of the variable were increased.",
   "why": "It gives the learner the core idea of the differential coefficient as the index of how a function is changing at a given value of the variable.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/function",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a49939cbf2",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-differential-coefficient-considered-with-respect-to-its-magnitude",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "113",
   "location": "The Differential Coefficient Considered with Respect to its Magnitude",
   "latex": "We do not take the increments themselves, but the proportion they bear to the changes in the variable which gave rise to them; so in estimating the rate of motion of two points, we either consider lengths described in the same time, or if that cannot be done, we judge, not by the lengths described in different times, but by the proportion of those lengths to the times, or the proportions of the units which express them.",
   "markdown": "We do not take the increments themselves, but the proportion they bear to the changes in the variable which gave rise to them; so in estimating the rate of motion of two points, we either consider lengths described in the same time, or if that cannot be done, we judge, not by the lengths described in different times, but by the proportion of those lengths to the times, or the proportions of the units which express them.",
   "why": "It shows through motion why the ratio of change, not the raw change, is what measures rate.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/increment",
    "concept/rate-of-change",
    "concept/ratio",
    "quantity/length",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-d24df0bafe",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-differential-coefficient-considered-with-respect-to-its-magnitude",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "113",
   "location": "The Differential Coefficient Considered with Respect to its Magnitude",
   "latex": "In passing from $1000$ to~$1003$, we have the logarithms $3$ and~$3.0013009$, the above-mentioned ratio being~$.0004336$, little more than a tenth of the former.",
   "markdown": "In passing from $1000$ to $1003$, we have the logarithms $3$ and $3.0013009$, the above-mentioned ratio being $.0004336$, little more than a tenth of the former.",
   "why": "It is a checkable worked example showing that the same kind of ratio shrinks as the logarithm's rate of increase changes with x.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/common-logarithm",
    "concept/function",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-cf79c267e2",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-differential-coefficient-considered-with-respect-to-its-magnitude",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "114",
   "location": "The Differential Coefficient Considered with Respect to its Magnitude",
   "latex": "In the same way, if a point is moving, so that at the end of $1$~second it is at $3$~feet from a fixed point, and at the end of $2$~seconds it is at $5$~feet from the fixed point, we cannot say which way it is moving at the end of one second.",
   "markdown": "In the same way, if a point is moving, so that at the end of $1$ second it is at $3$ feet from a fixed point, and at the end of $2$ seconds it is at $5$ feet from the fixed point, we cannot say which way it is moving at the end of one second.",
   "why": "A concrete moving-point picture that shows why a change over a finite interval cannot tell a learner the direction of motion at an instant.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/interval",
    "concept/point",
    "concept/rate-of-change"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-cbee4dc0d1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-differential-coefficient-considered-with-respect-to-its-magnitude",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "114",
   "location": "The Differential Coefficient Considered with Respect to its Magnitude",
   "latex": "But if on adding any interval, \\emph{however small}, to the first second, the moving point does, during that interval, increase its distance from the fixed point, we can then certainly say that at the end of the first second the point is moving from the fixed point.",
   "markdown": "But if on adding any interval, *however small*, to the first second, the moving point does, during that interval, increase its distance from the fixed point, we can then certainly say that at the end of the first second the point is moving from the fixed point.",
   "why": "It states the condition under which an instantaneous direction of change can be asserted, which is the idea behind the limit.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/interval",
    "concept/point"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-169285f177",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-differential-coefficient-considered-with-respect-to-its-magnitude",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "114",
   "location": "The Differential Coefficient Considered with Respect to its Magnitude",
   "latex": "The objection becomes of less force as the increment diminishes, but always exists unless we take the limit of the ratio of the increments, instead of that ratio.",
   "markdown": "The objection becomes of less force as the increment diminishes, but always exists unless we take the limit of the ratio of the increments, instead of that ratio.",
   "why": "It names the step that resolves the difficulty: passing from the ratio of increments to its limit.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/increment",
    "concept/limit",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-1213753f8b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-differential-coefficient-considered-with-respect-to-its-magnitude",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "112",
   "location": "The Differential Coefficient Considered with Respect to its Magnitude",
   "latex": "Every value of the variable, gives not only a different value to the function, but a different quantity of increase or decrease in passing to what we may call \\emph{contiguous} values, obtained by a given increase of the variable.",
   "markdown": "Every value of the variable, gives not only a different value to the function, but a different quantity of increase or decrease in passing to what we may call *contiguous* values, obtained by a given increase of the variable.",
   "why": "It introduces the idea of nearby values of the variable, each with its own rate of change, which the rest of the chapter depends on.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/contiguous-values",
    "concept/function",
    "concept/increment",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5a0f526fd4",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-differential-coefficient-considered-with-respect-to-its-magnitude",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "115",
   "location": "The Differential Coefficient Considered with Respect to its Magnitude",
   "latex": "How well this answers to our previously formed ideas on such subjects as direction, velocity, and force, has already appeared.",
   "markdown": "How well this answers to our previously formed ideas on such subjects as direction, velocity, and force, has already appeared.",
   "why": "A short historical remark that links the new idea back to the familiar notions of direction, velocity and force from mechanics.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/derivative",
    "quantity/force",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-cf806a9629",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "115",
   "location": "The Integral Calculus",
   "latex": "We have already shown, that when two functions \\emph{increase} or \\emph{decrease} without limit, their \\emph{ratio} may either increase or decrease without limit, or may tend to some finite limit.",
   "markdown": "We have already shown, that when two functions *increase* or *decrease* without limit, their *ratio* may either increase or decrease without limit, or may tend to some finite limit.",
   "why": "It states plainly that a ratio of two quantities growing without bound need not grow without bound, which is the puzzle the whole chapter resolves.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "concept/ratio",
    "concept/tends-to-infinity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-bfdf606489",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "115",
   "location": "The Integral Calculus",
   "latex": "Nevertheless the product $\\cos\\theta × \\tan\\theta$, of which the first factor diminishes without limit, while the second increases without limit, is always finite, and tends towards the limit~$1$; for $\\cos\\theta × \\tan\\theta$ is always~$\\sin\\theta$, which last approaches to~$1$ as $\\theta$~approaches to a right angle, and is~$1$ when $\\theta$~\\emph{is} a right angle.",
   "markdown": "Nevertheless the product $\\cos\\theta × \\tan\\theta$, of which the first factor diminishes without limit, while the second increases without limit, is always finite, and tends towards the limit $1$; for $\\cos\\theta × \\tan\\theta$ is always $\\sin\\theta$, which last approaches to $1$ as $\\theta$ approaches to a right angle, and is $1$ when $\\theta$ *is* a right angle.",
   "why": "A concrete trigonometric example shows a product of a vanishing factor and an infinite factor settling to a finite limit, which a learner can check by hand.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/cosine",
    "concept/limit",
    "concept/product",
    "concept/right-angle",
    "concept/sine",
    "concept/tangent-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c5547d9bbe",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "117",
   "location": "The Integral Calculus",
   "latex": "But though the two sums increase without limit when $m$~increases without limit, it does not therefore follow that their ratio increases without limit; indeed we can show that this cannot be the case when all the separate terms of~\\Eq{(2)} remain finite.",
   "markdown": "But though the two sums increase without limit when $m$ increases without limit, it does not therefore follow that their ratio increases without limit; indeed we can show that this cannot be the case when all the separate terms of (2) remain finite.",
   "why": "It warns the learner against concluding that two sums growing without bound must have a ratio growing without bound.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "concept/ratio",
    "concept/sum"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-8450615d69",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "116",
   "location": "The Integral Calculus",
   "latex": "If we take any numbers, such as $1$~and~$2$, it is evident that between the two we may interpose any number of fractions, however great, either in arithmetical progression, or according to any other law.",
   "markdown": "If we take any numbers, such as $1$ and $2$, it is evident that between the two we may interpose any number of fractions, however great, either in arithmetical progression, or according to any other law.",
   "why": "It gives an intuitive starting picture for subdividing an interval into as many small steps as one likes.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/arithmetical-progression",
    "concept/interval"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-29a944d5ea",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "116",
   "location": "The Integral Calculus",
   "latex": "Generally, if $A$~diminishes without limit at the same time as $B$~increases without limit, the product~$AB$ may, and often will, tend towards a finite limit.",
   "markdown": "Generally, if $A$ diminishes without limit at the same time as $B$ increases without limit, the product $AB$ may, and often will, tend towards a finite limit.",
   "why": "It states the general principle that a vanishing quantity times an infinite one may still have a finite limit, so the learner should not assume the product is zero or infinite.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "concept/product",
    "concept/tends-to-infinity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-23a2644d2e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "119",
   "location": "The Integral Calculus",
   "latex": "There is as yet no general agreement on this point of notation.",
   "markdown": "There is as yet no general agreement on this point of notation.",
   "why": "De Morgan's remark from 1832 shows that the notation for limits of integration was still being debated, which gives a historical sense of the symbols.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/integral",
    "concept/limits-of-integration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-7d691f8d65",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-connexion-of-the-integral-with-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "120",
   "location": "Connexion of the Integral with the Differential Calculus",
   "latex": "Let $x$ have the successive values $a$, $a + dx$, $a + 2\\, dx$, etc.,~\\dots\\ up to $a + m\\, dx$, or $a + h$, $h$~being a given quantity, and $dx$ the $m$\\th~part of~$h$, so that as $m$~is increased without limit, $dx$~is diminished without limit.",
   "markdown": "Let $x$ have the successive values $a$, $a + dx$, $a + 2\\, dx$, etc., … up to $a + m\\, dx$, or $a + h$, $h$ being a given quantity, and $dx$ the $m$th part of $h$, so that as $m$ is increased without limit, $dx$ is diminished without limit.",
   "why": "It sets up the idea of a small increment dx shrinking without limit as the number of steps grows, which is the basis of the integral.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c86d97de85",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-connexion-of-the-integral-with-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "121",
   "location": "Connexion of the Integral with the Differential Calculus",
   "latex": "That is, the integral of $\\phi x\\, dx$ between the limits $a$~and~$a + h$, is $\\psi(a + h) - \\psi a$, where $\\psi x$~is the function, which, when differentiated, gives~$\\phi x$.",
   "markdown": "That is, the integral of $\\phi x\\, dx$ between the limits $a$ and $a + h$, is $\\psi(a + h) - \\psi a$, where $\\psi x$ is the function, which, when differentiated, gives $\\phi x$.",
   "why": "It states the central result: the integral is found by subtracting the function's values at the limits.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/antiderivative",
    "concept/limits-of-integration",
    "theorem/fundamental-theorem-of-calculus"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a95b3b9aac",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-connexion-of-the-integral-with-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "121",
   "location": "Connexion of the Integral with the Differential Calculus",
   "latex": "Let us suppose that $\\psi a$~is the function of which $\\phi a$~is the differential coefficient, that is, that $\\psi' a = \\phi a$.",
   "markdown": "Let us suppose that $\\psi a$ is the function of which $\\phi a$ is the differential coefficient, that is, that $\\psi' a = \\phi a$.",
   "why": "It shows the learner how to pick the function whose differential coefficient is the given one, the step that makes the integral computable.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/antiderivative",
    "concept/derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-fb07faea49",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-connexion-of-the-integral-with-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "122",
   "location": "Connexion of the Integral with the Differential Calculus",
   "latex": "which is said to be the integral of~$\\phi x\\, dx$, beginning when $x = a$, the summation being supposed to be continued from $x = a$ until $x$~has the value which it may be convenient to give it.",
   "markdown": "which is said to be the integral of $\\phi x\\, dx$, beginning when $x = a$, the summation being supposed to be continued from $x = a$ until $x$ has the value which it may be convenient to give it.",
   "why": "It shows the older phrasing of an indefinite integral as a summation that may run to any convenient upper value.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/integral",
    "concept/limits-of-integration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ba4ecf9f65",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-connexion-of-the-integral-with-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "121",
   "location": "Connexion of the Integral with the Differential Calculus",
   "latex": "which is the limit arising from supposing $x$ to increase from~$a$ through $a + dx$, $a + 2\\, dx$, etc., up to~$a + h$, multiplying every value of~$\\phi x$ so obtained by~$dx$, summing the results, and decreasing~$dx$ without limit.",
   "markdown": "which is the limit arising from supposing $x$ to increase from $a$ through $a + dx$, $a + 2\\, dx$, etc., up to $a + h$, multiplying every value of $\\phi x$ so obtained by $dx$, summing the results, and decreasing $dx$ without limit.",
   "why": "It spells out the integral as a step-by-step sum that becomes exact as dx decreases, which a learner can follow concretely.",
   "use": [
    "website"
   ],
   "concepts": [
    "concept/integral",
    "concept/limit",
    "concept/sum"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-bb2b1399be",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-connexion-of-the-integral-with-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "121",
   "location": "Connexion of the Integral with the Differential Calculus",
   "latex": "It is evident that this series bears a great resemblance to the development in \\PageRef{21}, deprived of its first term.",
   "markdown": "It is evident that this series bears a great resemblance to the development in 21, deprived of its first term.",
   "why": "It shows that the integral series is the earlier expansion with its first term removed, linking two results a learner should keep together.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/integral",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-8d4c9bca98",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "124",
   "location": "Nature of Integration",
   "latex": "bringing before him modes of speech, which, taken quite literally, are absurd.",
   "markdown": "bringing before him modes of speech, which, taken quite literally, are absurd.",
   "why": "It candidly tells learners that the infinitesimal language they will meet is informal and should not be read literally.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/infinitesimal",
    "person/gottfried-wilhelm-leibniz"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-9fb6bbbde5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "122",
   "location": "Nature of Integration",
   "latex": "Hence results a new branch of the inquiry, the reverse of the Differential Calculus, the object of which is, not to find the differential coefficient, having given the function, but to find the function, having given the differential coefficient. This is called the Integral Calculus.",
   "markdown": "Hence results a new branch of the inquiry, the reverse of the Differential Calculus, the object of which is, not to find the differential coefficient, having given the function, but to find the function, having given the differential coefficient. This is called the Integral Calculus.",
   "why": "It states plainly that integration is the reverse of differentiation, the frame a learner needs before anything else in the chapter.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/integral",
    "method/differentiation",
    "method/integration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-1792c4c84d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "122",
   "location": "Nature of Integration",
   "latex": "For whatever differential coefficient~$\\psi x$ gives, $C + \\psi x$ will give the same, if $C$~be a constant, that is, not varying when $x$~varies.",
   "markdown": "For whatever differential coefficient $\\psi x$ gives, $C + \\psi x$ will give the same, if $C$ be a constant, that is, not varying when $x$ varies.",
   "why": "It explains why an integral is never unique: adding a constant leaves the differential coefficient unchanged.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/antiderivative",
    "concept/constant"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-8fdd73711d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "123",
   "location": "Nature of Integration",
   "latex": "Thus, since $x^{2}$, when differentiated, gives~$2x$, $x^{2}$~is the integral of~$2x$, beginning at $x = 0$; and $x^{2} - 4$~is the integral beginning at~$x = 2$.",
   "markdown": "Thus, since $x^{2}$, when differentiated, gives $2x$, $x^{2}$ is the integral of $2x$, beginning at $x = 0$; and $x^{2} - 4$ is the integral beginning at $x = 2$.",
   "why": "A worked pair of examples shows how the starting value of x fixes which integral is meant.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limits-of-integration",
    "method/integration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-d1488d3b6e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "123",
   "location": "Nature of Integration",
   "latex": "the sum of an infinite number of infinitely small quantities, which are the differentials or infinitely small increments of a function.",
   "markdown": "the sum of an infinite number of infinitely small quantities, which are the differentials or infinitely small increments of a function.",
   "why": "It gives Leibnitz's picture of an integral as a sum of infinitesimal differentials, which the chapter then interprets with limits.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/differential",
    "concept/infinitesimal",
    "person/gottfried-wilhelm-leibniz"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-0d65fa3132",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "124",
   "location": "Nature of Integration",
   "latex": "so that $mC$, or the sum of all the sides of the polygon, can be made as nearly equal to the circumference as we please.",
   "markdown": "so that $mC$, or the sum of all the sides of the polygon, can be made as nearly equal to the circumference as we please.",
   "why": "Inscribed polygons with more and more sides approach the circumference, giving a concrete reading of the limit.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/circle",
    "concept/limit",
    "concept/regular-polygon",
    "quantity/circumference"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-9fc5b1f99e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "122",
   "location": "Nature of Integration",
   "latex": "the value of an integral is not to be determined, unless we know the values of~$x$ corresponding to the beginning and end of the summation, whose limit furnishes the integral.",
   "markdown": "the value of an integral is not to be determined, unless we know the values of $x$ corresponding to the beginning and end of the summation, whose limit furnishes the integral.",
   "why": "It warns that an integral without its limits is incomplete, a common mistake for beginners.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/integral",
    "concept/limit",
    "concept/limits-of-integration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-35f3dad49a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "122",
   "location": "Nature of Integration",
   "latex": "if $\\phi x$~be the differential coefficient of~$\\psi x$, $\\psi x$~might have been called the integral of~$\\phi x\\, dx$.",
   "markdown": "if $\\phi x$ be the differential coefficient of $\\psi x$, $\\psi x$ might have been called the integral of $\\phi x\\, dx$.",
   "why": "It defines the integral as the converse of the differential coefficient, the alternative definition the chapter weighs against the summation.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/antiderivative",
    "concept/integral",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-ee683b9820",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-determination-of-curvilinear-areas-the-parabola",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "126",
   "location": "Determination of Curvilinear Areas. The Parabola",
   "latex": "Hence the curvilinear area~$MPP'M'$ is the limit towards which we continually approach, but which we never reach, by dividing $MM'$ into a greater and greater number of equal parts, and adding the parallelograms $Mr$,~$mr'$,~etc., so obtained.",
   "markdown": "Hence the curvilinear area $MPP'M'$ is the limit towards which we continually approach, but which we never reach, by dividing $MM'$ into a greater and greater number of equal parts, and adding the parallelograms $Mr$, $mr'$, etc., so obtained.",
   "why": "It explains the idea of a limit of successively finer approximations, which is the foundation of the integral.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/area",
    "concept/limit",
    "concept/parallelogram"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-eb12b3705c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-determination-of-curvilinear-areas-the-parabola",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "126",
   "location": "Determination of Curvilinear Areas. The Parabola",
   "latex": "These are the altitudes of a set of parallelograms, the base of each of which is~$dx$; hence the sum of their area is",
   "markdown": "These are the altitudes of a set of parallelograms, the base of each of which is $dx$; hence the sum of their area is",
   "why": "It shows the strips of width dx as the building blocks of the area sum, which a learner can picture directly.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/area",
    "concept/differential",
    "concept/parallelogram"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-c9c24e93e7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-determination-of-curvilinear-areas-the-parabola",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "127",
   "location": "Determination of Curvilinear Areas. The Parabola",
   "latex": "If we take the function~$cx^{n}$, $c$~being independent of~$x$, and substitute $x + h$ for~$x$, we have for the development $cx^{n} + cnx^{n-1}\\, h + \\etc$.",
   "markdown": "If we take the function $cx^{n}$, $c$ being independent of $x$, and substitute $x + h$ for $x$, we have for the development $cx^{n} + cnx^{n-1}\\, h + \\etc$.",
   "why": "It works a complete method for differentiating a power, showing where the coefficient and exponent come from.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/exponent",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-5fdd8b024c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-determination-of-curvilinear-areas-the-parabola",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "127",
   "location": "Determination of Curvilinear Areas. The Parabola",
   "latex": "Here $y = p^{\\efrac{1}{2}} x^{\\efrac{1}{2}}$, and we must find the integral of~$p^{\\efrac{1}{2}} x^{\\efrac{1}{2}}\\, dx$, or the function whose differential coefficient is~$p^{\\efrac{1}{2}} x^{\\efrac{1}{2}}$, $p^{\\efrac{1}{2}}$~being a constant.",
   "markdown": "Here $y = p^{\\efrac{1}{2}} x^{\\efrac{1}{2}}$, and we must find the integral of $p^{\\efrac{1}{2}} x^{\\efrac{1}{2}}\\, dx$, or the function whose differential coefficient is $p^{\\efrac{1}{2}} x^{\\efrac{1}{2}}$, $p^{\\efrac{1}{2}}$ being a constant.",
   "why": "It shows how the parabola's equation becomes a concrete integral to be solved, with the constant pulled out.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant",
    "concept/exponent",
    "concept/integral",
    "concept/parabola"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-326537879a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-determination-of-curvilinear-areas-the-parabola",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "126",
   "location": "Determination of Curvilinear Areas. The Parabola",
   "latex": "Hence, $y$~being the ordinate, the area included between the axis of~$x$, any two values of~$y$, and the portion of the curve they cut off, is $\\int y\\, dx$, beginning at the one ordinate and ending at the other.",
   "markdown": "Hence, $y$ being the ordinate, the area included between the axis of $x$, any two values of $y$, and the portion of the curve they cut off, is $\\int y\\, dx$, beginning at the one ordinate and ending at the other.",
   "why": "It states plainly what area the integral of the ordinate measures, the central idea of the chapter.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/area",
    "concept/integral",
    "concept/ordinate"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-92631a44d1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-determination-of-curvilinear-areas-the-parabola",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "127",
   "location": "Determination of Curvilinear Areas. The Parabola",
   "latex": "at a time when such a step was one of no small magnitude.",
   "markdown": "at a time when such a step was one of no small magnitude.",
   "why": "The old phrase shows how a result now routine once counted as a major advance, which gives learners a sense of the history.",
   "use": [
    "history"
   ],
   "concepts": [
    "person/archimedes",
    "theorem/area-of-a-parabolic-segment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-925739d155",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "127",
   "location": "Method of Indivisibles",
   "latex": "A line is considered as the sum of an infinite number of points, a surface of an infinite number of lines, and a solid of an infinite number of surfaces.",
   "markdown": "A line is considered as the sum of an infinite number of points, a surface of an infinite number of lines, and a solid of an infinite number of surfaces.",
   "why": "It states the central idea of the indivisibles method in one sentence, so a learner can see exactly what is being rejected.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/line",
    "concept/point",
    "concept/surface",
    "concept/three-dimensional-figure",
    "method/method-of-indivisibles"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-480d9916a3",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "128",
   "location": "Method of Indivisibles",
   "latex": "If twenty points be taken on a straight line, the sum of the twenty-one lines which lie between point and point is equal to the whole line; which cannot be if the points by themselves constitute any part of the line, however small.",
   "markdown": "If twenty points be taken on a straight line, the sum of the twenty-one lines which lie between point and point is equal to the whole line; which cannot be if the points by themselves constitute any part of the line, however small.",
   "why": "It gives a concrete counting argument that shows why adding up points cannot build a line.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/line",
    "concept/point",
    "method/method-of-indivisibles"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-bb0d03fa7d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "128",
   "location": "Method of Indivisibles",
   "latex": "We would therefore recommend to the student not to regard any proposition derived from this method as true on that account; for falsehoods, as well as truths, may be deduced from it.",
   "markdown": "We would therefore recommend to the student not to regard any proposition derived from this method as true on that account; for falsehoods, as well as truths, may be deduced from it.",
   "why": "It warns learners that a method can give true and false results alike, so each result needs checking.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "method/method-of-indivisibles"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-7874592d87",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "129",
   "location": "Method of Indivisibles",
   "latex": "Though a point is not treated as a length, or as any part of space whatever, it is considered as having weight; and two points are spoken of as having different weights.",
   "markdown": "Though a point is not treated as a length, or as any part of space whatever, it is considered as having weight; and two points are spoken of as having different weights.",
   "why": "It points out a common way of speaking in mechanics that looks like the indivisibles idea and is not strictly correct.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/mechanics",
    "concept/point",
    "concept/weight"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-400324c8c0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "130",
   "location": "Method of Indivisibles",
   "latex": "A cubic inch of gold weighs more than a cubic inch of water; hence gold is \\emph{denser} than water. If the first weighs $19$~times as much as the second, gold is said to be $19$~times more dense than water, or the density of gold is $19$~times that of water.",
   "markdown": "A cubic inch of gold weighs more than a cubic inch of water; hence gold is *denser* than water. If the first weighs $19$ times as much as the second, gold is said to be $19$ times more dense than water, or the density of gold is $19$ times that of water.",
   "why": "It builds the idea of density from a familiar comparison of gold and water before any formula appears.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/specific-gravity",
    "quantity/density"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-a3d7b4ad13",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "129",
   "location": "Method of Indivisibles",
   "latex": "But if the weight of every two cubic inches is different, we can only find the weight of the whole by the integral calculus.",
   "markdown": "But if the weight of every two cubic inches is different, we can only find the weight of the whole by the integral calculus.",
   "why": "It motivates the integral by showing a case where simple multiplication of volume by weight fails.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/calculus",
    "concept/integral",
    "concept/weight"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-905861baaa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "132",
   "location": "Method of Indivisibles",
   "latex": "The integral of~$bwx^{2}\\, dx$ is~$\\frac{1}{3}bwx^{3}$, which taken from $x = 0$ to $x = a$ is~$\\frac{1}{3}bwa^{3}$.",
   "markdown": "The integral of $bwx^{2}\\, dx$ is $\\frac{1}{3}bwx^{3}$, which taken from $x = 0$ to $x = a$ is $\\frac{1}{3}bwa^{3}$.",
   "why": "It shows the finished worked result of the chapter, where the integral gives the total weight of the bar.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/integral",
    "method/integration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-dee9e1b6ba",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-concluding-remarks-on-the-study-of-the-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "132",
   "location": "Concluding Remarks on the Study of the Calculus",
   "latex": "Thus, if he has the area of a curve to find, instead of merely saying that~$y$, the ordinate, being a certain function of the abscissa~$x$, $\\int y\\, dx$ within the given limits would be the area required;",
   "markdown": "Thus, if he has the area of a curve to find, instead of merely saying that $y$, the ordinate, being a certain function of the abscissa $x$, $\\int y\\, dx$ within the given limits would be the area required;",
   "why": "It shows the learner exactly what the integral means geometrically before any mechanical manipulation begins.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/abscissa",
    "concept/area",
    "concept/function",
    "concept/integral",
    "concept/ordinate"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-b9f5c8ee73",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-concluding-remarks-on-the-study-of-the-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "133",
   "location": "Concluding Remarks on the Study of the Calculus",
   "latex": "let him remark that if an approximate solution only were required, it might be obtained by dividing the curvilinear area into a number of four-sided figures, as in \\Fig[Figure]{10}, one side of which only is curvilinear, and embracing so small an arc that it may, without visible error, be considered as rectilinear.",
   "markdown": "let him remark that if an approximate solution only were required, it might be obtained by dividing the curvilinear area into a number of four-sided figures, as in [Figure]10, one side of which only is curvilinear, and embracing so small an arc that it may, without visible error, be considered as rectilinear.",
   "why": "It gives a concrete picture of approximating a curved area by straight-sided pieces that a learner can draw and check.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/approximation",
    "concept/curvilinear-figure",
    "concept/rectilinear-figure"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-3210994472",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-concluding-remarks-on-the-study-of-the-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "133",
   "location": "Concluding Remarks on the Study of the Calculus",
   "latex": "The mathematical method begins with the same principle, investigating upon this supposition, not the sum of these rectilinear areas, but the limit towards which this sum approaches, as the subdivision is rendered more minute.",
   "markdown": "The mathematical method begins with the same principle, investigating upon this supposition, not the sum of these rectilinear areas, but the limit towards which this sum approaches, as the subdivision is rendered more minute.",
   "why": "It explains in plain terms why the exact method is a limit of approximate sums rather than a different idea.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "concept/rectilinear-figure",
    "method/integration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-3dc3432fd0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-concluding-remarks-on-the-study-of-the-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "133",
   "location": "Concluding Remarks on the Study of the Calculus",
   "latex": "This limit is shown to be that of which we are in search, since it is proved that the error diminishes without limit, as the subdivision is indefinitely continued.",
   "markdown": "This limit is shown to be that of which we are in search, since it is proved that the error diminishes without limit, as the subdivision is indefinitely continued.",
   "why": "It tells the learner why the limit gives the true area: the error of the approximation can be made as small as wished.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/error-of-measurement",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/x-7582183c36",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-concluding-remarks-on-the-study-of-the-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "133",
   "location": "Concluding Remarks on the Study of the Calculus",
   "latex": "The method so generally followed in our elementary works, of leading the student at once into the mechanical processes of the science, postponing entirely all other considerations, is to many students a source of obscurity at least, if not an absolute impediment to their progress; since they cannot imagine what is the object of that which they are required to do.",
   "markdown": "The method so generally followed in our elementary works, of leading the student at once into the mechanical processes of the science, postponing entirely all other considerations, is to many students a source of obscurity at least, if not an absolute impediment to their progress; since they cannot imagine what is the object of that which they are required to do.",
   "why": "It warns teachers that drilling procedures before giving their purpose leaves many students unable to see why they are working.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/calculus",
    "concept/mathematics"
   ]
  }
 ],
 "equations": [
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-b5508a74e5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "4",
   "location": "On the Ratio or Proportion of Two Magnitudes",
   "latex": "\\dfrac{x + a}{x} = 1 + \\dfrac{a}{x}",
   "name": null,
   "statement": "The ratio of the two quantities x + a and x equals 1 plus a/x, which expresses how far that ratio is from unity.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "a quantity (the variable quantity x)"
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    {
     "unit": null,
     "symbol": "a",
     "meaning": "the difference between the two quantities x and x + a"
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   ],
   "sympy": "Eq((x + a)/x, 1 + a/x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/common-ratio",
    "concept/difference",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-0eb46ee9ff",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "4",
   "location": "On the Ratio or Proportion of Two Magnitudes",
   "latex": "\\dfrac{x + m + a}{x + m} = 1 + \\dfrac{a}{x + m}",
   "name": null,
   "statement": "After an increase m is given to x, the ratio of x + m + a to x + m equals 1 plus a/(x + m), which lies nearer to unity than before because the same difference a is divided by a larger quantity.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "a quantity (the variable quantity x)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the difference between the two quantities x and x + a, unchanged by the increase"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "the increase given to x"
    }
   ],
   "sympy": "Eq((x + m + a)/(x + m), 1 + a/(x + m))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/common-ratio",
    "concept/difference",
    "concept/equality",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-1456e3d9a6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "6",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "\\dfrac{2}{x(x + 1)}",
   "name": null,
   "statement": "The x-th value of M in the second table is 2 divided by x(x+1), where M and N are the two decreasing quantities in the example.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "M",
     "meaning": "the first of the two decreasing quantities (the x-th value of M)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "position of the term in the sequence (a positive integer)"
    }
   ],
   "sympy": "Eq(M, 2/(x*(x + 1)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/infinite-sequence",
    "concept/real-number",
    "concept/term",
    "concept/variable"
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  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-344f2569cd",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "6",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "\\dfrac{1}{x^{2}}",
   "name": null,
   "statement": "The x-th value of N in the second table is 1 divided by x squared.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "N",
     "meaning": "the second of the two decreasing quantities (the x-th value of N)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "position of the term in the sequence (a positive integer)"
    }
   ],
   "sympy": "Eq(N, 1/x**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/real-number",
    "concept/term",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2aef2fdd2d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "6",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "\\dfrac{M}{N} = \\dfrac{2x^{2}}{x(x + 1)}",
   "name": null,
   "statement": "The x-th value of the ratio M to N equals 2x squared over x(x+1), obtained by dividing the two sequence values above.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "M",
     "meaning": "the first quantity"
    },
    {
     "unit": null,
     "symbol": "N",
     "meaning": "the second quantity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "position of the term in the sequence (a positive integer)"
    }
   ],
   "sympy": "Eq(M/N, 2*x**2/(x*(x + 1)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/ratio",
    "concept/term",
    "concept/variable"
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  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-7fe6d9b045",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "6",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "\\dfrac{2x}{x + 1}",
   "name": null,
   "statement": "The ratio M to N simplifies to 2x divided by (x+1); it is always less than 2 yet approaches 2 as x grows.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "position of the term in the sequence (a positive integer)"
    }
   ],
   "sympy": "Eq(M/N, 2*x/(x + 1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/common-ratio",
    "concept/limit",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-4bb1604b75",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "7",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "1 - \\dfrac{1}{x + 1}",
   "name": null,
   "statement": "x/(x+1) equals 1 minus 1/(x+1), so it differs from 1 by 1/(x+1), which can be made as small as we please.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "position of the term in the sequence (a positive integer)"
    }
   ],
   "sympy": "Eq(x/(x + 1), 1 - 1/(x + 1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/rational-number",
    "concept/variable"
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  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a496cdeed7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "6",
   "location": "On the Ratio of Magnitudes that Vanish Together",
   "latex": "1 + 2 + 3 + \\dots + x,\\quad\\text{or}\\quad \\frac{x(x + 1)}{2}",
   "name": null,
   "statement": "The denominator of the x-th value of M is the sum of the integers 1 through x, which equals x(x+1)/2.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "position of the term in the sequence (a positive integer)"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/real-number",
    "concept/term",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-178b195afa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "9",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "OD ÷ OA = BM ÷ BA",
   "name": null,
   "statement": "The ratio of OD to OA equals the ratio of BM to BA, because the right triangles ODA and BMA are similar.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "OD",
     "meaning": "segment from the centre O to the midpoint D of the chord AB"
    },
    {
     "unit": null,
     "symbol": "OA",
     "meaning": "radius of the circle"
    },
    {
     "unit": null,
     "symbol": "BM",
     "meaning": "perpendicular from B to OA"
    },
    {
     "unit": null,
     "symbol": "BA",
     "meaning": "chord from B to A"
    }
   ],
   "sympy": "Eq(OD/OA, BM/BA)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/limit",
    "concept/proportion",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-b47e845802",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "9",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "\\sin\\theta = .0174524",
   "name": null,
   "statement": "Table value: the sine of one degree is about 0.0174524.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": "degree of angle",
     "symbol": "theta",
     "meaning": "the angle BOA, here 1 degree"
    }
   ],
   "sympy": "Eq(sin(theta), 0.0174524)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/plane-angle",
    "concept/sine",
    "unit/degree-of-angle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-3f581fc90d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "9",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "2\\sin\\frac{1}{2}\\theta ÷ \\sin\\theta = 1.00003",
   "name": null,
   "statement": "For an angle of one degree, twice the sine of half the angle divided by the sine of the angle is very nearly 1.00003, so the chord BA differs from BM by less than four hundred-thousandths of itself.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": "degree of angle",
     "symbol": "theta",
     "meaning": "the angle BOA"
    }
   ],
   "sympy": "Eq(2*sin(theta/2)/sin(theta), 1.00003)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/limit",
    "concept/plane-angle",
    "concept/sine"
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  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c57fc5661a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "9",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "BM ÷ MA = 114.589",
   "name": null,
   "statement": "For an angle BOA of one degree, the ratio of BM to MA is very nearly 114.589, so BM contains MA more than 114 times.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "BM",
     "meaning": "perpendicular from B to OA"
    },
    {
     "unit": null,
     "symbol": "MA",
     "meaning": "segment from M to A along OA"
    }
   ],
   "sympy": "Eq(BM/MA, 114.589)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/limit",
    "concept/ratio",
    "unit/unit"
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  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-9f68caeb53",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "9",
   "location": "On the Ratios of Continuously Increasing or Decreasing Quantities",
   "latex": "\\angle BOA = \\theta",
   "name": null,
   "statement": "The angle at the centre O subtended by the arc AB is denoted theta.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "degree of angle",
     "symbol": "theta",
     "meaning": "the angle BOA at the centre O"
    }
   ],
   "sympy": "Eq(angle_BOA, theta)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/centre-of-a-circle",
    "concept/plane-angle",
    "quantity/arc-of-a-circle"
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  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-9f105d9495",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "11",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "(a  +  h)^{2} = a^{2} + 2ah + h^{2}",
   "name": null,
   "statement": "Squaring a sum a + h expands to a^2 + 2ah + h^2, so increasing a by h increases a^2 by 2ah + h^2.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a magnitude (number) that is increased by h"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "the increment added to a"
    }
   ],
   "sympy": "Eq((a + h)**2, a**2 + 2*a*h + h**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/increment",
    "concept/real-number",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-cc1f10d393",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notion-of-infinitely-small-quantities",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "11",
   "location": "The Notion of Infinitely Small Quantities",
   "latex": "1:h :: h:h^{2}",
   "name": null,
   "statement": "As 1 contains h, so many times, h contains h^2; the four quantities 1, h, h, h^2 are in proportion.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small quantity (the increment), taken as a magnitude less than the unit"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/infinitesimal",
    "concept/proportion",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-869ac9c8cf",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "15",
   "location": "On Functions",
   "latex": "\\phi x = x + x^{2}",
   "name": null,
   "statement": "An illustrative function: φx is defined as x plus x squared, so φ is the direction to add x to its square.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "φ",
     "meaning": "the abbreviated direction to perform the operation (here: add x to its square); not a number multiplying x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable on which the function acts"
    }
   ],
   "sympy": "Eq(phi(x), x + x**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/function-notation",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-d67b06fdfa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "16",
   "location": "Infinite Series",
   "latex": "\\phi x +  ph + qh^{2} + rh^{3} + \\etc.,\\quad \\textit{ad infinitum}",
   "name": "Taylor's theorem (series expansion)",
   "statement": "A function of x + h can generally be expanded as an infinite series in whole, positive powers of h, with the constant term phi x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "any function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the value of the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "the increment of x"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "coefficient of h in the expansion"
    },
    {
     "unit": null,
     "symbol": "q",
     "meaning": "coefficient of h^2 in the expansion"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "coefficient of h^3 in the expansion"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [
    "theorem/taylor-s-theorem-series-expansion"
   ],
   "concepts": [
    "concept/function",
    "concept/increment",
    "concept/infinite-sequence",
    "concept/power",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6d168e5d41",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "16",
   "location": "Infinite Series",
   "latex": "\\phi(x +  h) - \\phi x",
   "name": null,
   "statement": "The increment of phi x is the difference between phi(x + h) and phi x, negative when phi(x + h) is less than phi x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "any function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the value of the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "the increment of x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/increment",
    "concept/value-of-a-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-91e88ae6d4",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "16",
   "location": "Infinite Series",
   "latex": "\\phi(x + h) < \\phi x",
   "name": null,
   "statement": "The increment of phi x is negative exactly when phi(x + h) is less than phi x.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "any function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the value of the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "the increment of x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/decrement",
    "concept/function",
    "concept/increment",
    "concept/inequality"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-1135d4f0ca",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "17",
   "location": "Infinite Series",
   "latex": "\\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \\etc.,\\quad\\textit{ad infinitum}",
   "name": null,
   "statement": "For x less than unity, the infinite geometric series 1 + x + x^2 + ... can be brought as near as we please to 1/(1 - x), so the series is said to equal it.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "a number supposed less than unity"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/infinite-sequence",
    "concept/limit",
    "concept/power",
    "theorem/limit-of-a-sum"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-e56c305587",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "18",
   "location": "Convergent and Divergent Series",
   "latex": "a\\left(1 + \\frac{b}{a} + \\frac{c}{b}\\, \\frac{b}{a} + \\frac{d}{c}\\, \\frac{c}{b}\\, \\frac{b}{a} + \\etc.\\right)",
   "name": null,
   "statement": "The series a + b + c + d + ... can be rewritten as a times a series whose terms are successive ratios of the original terms.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "first term of the series"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "second term of the series"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "third term of the series"
    },
    {
     "unit": null,
     "symbol": "d",
     "meaning": "fourth term of the series"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/infinite-sequence",
    "concept/sum",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a148c73f49",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "18",
   "location": "Convergent and Divergent Series",
   "latex": "k + l + m + \\etc. = k\\left(1 + \\frac{l}{k} + \\frac{m}{l}\\, \\frac{l}{k} + \\etc.\\right)",
   "name": null,
   "statement": "The same rewriting holds when the series is taken from any term k onward: the tail sum equals k times a series of successive ratios.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "k",
     "meaning": "the term of the series from which the rewriting starts"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "the term immediately after k"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "the term immediately after l"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/infinite-sequence",
    "concept/sum",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-cdd13e9a99",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "18",
   "location": "Convergent and Divergent Series",
   "latex": "\\dfrac{l}{k} > \\dfrac{m}{l} > \\dfrac{n}{m}",
   "name": null,
   "statement": "The successive ratios of the terms are assumed to decrease, so each ratio is greater than the next.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "k",
     "meaning": "the term from which the series is taken"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "the term immediately after k"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "the term immediately after l"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the term immediately after m"
    }
   ],
   "sympy": "And(Gt(l/k, m/l), Gt(m/l, n/m))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/inequality",
    "concept/infinite-sequence",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-b5720c9ca6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "20",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "(x + h)^{n} = x^{n} + nx^{n-1}h + n(n - 1)x^{n-2} \\frac{h^{2}}{2} + n(n - 1)(n - 2)x^{n-3} \\frac{h^{3}}{2·3}",
   "name": null,
   "statement": "The power (x + h)^n is developed in ascending powers of the increment h, with the coefficients of h, h^2/2 and h^3/(2·3) given.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable (the starting value)"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small increment added to x"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the exponent"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/power",
    "concept/variable",
    "theorem/binomial-theorem",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2041e1ecc9",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "20",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\log(x + h) = \\log x + \\frac{1}{x}\\, h - \\frac{1}{x^{2}}\\, \\frac{h^{2}}{2} + \\frac{2}{x^{3}}\\, \\frac{h^{3}}{2·3}",
   "name": null,
   "statement": "The logarithm of x + h is developed in powers of the increment h.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small increment added to x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/logarithm",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-17c3a81421",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "20",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\cos(x + h) = \\cos x - \\sin x\\, h - \\cos x\\, \\frac{h^{2}}{2} + \\sin x\\, \\frac{h^{3}}{2·3}",
   "name": null,
   "statement": "The cosine of x + h is developed in powers of h; the terms are positive and negative in pairs.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable (an angle)"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small increment added to x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/sine",
    "concept/variable",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a4a82055ae",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\phi(x + h) = \\phi x + \\phi' x\\, h + \\phi''x\\, \\frac{h^{2}}{2} + \\phi''' x\\, \\frac{h^{3}}{2·3} + \\etc.",
   "name": "Taylor's theorem",
   "statement": "A function phi(x + h) is expanded as a series in powers of h whose coefficients are the successive derived functions of phi at x, divided by factorials.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "the function (general symbol for a function)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small increment added to x"
    },
    {
     "unit": null,
     "symbol": "phi'",
     "meaning": "the first derived function (derivative) of phi"
    },
    {
     "unit": null,
     "symbol": "phi''",
     "meaning": "the second derived function of phi"
    },
    {
     "unit": null,
     "symbol": "phi'''",
     "meaning": "the third derived function of phi"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [
    "theorem/taylor-s-theorem"
   ],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/differential",
    "concept/function",
    "concept/function-notation",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-fe73ebca9e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\phi' x = nx^{n-1}",
   "name": null,
   "statement": "When phi x = x^n, its first derived function is n x^(n-1).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "the function (here x^n)"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the exponent"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(diff(x**n, x), n*x**(n - 1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/exponent",
    "concept/function-notation",
    "concept/power"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-cd1a255f5a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\phi'' x = n(n - 1)x^{n-2}",
   "name": null,
   "statement": "When phi x = x^n, its second derived function is n(n - 1) x^(n-2).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "the function (here x^n)"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the exponent"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(diff(x**n, x, 2), n*(n - 1)*x**(n - 2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/exponent",
    "concept/higher-order-derivative",
    "concept/power"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-9c0c495ec8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\phi' x = \\cos x",
   "name": null,
   "statement": "When phi x = sin x, its first derived function is cos x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "the function (here sin x)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable (an angle)"
    }
   ],
   "sympy": "Eq(diff(sin(x), x), cos(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/derivative",
    "concept/function-notation",
    "concept/sine"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6e24c4adb9",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\phi'' x = -\\sin x",
   "name": null,
   "statement": "When phi x = sin x, its second derived function is minus sin x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "the function (here sin x)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable (an angle)"
    }
   ],
   "sympy": "Eq(diff(sin(x), x, 2), -sin(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative",
    "concept/sine"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-575dcb219a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\phi' x = ka^{x}",
   "name": null,
   "statement": "When phi x = a^x, its first derived function is k a^x, where k is the natural logarithm of a.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "the function (here a^x)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the base of the exponential"
    },
    {
     "unit": null,
     "symbol": "k",
     "meaning": "the natural (Naperian or hyperbolic) logarithm of a"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(diff(a**x, x), k*a**x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/exponent",
    "concept/function-notation",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-197e96a074",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "21",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\phi'(x + h) = ka^{x+h} = k(a^{x} + ka^{x}\\, h + \\etc.)",
   "name": null,
   "statement": "The first derived function of a^x, evaluated at x + h, is k a^(x+h), developed in powers of h.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the base of the exponential"
    },
    {
     "unit": null,
     "symbol": "k",
     "meaning": "the natural logarithm of a"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small increment added to x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/natural-logarithm",
    "concept/value-of-a-function",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a7dac87ce5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "22",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\dfrac{1}{x + h} = \\dfrac{1}{x} - \\dfrac{h}{x^{2}} + \\etc.",
   "name": null,
   "statement": "The reciprocal 1/(x + h) is developed in powers of h, found by common division.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small increment added to x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/variable",
    "theorem/taylor-s-theorem"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8d2205e664",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "22",
   "location": "Taylor's Theorem. Derived Functions",
   "latex": "\\phi''(x + h) = -\\dfrac{1}{(x + h)^{2}} = -(x + h)^{-2}",
   "name": null,
   "statement": "The second derived function of log x, evaluated at x + h, equals minus one over (x + h) squared.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "the function (here log x)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small increment added to x"
    }
   ],
   "sympy": "Eq(diff(log(x), x, 2), -1/(x**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/exponent",
    "concept/higher-order-derivative",
    "concept/logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6dc61d02bb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "22",
   "location": "Differential Coefficients",
   "latex": "\\phi x + \\phi' x\\, h + \\phi'' x\\, \\frac{h^{2}}{2} + \\phi''' x\\, \\frac{h^{3}}{2·3} + \\etc.",
   "name": null,
   "statement": "The function of x+h, expanded in powers of the increment h, has as its coefficients the function, its first, second and third differential coefficients divided by the factorials 1, 2, 2·3 (the displayed right-hand side; the chapter does not write the left-hand side).",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "φ",
     "meaning": "any function of x"
    },
    "x",
    "h",
    "φ' x",
    "φ'' x",
    "φ''' x"
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-expression",
    "concept/coefficient",
    "concept/derivative",
    "concept/function",
    "concept/higher-order-derivative",
    "concept/increment",
    "concept/infinite-sequence",
    "concept/power"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-4819483f76",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "23",
   "location": "Differential Coefficients",
   "latex": "\\phi' x\\, h + \\phi'' x\\, \\frac{h^{2}}{2} x + \\phi''' x\\, \\frac{h^{3}}{2·3} + \\etc.",
   "name": null,
   "statement": "The increment of the function produced by changing x into x+h is the series in h whose first term is φ'x·h. Note: the chapter prints a stray letter x after h²/2 in this line; this is recorded as printed and flagged as a probable typesetting error in the book.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "φ",
     "meaning": "any function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "the increment of x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/function",
    "concept/increment",
    "concept/infinite-sequence",
    "concept/power",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a4c8aad86c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "23",
   "location": "Differential Coefficients",
   "latex": "\\frac{\\emph{increment of } \\phi x}{\\emph{increment of } x} = \\phi' x + \\phi'' x\\, \\frac{h}{2} x + \\phi''' x\\, \\frac{h^{2}}{2·3} + \\etc.",
   "name": null,
   "statement": "The ratio of the increment of the function to the increment of its variable equals φ'x plus a series in h that vanishes as h diminishes. Note: the chapter prints a stray letter x after h/2 in this line; recorded as printed and flagged as a probable typesetting error in the book.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "φ",
     "meaning": "any function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "the increment of x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/function",
    "concept/increment",
    "concept/infinite-sequence",
    "concept/limit",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2f07fe1e72",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "23",
   "location": "Differential Coefficients",
   "latex": "h\\left(\\phi'' x\\, \\frac{1}{2} + \\phi''' x\\, \\frac{h}{2·3} + \\etc.\\right)",
   "name": null,
   "statement": "The part of the ratio other than its first term φ'x is written as h times a series; this is an expression for that remainder (no equality sign in the chapter).",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "φ''x",
     "meaning": "second differential coefficient of φx"
    },
    {
     "unit": null,
     "symbol": "φ'''x",
     "meaning": "third differential coefficient of φx"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "the increment of x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-expression",
    "concept/higher-order-derivative",
    "concept/increment",
    "concept/infinite-sequence",
    "concept/remainder"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-812bb93f4b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "24",
   "location": "Differential Coefficients",
   "latex": "kh^{n} : lh^{n+1} + mh^{n+2} + \\etc.,\\ ::\\ k : lh + mh^{2} + \\etc.,",
   "name": null,
   "statement": "The ratio of the term kh^n to the tail lh^(n+1)+mh^(n+2)+… is the same as the ratio of k to lh+mh²+…; a proportion used to show the tail can be made a small part of the term kh^n.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "k",
     "meaning": "a given quantity independent of h"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "the increment"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the power index of the term"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "coefficient of the next power of h"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "coefficient of the following power of h"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/coefficient",
    "concept/common-ratio",
    "concept/infinite-sequence",
    "concept/power",
    "concept/proportion",
    "concept/remainder",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8b05716c99",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "25",
   "location": "The Notation of the Differential Calculus",
   "latex": "d.x^{2} = 2x\\, dx + (dx)^{2}",
   "name": null,
   "statement": "The increment of x squared, found by expanding (x + dx)^2, equals 2x dx plus the square of dx.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "d.x^2",
     "meaning": "increment of x^2"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    }
   ],
   "sympy": "Eq(d_x2, 2*x*dx + dx**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/increment",
    "concept/power",
    "concept/square"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-95f6f0bd39",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "26",
   "location": "The Notation of the Differential Calculus",
   "latex": "\\dfrac{d.x^{2}}{dx} = 2x + dx",
   "name": null,
   "statement": "Dividing the increment of x^2 by the increment of x gives 2x plus dx, which tends to 2x as dx diminishes.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "d.x^2",
     "meaning": "increment of x^2"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(d_x2/dx, 2*x + dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/differential",
    "concept/increment",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f812adf496",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "26",
   "location": "The Notation of the Differential Calculus",
   "latex": "\\dfrac{d.x^{2}}{dx} = 2x",
   "name": null,
   "statement": "The ratio of the increment of x^2 to the increment of x is, in the limit, 2x; in the first explanation this equation is strictly true.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "d.x^2",
     "meaning": "increment of x^2"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(d_x2/dx, 2*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/derivative",
    "concept/differential",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-54038ee0b9",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "27",
   "location": "The Notation of the Differential Calculus",
   "latex": "\\dfrac{dy}{dx} = 2x + dx",
   "name": null,
   "statement": "For y = x^2 the ratio of the increment of y to the increment of x equals 2x plus dx.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(dy/dx, 2*x + dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/derivative",
    "concept/differential",
    "concept/increment",
    "concept/tangent"
   ],
   "pages": [
    "27",
    "37"
   ],
   "chapters": [
    "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
    "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-4df3fc90aa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "27",
   "location": "The Notation of the Differential Calculus",
   "latex": "\\dfrac{dy}{dx} = 2x",
   "name": null,
   "statement": "The ratio of the increment of y to the increment of x, with dx made zero, is 2x; this is the limit of the ratio, not a ratio of two actual quantities.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(dy/dx, 2*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/coefficient",
    "concept/common-ratio",
    "concept/derivative",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c0e0693e66",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "28",
   "location": "The Notation of the Differential Calculus",
   "latex": "dy = 2x\\, dx + (dx)^{2}",
   "name": null,
   "statement": "The increment of y for y = x^2 is 2x dx plus the square of dx; the (dx)^2 term affects dy but not the limit of dy/dx.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    }
   ],
   "sympy": "Eq(dy, 2*x*dx + dx**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/function",
    "concept/increment",
    "concept/infinitesimal",
    "concept/limit"
   ],
   "pages": [
    "28",
    "37",
    "60"
   ],
   "chapters": [
    "de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",
    "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
    "de-morgan-elementary-illustrations-calculus-1899/ch-the-method-of-fluxions"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2f78e0fd4d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-algebraical-geometry",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "30",
   "location": "Algebraical Geometry",
   "latex": "y = x^{2}",
   "name": null,
   "statement": "The ordinate y of a moving point P is always equal to the square of its abscissa x, so P traces a parabola (the book's example of a curve given by a function).",
   "kind": "definition",
   "symbols": [
    {
     "unit": "linear unit",
     "symbol": "y",
     "meaning": "ordinate PM of the point P (distance measured parallel to OB, the axis of y)"
    },
    {
     "unit": "linear unit",
     "symbol": "x",
     "meaning": "abscissa OM of the point P (distance measured along OA, the axis of x)"
    }
   ],
   "sympy": "Eq(y, x**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/abscissa",
    "concept/cartesian-coordinates",
    "concept/curve",
    "concept/exponent",
    "concept/function",
    "concept/ordinate",
    "concept/parabola",
    "concept/relation-between-variables",
    "concept/square",
    "concept/value-of-a-function",
    "concept/variable"
   ],
   "pages": [
    "30",
    "35",
    "36",
    "60",
    "102"
   ],
   "chapters": [
    "de-morgan-elementary-illustrations-calculus-1899/ch-algebraical-geometry",
    "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
    "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
    "de-morgan-elementary-illustrations-calculus-1899/ch-the-method-of-fluxions",
    "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-3f89993684",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "31",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "x^{2} + y^{2} = r^{2}",
   "name": "equation of the circle",
   "statement": "A point whose co-ordinates are x and y lies on a circle of radius r centred at the origin O exactly when x squared plus y squared equals r squared.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "co-ordinate OM of the point P (abscissa)"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "co-ordinate MP of the point P (ordinate)"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the circle OA"
    }
   ],
   "sympy": "Eq(x**2 + y**2, r**2)",
   "physics": false,
   "states": [
    "theorem/equation-of-the-circle"
   ],
   "concepts": [
    "concept/abscissa",
    "concept/cartesian-coordinates",
    "concept/centre-of-a-circle",
    "concept/circle",
    "concept/equation",
    "concept/ordinate",
    "concept/radius-of-a-regular-polygon"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f425d5cb6f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "32",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "ay + bx = ab",
   "name": null,
   "statement": "The straight line EF, cutting the axes at distances a and b from O, satisfies ay + bx = ab for every one of its points, with the signs of x and y taken in the stated directions.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OE, where the line EF cuts the axis of x"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OF, where the line EF cuts the axis of y"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "co-ordinate OM (abscissa)"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "co-ordinate MP (ordinate)"
    }
   ],
   "sympy": "Eq(a*y + b*x, a*b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/equation",
    "concept/inequality",
    "concept/line",
    "concept/negative-direction",
    "concept/sign"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2e1ac207a9",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "32",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "ay - bx = ab",
   "name": null,
   "statement": "For a point P' on EF produced, with x taken as the signed co-ordinate, the relation is ay - bx = ab, which is the earlier equation with the sign of x changed.",
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    {
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    },
    {
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     "symbol": "b",
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    },
    {
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    },
    {
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   ],
   "sympy": "Eq(a*y - b*x, a*b)",
   "physics": false,
   "states": [],
   "concepts": [
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    "concept/equation",
    "concept/line",
    "concept/negative-direction",
    "concept/sign"
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  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-210c626c47",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "32",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "bx - ay = ab",
   "name": null,
   "statement": "For a point P'' on FE produced, the relation is bx - ay = ab, which is the earlier equation with the sign of y changed.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
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    },
    {
     "unit": null,
     "symbol": "b",
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    },
    {
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    },
    {
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   ],
   "sympy": "Eq(b*x - a*y, a*b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/equation",
    "concept/line",
    "concept/negative-direction",
    "concept/sign"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-be98437520",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "33",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "x^{2} + b^{2}\\, \\frac{(a - x)^{2}}{a^{2}} = r^{2}",
   "name": null,
   "statement": "Substituting the value of y from the line equation into the circle equation gives this equation in x alone.",
   "kind": "result",
   "symbols": [
    {
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     "symbol": "x",
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    {
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    {
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   ],
   "sympy": "Eq(x**2 + b**2*(a - x)**2/a**2, r**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/circle",
    "concept/equation",
    "concept/line",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-43e4701644",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "33",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "(a^{2} + b^{2}) x^{2} - 2ab^{2}x + a^{2}(b^{2} - r^{2}) = 0",
   "name": null,
   "statement": "The abscissas x of the points where the line meets the circle are the roots of this quadratic equation in x.",
   "kind": "result",
   "symbols": [
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    {
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    {
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    {
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   ],
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   "physics": false,
   "states": [],
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    "concept/circle",
    "concept/equation",
    "concept/line",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a74922b914",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "33",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "(a^{2} + b^{2}) y^{2} - 2a^{2}by + b^{2}(a^{2} - r^{2}) = 0",
   "name": null,
   "statement": "The ordinates y of the points where the line meets the circle are the roots of this quadratic equation in y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
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    {
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    },
    {
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    {
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   ],
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   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/circle",
    "concept/equation",
    "concept/line",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-9e1036605a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "33",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "x = a\\, \\frac{b^{2} ± \\sqrt{(a^{2} + b^{2})r^{2} - a^{2}b^{2}}}{a^{2} + b^{2}}",
   "name": null,
   "statement": "The abscissas of the two intersection points of the line and the circle, taking the upper or the lower sign throughout.",
   "kind": "result",
   "symbols": [
    {
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     "symbol": "x",
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    },
    {
     "unit": null,
     "symbol": "a",
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    },
    {
     "unit": null,
     "symbol": "b",
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    },
    {
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     "symbol": "r",
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    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
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    "concept/cartesian-coordinates",
    "concept/circle",
    "concept/line",
    "concept/sign",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-3b422fe3af",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "33",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "y = b\\, \\frac{a^{2} \\mp \\sqrt{(a^{2} + b^{2})r^{2} - a^{2}b^{2}}}{a^{2} + b^{2}}",
   "name": null,
   "statement": "The ordinates of the two intersection points of the line and the circle, with the sign of the root taken opposite to that in the formula for x.",
   "kind": "result",
   "symbols": [
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     "unit": null,
     "symbol": "y",
     "meaning": "co-ordinate MP of the intersection point"
    },
    {
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     "symbol": "a",
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    },
    {
     "unit": null,
     "symbol": "b",
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    {
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     "meaning": "radius of the circle"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/circle",
    "concept/line",
    "concept/ordinate",
    "concept/sign",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8cdd831823",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "33",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "(a^{2} + b^{2})r^{2} > a^{2}b^{2}",
   "name": null,
   "statement": "The line meets the circle in two points exactly when r is greater than the perpendicular from O to EF, whose length is ab over the square root of a squared plus b squared.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OE"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OF"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Gt((a**2 + b**2)*r**2, a**2*b**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "concept/inequality",
    "concept/line",
    "concept/number",
    "concept/perpendicular"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-3da072c933",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "34",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "(a^{2} + b^{2})r^{2} = a^{2}b^{2}",
   "name": null,
   "statement": "The two intersection points coincide and the straight line EF is a tangent to the circle.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OE"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OF"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq((a**2 + b**2)*r**2, a**2*b**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "concept/equation",
    "concept/line",
    "concept/perpendicular",
    "concept/tangent"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-afa4a026ac",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "34",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "(a^{2} + b^{2})r^{2} < a^{2}b^{2}",
   "name": null,
   "statement": "The values of x and y are impossible and the straight line does not meet the circle.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OE"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OF"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Lt((a**2 + b**2)*r**2, a**2*b**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "concept/inequality",
    "concept/line",
    "concept/real-number"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a9e3feda6d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "35",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "(-x)^{2} = x^{2}",
   "name": null,
   "statement": "Squaring a negative value of x gives the same result as squaring the positive value, so the left branch of the curve mirrors the right.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the curve"
    }
   ],
   "sympy": "Eq((-x)**2, x**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/curve",
    "concept/function",
    "concept/negative-number",
    "concept/sign",
    "concept/value-of-a-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-3f3222b0c3",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "31",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "OE = a",
   "name": null,
   "statement": "The length OE, measured along the axis of x, is called a and is the intercept of the straight line EF on that axis.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OE, intercept of the line EF on the axis of x"
    }
   ],
   "sympy": "Eq(OE, a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/line",
    "concept/point",
    "quantity/length"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ad87ee0ddb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "31",
   "location": "On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes",
   "latex": "OF = b",
   "name": null,
   "statement": "The length OF, measured along the axis of y, is called b and is the intercept of the straight line EF on that axis.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OF, intercept of the line EF on the axis of y"
    }
   ],
   "sympy": "Eq(OF, b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/line",
    "concept/point",
    "quantity/length"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ad8fd0ddd0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "36",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "y + dy = (x +  dx)^{2}",
   "name": null,
   "statement": "When x increases by dx, the point on the curve has co-ordinates x + dx and y + dy, and the curve relation still holds for them.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x (MM')"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y (QP')"
    }
   ],
   "sympy": "Eq(y + dy, (x + dx)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/curve",
    "concept/differential",
    "concept/increment",
    "concept/relation-between-variables"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-dcf7f0ca40",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "37",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "y - dy = (x - dx)^{2}",
   "name": null,
   "statement": "When P' is placed on the other side of P, at co-ordinates x - dx and y - dy, the curve relation holds for those co-ordinates.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x, taken backwards"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y, taken backwards"
    }
   ],
   "sympy": "Eq(y - dy, (x - dx)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/curve",
    "concept/increment",
    "concept/relation-between-variables"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-b31bfc3890",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "37",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "dy = 2x\\, dx - (dx)^{2}",
   "name": null,
   "statement": "Subtracting the backward relation from y = x^2 gives the increment of y for the backward step.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x, taken backwards"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y for the backward step"
    }
   ],
   "sympy": "Eq(dy, 2*x*dx - dx**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-fbcd5110c0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "37",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "\\dfrac{dy}{dx} = 2x - dx",
   "name": null,
   "statement": "For the backward step, the ratio of the increment of y to the increment of x equals 2x minus dx, which approaches 2x as dx diminishes.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "ratio of the increment of y to the increment of x for the backward step"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of P"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x, taken backwards"
    }
   ],
   "sympy": "Eq(dy/dx, 2*x - dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/derivative",
    "concept/limit",
    "concept/tangent"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6b2b7a8ffa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "38",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "y = \\phi x",
   "name": null,
   "statement": "The ordinate y is some function of the abscissa x, written as phi x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the curve"
    },
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "a function of x"
    }
   ],
   "sympy": "Eq(y, phi(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/curve",
    "concept/function",
    "concept/function-notation",
    "concept/relation-between-variables",
    "concept/variable"
   ],
   "pages": [
    "38",
    "102"
   ],
   "chapters": [
    "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
    "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-5112129174",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "38",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "y = \\log x",
   "name": null,
   "statement": "For the curve whose ordinates are the Naperian logarithms of the abscissae, y equals the natural logarithm of x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the curve"
    },
    {
     "unit": null,
     "symbol": "log",
     "meaning": "Naperian (natural) logarithm"
    }
   ],
   "sympy": "Eq(y, log(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/curve",
    "concept/function",
    "concept/logarithm",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c52798580e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "38",
   "location": "The Drawing of a Tangent to a Curve",
   "latex": "y + dy = \\log x + \\dfrac{1}{x}\\, dx - \\dfrac{1}{2x^{2}}\\, dx^{2}",
   "name": null,
   "statement": "The incremented ordinate of the logarithmic curve, expanded in powers of dx, is log x plus dx/x minus dx^2/(2x^2), with further terms omitted (etc.).",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the point"
    },
    {
     "unit": null,
     "symbol": "log",
     "meaning": "Naperian (natural) logarithm"
    }
   ],
   "sympy": "Eq(y + dy, log(x) + dx/x - dx**2/(2*x**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/differential",
    "concept/function",
    "concept/increment",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-4ae42ee0fc",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "40",
   "location": "Rational Explanation of the Language of Leibnitz",
   "latex": "y = \\phi(x)",
   "name": null,
   "statement": "The curve PP' is the graph of y equal to a given function of x, so PM is found by substituting OM into that function.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate PM of a point on the curve"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa OM"
    },
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "the given function of x whose graph is the curve"
    }
   ],
   "sympy": "Eq(y, phi(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/curve",
    "concept/function",
    "concept/function-notation",
    "concept/variable"
   ],
   "pages": [
    "40",
    "74"
   ],
   "chapters": [
    "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
    "de-morgan-elementary-illustrations-calculus-1899/ch-recapitulation-of-results-reached-in-the-theory-of-functions"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-cef0aaf15d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "40",
   "location": "Rational Explanation of the Language of Leibnitz",
   "latex": "MM' = dx",
   "name": null,
   "statement": "The increment of x, taken as the segment MM', is written dx; it may be supposed as small as we please.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "MM'",
     "meaning": "the increment of OM from M to M'"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of the variable x"
    }
   ],
   "sympy": "Eq(MMp, dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/increment",
    "concept/infinitesimal",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8dd22968a7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "41",
   "location": "Rational Explanation of the Language of Leibnitz",
   "latex": "P'Q = \\phi' x\\, dx + \\phi'' x\\, \\frac{(dx)^{2}}{2} + \\phi''' x\\, \\frac{(dx)^{3}}{2·3} + \\etc.",
   "name": null,
   "statement": "The increment P'Q of the ordinate equals a series in powers of dx, whose first term is phi'(x) dx, with higher derivatives divided by factorials.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "P'Q",
     "meaning": "increment of the ordinate y, equal to dy = phi(x + dx) - phi(x)"
    },
    {
     "unit": null,
     "symbol": "\\phi' x",
     "meaning": "first differential coefficient of phi at x"
    },
    {
     "unit": null,
     "symbol": "\\phi'' x",
     "meaning": "second differential coefficient of phi at x"
    },
    {
     "unit": null,
     "symbol": "\\phi''' x",
     "meaning": "third differential coefficient of phi at x"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of the variable x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/first-term",
    "concept/function",
    "concept/higher-order-derivative",
    "concept/increment",
    "concept/infinite-sequence",
    "concept/infinitesimal",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c845c6f79f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "41",
   "location": "Rational Explanation of the Language of Leibnitz",
   "latex": "dx \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^{2}}",
   "name": null,
   "statement": "The chord PP' equals dx times the square root of one plus the square of dy over dx.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "PP'",
     "meaning": "chord joining P and P' on the curve"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y"
    }
   ],
   "sympy": "Eq(chord, dx*sqrt(1 + (dy/dx)**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/arc-of-a-curve",
    "concept/chord",
    "concept/derivative",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a9df55201a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rational-explanation-of-the-language-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "41",
   "location": "Rational Explanation of the Language of Leibnitz",
   "latex": "\\tan VPQ·PQ = VQ",
   "name": null,
   "statement": "The tangent of the angle VPQ times PQ gives VQ, so the first term of the series is the segment VQ along the tangent.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "VPQ",
     "meaning": "angle between the tangent PV and the line PQ at P"
    },
    {
     "unit": null,
     "symbol": "PQ",
     "meaning": "segment PQ, equal to dx"
    },
    {
     "unit": null,
     "symbol": "VQ",
     "meaning": "segment of the tangent from V to Q"
    }
   ],
   "sympy": "Eq(tan(VPQ)*PQ, VQ)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/line",
    "concept/tangent",
    "concept/tangent-function",
    "quantity/angle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-cf5c482601",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-orders-of-infinity",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "44",
   "location": "Orders of Infinity",
   "latex": "\\phi' x\\, dx + \\phi'' x\\, \\dfrac{(dx)^{2}}{2}",
   "name": null,
   "statement": "The increment QP' along the curve, to the first two orders in dx, is the first derivative of phi at x times dx plus the second derivative times half (dx) squared, with further terms omitted.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\phi' x",
     "meaning": "first differential coefficient (first derivative) of phi at x"
    },
    {
     "unit": null,
     "symbol": "\\phi'' x",
     "meaning": "second differential coefficient (second derivative) of phi at x"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "the small increment PQ of the variable x, put in place of h"
    },
    {
     "unit": null,
     "symbol": "QP'",
     "meaning": "the segment QP' of the curve in Fig. 4, shown in the chapter as the expansion itself (the chapter gives the expression without an explicit equals sign)"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/higher-order-derivative",
    "concept/infinitesimal",
    "concept/order-of-smallness",
    "concept/power"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-5ef55225ff",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "46",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "a^{2} + b^{2} = l^{2}",
   "name": "Pythagorean theorem",
   "statement": "The squares of the two axial segments OA and OB add up to the square of the length of the sliding line AB.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA, the length of the segment from O to A on one axis"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB, the length of the segment from O to B on the other axis"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "AB, the given fixed length of the sliding line"
    }
   ],
   "sympy": "Eq(a**2 + b**2, l**2)",
   "physics": false,
   "states": [
    "law/pythagorean-relation",
    "theorem/pythagorean-theorem"
   ],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/geometry",
    "concept/triangle",
    "method/solving-a-right-spherical-triangle",
    "quantity/length"
   ],
   "pages": [
    "46",
    "50"
   ],
   "chapters": [
    "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
    "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f221481e7c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "46",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "(a + da)^{2} + (b - db)^{2} = l^{2}",
   "name": null,
   "statement": "In the displaced position the line still has length l, with its ends at a + da and b - db along the axes.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    },
    {
     "unit": null,
     "symbol": "da",
     "meaning": "AA', the increment of OA when the line moves"
    },
    {
     "unit": null,
     "symbol": "db",
     "meaning": "BB', the decrement of OB when the line moves"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "AB, the fixed length of the line"
    }
   ],
   "sympy": "Eq((a + da)**2 + (b - db)**2, l**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/limit-of-intersections",
    "concept/line",
    "quantity/length"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-7586f33bb7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "46",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "2a\\, da + (da)^{2} - 2b\\, db + (db)^{2} = 0\\Add{,}",
   "name": null,
   "statement": "Subtracting the first length equation from the displaced one gives a relation between the increments da and db.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    },
    {
     "unit": null,
     "symbol": "da",
     "meaning": "increment AA' of OA"
    },
    {
     "unit": null,
     "symbol": "db",
     "meaning": "decrement BB' of OB"
    }
   ],
   "sympy": "Eq(2*a*da + da**2 - 2*b*db + db**2, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/limit-of-intersections",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8b54a8bd74",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "46",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "\\frac{db}{da} = \\frac{2a + da}{2b - db}\\Add{.}",
   "name": null,
   "statement": "The ratio of the increment db to the increment da equals (2a + da)/(2b - db); this is equation (1).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "da",
     "meaning": "increment AA' of OA"
    },
    {
     "unit": null,
     "symbol": "db",
     "meaning": "decrement BB' of OB"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    }
   ],
   "sympy": "Eq(db/da, (2*a + da)/(2*b - db))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/limit",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f0f7008416",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "47",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "ay + bx = ab\\Add{.}",
   "name": null,
   "statement": "Every point (x, y) of the line AB satisfies this equation in the coordinate axes; this is equation (2).",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "co-ordinate of a point measured along the axis OC (OM for P')"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "co-ordinate of a point measured parallel to OC (MP for P)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    }
   ],
   "sympy": "Eq(a*y + b*x, a*b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/equation",
    "concept/line"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-db9055b953",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "47",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "(a + da)y + (b - db)x = (a + da)(b - db)\\Add{;}",
   "name": null,
   "statement": "The line of the displaced position, which cuts off a + da and b - db from the axes, has this equation; this is equation (3).",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "co-ordinate measured along OC"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "co-ordinate measured parallel to OC"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    },
    {
     "unit": null,
     "symbol": "da",
     "meaning": "increment AA' of OA"
    },
    {
     "unit": null,
     "symbol": "db",
     "meaning": "decrement BB' of OB"
    }
   ],
   "sympy": "Eq((a + da)*y + (b - db)*x, (a + da)*(b - db))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/line",
    "concept/point"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f669eb9351",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "47",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "y\\, da - x\\, db = b\\, da - a\\, db - da\\, db\\Add{.}",
   "name": null,
   "statement": "Subtracting equation (2) from equation (3) gives this relation for the point P' of intersection; this is equation (4).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "co-ordinate of P' along OC"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "co-ordinate of P' parallel to OC"
    },
    {
     "unit": null,
     "symbol": "da",
     "meaning": "increment AA' of OA"
    },
    {
     "unit": null,
     "symbol": "db",
     "meaning": "decrement BB' of OB"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    }
   ],
   "sympy": "Eq(y*da - x*db, b*da - a*db - da*db)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/limit-of-intersections",
    "concept/point"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c538f03ded",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "47",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "y - x\\, \\frac{2a + da}{2b - db} = b - a\\, \\frac{2a + da}{2b - db} - db\\Add{.}",
   "name": null,
   "statement": "Dividing equation (4) by da and substituting db/da from equation (1) gives this relation; this is equation (5).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "co-ordinate of P' along OC"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "co-ordinate of P' parallel to OC"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    },
    {
     "unit": null,
     "symbol": "da",
     "meaning": "increment AA' of OA"
    },
    {
     "unit": null,
     "symbol": "db",
     "meaning": "decrement BB' of OB"
    }
   ],
   "sympy": "Eq(y - x*(2*a + da)/(2*b - db), b - a*(2*a + da)/(2*b - db) - db)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/limit-of-intersections",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-10bbd0fdea",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "47",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "y - \\frac{a}{b}\\, x = b - \\frac{a^{2}}{b}",
   "name": null,
   "statement": "Equation (5) with da and db diminished without limit gives this line through the limit point P; the same line in cleared form is equation (6).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "co-ordinate of the limit point P along OC"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "co-ordinate of the limit point P parallel to OC"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    }
   ],
   "sympy": "Eq(y - a/b*x, b - a**2/b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/limit",
    "concept/limit-of-intersections"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-9f03fe101d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "47",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "by - ax = b^{2} - a^{2}\\Add{.}",
   "name": null,
   "statement": "Cleared of fractions, the limit of the intersections satisfies this equation; this is equation (6).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "co-ordinate of the limit point P along OC"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "co-ordinate of the limit point P parallel to OC"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    }
   ],
   "sympy": "Eq(b*y - a*x, b**2 - a**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/limit",
    "concept/limit-of-intersections"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8ebcb7b59b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "47",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "x = OM = \\frac{a^{3}}{a^{2} + b^{2}} = \\frac{a^{3}}{l^{2}}",
   "name": null,
   "statement": "Solving equations (6) and (2) gives the co-ordinate x of the limit point P as a^3 over l^2.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "OM, co-ordinate of the limit point P along OC"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "AB, length of the line"
    }
   ],
   "sympy": "Eq(x, a**3/(a**2 + b**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/limit-of-intersections"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-b31d037a60",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "47",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "y = MP = \\frac{b^{3}}{a^{2} + b^{2}} = \\frac{b^{3}}{l^{2}}",
   "name": null,
   "statement": "Solving equations (6) and (2) gives the co-ordinate y of the limit point P as b^3 over l^2.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "MP, co-ordinate of the limit point P parallel to OC"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "AB, length of the line"
    }
   ],
   "sympy": "Eq(y, b**3/(a**2 + b**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/limit-of-intersections"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-e898c07ec8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "48",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "BP = OM\\, \\dfrac{BA}{AO} = \\dfrac{a^{3}}{l^{2}}\\, \\dfrac{l}{a} = \\dfrac{a^{2}}{l}",
   "name": null,
   "statement": "By similar triangles the distance BP from B to the limit point equals a^2 over l.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "BP",
     "meaning": "distance from B to the limit point P"
    },
    {
     "unit": null,
     "symbol": "OM",
     "meaning": "co-ordinate x of P"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "AB, length of the line"
    }
   ],
   "sympy": "Eq(BP, a**2/l)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/limit-of-intersections",
    "quantity/length"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-0c02015819",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "48",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "PA = \\dfrac{b^{2}}{l}",
   "name": null,
   "statement": "Similarly, the distance PA from A to the limit point equals b^2 over l.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "PA",
     "meaning": "distance from A to the limit point P"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "AB, length of the line"
    }
   ],
   "sympy": "Eq(PA, b**2/l)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/limit-of-intersections",
    "quantity/length"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ce212c1be0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "48",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "AQ = \\dfrac{a^{2}}{l}",
   "name": null,
   "statement": "Since OA is a mean proportional between AQ and AB (OQ perpendicular to BA), AQ equals a^2 over l.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "AQ",
     "meaning": "distance from A to the foot Q of the perpendicular from O to AB"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "OA"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "AB, length of the line"
    }
   ],
   "sympy": "Eq(AQ, a**2/l)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/construction",
    "concept/geometrical-mean",
    "concept/perpendicular"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c8717ae212",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "48",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "BQ = \\dfrac{b^{2}}{l}",
   "name": null,
   "statement": "Similarly, BQ equals b^2 over l, the distance from B to the foot Q.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "BQ",
     "meaning": "distance from B to the foot Q of the perpendicular from O to AB"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "OB"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "AB, length of the line"
    }
   ],
   "sympy": "Eq(BQ, b**2/l)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/construction",
    "concept/geometrical-mean",
    "concept/perpendicular"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-d26b964259",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "48",
   "location": "A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines",
   "latex": "BP = AQ",
   "name": null,
   "statement": "The limit point P lies as far from B as Q lies from A, so the limit of the intersections is found by this construction.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "BP",
     "meaning": "distance from B to the limit point P"
    },
    {
     "unit": null,
     "symbol": "AQ",
     "meaning": "distance from A to the foot Q"
    }
   ],
   "sympy": "Eq(BP, AQ)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/construction",
    "concept/limit-of-intersections",
    "concept/point"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-401d77d185",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "BP = q",
   "name": null,
   "statement": "The length BP is denoted q.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "q",
     "meaning": "length BP"
    }
   ],
   "sympy": "Eq(BP, q)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/radius-of-a-regular-polygon"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-59aa1980ad",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "Aa = dp",
   "name": null,
   "statement": "The infinitely small increment Aa is denoted dp.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "dp",
     "meaning": "infinitely small increment Aa"
    }
   ],
   "sympy": "Eq(Aa, dp)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c09da155b8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "50",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "Aa : A'a :: OA : OB :: a : b",
   "name": null,
   "statement": "Corresponding sides of the small triangles at A are in the same ratio as the radii OA and OB.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "A",
     "meaning": "point A on the line AB"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal",
    "concept/proportion",
    "concept/radius-of-a-regular-polygon",
    "concept/similar-triangles"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-5ff27c6a1e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "50",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "Bb : B'b :: OA : OB :: a : b",
   "name": null,
   "statement": "The small arcs at B are in the same proportion as the radii, so this ratio matches the one at A.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "B",
     "meaning": "point B on the line AB"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/proportion",
    "concept/radius-of-a-regular-polygon",
    "quantity/arc-of-a-circle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-121859064c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "50",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "\\PadTo[r]{BP + Pa}{Bb} : A'a :: \\PadTo[l]{a^{2} + b^{2}}{a^{2}} : b^{2}",
   "name": null,
   "statement": "The composed proportion obtained from the two proportions above, with Aa equal to B'b, relates the arc Bb to A'a.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    },
    {
     "unit": null,
     "symbol": "BP",
     "meaning": "distance from B to the point P"
    },
    {
     "unit": null,
     "symbol": "Pa",
     "meaning": "distance from P to a"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal",
    "concept/proportion",
    "method/composition-of-proportions"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-7b1bf0f4df",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "50",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "BP + Pa : Pa :: a^{2} + b^{2} : b^{2}",
   "name": null,
   "statement": "After cancelling the common factor, the segments BP and Pa stand in the same proportion as a² + b² and b².",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    },
    {
     "unit": null,
     "symbol": "BP",
     "meaning": "distance from B to the point P"
    },
    {
     "unit": null,
     "symbol": "Pa",
     "meaning": "distance from P to a"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/proportion",
    "method/composition-of-proportions"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f715bd42e2",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "50",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "BP + PA = l",
   "name": null,
   "statement": "The segments BP and PA together make up the whole line AB, of length l.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "BP",
     "meaning": "distance from B to P"
    },
    {
     "unit": null,
     "symbol": "PA",
     "meaning": "distance from P to A"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "length AB"
    }
   ],
   "sympy": "Eq(BP + PA, l)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/limit",
    "concept/line"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-703497383f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "50",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "\\PadTo[r]{BP + Pa}{BP} : Pa :: \\PadTo[l]{a^{2} + b^{2}}{a^{2}} : b^{2}",
   "name": null,
   "statement": "Rearranging the composed proportion gives a proportion between the segments BP and Pa and the squares a² and b².",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    },
    {
     "unit": null,
     "symbol": "BP",
     "meaning": "distance from B to P"
    },
    {
     "unit": null,
     "symbol": "Pa",
     "meaning": "distance from P to a"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/proportion",
    "method/composition-of-proportions"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-1549df84a0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "50",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "PA = \\frac{b^{2}}{l}",
   "name": null,
   "statement": "The distance PA from A to the sought point P equals b² divided by l, the result already obtained.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "PA",
     "meaning": "distance from P to A"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "length AB"
    }
   ],
   "sympy": "Eq(PA, b**2/l)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/point",
    "concept/proportion"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6f1b3f6b56",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "\\angle A'PA = d\\theta",
   "name": null,
   "statement": "The infinitely small angle A'PA is denoted dθ.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "radian (angle unit as in the book's footnote)",
     "symbol": "dθ",
     "meaning": "infinitely small angle A'PA"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal",
    "concept/plane-angle",
    "quantity/angle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-42e6d48b6f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "A'a = (p - dp)\\, d\\theta",
   "name": null,
   "statement": "The small arc A'a equals the radius PA' times the small angle dθ.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "A'a",
     "meaning": "small arc from A' to a"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "length PA"
    },
    {
     "unit": null,
     "symbol": "dp",
     "meaning": "infinitely small increment Aa of PA"
    },
    {
     "unit": "radian",
     "symbol": "dθ",
     "meaning": "infinitely small angle A'PA"
    }
   ],
   "sympy": "Eq(A_a_prime, (p - dp)*dtheta)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal",
    "concept/radius-of-a-regular-polygon",
    "quantity/arc-of-a-circle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-d68180241d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "Bb = q\\, d\\theta",
   "name": null,
   "statement": "The small arc Bb equals the radius PB times the small angle dθ.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "Bb",
     "meaning": "small arc from B to b"
    },
    {
     "unit": null,
     "symbol": "q",
     "meaning": "length BP"
    },
    {
     "unit": "radian",
     "symbol": "dθ",
     "meaning": "infinitely small angle A'PA"
    }
   ],
   "sympy": "Eq(Bb, q*dtheta)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal",
    "concept/radius-of-a-regular-polygon",
    "quantity/arc-of-a-circle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-99fd7b356a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "PA = p",
   "name": null,
   "statement": "The length PA is denoted p.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "p",
     "meaning": "length PA"
    }
   ],
   "sympy": "Eq(PA, p)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/radius-of-a-regular-polygon"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-63e5a37a91",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "B'b = dq",
   "name": null,
   "statement": "The infinitely small increment B'b is denoted dq.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "dq",
     "meaning": "infinitely small increment B'b"
    }
   ],
   "sympy": "Eq(B_b_prime, dq)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f30db6aab5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "OA = a",
   "name": null,
   "statement": "The length OA is denoted a.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    }
   ],
   "sympy": "Eq(OA, a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/radius-of-a-regular-polygon"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-327c49b6d7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "OB = b",
   "name": null,
   "statement": "The length OB is denoted b.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    }
   ],
   "sympy": "Eq(OB, b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/radius-of-a-regular-polygon"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-9f7090a0d7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "AB = l",
   "name": null,
   "statement": "The length AB is denoted l.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "l",
     "meaning": "length AB"
    }
   ],
   "sympy": "Eq(AB, l)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/line"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-e776827fac",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "a\\alpha = \\mu",
   "name": null,
   "statement": "The small perpendicular offset a α is denoted μ.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "μ",
     "meaning": "small offset a α, which vanishes in the limit"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal",
    "concept/perpendicular"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-80281bfea2",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "51",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "b\\beta = \\nu",
   "name": null,
   "statement": "The small perpendicular offset b β is denoted ν.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "ν",
     "meaning": "small offset b β, which vanishes in the limit"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal",
    "concept/perpendicular"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a45a1f8223",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "52",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "dp = dq",
   "name": null,
   "statement": "The infinitely small increments dp and dq are equal, which the book states is rigorously true.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dp",
     "meaning": "infinitely small increment Aa of PA"
    },
    {
     "unit": null,
     "symbol": "dq",
     "meaning": "infinitely small increment B'b of BP"
    }
   ],
   "sympy": "Eq(dp, dq)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equality",
    "concept/infinitesimal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-e5e8aba377",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "52",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "dp + \\mu &: (p - dp) \\sin d\\theta &&:: a &&: b",
   "name": null,
   "statement": "The exact proportion at A: the small segment A α plus μ, compared with the perpendicular, is in the ratio a : b.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dp",
     "meaning": "infinitely small increment Aa"
    },
    {
     "unit": null,
     "symbol": "μ",
     "meaning": "small offset a α"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "length PA"
    },
    {
     "unit": "radian",
     "symbol": "dθ",
     "meaning": "infinitely small angle A'PA"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal",
    "concept/proportion",
    "concept/sine"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ed2f7564f5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "52",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "q \\sin d\\theta &: \\PadTo{(p - dp) \\sin d\\theta}{dq + \\nu} &&:: a - da &&: b + db",
   "name": null,
   "statement": "The exact companion proportion at B, with the increments of a and b included.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "q",
     "meaning": "length BP"
    },
    {
     "unit": "radian",
     "symbol": "dθ",
     "meaning": "infinitely small angle A'PA"
    },
    {
     "unit": null,
     "symbol": "dq",
     "meaning": "infinitely small increment B'b"
    },
    {
     "unit": null,
     "symbol": "ν",
     "meaning": "small offset b β"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    },
    {
     "unit": null,
     "symbol": "da",
     "meaning": "infinitely small increment of a"
    },
    {
     "unit": null,
     "symbol": "db",
     "meaning": "infinitely small increment of b"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/infinitesimal",
    "concept/proportion",
    "concept/sine"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-63be0f0666",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "52",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "q\\left(1 + \\frac{\\mu}{dp}\\right) : (p - dp)\\left(1 + \\frac{\\nu}{dp}\\right) :: a(a - da) : b(b + db)",
   "name": null,
   "statement": "After composition and division by dp, the two sides of the proportion are written with the ratios μ/dp and ν/dp.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "q",
     "meaning": "length BP"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "length PA"
    },
    {
     "unit": null,
     "symbol": "dp",
     "meaning": "infinitely small increment Aa"
    },
    {
     "unit": null,
     "symbol": "μ",
     "meaning": "small offset a α"
    },
    {
     "unit": null,
     "symbol": "ν",
     "meaning": "small offset b β"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    },
    {
     "unit": null,
     "symbol": "da",
     "meaning": "infinitely small increment of a"
    },
    {
     "unit": null,
     "symbol": "db",
     "meaning": "infinitely small increment of b"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/limit",
    "concept/proportion",
    "method/composition-of-proportions"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-9244f5e9ef",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "52",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "q &: p &&:: a^{2} &&: b^{2}",
   "name": null,
   "statement": "In the limit as dθ vanishes, the segments BP and PA stand in the ratio of a² to b².",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "q",
     "meaning": "length BP"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "length PA"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/limit",
    "concept/limits-of-integration",
    "concept/proportion"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-1f7848bcf0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "52",
   "location": "The Same Problem Solved by the Principles of Leibnitz",
   "latex": "q + p = l &: p &&:: a^{2} + b^{2} = l^{2} &&: b^{2}",
   "name": null,
   "statement": "Since BP + PA = l and a² + b² = l², the limiting proportion gives PA in terms of l and b², the same result as before.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "q",
     "meaning": "length BP"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "length PA"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "length AB"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "length OA"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "length OB"
    }
   ],
   "sympy": "Eq(q + p, l)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/limit",
    "concept/proportion",
    "method/composition-of-proportions"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c696d116d9",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-an-illustration-from-dynamics-velocity-acceleration-etc",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "56",
   "location": "An Illustration from Dynamics: Velocity, Acceleration, etc",
   "latex": "a = 16\\frac{1}{12}",
   "name": null,
   "statement": "The constant a in the falling-body law (distance = a t^2, with time in seconds and distance in feet) is taken as 16 1/12, very nearly.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": "foot per second squared",
     "symbol": "a",
     "meaning": "constant in the falling-body law, so that the distance fallen in t seconds is a t^2"
    }
   ],
   "sympy": "Eq(a, Rational(193, 12))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/falling-bodies",
    "quantity/acceleration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8d580270bb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "57",
   "location": "Simple Harmonic Motion",
   "latex": "x = r \\cos\\theta",
   "name": null,
   "statement": "The abscissa of the moving point P is r times the cosine of the angle θ it has described from A.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa OM of the point P"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius OA of the circle"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "angle AOP described by the moving point"
    }
   ],
   "sympy": "Eq(x, r*cos(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/abscissa",
    "concept/circle",
    "concept/cosine",
    "concept/geometrical-figure",
    "quantity/angle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f6e49eb6c4",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "57",
   "location": "Simple Harmonic Motion",
   "latex": "y = r \\sin\\theta",
   "name": null,
   "statement": "The ordinate of the moving point P is r times the sine of the angle θ it has described from A.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate MP of the point P"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius OA of the circle"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "angle AOP described by the moving point"
    }
   ],
   "sympy": "Eq(y, r*sin(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "concept/ordinate",
    "concept/sine",
    "quantity/angle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-444ce5df8b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "58",
   "location": "Simple Harmonic Motion",
   "latex": "dx = r \\sin\\theta\\, d\\theta",
   "name": null,
   "statement": "For a sufficiently small increment dθ, the infinitely small increment dx of the abscissa is approximately r sin θ times dθ; the error can be made a vanishingly small part of dθ.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "infinitely small increment of the abscissa x"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius OA of the circle"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "angle AOP"
    },
    {
     "unit": null,
     "symbol": "dθ",
     "meaning": "infinitely small increment of the angle θ"
    }
   ],
   "sympy": "Eq(dx, r*sin(theta)*dtheta)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/abscissa",
    "concept/error-of-measurement",
    "concept/infinitesimal",
    "concept/sine",
    "quantity/angle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c06d810e30",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "58",
   "location": "Simple Harmonic Motion",
   "latex": "dy = r \\cos\\theta\\, d\\theta",
   "name": null,
   "statement": "For a sufficiently small increment dθ, the infinitely small increment dy of the ordinate is approximately r cos θ times dθ; the error can be made a vanishingly small part of dθ.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "infinitely small increment of the ordinate y"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius OA of the circle"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "angle AOP"
    },
    {
     "unit": null,
     "symbol": "dθ",
     "meaning": "infinitely small increment of the angle θ"
    }
   ],
   "sympy": "Eq(dy, r*cos(theta)*dtheta)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/error-of-measurement",
    "concept/infinitesimal",
    "concept/ordinate",
    "quantity/angle"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-634dbc45b5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "59",
   "location": "Simple Harmonic Motion",
   "latex": "\\theta = \\phi t",
   "name": null,
   "statement": "If the angle described grows at a non-uniform rate, the angle at time t is φ times t, and the angular velocity is then the rate φ.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "angle described"
    },
    {
     "unit": null,
     "symbol": "φ",
     "meaning": "constant rate of increase of the angle (angular velocity)"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(theta, phi*t)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/uniform-motion",
    "quantity/angle",
    "quantity/angular-velocity",
    "quantity/time"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-4c765366e2",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "59",
   "location": "Simple Harmonic Motion",
   "latex": "\\frac{dx}{dt} = r \\sin\\theta\\, \\frac{d\\theta}{dt}",
   "name": null,
   "statement": "The rate of change of the abscissa with time equals r sin θ times the angular velocity dθ/dt; it becomes exact in the limit of the ratios.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dx/dt",
     "meaning": "velocity of the abscissa x"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius OA of the circle"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "angle AOP"
    },
    {
     "unit": null,
     "symbol": "dθ/dt",
     "meaning": "angular velocity of the point P"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(dx/dt, r*sin(theta)*(dtheta/dt))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/limit",
    "concept/ratio",
    "concept/sine",
    "quantity/angular-velocity",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-96d490aed6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "59",
   "location": "Simple Harmonic Motion",
   "latex": "\\frac{dy}{dt} = r \\cos\\theta\\, \\frac{d\\theta}{dt}",
   "name": null,
   "statement": "The rate of change of the ordinate with time equals r cos θ times the angular velocity dθ/dt; it becomes exact in the limit of the ratios.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dt",
     "meaning": "velocity of the ordinate y"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius OA of the circle"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "angle AOP"
    },
    {
     "unit": null,
     "symbol": "dθ/dt",
     "meaning": "angular velocity of the point P"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(dy/dt, r*cos(theta)*(dtheta/dt))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/limit",
    "concept/ratio",
    "quantity/angular-velocity",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a64391c055",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "59",
   "location": "Simple Harmonic Motion",
   "latex": "a \\sin\\theta",
   "name": null,
   "statement": "The velocity of the abscissa x is a sin θ, where a is the uniform speed of P, so the point M moves with a variable velocity that is sin θ of the velocity of P.",
   "kind": "result",
   "symbols": [
    {
     "unit": "inches per second",
     "symbol": "a",
     "meaning": "uniform velocity of the point P along the arc"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "angle AOP"
    }
   ],
   "sympy": "Eq(v_x, a*sin(theta))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/limit",
    "concept/point",
    "concept/simple-harmonic-motion",
    "concept/sine",
    "concept/uniform-motion",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-064bb6686a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "59",
   "location": "Simple Harmonic Motion",
   "latex": "a \\cos\\theta",
   "name": null,
   "statement": "The velocity of the ordinate y is a cos θ, so the point N moves from O with a velocity that is cos θ of the velocity of P.",
   "kind": "result",
   "symbols": [
    {
     "unit": "inches per second",
     "symbol": "a",
     "meaning": "uniform velocity of the point P along the arc"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "angle AOP"
    }
   ],
   "sympy": "Eq(v_y, a*cos(theta))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/point",
    "concept/simple-harmonic-motion",
    "concept/uniform-motion",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-cfaabb0e12",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-method-of-fluxions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "60",
   "location": "The Method of Fluxions",
   "latex": "\\dfrac{dy}{dt} = 2x\\, \\dfrac{dx}{dt} + \\dfrac{dx}{dt}\\, dx",
   "name": null,
   "statement": "Dividing the increment relation by the small interval of time dt gives the rate of change of y as 2x times the velocity of x plus a term that vanishes as dt shrinks.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y in the interval dt"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x in the interval dt"
    },
    {
     "unit": null,
     "symbol": "dt",
     "meaning": "small interval of time"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(dy/dt, 2*x*(dx/dt) + (dx/dt)*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/fluxional-notation",
    "concept/increment",
    "concept/limit",
    "quantity/time",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f080928982",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-method-of-fluxions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "60",
   "location": "The Method of Fluxions",
   "latex": "\\dot{y} = 2x\\, \\dot{x}",
   "name": null,
   "statement": "The fluxion (velocity) of y = x^2 equals 2x times the fluxion (velocity) of x, the Newtonian form of the derivative result.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\dot{y}",
     "meaning": "fluxion of y (velocity of y)"
    },
    {
     "unit": null,
     "symbol": "\\dot{x}",
     "meaning": "fluxion of x (velocity of x)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(ydot, 2*x*xdot)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/fluxional-notation",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6897ffb1b4",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-method-of-fluxions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "60",
   "location": "The Method of Fluxions",
   "latex": "dy = 2x\\, dx",
   "name": null,
   "statement": "The differential form of the result: dy equals 2x dx, the form used in the differential method in place of fluxions.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential (small increment) of y"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential (small increment) of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(dy, 2*x*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/fluxional-notation",
    "concept/increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-543f0584e1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "61",
   "location": "Accelerated Motion",
   "latex": "nt' = t",
   "name": null,
   "statement": "The time t is divided into n equal parts, each of length t', so n times t' equals t.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "number of equal parts"
    },
    {
     "unit": "second",
     "symbol": "t'",
     "meaning": "length of one equal part of the time t"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(n*tp, t)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/variable",
    "quantity/time"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c4e4f44716",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "61",
   "location": "Accelerated Motion",
   "latex": "nv' = v",
   "name": null,
   "statement": "The velocity v is divided into n equal parts, each of size v', so n times v' equals v.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "number of equal parts"
    },
    {
     "unit": "foot per second",
     "symbol": "v'",
     "meaning": "velocity increment of one equal part"
    },
    {
     "unit": "foot per second",
     "symbol": "v",
     "meaning": "velocity acquired"
    }
   ],
   "sympy": "Eq(n*vp, v)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/variable",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6981964a41",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "62",
   "location": "Accelerated Motion",
   "latex": "n · \\frac{(n + 1)}{2}\\, v't' = \\frac{n^{2} v't' + nv't'}{2}",
   "name": null,
   "statement": "The sum 1 + 2 + ... + n equals n(n+1)/2, so the total space from n equal steps of v't' is (n^2 + n)/2 times v't'.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "number of equal parts"
    },
    {
     "unit": "foot per second",
     "symbol": "v'",
     "meaning": "velocity increment of one equal part"
    },
    {
     "unit": "second",
     "symbol": "t'",
     "meaning": "length of one equal part of the time"
    }
   ],
   "sympy": "Eq(n*(n+1)/2*vp*tp, (n**2*vp*tp + n*vp*tp)/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/integral",
    "concept/number",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6e5039d9c6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "62",
   "location": "Accelerated Motion",
   "latex": "\\frac{1}{2}v(t + t')",
   "name": null,
   "statement": "The space described in the stepwise motion, with n steps, is one half of v times (t + t').",
   "kind": "result",
   "symbols": [
    {
     "unit": "foot per second",
     "symbol": "v",
     "meaning": "velocity acquired at the end of the time t"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": "second",
     "symbol": "t'",
     "meaning": "length of one equal part of the time"
    }
   ],
   "sympy": null,
   "physics": true,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/uniformly-accelerated-motion",
    "quantity/length"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-7676464158",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "62",
   "location": "Accelerated Motion",
   "latex": "\\frac{1}{2}vt",
   "name": null,
   "statement": "As the interval t' is diminished without limit, the space described tends to one half of v times t, the length of uniformly accelerated motion from rest to velocity v in time t.",
   "kind": "result",
   "symbols": [
    {
     "unit": "foot per second",
     "symbol": "v",
     "meaning": "velocity acquired at the end of the time t"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": null,
   "physics": true,
   "states": [],
   "concepts": [
    "concept/limit",
    "concept/uniformly-accelerated-motion",
    "quantity/length",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-db9ced9a57",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "62",
   "location": "Accelerated Motion",
   "latex": "v = gt",
   "name": null,
   "statement": "With accelerating force g, the velocity acquired in t seconds from rest is g times t.",
   "kind": "law",
   "symbols": [
    {
     "unit": "foot per second",
     "symbol": "v",
     "meaning": "velocity acquired"
    },
    {
     "unit": "foot per second per second",
     "symbol": "g",
     "meaning": "accelerating force, measured as the velocity acquired in one second"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(v, g*t)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/uniformly-accelerated-motion",
    "quantity/acceleration",
    "quantity/force",
    "quantity/time",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-d0eabfc443",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "62",
   "location": "Accelerated Motion",
   "latex": "\\frac{1}{2}gt^{2}",
   "name": null,
   "statement": "The space described from rest under uniform acceleration g in time t is one half of g times t squared.",
   "kind": "law",
   "symbols": [
    {
     "unit": "foot per second per second",
     "symbol": "g",
     "meaning": "accelerating force, measured as the velocity acquired in one second"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": null,
   "physics": true,
   "states": [],
   "concepts": [
    "concept/uniformly-accelerated-motion",
    "quantity/acceleration",
    "quantity/force",
    "quantity/length"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6b1007824d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "63",
   "location": "Accelerated Motion",
   "latex": "at + \\frac{1}{2}gt^{2}",
   "name": null,
   "statement": "With initial velocity a and uniform accelerating force g, the length described in time t is a·t plus one half of g·t².",
   "kind": "formula",
   "symbols": [
    {
     "unit": "foot per second",
     "symbol": "a",
     "meaning": "initial velocity"
    },
    {
     "unit": "foot per second per second",
     "symbol": "g",
     "meaning": "accelerating force, measured as the velocity acquired in one second"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(s, a*t + g*t**2/2)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/uniformly-accelerated-motion",
    "quantity/force",
    "quantity/length",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-94325beae8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "63",
   "location": "Accelerated Motion",
   "latex": "at - \\frac{1}{2}gt^{2}",
   "name": null,
   "statement": "With initial velocity a and uniform retarding force g, the length described in time t is a·t minus one half of g·t².",
   "kind": "formula",
   "symbols": [
    {
     "unit": "foot per second",
     "symbol": "a",
     "meaning": "initial velocity"
    },
    {
     "unit": "foot per second per second",
     "symbol": "g",
     "meaning": "uniform retarding force, measured as the velocity lost in one second"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(s, a*t - g*t**2/2)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/uniformly-accelerated-motion",
    "quantity/force",
    "quantity/length",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-43588708dc",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "64",
   "location": "Accelerated Motion",
   "latex": "a + gt",
   "name": null,
   "statement": "The whole velocity after time t, with initial velocity a under uniform accelerating force g, is a plus g times t.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "foot per second",
     "symbol": "a",
     "meaning": "initial velocity"
    },
    {
     "unit": "foot per second per second",
     "symbol": "g",
     "meaning": "accelerating force"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(v, a + g*t)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/uniformly-accelerated-motion",
    "quantity/force",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-af284bdd27",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "63",
   "location": "Accelerated Motion",
   "latex": "\\phi t = t^{3}",
   "name": null,
   "statement": "The length described by the point in time t is t cubed, so the motion is defined by phi(t) equal to t cubed.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "inch",
     "symbol": "phi",
     "meaning": "function giving the length described in time t"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(phi(t), t**3)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/indirect-function",
    "concept/variable",
    "quantity/length"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8ad076e259",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "63",
   "location": "Accelerated Motion",
   "latex": "3t^{2}",
   "name": null,
   "statement": "The velocity of the point at the end of time t, for the motion phi(t) = t³, is 3t² inches per second, the coefficient of dt in the expansion of (t + dt)³.",
   "kind": "result",
   "symbols": [
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(v, 3*t**2)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "quantity/velocity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8ad6e02cb8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "65",
   "location": "Accelerated Motion",
   "latex": "\\phi' t\\, dt + \\phi'' t\\, \\frac{(dt)^{2}}{2} + \\phi''' t\\, \\frac{(dt)^{3}}{2·3} + \\etc.",
   "name": null,
   "statement": "The length described in the interval dt equals phi(t + dt) minus phi(t), expanded as a series in dt with derivative coefficients.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "inch",
     "symbol": "phi",
     "meaning": "function giving the length described in time t"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": "second",
     "symbol": "dt",
     "meaning": "small interval of time"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/derivative",
    "concept/differential",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-4eb619ecc8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "65",
   "location": "Accelerated Motion",
   "latex": "\\phi' t\\, dt + \\frac{1}{2}\\phi'' t (dt)^{2}",
   "name": null,
   "statement": "The first two terms of the expansion represent the length described in dt with uniform velocity phi'(t) and accelerating force phi''(t), and approximate the motion for small dt as closely as we please.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": "inch per second",
     "symbol": "phi'",
     "meaning": "first derivative of phi, the velocity at time t"
    },
    {
     "unit": "inch per second per second",
     "symbol": "phi''",
     "meaning": "second derivative of phi, the accelerating force at time t"
    },
    {
     "unit": "second",
     "symbol": "dt",
     "meaning": "small interval of time"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": null,
   "physics": true,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/derivative",
    "concept/differential",
    "concept/uniform-motion",
    "concept/uniformly-accelerated-motion",
    "quantity/force"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-dba7d0871b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "67",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "\\dfrac{(x + 1)^{m}}{x^{m}} = 1 + \\dfrac{mx^{m-1} + \\etc.}{x^{m}}",
   "name": null,
   "statement": "The ratio (x+1)^m / x^m equals 1 plus a remainder whose numerator becomes negligible next to x^m as x grows without limit, so the ratio tends to unity.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable that is increased without limit"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "the exponent (a given power)"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/exponent",
    "concept/infinity",
    "concept/limit",
    "concept/power",
    "concept/tends-to-infinity"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-cf520c6df8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "67",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "(x + 1)^{m+1} = x^{m+1} + (m + 1)x^{m} + \\frac{1}{2}(m + 1)m x^{m-1} + \\etc.",
   "name": null,
   "statement": "The binomial expansion of (x+1) raised to the power m+1, with its leading terms written out.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "the exponent"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/power",
    "concept/sum",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c89962e0ff",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "71",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "\\frac{a}{pa + b} = \\frac{1}{p + \\dfrac{b}{a}}",
   "name": null,
   "statement": "A fraction with numerator a and denominator pa + b equals 1 divided by p + b/a, so it is near 1/p when b/a is small.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a numerator quantity that increases without limit"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "a quantity small compared with a"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "a given number (coefficient)"
    }
   ],
   "sympy": "Eq(a/(p*a + b), 1/(p + b/a))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-fraction",
    "concept/approximation",
    "concept/common-ratio",
    "concept/infinity",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-49d8c2ef0c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "71",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "\\frac{A + (a + a' + a'' + \\etc.)}{B + p(a + a' + a'' + \\etc.) + b + b' + b'' + \\etc.}",
   "name": null,
   "statement": "The summed fraction with given quantities A and B added still approaches 1/p, provided the summed numerators become large compared with A and B.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "A",
     "meaning": "a given quantity added to the numerator"
    },
    {
     "unit": null,
     "symbol": "B",
     "meaning": "a given quantity added to the denominator"
    },
    {
     "unit": null,
     "symbol": "a, a', a''",
     "meaning": "numerators of the original fractions"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "a given number"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/common-ratio",
    "concept/infinity",
    "concept/limit",
    "concept/sum"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2b10c9de0e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "72",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "\\frac{(x + 1)^{3} + (x + 2)^{3} + \\dots + (x + n)^{3}}{(x + 1)^{4} - x^{4}}",
   "name": null,
   "statement": "The sum of the numerators of the cube fractions, divided by the telescoped denominator (x+1)^4 - x^4, equal to the summed fraction in the argument.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable increased without limit"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the number of fractions summed"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/infinity",
    "concept/power",
    "concept/sum",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-41581a5c44",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "73",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "\\frac{1^{3} + 2^{3} + 3^{3} + \\dots + x^{3} + (x + 1)^{3} + \\dots + (x + n)^{3}}{(x + n)^{4}}",
   "name": null,
   "statement": "Adding x^4 to the denominator and the sum of cubes 1^3 to x^3 to the numerator leaves the ratio within the same nearness of 1/4, so the ratio of the sum of cubes to x^4 tends to 1/4.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable increased without limit"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "a number chosen large compared with x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/limit",
    "concept/power",
    "concept/sum",
    "concept/tends-to-infinity",
    "theorem/limit-of-a-sum"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8cb6210e2c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "73",
   "location": "Limiting Ratios of Magnitudes that Increase Without Limit",
   "latex": "x\\, \\dfrac{x - 1}{2} ÷ x^{2} = \\dfrac{x - 1}{2x}",
   "name": null,
   "statement": "The ratio of x(x-1)/2 to x^2 simplifies to (x-1)/(2x), whose limit as x increases without limit is 1/2.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable increased without limit"
    }
   ],
   "sympy": "Eq(x*(x - 1)/2/x**2, (x - 1)/(2*x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/infinity",
    "concept/limit",
    "concept/power",
    "concept/term"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-9151e9ad10",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-recapitulation-of-results-reached-in-the-theory-of-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "74",
   "location": "Recapitulation of Results Reached in the Theory of Functions",
   "latex": "\\phi' x\\, dx + \\phi'' x\\, \\frac{(dx)^{2}}{2} + \\phi''' x\\, \\frac{(dx)^{3}}{2·3} + \\etc.",
   "name": null,
   "statement": "When x receives an increment dx, the increase in y is given by a series in powers of dx whose coefficients are the successive derived functions of phi at x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\phi' x",
     "meaning": "first derivative of the function phi at x, the coefficient of dx"
    },
    {
     "unit": null,
     "symbol": "\\phi'' x",
     "meaning": "second derivative of the function phi at x"
    },
    {
     "unit": null,
     "symbol": "\\phi''' x",
     "meaning": "third derivative of the function phi at x"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "the increment given to the variable x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/differential",
    "concept/higher-order-derivative",
    "concept/increment",
    "theorem/expansion-of-a-function-by-increments"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-040a23ef8e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "75",
   "location": "Approximations by the Differential Calculus",
   "latex": "\\phi x + \\phi' x\\, dx",
   "name": null,
   "statement": "If x is changed into x + dx with dx very small, the function phi x changes to phi x plus phi' x times dx, very nearly.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "a very small increment of x"
    },
    {
     "unit": null,
     "symbol": "\\phi' x",
     "meaning": "differential coefficient (derivative) of phi x with respect to x"
    }
   ],
   "sympy": "Eq(phi(x + dx), phi(x) + Derivative(phi(x), x)*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/derivative",
    "concept/differential",
    "concept/function",
    "concept/increment",
    "concept/small-variation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-839bad55f8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "75",
   "location": "Approximations by the Differential Calculus",
   "latex": "\\phi x + \\phi' x\\, h",
   "name": null,
   "statement": "With the value x replaced by x + h, where h is a small error, phi(x + h) is very nearly phi x plus phi' x times h.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the correct value of the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small error committed in the valuation of x"
    },
    {
     "unit": null,
     "symbol": "\\phi' x",
     "meaning": "differential coefficient (derivative) of phi x with respect to x"
    }
   ],
   "sympy": "Eq(phi(x + h), phi(x) + Derivative(phi(x), x)*h)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/derivative",
    "concept/error-of-measurement",
    "concept/function",
    "concept/small-variation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-fe29db1e47",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "75",
   "location": "Approximations by the Differential Calculus",
   "latex": "\\phi x + \\phi' x\\, k",
   "name": null,
   "statement": "With the value x replaced by x + k, where k is a small error, phi(x + k) is very nearly phi x plus phi' x times k.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the correct value of the variable"
    },
    {
     "unit": null,
     "symbol": "k",
     "meaning": "a second small error committed in the valuation of x"
    },
    {
     "unit": null,
     "symbol": "\\phi' x",
     "meaning": "differential coefficient (derivative) of phi x with respect to x"
    }
   ],
   "sympy": "Eq(phi(x + k), phi(x) + Derivative(phi(x), x)*k)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/derivative",
    "concept/error-of-measurement",
    "concept/function",
    "concept/small-variation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-d8c1d9e29f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "75",
   "location": "Approximations by the Differential Calculus",
   "latex": "\\phi' x\\, h",
   "name": null,
   "statement": "The error committed in taking phi x, when the error h has been made in x, is very nearly phi' x times h, so errors of the result are in the proportion of the errors of x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\phi' x",
     "meaning": "differential coefficient (derivative) of phi x with respect to x"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small error committed in the valuation of x"
    }
   ],
   "sympy": "Eq(delta_phi, Derivative(phi(x), x)*h)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/error-of-measurement",
    "concept/proportion",
    "theorem/propagation-of-small-errors"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-fed5548ff2",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "75",
   "location": "Approximations by the Differential Calculus",
   "latex": "\\sqrt{3^{2} + 4^{2}}",
   "name": "Pythagorean theorem (hypotenuse of the right triangle with base 3 and other side 4)",
   "statement": "The hypotenuse of a right-angled triangle with base 3 and other side 4 is the square root of 3 squared plus 4 squared, which is 5.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "3",
     "meaning": "the base of the triangle"
    },
    {
     "unit": null,
     "symbol": "4",
     "meaning": "the other side of the triangle"
    }
   ],
   "sympy": "Eq(hypotenuse, sqrt(3**2 + 4**2))",
   "physics": false,
   "states": [
    "theorem/pythagorean-theorem-hypotenuse-of-the-right-triangle-with-base-3-and-other-side-4"
   ],
   "concepts": [
    "concept/base-of-a-triangle",
    "concept/hypotenuse",
    "concept/number",
    "concept/right-triangle",
    "concept/side"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-c714e5cdd1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "\\phi x = 0",
   "name": null,
   "statement": "The equation whose root x is sought is a function of x set equal to zero.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the unknown whose value is sought"
    },
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "a function of x, written phi x"
    }
   ],
   "sympy": "Eq(phi(x), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/function",
    "concept/function-notation",
    "concept/solution"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f9d4f70958",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "\\phi(a + h) = 0",
   "name": null,
   "statement": "Since a + h is the real root, the function vanishes at a + h.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a near approximation to the required value of x"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small quantity, the correction to a"
    },
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "the function phi"
    }
   ],
   "sympy": "Eq(phi(a + h), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/function",
    "concept/function-notation",
    "concept/solution"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-98878ca9fb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "\\phi a + \\phi' a\\, h = 0",
   "name": null,
   "statement": "Replacing phi(a + h) by its first-order expansion gives a nearly true linear relation for the small correction h.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a near approximation to the required value of x"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small quantity, the correction to a"
    },
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "the function phi"
    },
    {
     "unit": null,
     "symbol": "\\phi'",
     "meaning": "the derivative of phi"
    }
   ],
   "sympy": "Eq(phi(a) + Subs(Derivative(phi(x), x), x, a)*h, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/derivative",
    "concept/differential",
    "concept/function",
    "method/newton-s-method"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-10f85963ec",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "a - \\dfrac{\\phi a}{\\phi' a}",
   "name": "Newton's method",
   "statement": "The value a - phi(a)/phi'(a) is a nearer approximation to the root x than a, since h is nearly -phi(a)/phi'(a).",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a near approximation to the required value of x"
    },
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "the function phi"
    },
    {
     "unit": null,
     "symbol": "\\phi'",
     "meaning": "the derivative of phi"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [
    "method/newton-s-method"
   ],
   "concepts": [
    "concept/approximation",
    "concept/derivative",
    "concept/solution"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-e1d923dc8e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "\\phi' x = 2x + 1",
   "name": null,
   "statement": "For phi x = x^2 + x - 4, the derivative phi' x equals 2x + 1.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "\\phi'",
     "meaning": "the derivative of phi with respect to x"
    }
   ],
   "sympy": "Eq(diff(phi(x), x), 2*x + 1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-4554e2d1be",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "\\phi(x + h) = (x + h)^{2} + x + h - 4 = x^{2} + x - 4 + (2x + 1)h + h^{2}",
   "name": null,
   "statement": "For phi x = x^2 + x - 4, expanding phi(x + h) gives the original value plus (2x + 1)h plus h^2.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "a small quantity added to x"
    },
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "the function phi, here x^2 + x - 4"
    }
   ],
   "sympy": "Eq(phi(x + h), x**2 + x - 4 + (2*x + 1)*h + h**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/function",
    "concept/function-notation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-b4aa171034",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "77",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "x^{2} + x - 4 = 0",
   "name": null,
   "statement": "The example equation of the second degree whose root is sought by the method.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the unknown whose value is sought"
    }
   ],
   "sympy": "Eq(x**2 + x - 4, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/equation-of-the-second-degree",
    "concept/solution"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-bbfbf427cb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-solution-of-equations-by-the-differential-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "78",
   "location": "Solution of Equations by the Differential Calculus",
   "latex": "\\tan x = ax",
   "name": null,
   "statement": "An equation not treatable by common algebra, to which the same method of successive approximation applies.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the unknown whose value is sought"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant coefficient"
    }
   ],
   "sympy": "Eq(tan(x), a*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/solution",
    "concept/tangent-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a6f23902d1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "78",
   "location": "Partial and Total Differentials",
   "latex": "u = x^{2} y + 2xy^{3}",
   "name": null,
   "statement": "Defines the function u as x squared times y plus twice x times y cubed, the example used throughout the chapter.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "the function of x and y under discussion"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable"
    }
   ],
   "sympy": "Eq(u, x**2*y + 2*x*y**3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-e95faec009",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "78",
   "location": "Partial and Total Differentials",
   "latex": "du = (2xy + 2y^{3})\\, dx + \\etc.",
   "name": null,
   "statement": "When only x varies, the increment of u is (2xy + 2y^3) dx plus terms that become negligible as dx diminishes.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du",
     "meaning": "increment of u arising from a change in x only"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of the variable x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable held fixed in this supposition"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/increment",
    "concept/infinitesimal",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f31ad677eb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "79",
   "location": "Partial and Total Differentials",
   "latex": "du = (x^{2} + 6xy^{2})\\, dy + \\etc.",
   "name": null,
   "statement": "When only y varies, the increment of u is (x^2 + 6xy^2) dy plus terms negligible as dy diminishes.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du",
     "meaning": "increment of u arising from a change in y only"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of the variable y"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable held fixed in this supposition"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/increment",
    "concept/infinitesimal",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a350f39a3f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "79",
   "location": "Partial and Total Differentials",
   "latex": "du = (2xy + 2y^{3})\\, dx + (x^{2} + 6xy^{2})\\, dy + \\etc.",
   "name": null,
   "statement": "When x and y both vary, the increment of u is the sum of the two partial contributions plus negligible terms.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du",
     "meaning": "increment of u arising from changes in both x and y"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/increment",
    "concept/infinitesimal",
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6d4ebb0a79",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "81",
   "location": "Partial and Total Differentials",
   "latex": "\\dfrac{du}{dx}\\, dx = (2xy + 2y^{3})\\, dx",
   "name": null,
   "statement": "The x-part of the increment of u is the partial differential coefficient of u with respect to x times dx, here equal to (2xy + 2y^3) dx.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du/dx",
     "meaning": "differential coefficient of u with respect to x, taken as a single symbol with its numerator and denominator inseparable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/coefficient",
    "concept/increment",
    "concept/mathematical-notation",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-290a27a568",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "81",
   "location": "Partial and Total Differentials",
   "latex": "\\frac{du}{dx} = 2xy + 2y^{3}",
   "name": null,
   "statement": "The partial differential coefficient of u with respect to x equals 2xy + 2y^3 for this example.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du/dx",
     "meaning": "partial differential coefficient of u with respect to x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable"
    }
   ],
   "sympy": "Eq(Derivative(u, x), 2*x*y + 2*y**3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/function",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ca2f22cd08",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "81",
   "location": "Partial and Total Differentials",
   "latex": "\\dfrac{du}{dy}\\, dy = (x^{2} + 6xy^{2})\\, dy",
   "name": null,
   "statement": "The y-part of the increment of u is the partial differential coefficient of u with respect to y times dy, here equal to (x^2 + 6xy^2) dy.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du/dy",
     "meaning": "differential coefficient of u with respect to y, taken as a single symbol with its numerator and denominator inseparable"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/coefficient",
    "concept/increment",
    "concept/mathematical-notation",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-df8b8e80f3",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "81",
   "location": "Partial and Total Differentials",
   "latex": "\\frac{du}{dy} = x^{2} + 6xy^{2}",
   "name": null,
   "statement": "The partial differential coefficient of u with respect to y equals x^2 + 6xy^2 for this example.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du/dy",
     "meaning": "partial differential coefficient of u with respect to y"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable"
    }
   ],
   "sympy": "Eq(Derivative(u, y), x**2 + 6*x*y**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/function",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ebcb732414",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "81",
   "location": "Partial and Total Differentials",
   "latex": "d.u = \\frac{du}{dx}\\, dx + \\frac{du}{dy}\\,dy",
   "name": null,
   "statement": "The total differential of u, the increment from both suppositions at once, is the sum of its two partial contributions.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "d.u",
     "meaning": "total differential of u: the increment derived from changes in x and y together"
    },
    {
     "unit": null,
     "symbol": "du/dx",
     "meaning": "partial differential coefficient of u with respect to x"
    },
    {
     "unit": null,
     "symbol": "du/dy",
     "meaning": "partial differential coefficient of u with respect to y"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/increment",
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-66a9ed8fc6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "82",
   "location": "Partial and Total Differentials",
   "latex": "\\ux\\, dx + \\uy\\, dy + \\etc",
   "name": null,
   "statement": "The increment of u when x and y both receive increments is the sum of the partial terms in dx and dy, with the remaining terms negligible in the limit.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "u_x",
     "meaning": "differential coefficient of u with respect to x (macro \\ux)"
    },
    {
     "unit": null,
     "symbol": "u_y",
     "meaning": "differential coefficient of u with respect to y (macro \\uy)"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/increment",
    "concept/infinitesimal",
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-375a3051d8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "83",
   "location": "Partial and Total Differentials",
   "latex": "dz = \\frac{dz}{dp}\\, dp + \\frac{dz}{dq}\\, dq + \\frac{dz}{dr}\\, dr + \\frac{dz}{ds}\\, ds + \\etc.",
   "name": null,
   "statement": "For z a function of p, q, r and s, the total differential of z is the sum of the partial contributions from each variable, plus terms negligible in the limit.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "function of the variables p, q, r, s"
    },
    {
     "unit": null,
     "symbol": "dz",
     "meaning": "total differential (increment) of z"
    },
    {
     "unit": null,
     "symbol": "dz/dp",
     "meaning": "partial differential coefficient of z with respect to p"
    },
    {
     "unit": null,
     "symbol": "dp",
     "meaning": "increment of p"
    },
    {
     "unit": null,
     "symbol": "dq",
     "meaning": "increment of q"
    },
    {
     "unit": null,
     "symbol": "dr",
     "meaning": "increment of r"
    },
    {
     "unit": null,
     "symbol": "ds",
     "meaning": "increment of s"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/function",
    "concept/increment",
    "concept/limit",
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-09c4a3b473",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-application-of-the-theorem-for-total-differentials-to-the-determination-of-total-resultant-errors",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "84",
   "location": "Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors",
   "latex": "\\dfrac{dz}{dp}\\, dp + \\dfrac{dz}{dq}\\, dq + \\dfrac{dz}{dr}\\, dr + \\dfrac{dz}{ds}\\, ds",
   "name": null,
   "statement": "The function z, when its data p, q, r, s each carry a small error dp, dq, dr, ds, is corrected very nearly by increasing z by the sum of the separate effects of each error, taken one at a time.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the function required to be found, computed from the data"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "presumed value of the first datum"
    },
    {
     "unit": null,
     "symbol": "q",
     "meaning": "presumed value of the second datum"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "presumed value of the third datum"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "presumed value of the fourth datum"
    },
    {
     "unit": null,
     "symbol": "dp",
     "meaning": "error (small increment) in the datum p"
    },
    {
     "unit": null,
     "symbol": "dq",
     "meaning": "error (small increment) in the datum q"
    },
    {
     "unit": null,
     "symbol": "dr",
     "meaning": "error (small increment) in the datum r"
    },
    {
     "unit": null,
     "symbol": "ds",
     "meaning": "error (small increment) in the datum s"
    }
   ],
   "sympy": "Eq(dz, Derivative(z, p)*dp + Derivative(z, q)*dq + Derivative(z, r)*dr + Derivative(z, s)*ds)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/error-of-measurement",
    "concept/function",
    "concept/increment",
    "concept/partial-derivative",
    "concept/theorem-total-differential",
    "concept/value-of-a-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a7cf54886f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "dy = mx^{m-1}\\, dx",
   "name": "power rule for differentiation",
   "statement": "For y = x^m with m whole or fractional, positive or negative, the differential dy equals m x^(m-1) dx.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential of y"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "the constant exponent of x (whole or fractional, positive or negative)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential of x (the increment of x)"
    }
   ],
   "sympy": "Eq(dy, m*x**(m-1)*dx)",
   "physics": false,
   "states": [
    "theorem/power-rule-for-differentiation"
   ],
   "concepts": [
    "concept/differential",
    "concept/exponent",
    "concept/power",
    "concept/rule",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a2fc102871",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "dy = -\\dfrac{m\\, dx}{x^{m+1}}",
   "name": null,
   "statement": "When y = 1/x^m, the differential dy equals minus m dx divided by x^(m+1), the negative sign showing that increasing x decreases y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential of y"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "positive exponent in the denominator x^m"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential of x"
    }
   ],
   "sympy": "Eq(dy, -m*dx/x**(m+1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/exponent",
    "concept/power",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2c70f078fa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "dy = -mx^{-m-1}\\, dx",
   "name": null,
   "statement": "When y = x^(-m), the differential dy equals minus m x^(-m-1) dx, consistent with the power rule.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential of y"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "positive constant, so the exponent of x is -m"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential of x"
    }
   ],
   "sympy": "Eq(dy, -m*x**(-m-1)*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/exponent",
    "concept/power",
    "concept/rule",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6f949df9af",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "dy = a^{x}\\log a\\, dx",
   "name": null,
   "statement": "For y = a^x, the differential dy equals a^x times the natural logarithm of a, times dx.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential of y"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the base of the power a^x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable exponent"
    },
    {
     "unit": null,
     "symbol": "log",
     "meaning": "the Naperian or hyperbolic (natural) logarithm, as is always the case in this book unless stated otherwise"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential of x"
    }
   ],
   "sympy": "Eq(dy, a**x*log(a)*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/exponent",
    "concept/general-power",
    "concept/logarithm",
    "concept/rule",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-221f780fb1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "dy = e^{x}\\, dx",
   "name": null,
   "statement": "For y = e^x, where e is the base of the Naperian logarithms, the differential dy equals e^x dx.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential of y"
    },
    {
     "unit": null,
     "symbol": "e",
     "meaning": "the base of the Naperian (hyperbolic) logarithms, 2.7182818"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable exponent"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential of x"
    }
   ],
   "sympy": "Eq(dy, E**x*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/euler-s-number",
    "concept/general-power",
    "concept/logarithm",
    "concept/rule",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-b2c37e18de",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "a = 2.7182818 = e",
   "name": null,
   "statement": "The number whose Naperian logarithm is 1 is taken to be the base e of the hyperbolic logarithms, approximately 2.7182818.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the base of the logarithms being considered (here equal to e)"
    },
    {
     "unit": null,
     "symbol": "e",
     "meaning": "the base of the Naperian (hyperbolic) logarithms"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/euler-s-number",
    "concept/logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ae66fbe86f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "dy = \\dfrac{dx}{x}",
   "name": null,
   "statement": "For y equal to the Naperian logarithm of x, the differential dy equals dx divided by x.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential of y = log x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential of x"
    }
   ],
   "sympy": "Eq(dy, dx/x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/logarithm",
    "concept/rule",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-0c55da9012",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "dy = -.4342944\\, \\dfrac{dx}{x}",
   "name": null,
   "statement": "For y equal to the common logarithm of x, the differential dy equals minus 0.4342944 times dx divided by x; the constant is the decimal modulus 1/ln 10 given to seven places.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential of the common logarithm of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential of x"
    }
   ],
   "sympy": "Eq(dy, -0.4342944*dx/x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/differential",
    "concept/logarithm",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8b36f5e495",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "dy = \\cos x\\, dx",
   "name": null,
   "statement": "For y equal to sin x, the differential dy equals cos x times dx.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential of y = sin x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the angle, the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential of x"
    }
   ],
   "sympy": "Eq(dy, cos(x)*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/differential",
    "concept/rule",
    "concept/sine",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-54a8a3b655",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "dy = -\\sin x\\, dx",
   "name": null,
   "statement": "For y equal to cos x, the differential dy equals minus sin x times dx.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential of y = cos x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the angle, the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential of x"
    }
   ],
   "sympy": "Eq(dy, -sin(x)*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/differential",
    "concept/rule",
    "concept/sine",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-1322ef33cf",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "86",
   "location": "Rules for Differentiation",
   "latex": "dy = \\dfrac{dx}{\\cos^{2} x}",
   "name": null,
   "statement": "For y equal to tan x, the differential dy equals dx divided by the square of cos x.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "differential of y = tan x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the angle, the variable"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "differential of x"
    }
   ],
   "sympy": "Eq(dy, dx/cos(x)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/differential",
    "concept/rule",
    "concept/tangent-function",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-3ff2a7fc5c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-illustration-of-the-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "87",
   "location": "Illustration of the Rules for Differentiation",
   "latex": "y = \\text{common log}~x",
   "name": null,
   "statement": "The variable y is defined as the common logarithm of x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the common logarithm of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the number whose common logarithm is taken"
    }
   ],
   "sympy": "Eq(y, log(x, 10))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-4b519361f6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-illustration-of-the-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "87",
   "location": "Illustration of the Rules for Differentiation",
   "latex": ".4342944 \\left(\\frac{dx}{x} - \\tfrac{1}{2}\\, \\frac{(dx)^{2}}{x^{2}} + \\tfrac{1}{3}\\, \\frac{(dx)^{3}}{x^{3}} - \\etc.\\right)",
   "name": null,
   "statement": "The real increment of the common logarithm y when x becomes x + dx is given by an infinite series in dx/x, whose first term is .4342944 dx/x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of the variable x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable of the common logarithm"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the common logarithm of x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/increment",
    "concept/infinite-sequence",
    "concept/ratio"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-85808bb1f1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-illustration-of-the-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "87",
   "location": "Illustration of the Rules for Differentiation",
   "latex": "y = \\sin x",
   "name": null,
   "statement": "The variable y is defined as the sine of x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the sine of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the angle whose sine is taken"
    }
   ],
   "sympy": "Eq(y, sin(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/sine",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-9ede6b63ec",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-illustration-of-the-rules-for-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "87",
   "location": "Illustration of the Rules for Differentiation",
   "latex": "\\cos x\\, dx - \\frac{1}{2}\\sin x\\, (dx)^{2} - \\etc.",
   "name": null,
   "statement": "When x is increased by dx, sin x is increased by a series whose first term is cos x dx, of which only the first term is taken in the example.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment of the variable x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the angle"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/increment",
    "concept/infinite-sequence",
    "concept/sine"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-924ea37426",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients-of-differential-coefficients",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "88",
   "location": "Differential Coefficients of Differential Coefficients",
   "latex": "\\frac{d^{2} y}{dx^{2}}",
   "name": null,
   "statement": "The usual notation for the second differential coefficient of y with respect to x, written in preference to the cumbersome nested d.(dy/dx)/dx form; this is a notation convention, not an equation with an equals sign.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "function of x whose differential coefficients are taken"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/higher-order-derivative",
    "concept/mathematical-notation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-349b6d3752",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "91",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "\\Delta^{2} y_{1} - \\Delta^{2} y = \\Delta^{3} y",
   "name": null,
   "statement": "The third difference of y at the first step equals the second difference at the first step minus the second difference at the start, so each order of difference is taken from the one before it.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value of the function phi x at the first value of x"
    },
    {
     "unit": null,
     "symbol": "y_{1}",
     "meaning": "value of the function at x + \\Delta x"
    },
    {
     "unit": null,
     "symbol": "\\Delta",
     "meaning": "the difference operation (increment taken when x is replaced by x + \\Delta x)"
    },
    {
     "unit": null,
     "symbol": "\\Delta^{2}",
     "meaning": "second difference: the operation \\Delta repeated upon its own result"
    },
    {
     "unit": null,
     "symbol": "\\Delta^{3}",
     "meaning": "third difference: the operation \\Delta repeated three times"
    }
   ],
   "sympy": "Eq(D2y_1 - D2y, D3y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/calculus-of-finite-differences",
    "concept/difference",
    "concept/function",
    "concept/higher-order-difference"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-eb85e3d917",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "92",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "y_{2} = y_{1} + \\Delta y_{1}",
   "name": null,
   "statement": "Each value of the function is the previous value plus the first difference at the previous value, which generates the table of successive values.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y_{2}",
     "meaning": "value of the function phi x at x + 2\\Delta x"
    },
    {
     "unit": null,
     "symbol": "y_{1}",
     "meaning": "value of the function phi x at x + \\Delta x"
    },
    {
     "unit": null,
     "symbol": "\\Delta y_{1}",
     "meaning": "first difference of the function at x + \\Delta x, that is y_{2} - y_{1}"
    }
   ],
   "sympy": "Eq(y_2, y_1 + Dy_1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/calculus-of-finite-differences",
    "concept/difference",
    "concept/function",
    "concept/value-of-a-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a90b8b3363",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "92",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "y_{1} = y + \\Delta y",
   "name": null,
   "statement": "The value of the function after one step equals its starting value plus the first difference.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y_{1}",
     "meaning": "value of the function phi x at x + \\Delta x"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value of the function phi x at the starting value of x"
    },
    {
     "unit": null,
     "symbol": "\\Delta y",
     "meaning": "first difference of the function, y_{1} - y"
    }
   ],
   "sympy": "Eq(y_1, y + Dy)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/calculus-of-finite-differences",
    "concept/difference",
    "concept/increment",
    "concept/value-of-a-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6e3fc40df8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "92",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "\\Delta y_{2} = \\Delta y + 2\\Delta^{2} y + \\Delta^{3} y",
   "name": null,
   "statement": "The first difference two steps on equals the first difference at the start plus twice the second difference plus the third difference.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\Delta y_{2}",
     "meaning": "first difference of the function at x + 2\\Delta x, that is y_{3} - y_{2}"
    },
    {
     "unit": null,
     "symbol": "\\Delta y",
     "meaning": "first difference of the function at x, that is y_{1} - y"
    },
    {
     "unit": null,
     "symbol": "\\Delta^{2} y",
     "meaning": "second difference of the function at x"
    },
    {
     "unit": null,
     "symbol": "\\Delta^{3} y",
     "meaning": "third difference of the function at x"
    }
   ],
   "sympy": "Eq(Dy_2, Dy + 2*D2y + D3y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/calculus-of-finite-differences",
    "concept/coefficient",
    "concept/difference",
    "concept/higher-order-difference"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-747d61a173",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "93",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "n\\Delta x = h",
   "name": null,
   "statement": "Taking n equal steps of size \\Delta x to go from x to x + h means n times the step equals h.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "number of equal steps between x and x + h"
    },
    {
     "unit": null,
     "symbol": "\\Delta x",
     "meaning": "the difference (size of each step) given to x"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "total increase given to x, made in n equal steps"
    }
   ],
   "sympy": "Eq(n*Dx, h)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/difference",
    "concept/increment",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6c7d2d00b0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "93",
   "location": "Calculus of Finite Differences. Successive Differentiation",
   "latex": "\\phi(x + h) = y + \\frac{dy}{dx}\\, h + \\frac{d^{2} y}{dx^{2}}\\, \\frac{h^{2}}{2} + \\frac{d^{3} y}{dx^{3}}\\, \\frac{h^{3}}{2·3} + \\etc.\\Add{,}",
   "name": "Taylor's series",
   "statement": "The value of the function at x + h is expanded as the value at x plus successive higher-order derivatives multiplied by powers of h over factorials.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "the function"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "the increase given to x"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value of the function phi x at x"
    },
    {
     "unit": null,
     "symbol": "\\frac{dy}{dx}",
     "meaning": "first differential coefficient, the limit of the ratio of the increments, equal to \\phi' x"
    },
    {
     "unit": null,
     "symbol": "\\frac{d^{2} y}{dx^{2}}",
     "meaning": "second differential coefficient, equal to \\phi'' x"
    },
    {
     "unit": null,
     "symbol": "\\frac{d^{3} y}{dx^{3}}",
     "meaning": "third differential coefficient, equal to \\phi''' x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [
    "theorem/taylor-s-series"
   ],
   "concepts": [
    "concept/calculus-of-finite-differences",
    "concept/derivative",
    "concept/differential",
    "concept/function",
    "concept/higher-order-derivative",
    "concept/infinite-sequence",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ed44f1d834",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "95",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "d.z = \\frac{dz}{dx}\\, dx + \\frac{dz}{dy}\\, p\\, dx",
   "name": null,
   "statement": "When y is itself a function of x, the increment of z is the x-part plus the y-part, with dy replaced by p dx, where p is the differential coefficient of y with respect to x.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "d.z",
     "meaning": "the whole increment of z (total variation)"
    },
    {
     "unit": null,
     "symbol": "z",
     "meaning": "a function of x and y (and, indirectly, of x through y)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "another function of x"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "increment given to x"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "differential coefficient of y with respect to x (dy/dx)"
    }
   ],
   "sympy": "Eq(dz, dzdx*dx + dzdy*p*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/coefficient",
    "concept/derivative",
    "concept/differential",
    "concept/function",
    "concept/increment",
    "concept/indirect-function",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-da90e188c0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "95",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "\\frac{d.z}{dx} = \\frac{dz}{dx} + \\frac{dz}{dx}\\, p",
   "name": null,
   "statement": "The total differential coefficient of z with respect to x equals the partial coefficient plus the partial coefficient times p = dy/dx; as printed the second term reads dz/dx, though the surrounding text requires dz/dy (possible misprint in this edition, flagged for checking).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "d.z/dx",
     "meaning": "total (complete) differential coefficient of z with respect to x"
    },
    {
     "unit": null,
     "symbol": "dz/dx",
     "meaning": "partial differential coefficient of z with respect to x, y held fixed"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(Dz_Dx, dzdx + dzdx*p)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/partial-derivative",
    "concept/total-differential",
    "theorem/chain-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ceb507d891",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "98",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "d.z = \\frac{dz}{dx}\\, dx + \\frac{dz}{dy}\\, dy + \\frac{dz}{da}\\, da + \\etc.",
   "name": null,
   "statement": "With x, y and a varying independently, the increment of z is the sum of the partial increments with respect to each variable, continued for further variables.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "d.z",
     "meaning": "the whole increment of z"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a quantity contained in z, itself a function of y and x"
    },
    {
     "unit": null,
     "symbol": "da",
     "meaning": "increment of a"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "increment of y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/increment",
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-48c19557c0",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "98",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "da = \\frac{da}{dx}\\, dx + \\frac{da}{dy}\\, dy",
   "name": null,
   "statement": "Since a varies as a function of y and x, its increment is the sum of its partial increments in x and in y.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "da",
     "meaning": "increment of a"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a quantity that is a function of y and x"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "a function of x"
    }
   ],
   "sympy": "Eq(da, dadx*dx + dady*dy)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/increment",
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2097842d11",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "96",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "\\frac{d.z}{dx} = \\frac{dz}{dx} + \\frac{dz}{dy}\\, \\frac{dy}{dx} + \\frac{dz}{da}\\, \\frac{da}{dy}\\, \\frac{dy}{dx} + \\frac{dz}{da}\\, \\frac{da}{dx}",
   "name": null,
   "statement": "The complete differential coefficient of z with respect to x is the sum over every way z contains x: direct, through y, through a and y, and through a directly.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "d.z/dx",
     "meaning": "complete (total) differential coefficient of z with respect to x"
    },
    {
     "unit": null,
     "symbol": "z",
     "meaning": "a function of x, y and a"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a quantity containing y and x"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "a function of x"
    }
   ],
   "sympy": "Eq(Dz_Dx, dzdx + dzdy*dydx + dzda*dady*dydx + dzda*dadx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/indirect-function",
    "concept/partial-derivative",
    "concept/total-differential",
    "theorem/chain-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-bbddd9a260",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "100",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "\\frac{dz}{dx} = \\frac{dz}{da}\\, \\frac{da}{dy}\\, \\frac{dy}{dx}",
   "name": null,
   "statement": "For z = log a, a = log y, y = sin x, the derivative of z with respect to x is the product of the three successive coefficients.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "log a (the logarithm of the logarithm of sin x)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "log y"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "sin x"
    }
   ],
   "sympy": "Eq(Derivative(z, x), Derivative(z, a)*Derivative(a, y)*Derivative(y, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/logarithm",
    "concept/sine",
    "theorem/chain-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-878613750f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "100",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "\\dfrac{dz}{dy} = \\dfrac{1}{y}",
   "name": null,
   "statement": "For z = log y, the partial derivative of z with respect to y is 1/y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "log y"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the variable of the logarithm"
    }
   ],
   "sympy": "Eq(Derivative(z, y), 1/y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/logarithm",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a9b758ffee",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "100",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "\\frac{dz}{da} = \\frac{1}{a}",
   "name": null,
   "statement": "For z = log a, the derivative of z with respect to a is 1/a.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "log a"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the variable of the logarithm"
    }
   ],
   "sympy": "Eq(Derivative(z, a), 1/a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-cdbf9e2f31",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "101",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "\\frac{da}{dy} = \\frac{1}{y}",
   "name": null,
   "statement": "For a = log y, the derivative of a with respect to y is 1/y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "log y"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the variable of the logarithm"
    }
   ],
   "sympy": "Eq(Derivative(a, y), 1/y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-4dbfd578e9",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "101",
   "location": "Total and Partial Differential Coefficients. Implicit Differentiation",
   "latex": "\\frac{dy}{dx} = \\cos x",
   "name": null,
   "statement": "For y = sin x, the derivative of y with respect to x is cos x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "sin x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), cos(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/derivative",
    "concept/sine"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-d2b3c906d5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "101",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "z = ab",
   "name": null,
   "statement": "Defines z as the product of the two functions a and b of x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the function formed as the product of a and b"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a function of x"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "a function of x"
    }
   ],
   "sympy": "Eq(z, a*b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a45ffc88c1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "101",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "\\frac{dz}{dx} = \\frac{dz}{da}\\, \\frac{da}{dx} + \\frac{dz}{db}\\, \\frac{db}{dx}",
   "name": null,
   "statement": "The derivative of z with respect to x, where z depends on x indirectly through a and b, is the sum of the partial rates through each intermediate function.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the dependent function, depending on x indirectly through a and b"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "intermediate function of x"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "intermediate function of x"
    }
   ],
   "sympy": "Eq(Derivative(z, x), Derivative(z, a)*Derivative(a, x) + Derivative(z, b)*Derivative(b, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/formula",
    "concept/increment",
    "method/differentiation",
    "method/implicit-differentiation",
    "theorem/chain-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-614cda0c07",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "101",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "\\dfrac{dz}{db} = b",
   "name": null,
   "statement": "For z = ab, the ratio dz/db equals b. The book writes this as dz/db = b, but the argument given (z becomes ab + b da when a becomes a + da) yields dz/da = b; the subscript appears to be an erratum, flagged here and not corrected.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the product ab"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "the second function of x"
    }
   ],
   "sympy": "Eq(Derivative(z, b), b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/limit",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f8c2ff84cc",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "101",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "\\dfrac{dz}{db} = a",
   "name": null,
   "statement": "For z = ab, the ratio dz/db equals a (stated as similar to the previous case).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the product ab"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the first function of x"
    }
   ],
   "sympy": "Eq(Derivative(z, b), a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/limit",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-67ef441bc6",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "101",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "\\frac{dz}{dx} = b\\, \\frac{da}{dx} + a\\, \\frac{db}{dx}",
   "name": null,
   "statement": "The derivative of a product ab with respect to x is b times da/dx plus a times db/dx.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the product ab"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "function of x"
    }
   ],
   "sympy": "Eq(Derivative(z, x), b*Derivative(a, x) + a*Derivative(b, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/formula",
    "method/differentiation",
    "theorem/chain-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-379482b40b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "z = \\dfrac{a}{b}",
   "name": null,
   "statement": "Defines z as the quotient of a and b.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the quotient a/b"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "numerator function"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "denominator function"
    }
   ],
   "sympy": "Eq(z, a/b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-e29efdf6ef",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "\\frac{dz}{dx} = \\frac{1}{b}\\, \\frac{da}{dx} - \\frac{a}{b^{2}}\\, \\frac{db}{dx}",
   "name": null,
   "statement": "The derivative of the quotient a/b with respect to x, with the partial rates dz/da = 1/b and dz/db = -a/b^2 substituted into the general formula.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the quotient a/b"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "numerator function of x"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "denominator function of x"
    }
   ],
   "sympy": "Eq(Derivative(z, x), Derivative(a, x)/b - a*Derivative(b, x)/b**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/formula",
    "method/differentiation",
    "theorem/chain-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-fab0ba8d60",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "z = a^{b}",
   "name": null,
   "statement": "Defines z as a raised to the power b, where both a and b may be functions of x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the power a^b"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "base function of x"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "exponent function of x"
    }
   ],
   "sympy": "Eq(z, a**b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-e210e4f558",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "(a + da)^{b} = a^{b} + ba^{b-1}\\, da + \\etc.",
   "name": null,
   "statement": "Binomial expansion to first order of (a + da)^b, giving the increment of a^b when a is increased by da.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "base of the power"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "exponent"
    },
    {
     "unit": null,
     "symbol": "da",
     "meaning": "increment of a"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/increment",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-65ab71632b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "\\dfrac{dz}{da} = ba^{b-1}",
   "name": null,
   "statement": "For z = a^b, the ratio dz/da equals b times a to the power b minus 1.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the power a^b"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "base"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "exponent"
    }
   ],
   "sympy": "Eq(Derivative(z, a), b*a**(b - 1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/limit",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-bd356a8f96",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "a^{b+db} = a^{b}\\, a^{db} = a^{b}(1 + \\log a\\, db + \\etc.)",
   "name": null,
   "statement": "Expansion of a^(b+db) as a^b times a^db, with a^db approximated to first order by 1 + log a times db.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "base"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "exponent"
    },
    {
     "unit": null,
     "symbol": "db",
     "meaning": "increment of b"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/increment",
    "concept/limit",
    "concept/logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-d48dc49a7a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "\\dfrac{dz}{db} = a^{b} \\log a",
   "name": null,
   "statement": "For z = a^b, the ratio dz/db equals a^b times the logarithm of a.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the power a^b"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "base"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "exponent"
    },
    {
     "unit": null,
     "symbol": "log",
     "meaning": "logarithm (natural, as used in the expansion)"
    }
   ],
   "sympy": "Eq(Derivative(z, b), a**b*log(a))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/logarithm",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-bb9f8e5759",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Applications of the Theorem for Implicit Differentiation",
   "latex": "\\frac{dz}{dx} = ba^{b-1}\\, \\frac{da}{dx} + a^{b} \\log a\\, \\frac{db}{dx}",
   "name": null,
   "statement": "The derivative of a^b with respect to x when both a and b are functions of x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the power a^b"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "base function of x"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "exponent function of x"
    },
    {
     "unit": null,
     "symbol": "log",
     "meaning": "logarithm"
    }
   ],
   "sympy": "Eq(Derivative(z, x), b*a**(b - 1)*Derivative(a, x) + a**b*log(a)*Derivative(b, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/formula",
    "concept/logarithm",
    "method/differentiation",
    "theorem/chain-rule"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-3e196bc390",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Inverse Functions",
   "latex": "x = \\psi y",
   "name": null,
   "statement": "Solving the first relation for x gives x as another function psi of y, the inverse relation.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable, expressed in terms of y"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable of the inverse relation"
    },
    {
     "unit": null,
     "symbol": "\\psi",
     "meaning": "the inverse function of phi"
    }
   ],
   "sympy": "Eq(x, psi(y))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function-notation",
    "concept/inverse-function",
    "concept/relation-between-variables"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-62914ba88a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "102",
   "location": "Inverse Functions",
   "latex": "x = y^{\\efrac{1}{2}}",
   "name": null,
   "statement": "The inverse of y = x squared is x equal to y to the power one half.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable, expressed in terms of y"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable of the inverse relation"
    }
   ],
   "sympy": "Eq(x, y**(1/2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/inverse-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-402a2a290e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "103",
   "location": "Inverse Functions",
   "latex": "x = \\psi(\\phi x)",
   "name": null,
   "statement": "Substituting y = phi x into x = psi y gives x = psi(phi x): psi composed with phi returns x, so the operations of psi undo those of phi.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "the original function"
    },
    {
     "unit": null,
     "symbol": "\\psi",
     "meaning": "the inverse function"
    }
   ],
   "sympy": "Eq(x, psi(phi(x)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/composition-of-functions",
    "concept/function-notation",
    "concept/inverse-function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-65dc05be67",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "103",
   "location": "Inverse Functions",
   "latex": "\\dfrac{dy}{dx} = \\phi' x",
   "name": null,
   "statement": "The derivative of y with respect to x equals phi prime of x, obtained by differentiating y = phi x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    },
    {
     "unit": null,
     "symbol": "\\phi'",
     "meaning": "first derivative of phi"
    }
   ],
   "sympy": "Eq(Derivative(y, x), Derivative(phi(x), x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/function-notation",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6edf8a1461",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "103",
   "location": "Inverse Functions",
   "latex": "\\dfrac{dx}{dy} = \\psi' y",
   "name": null,
   "statement": "The derivative of x with respect to y equals psi prime of y, obtained by differentiating x = psi y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dx/dy",
     "meaning": "differential coefficient of x with respect to y"
    },
    {
     "unit": null,
     "symbol": "\\psi'",
     "meaning": "first derivative of psi"
    }
   ],
   "sympy": "Eq(Derivative(x, y), Derivative(psi(y), y))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/function-notation",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-205c1efb5d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "104",
   "location": "Inverse Functions",
   "latex": "\\frac{dy}{dx} = \\phi' x = \\frac{1}{\\psi' y} = \\frac{1}{p}",
   "name": null,
   "statement": "The derivative dy/dx equals phi prime of x, which is the reciprocal of psi prime of y, where p stands for psi prime of y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "psi prime of y, i.e. dx/dy"
    },
    {
     "unit": null,
     "symbol": "\\phi'",
     "meaning": "first derivative of phi"
    },
    {
     "unit": null,
     "symbol": "\\psi'",
     "meaning": "first derivative of psi"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 1/p)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/inverse-function",
    "concept/reciprocal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-088f9597aa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "105",
   "location": "Inverse Functions",
   "latex": "u = \\frac{1}{p}",
   "name": null,
   "statement": "Auxiliary quantity u is defined as the reciprocal of p.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "auxiliary quantity, equal to dy/dx"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "psi prime of y"
    }
   ],
   "sympy": "Eq(u, 1/p)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/reciprocal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-66984acf53",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "105",
   "location": "Inverse Functions",
   "latex": "\\frac{du}{dp} = -\\frac{1}{p^{2}}",
   "name": null,
   "statement": "The derivative of u = 1/p with respect to p is minus one over p squared.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du/dp",
     "meaning": "differential coefficient of u with respect to p"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "psi prime of y"
    }
   ],
   "sympy": "Eq(Derivative(u, p), -1/p**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/reciprocal",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-fb02532e85",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "105",
   "location": "Inverse Functions",
   "latex": "p = \\psi' y",
   "name": null,
   "statement": "The quantity p is defined as psi prime of y.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "p",
     "meaning": "psi prime of y"
    },
    {
     "unit": null,
     "symbol": "\\psi'",
     "meaning": "first derivative of psi"
    }
   ],
   "sympy": "Eq(p, Derivative(psi(y), y))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/function-notation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-1bbb3cb130",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "105",
   "location": "Inverse Functions",
   "latex": "\\frac{dp}{dy} = \\psi'' y",
   "name": null,
   "statement": "The derivative of p with respect to y equals psi double prime of y, the second derivative of psi.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dp/dy",
     "meaning": "differential coefficient of p with respect to y"
    },
    {
     "unit": null,
     "symbol": "\\psi''",
     "meaning": "second derivative of psi"
    }
   ],
   "sympy": "Eq(Derivative(p, y), Derivative(psi(y), (y, 2)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-3aa4580c3c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "104",
   "location": "Inverse Functions",
   "latex": "\\dfrac{d^{2} y}{dx^{2}} = \\phi'' x",
   "name": null,
   "statement": "The second differential coefficient of y with respect to x equals phi double prime of x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "d^2y/dx^2",
     "meaning": "second differential coefficient of y with respect to x"
    },
    {
     "unit": null,
     "symbol": "\\phi''",
     "meaning": "second derivative of phi"
    }
   ],
   "sympy": "Eq(Derivative(y, (x, 2)), Derivative(phi(x), (x, 2)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-31feb5f4bb",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "104",
   "location": "Inverse Functions",
   "latex": "\\dfrac{d^{2} x}{dy^{2}} = \\psi'' x",
   "name": null,
   "statement": "The second differential coefficient of x with respect to y is written as psi double prime of x. As printed, the variable should be y (psi double prime of y), since psi is applied to y; this is a candidate erratum to flag, not silently corrected.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "d^2x/dy^2",
     "meaning": "second differential coefficient of x with respect to y"
    },
    {
     "unit": null,
     "symbol": "\\psi''",
     "meaning": "second derivative of psi"
    }
   ],
   "sympy": "Eq(Derivative(x, (y, 2)), Derivative(psi(x), (x, 2)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ccb185fac7",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "104",
   "location": "Inverse Functions",
   "latex": "y = ax + b",
   "name": null,
   "statement": "The linear case: y is a linear function of x with constant a and b; only in this case are values of y equidistant when x is equidistant.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant slope"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant term"
    }
   ],
   "sympy": "Eq(y, a*x + b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/arithmetical-progression",
    "concept/constant",
    "concept/relation-between-variables"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-339708090e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "103",
   "location": "Inverse Functions",
   "latex": "y = e^{x}",
   "name": null,
   "statement": "Example: y equals e to the power x, whose inverse is x = log y.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, exp(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-b49090e645",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "103",
   "location": "Inverse Functions",
   "latex": "x = \\log y",
   "name": null,
   "statement": "The inverse of y = e^x is x equal to the logarithm of y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable, expressed in terms of y"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable of e^x"
    }
   ],
   "sympy": "Eq(x, log(y))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/inverse-function",
    "concept/logarithm"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-ce89c04f6c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "103",
   "location": "Inverse Functions",
   "latex": "\\dfrac{dy}{dx} = e^{x}",
   "name": null,
   "statement": "For y = e^x, the derivative of y with respect to x is e^x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(Derivative(y, x), exp(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-874cbf9d14",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "103",
   "location": "Inverse Functions",
   "latex": "\\dfrac{dx}{dy} = \\dfrac{1}{y}",
   "name": null,
   "statement": "For x = log y, the derivative of x with respect to y is 1/y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dx/dy",
     "meaning": "differential coefficient of x with respect to y"
    }
   ],
   "sympy": "Eq(Derivative(x, y), 1/y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/logarithm",
    "concept/reciprocal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-7c17c56f95",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "105",
   "location": "Inverse Functions",
   "latex": "\\dfrac{d^{2} y}{dx^{2}} = e^{x}",
   "name": null,
   "statement": "For y = e^x, the second differential coefficient of y with respect to x is e^x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "d^2y/dx^2",
     "meaning": "second differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(Derivative(y, (x, 2)), exp(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-7027b8e6f5",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "105",
   "location": "Inverse Functions",
   "latex": "\\dfrac{d^{2} x}{dy^{2}} = -\\dfrac{1}{y^{2}}",
   "name": null,
   "statement": "For x = log y, the second differential coefficient of x with respect to y is minus one over y squared.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "d^2x/dy^2",
     "meaning": "second differential coefficient of x with respect to y"
    }
   ],
   "sympy": "Eq(Derivative(x, (y, 2)), -1/y**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/logarithm",
    "concept/reciprocal"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-87411b1cad",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "108",
   "location": "Implicit Functions",
   "latex": "\\phi(x, y) = 0",
   "name": null,
   "statement": "The general implicit form of a relation between x and y, obtained by bringing all terms to one side of the equation.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "function of x and y"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable, implicitly a function of x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function-notation",
    "concept/implicit-function",
    "concept/relation-between-variables"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2603d53bda",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "107",
   "location": "Implicit Functions",
   "latex": "x^{2} - xy + y^{2} = a",
   "name": null,
   "statement": "An example equation in which y is implicitly, not explicitly, a function of x; for given x, y is found by solving a quadratic.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable (independent here)"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable, implicitly a function of x"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    }
   ],
   "sympy": "Eq(x**2 - x*y + y**2, a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation-of-the-second-degree",
    "concept/implicit-function",
    "concept/relation-between-variables"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-7c136ad3d8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "110",
   "location": "Implicit Functions",
   "latex": "xy - x = 1",
   "name": null,
   "statement": "A worked example relation between x and y, from which dy/dx is found by implicit differentiation and checked against the explicit solution.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable, implicitly a function of x"
    }
   ],
   "sympy": "Eq(x*y - x, 1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/implicit-function",
    "concept/relation-between-variables",
    "method/implicit-differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-15d6c67d72",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "108",
   "location": "Implicit Functions",
   "latex": "u = \\phi(x, y)",
   "name": null,
   "statement": "Defines the auxiliary quantity u as a function of the two variables x and y, used to derive the rule for dy/dx.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "auxiliary quantity, a function of x and y"
    },
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "function of x and y"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/function-notation",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-fab4d0a130",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "108",
   "location": "Implicit Functions",
   "latex": "du = \\ux\\, dx + \\uy\\, dy + \\etc.",
   "name": null,
   "statement": "The differential of u expressed through the partial derivatives of u with respect to x and y, with dx and dy as the small changes in x and y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du",
     "meaning": "small change in u"
    },
    {
     "unit": null,
     "symbol": "\\ux",
     "meaning": "partial derivative of u with respect to x (du/dx)"
    },
    {
     "unit": null,
     "symbol": "\\uy",
     "meaning": "partial derivative of u with respect to y (du/dy)"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "small change in x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small change in y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-80904a201c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "108",
   "location": "Implicit Functions",
   "latex": "\\ux\\, dx + \\uy\\, dy = 0",
   "name": null,
   "statement": "Since u stays zero along the curve, the small changes dx and dy satisfy this linear relation.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\ux",
     "meaning": "partial derivative of u with respect to x"
    },
    {
     "unit": null,
     "symbol": "\\uy",
     "meaning": "partial derivative of u with respect to y"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "small change in x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small change in y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/implicit-function",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-18aee15c26",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "108",
   "location": "Implicit Functions",
   "latex": "\\frac{dy}{dx} = -\\frac{\\ux}{\\uy}",
   "name": null,
   "statement": "The derivative of an implicit function y of x is minus the ratio of the partial derivatives of phi with respect to x and y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    },
    {
     "unit": null,
     "symbol": "\\ux",
     "meaning": "partial derivative of u with respect to x"
    },
    {
     "unit": null,
     "symbol": "\\uy",
     "meaning": "partial derivative of u with respect to y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/implicit-function",
    "concept/partial-derivative",
    "method/implicit-differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-249fbb79fa",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "108",
   "location": "Implicit Functions",
   "latex": "\\frac{dy}{dx} = -\\frac{\\;\\dfrac{du}{dx}\\;}{\\dfrac{du}{dy}}",
   "name": null,
   "statement": "Equation (1): the derivative of y with respect to x equals minus the partial derivative of u with respect to x divided by the partial derivative of u with respect to y, valid when u is held at zero.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    },
    {
     "unit": null,
     "symbol": "du/dx",
     "meaning": "partial derivative of u with respect to x"
    },
    {
     "unit": null,
     "symbol": "du/dy",
     "meaning": "partial derivative of u with respect to y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/partial-derivative",
    "method/implicit-differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6f88fad200",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "110",
   "location": "Implicit Functions",
   "latex": "\\dfrac{du}{dx} = y - 1",
   "name": null,
   "statement": "For u = xy - x - 1, the partial derivative of u with respect to x is y - 1.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du/dx",
     "meaning": "partial derivative of u with respect to x"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a9323a3005",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "110",
   "location": "Implicit Functions",
   "latex": "\\dfrac{du}{dy} = x",
   "name": null,
   "statement": "For u = xy - x - 1, the partial derivative of u with respect to y is x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du/dy",
     "meaning": "partial derivative of u with respect to y"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-dac4df72f3",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "110",
   "location": "Implicit Functions",
   "latex": "xy - x - 1 = 0",
   "name": null,
   "statement": "The example relation written in the form phi(x, y) = 0 with all terms on one side.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable, implicitly a function of x"
    }
   ],
   "sympy": "Eq(x*y - x - 1, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/implicit-function",
    "concept/relation-between-variables"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-becba1758a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "110",
   "location": "Implicit Functions",
   "latex": "\\frac{dy}{dx} = -\\frac{y - 1}{x}",
   "name": null,
   "statement": "Equation (3): the derivative of y with respect to x for the example xy - x = 1, expressed in x and y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "method/implicit-differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-dca2f35d6c",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "110",
   "location": "Implicit Functions",
   "latex": "y = 1 + \\dfrac{1}{x}",
   "name": null,
   "statement": "Solving xy - x = 1 for y gives y explicitly as a function of x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable, a function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, 1 + 1/x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/explicit-function",
    "concept/function",
    "method/solving-an-equation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-37df0c573a",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "110",
   "location": "Implicit Functions",
   "latex": "-\\dfrac{1}{x^{2}}",
   "name": null,
   "statement": "The limit of the ratio dy/dx for the example is minus one over x squared, which agrees with equation (3) after substituting y = 1 + 1/x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/limit-of-the-ratio-of-the-increment"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f200ed188d",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "109",
   "location": "Implicit Functions",
   "latex": "u = \\Chg{\\phi(x)}{\\phi x}",
   "name": null,
   "statement": "Defines u as a function of x alone, the first of two equations linking u, x and y.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "auxiliary quantity"
    },
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/function-notation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-8ca84612db",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "109",
   "location": "Implicit Functions",
   "latex": "u = \\psi y",
   "name": null,
   "statement": "Defines u as a function of y alone, the second of two equations linking u, x and y.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "auxiliary quantity"
    },
    {
     "unit": null,
     "symbol": "\\psi",
     "meaning": "function of y"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/function-notation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2beccc8947",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "109",
   "location": "Implicit Functions",
   "latex": "\\phi x = \\psi y",
   "name": null,
   "statement": "The third equation implied by the two equations u = phi x and u = psi y, relating x and y.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\phi",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "\\psi",
     "meaning": "function of y"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable, a function of x"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/implicit-function",
    "concept/relation-between-variables"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f4987d528b",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "109",
   "location": "Implicit Functions",
   "latex": "du = \\psi(y + dy) - \\psi y",
   "name": null,
   "statement": "The change in u when y changes by dy, computed from the second equation u = psi y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "du",
     "meaning": "small change in u"
    },
    {
     "unit": null,
     "symbol": "\\psi",
     "meaning": "function of y"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "variable"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small change in y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-2c0b9770b1",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "109",
   "location": "Implicit Functions",
   "latex": "\\phi' x\\, dx + \\etc. = \\psi' y\\, dy + \\etc.",
   "name": null,
   "statement": "Equating the two expressions for du gives a relation between dx and dy, the etc. terms vanishing in the limit.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\phi' x",
     "meaning": "derivative of phi at x"
    },
    {
     "unit": null,
     "symbol": "\\psi' y",
     "meaning": "derivative of psi at y"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "small change in x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small change in y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/function"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-70bf118f8e",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "109",
   "location": "Implicit Functions",
   "latex": "\\frac{dy}{dx} = \\frac{\\phi' x}{\\psi' y} = \\frac{\\;\\dfrac{du}{dx}\\;}{\\dfrac{du}{dy}}",
   "name": null,
   "statement": "Equation (2): dy/dx equals the ratio of the derivatives of phi and psi, which is also the ratio of the partial derivatives of u, in accordance with common algebra.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    },
    {
     "unit": null,
     "symbol": "\\phi' x",
     "meaning": "derivative of phi at x"
    },
    {
     "unit": null,
     "symbol": "\\psi' y",
     "meaning": "derivative of psi at y"
    },
    {
     "unit": null,
     "symbol": "du/dx",
     "meaning": "partial derivative of u with respect to x"
    },
    {
     "unit": null,
     "symbol": "du/dy",
     "meaning": "partial derivative of u with respect to y"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/partial-derivative",
    "method/implicit-differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-bdea9e9ab8",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "118",
   "location": "The Integral Calculus",
   "latex": "v = \\dfrac{h}{m + 1}",
   "name": null,
   "statement": "Sets v equal to the spacing h/(m+1) between successive interposed values of the variable.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "v",
     "meaning": "spacing between successive interposed values of x"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "total increment from a to a+h"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "number of fractions interposed between a and a+h"
    }
   ],
   "sympy": "Eq(v, h/(m + 1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/increment",
    "concept/interval",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-dd442119ca",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "118",
   "location": "The Integral Calculus",
   "latex": "\\frac{(m + 2)A}{(m + 2)a} = \\frac{A}{a}",
   "name": null,
   "statement": "The ratio (m+2)A over (m+2)a simplifies to A/a, which does not depend on m and is finite.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "A",
     "meaning": "a finite quantity greater than every term of the series (2)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the starting value of the interval"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "number of interposed values"
    }
   ],
   "sympy": "Eq((m + 2)*A/((m + 2)*a), A/a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/finite-set",
    "concept/limit"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-dc2d171b07",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "118",
   "location": "The Integral Calculus",
   "latex": "1 + 2 + \\dots + (m + 1) = \\frac{1}{2}(m + 1)(m + 2)",
   "name": null,
   "statement": "The sum of the consecutive whole numbers from 1 to m+1 equals half of (m+1)(m+2).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "m",
     "meaning": "number of interposed values, so the last summand is m+1"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/arithmetical-progression",
    "concept/real-number",
    "concept/sum"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-6f599eb610",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "119",
   "location": "The Integral Calculus",
   "latex": "\\frac{m + 2}{m + 1}\\, ha^{2} + \\frac{m + 2}{m + 1}\\, ha^{2} + (1 + \\alpha)\\, \\frac{h^{3}}{3}",
   "name": null,
   "statement": "The book's expression for the sum after substitution; its second term is printed as ha^2, but the chapter's own expansion gives ah^2 there, so the printed line is an erratum or a transcription error that is flagged here, not corrected.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "h",
     "meaning": "total increment of x from a to a+h"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "lower value of x"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "number of interposed values"
    },
    {
     "unit": null,
     "symbol": "alpha",
     "meaning": "a quantity that diminishes without limit as m increases without limit"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/integral",
    "concept/limit",
    "theorem/limit-of-a-sum"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-d5bd7f56e9",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "119",
   "location": "The Integral Calculus",
   "latex": "ha^{2} + ha^{2} + \\frac{h^{3}}{3} \\quad\\text{or}\\quad \\frac{(a + h)^{3} - a^{3}}{3}",
   "name": null,
   "statement": "The limit of the sum for x^2 dx is stated as ((a+h)^3 - a^3)/3, but the printed left side has ha^2 twice where the chapter's own expansion gives ah^2, so as printed the two sides do not agree; this is flagged as a possible erratum rather than corrected.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "h",
     "meaning": "total increment of x from a to a+h"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "lower limit of the interval"
    }
   ],
   "sympy": "Eq(h*a**2 + h*a**2 + h**3/3, ((a + h)**3 - a**3)/3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/integral",
    "theorem/limit-of-a-sum"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-d533667612",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "119",
   "location": "The Integral Calculus",
   "latex": "\\int_{a}^{a+h} x^{2}\\, dx",
   "name": null,
   "statement": "Defines the integral of x^2 dx between the limits a and a+h as the limit of the sum of x^2 dx over the interval.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "lower limit of integration"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "increment, so a+h is the upper limit of integration"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/integral",
    "concept/limits-of-integration",
    "method/integration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-16aa2d8765",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "123",
   "location": "Nature of Integration",
   "latex": "C = -\\psi a",
   "name": null,
   "statement": "The constant C of integration equals minus the value of the antiderivative at the starting point a, so the integral beginning at a is psi x minus psi a.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "C",
     "meaning": "a constant (not varying when x varies) added to an integral"
    },
    {
     "unit": null,
     "symbol": "\\psi",
     "meaning": "the integral (antiderivative) of the function being integrated"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the value of x from which the summation begins"
    }
   ],
   "sympy": "Eq(C, -Function('psi')(a))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/antiderivative",
    "concept/constant",
    "concept/definite-integral",
    "concept/integral",
    "concept/limits-of-integration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-774debd950",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "123",
   "location": "Nature of Integration",
   "latex": "\\psi x - \\psi a = 0",
   "name": null,
   "statement": "When x equals the starting value a, the integral psi x minus psi a vanishes, so the integral can be made to begin at a by equating it to zero and solving for x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\psi",
     "meaning": "the integral (antiderivative) of the function being integrated"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the upper limit, a variable value"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the value of x at which the integral begins"
    }
   ],
   "sympy": "Eq(Function('psi')(x) - Function('psi')(a), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/antiderivative",
    "concept/integral",
    "concept/limits-of-integration",
    "concept/value-of-a-function",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-b18ffa83d9",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "123",
   "location": "Nature of Integration",
   "latex": "\\psi x - \\psi a",
   "name": null,
   "statement": "The integral of psi' x taken from the starting value a up to x is psi x minus psi a.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\psi",
     "meaning": "the integral (antiderivative) of the function being integrated"
    },
    {
     "unit": null,
     "symbol": "\\psi'",
     "meaning": "the function to be integrated, the differential coefficient of psi"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the value of x from which the summation begins"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the upper limit of the summation, a variable value"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/antiderivative",
    "concept/definite-integral",
    "concept/integral",
    "concept/limits-of-integration",
    "method/differentiation"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f1f1cc4e27",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "131",
   "location": "Method of Indivisibles",
   "latex": "AB = a",
   "name": null,
   "statement": "The whole length of the bar AB is called a.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "AB",
     "meaning": "the line (bar) whose density varies"
    },
    {
     "unit": "feet",
     "symbol": "a",
     "meaning": "total length of the bar AB"
    }
   ],
   "sympy": "Eq(AB, a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/dimension",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-491358a244",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "131",
   "location": "Method of Indivisibles",
   "latex": "n\\, dx = a",
   "name": null,
   "statement": "Dividing the length a into n equal parts, each of width dx, makes the n parts together equal the whole length.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "number of equal parts"
    },
    {
     "unit": "feet",
     "symbol": "dx",
     "meaning": "width of each small part"
    },
    {
     "unit": "feet",
     "symbol": "a",
     "meaning": "total length of the bar"
    }
   ],
   "sympy": "Eq(n*dx, a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/dimension",
    "concept/infinitesimal",
    "concept/variable"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-a51b80cd06",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "131",
   "location": "Method of Indivisibles",
   "latex": "wb\\, dx",
   "name": null,
   "statement": "The weight of a bulk of water equal to a slice of volume b dx equals w times that volume.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "pounds",
     "symbol": "w",
     "meaning": "weight of a cubic foot of water"
    },
    {
     "unit": "square feet",
     "symbol": "b",
     "meaning": "area of the cross section of the bar"
    },
    {
     "unit": "feet",
     "symbol": "dx",
     "meaning": "width of the slice"
    }
   ],
   "sympy": "Eq(W_water, w*b*dx)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/weight",
    "quantity/density",
    "quantity/volume"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-85e33b8c02",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "131",
   "location": "Method of Indivisibles",
   "latex": "x^{2} × bw\\, dx",
   "name": null,
   "statement": "If the density at the slice were uniform and equal to x squared, the weight of the slice would be approximately x squared times b w dx; this is only approximately true.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": "feet",
     "symbol": "x",
     "meaning": "distance of the point from A"
    },
    {
     "unit": "square feet",
     "symbol": "b",
     "meaning": "area of the cross section"
    },
    {
     "unit": "pounds",
     "symbol": "w",
     "meaning": "weight of a cubic foot of water"
    },
    {
     "unit": "feet",
     "symbol": "dx",
     "meaning": "width of the slice"
    }
   ],
   "sympy": "Eq(W_slice, x**2*b*w*dx)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/infinitesimal",
    "concept/weight",
    "quantity/density"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-f489c02f1f",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "132",
   "location": "Method of Indivisibles",
   "latex": "bwx^{2}\\, dx + \\alpha",
   "name": null,
   "statement": "The real weight of the part pq is represented by the term bwx^2 dx plus a correction alpha, which can be made as small a part as we please of the first term.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": "feet",
     "symbol": "x",
     "meaning": "distance of the point from A"
    },
    {
     "unit": "square feet",
     "symbol": "b",
     "meaning": "area of the cross section"
    },
    {
     "unit": "pounds",
     "symbol": "w",
     "meaning": "weight of a cubic foot of water"
    },
    {
     "unit": "feet",
     "symbol": "dx",
     "meaning": "width of the slice"
    },
    {
     "unit": "pounds",
     "symbol": "alpha",
     "meaning": "error term, small compared with bwx^2 dx"
    }
   ],
   "sympy": "Eq(W_pq, b*w*x**2*dx + alpha)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/infinitesimal",
    "concept/weight",
    "quantity/density"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-3dfaefda20",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "132",
   "location": "Method of Indivisibles",
   "latex": "\\frac{1}{3}bwx^{3}",
   "name": null,
   "statement": "The integral of b w x^2 dx is one third of b w x cubed.",
   "kind": "result",
   "symbols": [
    {
     "unit": "square feet",
     "symbol": "b",
     "meaning": "area of the cross section"
    },
    {
     "unit": "pounds",
     "symbol": "w",
     "meaning": "weight of a cubic foot of water"
    },
    {
     "unit": "feet",
     "symbol": "x",
     "meaning": "distance of the point from A"
    }
   ],
   "sympy": "Eq(Integral(b*w*x**2, x), b*w*x**3/3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/integral",
    "concept/weight",
    "method/integration"
   ]
  },
  {
   "id": "de-morgan-elementary-illustrations-calculus-1899/eq-7352dd5663",
   "chapter": "de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",
   "book": "de-morgan-elementary-illustrations-calculus-1899",
   "edition": "Open Court Publishing Company, Chicago, new ed., 1899",
   "page": "132",
   "location": "Method of Indivisibles",
   "latex": "\\frac{1}{3}bwa^{3}",
   "name": null,
   "statement": "Taking the integral from x = 0 to x = a gives the weight of the whole bar of length a and section b when the density is x squared.",
   "kind": "result",
   "symbols": [
    {
     "unit": "pounds",
     "symbol": "W",
     "meaning": "weight of the bar in pounds"
    },
    {
     "unit": "feet",
     "symbol": "a",
     "meaning": "length of the bar"
    },
    {
     "unit": "square feet",
     "symbol": "b",
     "meaning": "area of the section of the bar"
    },
    {
     "unit": "pounds",
     "symbol": "w",
     "meaning": "weight of a cubic foot of water"
    }
   ],
   "sympy": "Eq(W, b*w*a**3/3)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/integral",
    "concept/weight",
    "quantity/density"
   ]
  }
 ],
 "exercise_sets": [],
 "problems": [],
 "errata_statuses": [
  "transcriber_marked",
  "candidate",
  "probable",
  "confirmed",
  "dismissed",
  "note"
 ],
 "errata": []
}