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XIV: Series", "pages": [ "194", "217" ], "concepts": [ "concept/approximation", "concept/arithmetical-progression", "concept/continuous-function", "concept/convergent-series", "concept/cosine", "concept/curve-of-normal-error", "concept/damped-vibration", "concept/divergent-series", "concept/exponential-function", "concept/factorial", "concept/geometrical-progression", "concept/infinite-sequence", "concept/limit", "concept/non-uniform-convergence", "concept/partial-sum", "concept/permutation", "concept/recurring-decimal", "concept/sine", "concept/standard-of-approximation", "concept/sum-to-infinity", "concept/type-of-order", "concept/uniform-convergence", "person/george-stokes", "person/seidel", "theorem/addition-theorem", "theorem/exponential-series" ], "excerpts": [ "whitehead-introduction-to-mathematics-1911/x-8ae0b01c36", "whitehead-introduction-to-mathematics-1911/x-288e0af93f", "whitehead-introduction-to-mathematics-1911/x-faba375778", "whitehead-introduction-to-mathematics-1911/x-d3a408d718", "whitehead-introduction-to-mathematics-1911/x-981a1254c4", "whitehead-introduction-to-mathematics-1911/x-68b09adc7a", "whitehead-introduction-to-mathematics-1911/x-2177f55248", "whitehead-introduction-to-mathematics-1911/x-462fbb605a", "whitehead-introduction-to-mathematics-1911/x-d66abb8814", "whitehead-introduction-to-mathematics-1911/x-29affacd83", "whitehead-introduction-to-mathematics-1911/x-cf9e730982", "whitehead-introduction-to-mathematics-1911/x-10ea9056ea" ], "equations": [ "whitehead-introduction-to-mathematics-1911/eq-0ce27a921e", "whitehead-introduction-to-mathematics-1911/eq-f25ee20249", "whitehead-introduction-to-mathematics-1911/eq-6d9d81f711", "whitehead-introduction-to-mathematics-1911/eq-02b51eb846", "whitehead-introduction-to-mathematics-1911/eq-38b8fe1c89", "whitehead-introduction-to-mathematics-1911/eq-b6e6e99deb", "whitehead-introduction-to-mathematics-1911/eq-e5c7020535", "whitehead-introduction-to-mathematics-1911/eq-f517f9b38a", "whitehead-introduction-to-mathematics-1911/eq-5718dd3c31", "whitehead-introduction-to-mathematics-1911/eq-5c5cb37b34", "whitehead-introduction-to-mathematics-1911/eq-7fe52db00b", "whitehead-introduction-to-mathematics-1911/eq-f283bbb29c", "whitehead-introduction-to-mathematics-1911/eq-eefb986697", "whitehead-introduction-to-mathematics-1911/eq-c7d8305ef1" ], "exercise_sets": [] }, { "id": "whitehead-introduction-to-mathematics-1911/ch-xv", "number": "XV", "title": "The Differential Calculus", "name": "Whitehead 1911, ch. XV: The Differential Calculus", "pages": [ "217", "236" ], "concepts": [ "concept/approximation", "concept/argument-of-a-function", "concept/calculus", "concept/constant", "concept/continuous-function", "concept/coordinate-geometry", "concept/derivative", "concept/fluxional-notation", "concept/function", "concept/infinitesimal", "concept/integral", "concept/interval", "concept/limit", "concept/mathematical-notation", "concept/mathematics", "concept/neighbourhood", "concept/parameter", "concept/rate-of-change", "concept/slope-of-a-curve", "concept/standard-of-approximation", "concept/tangent", "concept/value-of-a-function", "concept/variable", "person/gottfried-wilhelm-leibniz", "person/isaac-newton", "person/karl-weierstrass", "person/pierre-de-fermat", "person/ren-descartes", "quantity/acceleration", "quantity/distance", "quantity/velocity" ], "excerpts": [ "whitehead-introduction-to-mathematics-1911/x-9227aa7012", "whitehead-introduction-to-mathematics-1911/x-29bd387c8d", "whitehead-introduction-to-mathematics-1911/x-e8718671da", "whitehead-introduction-to-mathematics-1911/x-f56181d402", "whitehead-introduction-to-mathematics-1911/x-40ca4ae07b", "whitehead-introduction-to-mathematics-1911/x-b50a92e976", "whitehead-introduction-to-mathematics-1911/x-cc425bc87f" ], "equations": [], "exercise_sets": [] }, { "id": "whitehead-introduction-to-mathematics-1911/ch-xvi", "number": "XVI", "title": "Geometry", "name": "Whitehead 1911, ch. XVI: Geometry", "pages": [ "236", "245" ], "concepts": [ "concept/abstractness", "concept/algebra", "concept/congruent-figures", "concept/conic-section", "concept/coordinate-geometry", "concept/geometry", "concept/similarity", "concept/triangle", "concept/variable", "theorem/greater-angle-lies-opposite-greater-side-in-a-triangle", "theorem/sum-of-the-angles-of-a-triangle", "theorem/sum-of-two-sides-of-a-triangle-exceeds-the-third" ], "excerpts": [ "whitehead-introduction-to-mathematics-1911/x-9ba46eba54", "whitehead-introduction-to-mathematics-1911/x-9ac1b0cb5e", "whitehead-introduction-to-mathematics-1911/x-6e7dee6b77", "whitehead-introduction-to-mathematics-1911/x-5d223a0d9e", "whitehead-introduction-to-mathematics-1911/x-ebbee8947f", "whitehead-introduction-to-mathematics-1911/x-ad678ce0a3", "whitehead-introduction-to-mathematics-1911/x-2bd7053c9d", "whitehead-introduction-to-mathematics-1911/x-515371b2fa" ], "equations": [ "whitehead-introduction-to-mathematics-1911/eq-7aec5d73b3", "whitehead-introduction-to-mathematics-1911/eq-f4fb62a4cd", "whitehead-introduction-to-mathematics-1911/eq-8f3934de24", "whitehead-introduction-to-mathematics-1911/eq-b006ad1e88" ], "exercise_sets": [] }, { "id": "whitehead-introduction-to-mathematics-1911/ch-xvii", "number": "XVII", "title": "Quantity", "name": "Whitehead 1911, ch. XVII: Quantity", "pages": [ "245", "250" ], "concepts": [ "concept/axiom", "concept/continuous-quantity", "concept/equality", "concept/measure-of-time", "concept/number", "concept/periodicity", "concept/quantities-of-the-same-kind", "instrument/foot-rule", "law/laws-of-motion", "method/addition", "quantity/area", "quantity/length", "quantity/quantity", "quantity/time", "quantity/volume", "unit/unit" ], "excerpts": [ "whitehead-introduction-to-mathematics-1911/x-946352486e", "whitehead-introduction-to-mathematics-1911/x-7bf0fc5e88", "whitehead-introduction-to-mathematics-1911/x-672dcfcbf7", "whitehead-introduction-to-mathematics-1911/x-19fbb8d81d", "whitehead-introduction-to-mathematics-1911/x-9c66e36fc4", "whitehead-introduction-to-mathematics-1911/x-08b76209a5", "whitehead-introduction-to-mathematics-1911/x-d2e66bcecd" ], "equations": [], "exercise_sets": [] }, { "id": "whitehead-introduction-to-mathematics-1911/ch-notes", "number": "Notes", "title": "Notes", "name": "Whitehead 1911, Notes", "pages": [ "250", "251" ], "concepts": [ "concept/absolute-convergence", "concept/convergent-series", "concept/divergent-series", "concept/ellipse", "concept/hyperbola", "concept/parabola", "concept/parenthesis", "method/addition", "method/multiplication", "method/order-of-operations", "quantity/eccentricity" ], "excerpts": [ "whitehead-introduction-to-mathematics-1911/x-17e4c4ec6a", "whitehead-introduction-to-mathematics-1911/x-6b64ff05e2", "whitehead-introduction-to-mathematics-1911/x-5681a760cf", "whitehead-introduction-to-mathematics-1911/x-5c560cfcb8", "whitehead-introduction-to-mathematics-1911/x-a5a3942699", "whitehead-introduction-to-mathematics-1911/x-035cad72ec" ], "equations": [ "whitehead-introduction-to-mathematics-1911/eq-60fcddf9e0" ], "exercise_sets": [] }, { "id": "whitehead-introduction-to-mathematics-1911/ch-bibliography", "number": "Bibliography", "title": "Bibliography", "name": "Whitehead 1911, Bibliography", "pages": [ "251", "252" ], "concepts": [ "concept/algebra", "concept/arithmetic", "concept/calculus", "concept/conic-section", "concept/coordinate-geometry", "concept/differential-equation", "concept/geometry", "concept/mathematics", "concept/plane-trigonometry", "concept/solid-geometry" ], "excerpts": [ "whitehead-introduction-to-mathematics-1911/x-71ba536915", "whitehead-introduction-to-mathematics-1911/x-81c9e14c58", "whitehead-introduction-to-mathematics-1911/x-9c4b0f1da9", "whitehead-introduction-to-mathematics-1911/x-faf95e8c62" ], "equations": [], "exercise_sets": [] } ], "excerpts": [ { "id": "whitehead-introduction-to-mathematics-1911/x-12f728cdb4", "chapter": "whitehead-introduction-to-mathematics-1911/ch-i", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "8", "location": "The Abstract Nature of Mathematics", "latex": "The reason for this failure of the science to live up to its reputation is that its fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances.", "markdown": "The reason for this failure of the science to live up to its reputation is that its fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances.", "why": "It names the cause of the learner's frustration: fundamental ideas are taught tangled up with technique, before the student sees what the technique is for.", "use": [ "lesson", "website" ], "concepts": [ "concept/mathematics" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-10051c0cf1", "chapter": "whitehead-introduction-to-mathematics-1911/ch-i", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "9", "location": "The Abstract Nature of Mathematics", "latex": "Thus we write down as the leading characteristic of mathematics that it deals with properties and ideas which are applicable to things just because they are things, and apart from any particular feelings, or emotions, or sensations, in any way connected with them. This is what is meant by calling mathematics an abstract science.", "markdown": "Thus we write down as the leading characteristic of mathematics that it deals with properties and ideas which are applicable to things just because they are things, and apart from any particular feelings, or emotions, or sensations, in any way connected with them. This is what is meant by calling mathematics an abstract science.", "why": "It gives a plain definition of what 'abstract' means in mathematics, separating the property from any particular sensation.", "use": [ "lesson", "website" ], "concepts": [ "concept/abstractness", "concept/mathematics" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-fdc7548e5a", "chapter": "whitehead-introduction-to-mathematics-1911/ch-i", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "9", "location": "The Abstract Nature of Mathematics", "latex": "Now, the first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body.", "markdown": "Now, the first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body.", "why": "It gives a vivid, concrete example of generality, showing that arithmetic does not depend on what the things are.", "use": [ "lesson", "website" ], "concepts": [ "concept/arithmetic" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-004ac1bf71", "chapter": "whitehead-introduction-to-mathematics-1911/ch-i", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "8", "location": "The Abstract Nature of Mathematics", "latex": "The object of the following Chapters is not to teach mathematics, but to enable students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena.", "markdown": "The object of the following Chapters is not to teach mathematics, but to enable students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena.", "why": "It states the purpose of the whole book, so a learner knows what to look for before the technical work begins.", "use": [ "lesson", "website" ], "concepts": [ "concept/mathematics" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-f3e2549e93", "chapter": "whitehead-introduction-to-mathematics-1911/ch-i", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "8", "location": "The Abstract Nature of Mathematics", "latex": "But it is equally an error to confine attention to technical processes, excluding consideration of general ideas. Here lies the road to pedantry.", "markdown": "But it is equally an error to confine attention to technical processes, excluding consideration of general ideas. Here lies the road to pedantry.", "why": "It warns the learner against mistaking memorised procedure for understanding of the subject.", "use": [ "lesson" ], "concepts": [] }, { "id": "whitehead-introduction-to-mathematics-1911/x-501bc7829f", "chapter": "whitehead-introduction-to-mathematics-1911/ch-i", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "11", "location": "The Abstract Nature of Mathematics", "latex": "In the eye of science, the fall of an apple, the motion of a planet round a sun, and the clinging of the atmosphere to the earth are all seen as examples of the law of gravity.", "markdown": "In the eye of science, the fall of an apple, the motion of a planet round a sun, and the clinging of the atmosphere to the earth are all seen as examples of the law of gravity.", "why": "It shows how one general law explains very different everyday and astronomical events, which makes the idea of a law concrete.", "use": [ "lesson", "website" ], "concepts": [ "law/law-of-gravity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-a2a22dd6c1", "chapter": "whitehead-introduction-to-mathematics-1911/ch-i", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "13", "location": "The Abstract Nature of Mathematics", "latex": "Pythagoras had a glimpse of it when he proclaimed that number was the source of all things.", "markdown": "Pythagoras had a glimpse of it when he proclaimed that number was the source of all things.", "why": "It places the idea of mathematical explanation in a historical line that starts with Pythagoras.", "use": [ "history" ], "concepts": [ "concept/number", "person/pythagoras" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-b1afbecec7", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "19", "location": "Variables", "latex": "Thus the ``field'' of the relation for~$x$ is restricted to numbers less than~$1$, and similarly for the ``field'' open to~$y$.", "markdown": "Thus the “field” of the relation for $x$ is restricted to numbers less than $1$, and similarly for the “field” open to $y$.", "why": "It shows how a relation restricts the values each variable may take, using x + y = 1 with positive numbers.", "use": [ "lesson" ], "concepts": [ "concept/field-of-a-variable", "concept/relation-between-variables" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-f37d92a7a6", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "15", "location": "Variables", "latex": "The ideas of \\emph{any} and of \\emph{some} are introduced into algebra by the use of letters, instead of the definite numbers of arithmetic.", "markdown": "The ideas of *any* and of *some* are introduced into algebra by the use of letters, instead of the definite numbers of arithmetic.", "why": "It states in one sentence what letters add to arithmetic, which is the chapter's central idea.", "use": [ "lesson" ], "concepts": [ "concept/algebra", "concept/arithmetic", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-6bd5d1afd0", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "17", "location": "Variables", "latex": "Thus, as here used, \\emph{any} implies \\emph{some} and \\emph{some} does not exclude \\emph{any}.", "markdown": "Thus, as here used, *any* implies *some* and *some* does not exclude *any*.", "why": "It clarifies that 'some' includes the case of 'any', a point learners often misread.", "use": [ "lesson" ], "concepts": [ "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-29aec76035", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "17", "location": "Variables", "latex": "When we have asked the question implied in the statement of the equation $x + 2 = 3$, $x$~is called the unknown.", "markdown": "When we have asked the question implied in the statement of the equation $x + 2 = 3$, $x$ is called the unknown.", "why": "It shows how a statement becomes an equation to solve, and names the unknown.", "use": [ "lesson" ], "concepts": [ "concept/equation", "concept/unknown" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-69b52ebac8", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "18", "location": "Variables", "latex": "One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the text-books with the solution of equations.", "markdown": "One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the text-books with the solution of equations.", "why": "It warns that over-focus on solving equations hides the deeper role of variables.", "use": [ "lesson" ], "concepts": [ "concept/equation", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-77da967540", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "22", "location": "Variables", "latex": "Then the law, known as Boyle's law, expressing the relation between $p$ and~$v$ as both vary, is that the product~$pv$ is constant, always supposing that the temperature does not alter.", "markdown": "Then the law, known as Boyle’s law, expressing the relation between $p$ and $v$ as both vary, is that the product $pv$ is constant, always supposing that the temperature does not alter.", "why": "It gives a physical case where two varying quantities are bound by one fixed relation.", "use": [ "lesson", "history" ], "concepts": [ "concept/pressure", "law/boyle-s-law", "quantity/volume" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-6b1a67c688", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "23", "location": "Variables", "latex": "In other words the really fundamental idea is that of the pair of \\emph{variables} satisfying the relation $pv = 1$.", "markdown": "In other words the really fundamental idea is that of the pair of *variables* satisfying the relation $pv = 1$.", "why": "It states the chapter's main claim that variables, not unknowns, are the fundamental idea.", "use": [ "lesson", "website" ], "concepts": [ "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-204b9aa8e9", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "16", "location": "Variables", "latex": "The Romans would have stated the number of the year in which this is written in the form MDCCCCX., whereas we write it~1910, thus leaving the letters for the other usage.", "markdown": "The Romans would have stated the number of the year in which this is written in the form MDCCCCX., whereas we write it 1910, thus leaving the letters for the other usage.", "why": "A charming remark on why positional numerals freed letters for general use, told with a historical flourish.", "use": [ "history", "website" ], "concepts": [ "concept/arabic-numerals", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-8939a7e7f5", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "27", "location": "Methods of Application", "latex": "The conclusion of no argument can be more certain than the assumptions from which it starts. All mathematical calculations about the course of nature must start from some assumed law of nature, such, for instance, as the assumed law of the cost of building stated above.", "markdown": "The conclusion of no argument can be more certain than the assumptions from which it starts. All mathematical calculations about the course of nature must start from some assumed law of nature, such, for instance, as the assumed law of the cost of building stated above.", "why": "It shows a learner that a correct calculation cannot make an uncertain assumption certain, which is the central caution of applied mathematics.", "use": [ "lesson" ], "concepts": [ "concept/mathematical-physics" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-71197fce08", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "31", "location": "Methods of Application", "latex": "The vital point in the application of mathematical formulæ is to have clear ideas and a correct estimate of their relevance to the phenomena under observation.", "markdown": "The vital point in the application of mathematical formulæ is to have clear ideas and a correct estimate of their relevance to the phenomena under observation.", "why": "It states plainly why a formula must be matched to the phenomenon it describes before its results are trusted.", "use": [ "lesson", "website" ], "concepts": [ "concept/mathematical-physics" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-87afdf8b07", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "27", "location": "Methods of Application", "latex": "there is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain.", "markdown": "there is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain.", "why": "It warns learners against the common mistake of treating a precise calculation as proof that a physical prediction is exact.", "use": [ "lesson" ], "concepts": [ "concept/mathematical-physics" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-b83361155f", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "38", "location": "Methods of Application", "latex": "He saw that a body when immersed in water is pressed upwards by the surrounding water with a resultant force equal to the weight of the water it displaces.", "markdown": "He saw that a body when immersed in water is pressed upwards by the surrounding water with a resultant force equal to the weight of the water it displaces.", "why": "It gives the buoyancy law in one clear sentence that a learner can test against the crown argument that follows.", "use": [ "lesson", "website" ], "concepts": [ "law/archimedes-principle" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-755f4182a8", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "40", "location": "Methods of Application", "latex": "This ratio is the same for any lump of metal of the same material: it is now called the specific gravity of the material, and depends only on the intrinsic nature of the substance and not on its shape or quantity.", "markdown": "This ratio is the same for any lump of metal of the same material: it is now called the specific gravity of the material, and depends only on the intrinsic nature of the substance and not on its shape or quantity.", "why": "It defines specific gravity and explains why it identifies a material independently of the size of the sample.", "use": [ "lesson" ], "concepts": [ "concept/specific-gravity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-a9e02478e6", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "34", "location": "Methods of Application", "latex": "Faraday was asked: ``What is the use of this discovery?'' He answered: ``What is the use of a child---it grows to be a man.''", "markdown": "Faraday was asked: “What is the use of this discovery?” He answered: “What is the use of a child---it grows to be a man.”", "why": "It is a charming historical exchange that shows how a basic discovery can later become the foundation of practical applications.", "use": [ "history", "website" ], "concepts": [ "concept/electromagnetism" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-f937a75ca7", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "44", "location": "Dynamics", "latex": "The state of a body unacted on by force is that of uniform motion in a straight line, and no external force or influence is to be looked for as the cause, or, if you like to put it so, as the invariable accompaniment of this uniform rectilinear motion. Rest is merely a particular case of such motion, merely when the velocity is and remains zero.", "markdown": "The state of a body unacted on by force is that of uniform motion in a straight line, and no external force or influence is to be looked for as the cause, or, if you like to put it so, as the invariable accompaniment of this uniform rectilinear motion. Rest is merely a particular case of such motion, merely when the velocity is and remains zero.", "why": "It states the Newtonian picture of motion, in which rest is only the special case of uniform straight-line motion with zero velocity, so no force is needed to keep a body moving.", "use": [ "lesson", "website" ], "concepts": [ "law/first-law-of-motion", "quantity/velocity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-4adae97035", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "52", "location": "Dynamics", "latex": "For example, a velocity requires for its definition the assignment of a magnitude and of a direction. It must be of so many miles per hour in such and such a direction.", "markdown": "For example, a velocity requires for its definition the assignment of a magnitude and of a direction. It must be of so many miles per hour in such and such a direction.", "why": "It gives learners a concrete reason why velocity needs both a magnitude and a direction, which is the core idea of a vector.", "use": [ "lesson" ], "concepts": [ "concept/vector", "quantity/velocity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-f538b06879", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "55", "location": "Dynamics", "latex": "If the steamer were still, in one minute he would arrive at~$B$; but during that minute his starting point~$A$ on the deck has moved to~$D$, and his path on the deck has moved from $AB$ to~$DC$.", "markdown": "If the steamer were still, in one minute he would arrive at $B$; but during that minute his starting point $A$ on the deck has moved to $D$, and his path on the deck has moved from $AB$ to $DC$.", "why": "The steamer and the walking man show, in everyday terms, why a displacement can be split into two added displacements.", "use": [ "lesson", "website" ], "concepts": [ "concept/vector-of-transportation", "law/parallelogram-law-of-vector-addition" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-da44a725ea", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "46", "location": "Dynamics", "latex": "But according to the Newtonian law, apart from some force the planet would move for ever with its existing velocity in a straight line, and thus depart entirely from the sun.", "markdown": "But according to the Newtonian law, apart from some force the planet would move for ever with its existing velocity in a straight line, and thus depart entirely from the sun.", "why": "It explains why a planet needs a force toward the sun to stay in its orbit, which makes the first law of motion concrete.", "use": [ "lesson", "history" ], "concepts": [ "law/first-law-of-motion", "person/isaac-newton", "person/johannes-kepler" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-31c243ad28", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "43", "location": "Dynamics", "latex": "Galileo affirmed that they would fall in the same time, and proved his point by dropping weights from the top of the leaning tower.", "markdown": "Galileo affirmed that they would fall in the same time, and proved his point by dropping weights from the top of the leaning tower.", "why": "It is the historical example of an experiment overturning an authority, showing how careful observation tests a belief.", "use": [ "history" ], "concepts": [ "concept/dynamics", "person/aristotle", "person/galileo-galilei" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-258d7f085e", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "59", "location": "The Symbolism of Mathematics", "latex": "By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race.", "markdown": "By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race.", "why": "It gives learners the reason notation matters: good symbols free the mind for harder problems.", "use": [ "lesson", "history" ], "concepts": [ "concept/arabic-numerals", "concept/mathematical-notation" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-60126989c9", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "61", "location": "The Symbolism of Mathematics", "latex": "Operations of thought are like cavalry charges in a battle---they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.", "markdown": "Operations of thought are like cavalry charges in a battle---they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.", "why": "A memorable image that explains why routine operations should become automatic so that effort goes to decisive steps.", "use": [ "website" ], "concepts": [ "concept/mathematical-notation" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-ddcfc37f89", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "64", "location": "The Symbolism of Mathematics", "latex": "This service is performed by~$0$, the symbol for zero.", "markdown": "This service is performed by $0$, the symbol for zero.", "why": "It shows the place-holder role of zero: it fills an empty position so that the other digits keep their value.", "use": [ "lesson" ], "concepts": [ "concept/digit", "concept/place-value", "concept/zero" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-4fd6c8e340", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "65", "location": "The Symbolism of Mathematics", "latex": "Similarly the important way of writing the equation $x = 1$ is $x - 1 = 0$, and of representing the equation $3x - 2 = 2x^{2}$ is $2x^{2} - 3x + 2 = 0$.", "markdown": "Similarly the important way of writing the equation $x = 1$ is $x - 1 = 0$, and of representing the equation $3x - 2 = 2x^{2}$ is $2x^{2} - 3x + 2 = 0$.", "why": "It shows learners the standard habit of moving every term to the left so that the right side is zero.", "use": [ "lesson" ], "concepts": [ "concept/algebraic-form", "concept/equation", "concept/zero" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-30c30062b1", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "62", "location": "The Symbolism of Mathematics", "latex": "Mathematicians have chosen to make their symbolism more concise by defining $xy$ to stand for $x × y$.", "markdown": "Mathematicians have chosen to make their symbolism more concise by defining $xy$ to stand for $x × y$.", "why": "It explains the convention behind writing xy or 3x, which learners often find puzzling.", "use": [ "lesson" ], "concepts": [ "concept/implied-multiplication", "method/multiplication" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-2f7b6ac605", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "62", "location": "The Symbolism of Mathematics", "latex": "Thus when we substitute $2$~for~$x$ and $3$~for~$y$ in~$xy$, we must write $2 × 3$ for~$xy$, and not~$23$ which means $20 + 3$.", "markdown": "Thus when we substitute $2$ for $x$ and $3$ for $y$ in $xy$, we must write $2 × 3$ for $xy$, and not $23$ which means $20 + 3$.", "why": "It warns that dropping the multiplication sign after substitution changes the number, so the sign must be restored.", "use": [ "lesson" ], "concepts": [ "concept/arabic-numerals", "concept/implied-multiplication" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-0b50831bd9", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "69", "location": "The Symbolism of Mathematics", "latex": "Thus, by now varying $a$,~$b$, and~$c$, we arrive at the idea that $ax + by - c = 0$ represents a variable linear correlation between $x$ and~$y$.", "markdown": "Thus, by now varying $a$, $b$, and $c$, we arrive at the idea that $ax + by - c = 0$ represents a variable linear correlation between $x$ and $y$.", "why": "It shows how fixing letters as constants during study, then varying them, yields a result for any linear relation.", "use": [ "lesson" ], "concepts": [ "concept/constant", "concept/linear-equation", "concept/parameter" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-189259e71a", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "72", "location": "Generalizations of Number", "latex": "The Greeks thought of this subject rather in the form of ratio, so that a Greek would naturally say that a line of two feet in length bears to a line of three feet in length the ratio of $2$~to~$3$.", "markdown": "The Greeks thought of this subject rather in the form of ratio, so that a Greek would naturally say that a line of two feet in length bears to a line of three feet in length the ratio of $2$ to $3$.", "why": "It shows a learner that a fraction and a ratio are the same idea seen from two historical viewpoints.", "use": [ "lesson", "history" ], "concepts": [ "concept/ratio", "concept/rational-number" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-101f0f2b10", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "72", "location": "Generalizations of Number", "latex": "For example, the diagonal of a square cannot be expressed as any fraction of the side of the same square; in our modern notation the length of the diagonal is $\\sqrt{2}$~times the length of the side. But there is no fraction which exactly represents~$\\sqrt{2}$.", "markdown": "For example, the diagonal of a square cannot be expressed as any fraction of the side of the same square; in our modern notation the length of the diagonal is $\\sqrt{2}$ times the length of the side. But there is no fraction which exactly represents $\\sqrt{2}$.", "why": "It gives a concrete case where a length cannot be written as an exact fraction, which motivates the incommensurable numbers.", "use": [ "lesson", "website" ], "concepts": [ "concept/incommensurable-magnitudes", "concept/rational-number" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-8274a844dd", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "75", "location": "Generalizations of Number", "latex": "One very simple way of doing this is to add the fractions together and to halve the result.", "markdown": "One very simple way of doing this is to add the fractions together and to halve the result.", "why": "It gives a learner a simple method for finding a fraction between any two fractions, which shows why the fractions have no immediate neighbours.", "use": [ "lesson" ], "concepts": [ "concept/compact-series", "concept/rational-number" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-28e1548dc2", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "83", "location": "Generalizations of Number", "latex": "But if we now interpret our symbols as ``operations,'' all limitation vanishes like magic.", "markdown": "But if we now interpret our symbols as “operations,” all limitation vanishes like magic.", "why": "It explains in one line why treating symbols as operations removes the need for limits on the constants in an equation.", "use": [ "lesson", "website" ], "concepts": [ "concept/algebraic-form", "concept/equation", "concept/operation" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-7771622743", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "86", "location": "Generalizations of Number", "latex": "If a balance at the bank is positive, an overdraft is negative.", "markdown": "If a balance at the bank is positive, an overdraft is negative.", "why": "It gives a familiar everyday picture of a negative number as the opposite of a positive one.", "use": [ "lesson", "website" ], "concepts": [ "concept/negative-number", "concept/positive-number" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-1cc2c688fc", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "82", "location": "Generalizations of Number", "latex": "Any limitation whatsoever upon the generality of theorems, or of proofs, or of interpretation is abhorrent to the mathematical instinct.", "markdown": "Any limitation whatsoever upon the generality of theorems, or of proofs, or of interpretation is abhorrent to the mathematical instinct.", "why": "It states the aim of generality that drives the extension of number, giving learners a reason for the generalizations.", "use": [ "website" ], "concepts": [ "concept/generality" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-5f4ac51206", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "89", "location": "Imaginary Numbers", "latex": "Hence, if our symbols are to mean the ordinary positive or negative numbers, there is no solution to $x^{2} = -2$, and the equation is in fact nonsense.", "markdown": "Hence, if our symbols are to mean the ordinary positive or negative numbers, there is no solution to $x^{2} = -2$, and the equation is in fact nonsense.", "why": "It shows the learner exactly why a new kind of number is needed: the old symbols give no answer to x squared equals negative 2.", "use": [ "lesson" ], "concepts": [ "concept/complex-number", "concept/negative-number" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-7e26e20772", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "88", "location": "Imaginary Numbers", "latex": "The equation $x^{2} + 1 = 3$ becomes $x^{2} = 2$, and this has two solutions, either $x = +\\sqrt{2}$, or $x = -\\sqrt{2}$.", "markdown": "The equation $x^{2} + 1 = 3$ becomes $x^{2} = 2$, and this has two solutions, either $x = +\\sqrt{2}$, or $x = -\\sqrt{2}$.", "why": "It walks through a familiar quadratic with two real roots, the baseline the new numbers must extend.", "use": [ "lesson" ], "concepts": [ "concept/root" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-347838c23c", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "91", "location": "Imaginary Numbers", "latex": "Nothing can be proved by a succession of blots, except the existence of a bad pen or a careless writer.", "markdown": "Nothing can be proved by a succession of blots, except the existence of a bad pen or a careless writer.", "why": "It warns learners that symbols must be properly defined before any manipulation with them can prove anything.", "use": [ "lesson", "website" ], "concepts": [ "concept/complex-number" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-c05ff2faef", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "97", "location": "Imaginary Numbers", "latex": "All these requisites are satisfied by taking $(x, y) + (x', y')$ to mean the ordered couple $(x + x', y + y')$.", "markdown": "All these requisites are satisfied by taking $(x, y) + (x', y')$ to mean the ordered couple $(x + x', y + y')$.", "why": "It states the rule for adding ordered pairs and shows that the rule was chosen to keep ordinary addition laws working.", "use": [ "lesson" ], "concepts": [ "concept/ordered-pair", "method/adding-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-dc8584c3d9", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "100", "location": "Imaginary Numbers", "latex": "It is no paradox to say that in our most theoretical moods we may be nearest to our most practical applications.", "markdown": "It is no paradox to say that in our most theoretical moods we may be nearest to our most practical applications.", "why": "It gives learners a reason to value abstract reasoning, since the parallelogram law reappears in mechanics.", "use": [ "website", "history" ], "concepts": [ "concept/ordered-pair", "law/parallelogram-law-of-vector-addition" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-6963d7998d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "102", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "We came across equations of the form $x^{2} = -3$, to which no solutions could be assigned in terms of positive and negative real numbers.", "markdown": "We came across equations of the form $x^{2} = -3$, to which no solutions could be assigned in terms of positive and negative real numbers.", "why": "It shows the learner the difficulty that motivates the whole chapter: an ordinary equation with no real solution.", "use": [ "lesson" ], "concepts": [ "concept/imaginary-root" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-fc1c3134d2", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "102", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "This is the definition of the meaning of the symbol~$×$ when it is written between two ordered couples.", "markdown": "This is the definition of the meaning of the symbol $×$ when it is written between two ordered couples.", "why": "It states plainly that the rule for multiplying couples is a definition chosen to meet stated conditions, not something to be recalled.", "use": [ "lesson" ], "concepts": [ "concept/ordered-pair", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-085a6f17a4", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "103", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "Hence both for addition and for multiplication the couple $(0, 0)$ plays the part of zero in elementary arithmetic and algebra; compare the above equations with $x + 0 = x$, and $x × 0 = 0$.", "markdown": "Hence both for addition and for multiplication the couple $(0, 0)$ plays the part of zero in elementary arithmetic and algebra; compare the above equations with $x + 0 = x$, and $x × 0 = 0$.", "why": "It links the new zero couple to the zero the learner already knows from arithmetic.", "use": [ "lesson" ], "concepts": [ "concept/ordered-pair", "concept/zero" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-0285cd9d47", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "108", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "The product of the two vectors $OP$ and~$OQ$ is a vector~$OR$, whose length is the product of the lengths of $OP$ and~$OQ$ and whose direction~$OR$ is such that the angle~$XOR$ is equal to the sum of the angles $XOP$ and~$XOQ$.", "markdown": "The product of the two vectors $OP$ and $OQ$ is a vector $OR$, whose length is the product of the lengths of $OP$ and $OQ$ and whose direction $OR$ is such that the angle $XOR$ is equal to the sum of the angles $XOP$ and $XOQ$.", "why": "It gives the geometric rule for multiplying vectors in a single sentence: multiply lengths, add angles.", "use": [ "lesson", "website" ], "concepts": [ "concept/vector", "method/multiplying-vectors-in-a-plane" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-37178c8146", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "104", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "The answer is that it is perfectly indifferent which symbolism we adopt.", "markdown": "The answer is that it is perfectly indifferent which symbolism we adopt.", "why": "It tells the learner that the choice of which couple stands for plus or minus root is a convention, not a fact to be discovered.", "use": [ "lesson", "website" ], "concepts": [ "concept/imaginary-root" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-9458afe428", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "111", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "It was receiving its final form about the same time as when the steam engine was being perfected, and will remain a great and powerful weapon for the achievement of the victory of thought over things when curious specimens of that machine repose in museums in company with the helmets and breastplates of a slightly earlier epoch.", "markdown": "It was receiving its final form about the same time as when the steam engine was being perfected, and will remain a great and powerful weapon for the achievement of the victory of thought over things when curious specimens of that machine repose in museums in company with the helmets and breastplates of a slightly earlier epoch.", "why": "It places the development of algebra in its historical period in a memorable way for a general reader.", "use": [ "history", "website" ], "concepts": [ "concept/complex-number" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-826fef33c1", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "112", "location": "Coordinate Geometry", "latex": "This conception, simple as it looks, is the main idea of the great subject of coordinate geometry.", "markdown": "This conception, simple as it looks, is the main idea of the great subject of coordinate geometry.", "why": "It states in one sentence what coordinate geometry rests on: a point is a pair of numbers.", "use": [ "lesson", "website" ], "concepts": [ "concept/cartesian-coordinates", "concept/coordinate-geometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-9f106f14e0", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "121", "location": "Coordinate Geometry", "latex": "A locus is the curve (or surface, if we do not confine ourselves to a plane) formed by points, all of which possess some given property.", "markdown": "A locus is the curve (or surface, if we do not confine ourselves to a plane) formed by points, all of which possess some given property.", "why": "It gives the learner a clear definition of a locus as the set of points sharing a property.", "use": [ "lesson" ], "concepts": [ "concept/locus" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-363fba015b", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "124", "location": "Coordinate Geometry", "latex": "Consider $y - x = 1$: the corresponding locus does not pass through the origin. We therefore seek where it cuts the axes.", "markdown": "Consider $y - x = 1$: the corresponding locus does not pass through the origin. We therefore seek where it cuts the axes.", "why": "It shows the method of finding where a line meets the axes by setting one coordinate to zero.", "use": [ "lesson" ], "concepts": [ "concept/equation", "concept/line", "concept/origin" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-6ec6d53d26", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "125", "location": "Coordinate Geometry", "latex": "We each of us refer our sensible perceptions of things to an origin which we call ``here'': our location in a particular part of space round which we group the whole Universe is the essential fact of our bodily existence.", "markdown": "We each of us refer our sensible perceptions of things to an origin which we call “here”: our location in a particular part of space round which we group the whole Universe is the essential fact of our bodily existence.", "why": "It gives a memorable analogy for why an origin is chosen arbitrarily and then fixed.", "use": [ "website" ], "concepts": [ "concept/origin" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-ff303a7502", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "112", "location": "Coordinate Geometry", "latex": "Its discovery marks a momentous epoch in the history of mathematical thought.", "markdown": "Its discovery marks a momentous epoch in the history of mathematical thought.", "why": "It places the discovery of coordinate geometry as a turning point in the history of ideas.", "use": [ "history" ], "concepts": [ "concept/coordinate-geometry", "person/ren-descartes" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-3d917500dd", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "119", "location": "Coordinate Geometry", "latex": "Euclid always contemplates a straight line as drawn between two definite points, and is very careful to mention when it is to be produced beyond this segment.", "markdown": "Euclid always contemplates a straight line as drawn between two definite points, and is very careful to mention when it is to be produced beyond this segment.", "why": "It contrasts Euclid's bounded line with the unbounded line of modern geometry, which helps learners see why the idea changed.", "use": [ "history", "lesson" ], "concepts": [ "concept/geometry", "concept/line" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-cbf48a7731", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "117", "location": "Coordinate Geometry", "latex": "Variables, like $a$,~$b$, and~$c$ above, which are used to determine the correlation are called ``constants,'' or parameters.", "markdown": "Variables, like $a$, $b$, and $c$ above, which are used to determine the correlation are called “constants,” or parameters.", "why": "It warns learners that a constant here is really a variable fixed only relative to x and y.", "use": [ "lesson" ], "concepts": [ "concept/constant", "concept/parameter" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-a85f7f4f18", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "142", "location": "Conic Sections", "latex": "If $ab - h^{2}$ is a positive number, the curve is an ellipse; if $ab - h^{2} = 0$, the curve is a parabola: and if $ab - h^{2}$ is a negative number, the curve is a hyperbola.", "markdown": "If $ab - h^{2}$ is a positive number, the curve is an ellipse; if $ab - h^{2} = 0$, the curve is a parabola: and if $ab - h^{2}$ is a negative number, the curve is a hyperbola.", "why": "Gives a learner a mechanical test that sorts any second-degree equation into ellipse, parabola or hyperbola.", "use": [ "lesson" ], "concepts": [ "concept/ellipse", "concept/hyperbola", "concept/parabola", "theorem/discriminant-of-a-conic" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-9b7da76a5d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "135", "location": "Conic Sections", "latex": "The characteristic property of a focus,~$S$, and its corresponding directrix,~$XN$, for any one of the three types of curve, is that the ratio $SP$ to~$PN$ $\\left(\\ie\\ \\dfrac{SP}{PN}\\right)$ is constant, where $PN$~is the perpendicular on the directrix from~$P$, and $P$~is any point on the curve.", "markdown": "The characteristic property of a focus, $S$, and its corresponding directrix, $XN$, for any one of the three types of curve, is that the ratio $SP$ to $PN$ $\\left(\\ie\\ \\dfrac{SP}{PN}\\right)$ is constant, where $PN$ is the perpendicular on the directrix from $P$, and $P$ is any point on the curve.", "why": "States the focus-directrix property, which defines all three conics in one plane-based way.", "use": [ "lesson", "history" ], "concepts": [ "concept/directrix", "concept/ellipse", "concept/focus", "concept/hyperbola", "concept/parabola", "theorem/focus-directrix-property-of-conic-sections" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-71345d269b", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "138", "location": "Conic Sections", "latex": "The orbits of the planets are ellipses, the sun being in the focus.", "markdown": "The orbits of the planets are ellipses, the sun being in the focus.", "why": "Shows a concrete physical use of the ellipse: Kepler's first law of planetary motion.", "use": [ "lesson", "website" ], "concepts": [ "concept/ellipse", "concept/focus", "law/kepler-s-laws-of-planetary-motion" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-7016b1d6ae", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "129", "location": "Conic Sections", "latex": "There is a certain type of mathematician who is always rather impatient at delaying over the ideas of a subject: he is anxious at once to get on to the proofs of ``important'' problems. The history of the science is entirely against him.", "markdown": "There is a certain type of mathematician who is always rather impatient at delaying over the ideas of a subject: he is anxious at once to get on to the proofs of “important” problems. The history of the science is entirely against him.", "why": "A warning to learners that understanding the ideas of a subject saves time later.", "use": [ "website", "history" ], "concepts": [] }, { "id": "whitehead-introduction-to-mathematics-1911/x-8c2e41f55e", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "138", "location": "Conic Sections", "latex": "Novel ideas are more apt to spring from an unusual assortment of knowledge---not necessarily from vast knowledge, but from a thorough conception of the methods and ideas of distinct lines of thought.", "markdown": "Novel ideas are more apt to spring from an unusual assortment of knowledge---not necessarily from vast knowledge, but from a thorough conception of the methods and ideas of distinct lines of thought.", "why": "Shows Kepler, an astronomer who was also a geometer, as an example of cross-disciplinary insight.", "use": [ "website", "history" ], "concepts": [] }, { "id": "whitehead-introduction-to-mathematics-1911/x-93dacdce5f", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "129", "location": "Conic Sections", "latex": "Nothing illustrates better the gain in power which is obtained by the introduction of relevant ideas into a science than to observe the progressive shortening of proofs which accompanies the growth of richness in idea.", "markdown": "Nothing illustrates better the gain in power which is obtained by the introduction of relevant ideas into a science than to observe the progressive shortening of proofs which accompanies the growth of richness in idea.", "why": "Shows learners that a subject gets easier as its ideas grow richer, which motivates studying the ideas before the proofs.", "use": [ "lesson" ], "concepts": [ "concept/generality" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-c2b5f59c6c", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "136", "location": "Conic Sections", "latex": "No more impressive warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic sections were studied for eighteen hundred years merely as an abstract science, without a thought of any utility other than to satisfy the craving for knowledge on the part of mathematicians, and that then at the end of this long period of abstract study, they were found to be the necessary key with which to attain the knowledge of one of the most important laws of nature.", "markdown": "No more impressive warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic sections were studied for eighteen hundred years merely as an abstract science, without a thought of any utility other than to satisfy the craving for knowledge on the part of mathematicians, and that then at the end of this long period of abstract study, they were found to be the necessary key with which to attain the knowledge of one of the most important laws of nature.", "why": "Gives a historical example of a pure mathematical idea later proving essential to physics.", "use": [ "history", "website" ], "concepts": [ "concept/abstractness", "concept/conic-section", "law/law-of-gravity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-8f49b29eb2", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "131", "location": "Conic Sections", "latex": "There are accordingly three types of conic sections, namely, ellipses, parabolas, and hyperbolas.", "markdown": "There are accordingly three types of conic sections, namely, ellipses, parabolas, and hyperbolas.", "why": "States the three-way classification that the rest of the chapter builds on.", "use": [ "lesson" ], "concepts": [ "concept/conic-section", "concept/ellipse", "concept/hyperbola", "concept/parabola" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-cbf28ac952", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "136", "location": "Conic Sections", "latex": "Here we have finally found the desired property of the curves which does not require us to leave the plane, and is stated uniformly for all three curves.", "markdown": "Here we have finally found the desired property of the curves which does not require us to leave the plane, and is stated uniformly for all three curves.", "why": "Explains why the focus-directrix definition is the unifying idea that replaces the solid-cone definition.", "use": [ "lesson" ], "concepts": [ "concept/directrix", "concept/focus", "theorem/focus-directrix-property-of-conic-sections" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-d469d2784b", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "138", "location": "Conic Sections", "latex": "(1) The orbits of the planets are ellipses, the sun being in the focus.", "markdown": "(1) The orbits of the planets are ellipses, the sun being in the focus.", "why": "Links the ellipse directly to planetary motion, showing learners why conics matter in astronomy.", "use": [ "lesson", "website" ], "concepts": [ "concept/ellipse", "concept/focus", "law/kepler-s-laws" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-d357bb1c05", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "139", "location": "Conic Sections", "latex": "This sweeping general law, coupled with the three laws of motion which he put into their final general shape, proved adequate to explain all astronomical phenomena, including Kepler's laws, and has formed the basis of modern physics.", "markdown": "This sweeping general law, coupled with the three laws of motion which he put into their final general shape, proved adequate to explain all astronomical phenomena, including Kepler’s laws, and has formed the basis of modern physics.", "why": "Shows how Newton's general law explains Kepler's empirical laws, a clear example of generalisation in science.", "use": [ "history", "lesson" ], "concepts": [ "law/kepler-s-laws", "law/law-of-gravity", "law/laws-of-motion", "person/isaac-newton" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-a24d43a195", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "144", "location": "Conic Sections", "latex": "This fact is worth noting; for it is characteristic of modern mathematics to include among general forms all sorts of particular cases which would formerly have received special treatment.", "markdown": "This fact is worth noting; for it is characteristic of modern mathematics to include among general forms all sorts of particular cases which would formerly have received special treatment.", "why": "Explains why a circle or a pair of lines appears inside the general conic equation.", "use": [ "lesson", "website" ], "concepts": [ "concept/conic-section", "concept/generality" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-e28df660a5", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "147", "location": "Functions", "latex": "The essential point is that when $x$~is given, then $y$~is thereby definitely determined.", "markdown": "The essential point is that when $x$ is given, then $y$ is thereby definitely determined.", "why": "It states the one property a function must have, which is determinacy of the value, and so tells a learner what to check for.", "use": [ "lesson" ], "concepts": [ "concept/function", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-b37fafbee5", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "154", "location": "Functions", "latex": "For the value of the function on the negative (left) side of the origin becomes endlessly great, but negative, and the function reappears on the positive (right) side as endlessly great but positive.", "markdown": "For the value of the function on the negative (left) side of the origin becomes endlessly great, but negative, and the function reappears on the positive (right) side as endlessly great but positive.", "why": "It describes a concrete discontinuity at the origin for y = 1/x, showing what a jump in value looks like.", "use": [ "lesson" ], "concepts": [ "concept/discontinuous-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-c7272cdcb3", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "155", "location": "Functions", "latex": "The whole difference between the older and the newer mathematics lies in the fact that vague half-metaphorical terms like ``gradually'' are no longer tolerated in its exact statements.", "markdown": "The whole difference between the older and the newer mathematics lies in the fact that vague half-metaphorical terms like “gradually” are no longer tolerated in its exact statements.", "why": "It shows why the older word 'gradually' in the definition of continuity is being questioned, which prepares the learner for the rigorous treatment.", "use": [ "history", "website" ], "concepts": [ "concept/continuous-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-d4362625e3", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "155", "location": "Functions", "latex": "This is exactly the sort of definition which satisfied our mathematical forefathers and no longer satisfies modern mathematicians.", "markdown": "This is exactly the sort of definition which satisfied our mathematical forefathers and no longer satisfies modern mathematicians.", "why": "It is a dated historical remark that marks the older definition of continuity as a stage in the history of the subject.", "use": [ "history" ], "concepts": [ "concept/continuous-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-b304c4aa3a", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "152", "location": "Functions", "latex": "A man who, trusting that the mean height of the land above sea-level between London and Paris was a continuous function of the distance from London, walked at night on Shakespeare's Cliff by Dover in contemplation of the Milky Way, would be dead before he had had time to rearrange his ideas as to the necessity of caution in scientific conclusions.", "markdown": "A man who, trusting that the mean height of the land above sea-level between London and Paris was a continuous function of the distance from London, walked at night on Shakespeare’s Cliff by Dover in contemplation of the Milky Way, would be dead before he had had time to rearrange his ideas as to the necessity of caution in scientific conclusions.", "why": "A vivid and witty warning that assuming continuity without evidence is a dangerous mistake.", "use": [ "website", "lesson" ], "concepts": [ "concept/continuous-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-652d54770d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "145", "location": "Functions", "latex": "If a train has been travelling at the rate of twenty miles per hour, the distance ($s$~miles) gone after any number of hours, say~$t$, is given by $s = 20 × t$; and $s$~is called a function of~$t$.", "markdown": "If a train has been travelling at the rate of twenty miles per hour, the distance ($s$ miles) gone after any number of hours, say $t$, is given by $s = 20 × t$; and $s$ is called a function of $t$.", "why": "A concrete train journey shows a learner that a function is just one quantity determined by another, before any symbols are abstracted.", "use": [ "lesson" ], "concepts": [ "concept/argument-of-a-function", "concept/function", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-b1bd03f61f", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "147", "location": "Functions", "latex": "With these explanations and cautions, we write $y = f(x)$, to denote that $y$~is the value of some undetermined function of the argument~$x$; where $f(x)$ may stand for anything such as $x + 1$, $x^{2} - 2x + 1$, $\\sin x$, $\\log x$, or merely for $x$~itself.", "markdown": "With these explanations and cautions, we write $y = f(x)$, to denote that $y$ is the value of some undetermined function of the argument $x$; where $f(x)$ may stand for anything such as $x + 1$, $x^{2} - 2x + 1$, $\\sin x$, $\\log x$, or merely for $x$ itself.", "why": "It explains the general symbol f(x) and makes clear that f stands for any rule at all, including the trivial one.", "use": [ "lesson" ], "concepts": [ "concept/function", "concept/function-notation" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-b3b31327d4", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "147", "location": "Functions", "latex": "Thus in $y = f(x)$, we may determine, if we choose, $f(x)$~to mean that when $x$~is an integer, $f(x)$~is zero, and when $x$~has any other value, $f(x)$~is~$1$.", "markdown": "Thus in $y = f(x)$, we may determine, if we choose, $f(x)$ to mean that when $x$ is an integer, $f(x)$ is zero, and when $x$ has any other value, $f(x)$ is $1$.", "why": "This deliberately odd rule shows learners that a function may be defined by any correlation, not only by a formula.", "use": [ "lesson", "website" ], "concepts": [ "concept/function", "concept/integer" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-948d87850b", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "152", "location": "Functions", "latex": "The train certainly cannot be running at forty miles per hour from 11.45~a.m.\\ up to noon, and then suddenly, without any lapse of time, commence running at $50$~miles per~hour.", "markdown": "The train certainly cannot be running at forty miles per hour from 11.45 a.m. up to noon, and then suddenly, without any lapse of time, commence running at $50$ miles per hour.", "why": "A vivid everyday case shows why velocity is expected to change gradually, which motivates the idea of continuity.", "use": [ "lesson", "website" ], "concepts": [ "concept/continuous-function", "law/laws-of-motion", "quantity/velocity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-0b04c53d06", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "161", "location": "Functions", "latex": "This example brings out the fact that statements about a function~$f(x)$ in the neighbourhood of a number~$a$ are distinct from statements about the value of~$f(x)$ when $x = a$.", "markdown": "This example brings out the fact that statements about a function $f(x)$ in the neighbourhood of a number $a$ are distinct from statements about the value of $f(x)$ when $x = a$.", "why": "It warns learners that a statement about a neighbourhood is not the same as a statement about a single value, a common mistake.", "use": [ "lesson" ], "concepts": [ "concept/neighbourhood", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-f74b8f7495", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "164", "location": "Periodicity in Nature", "latex": "the existence of successive events so analogous to each other that, without any straining of language, they may be termed recurrences of the same event.", "markdown": "the existence of successive events so analogous to each other that, without any straining of language, they may be termed recurrences of the same event.", "why": "It gives the learner a plain definition of periodicity as recurrence of the same event, the idea the rest of the chapter builds on.", "use": [ "lesson", "website" ], "concepts": [ "concept/periodicity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-522c3dfb50", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "168", "location": "Periodicity in Nature", "latex": "Hence the velocity of the train has not been uniform, and on the average the velocity during the second period is twice that during the first period.", "markdown": "Hence the velocity of the train has not been uniform, and on the average the velocity during the second period is twice that during the first period.", "why": "It shows a learner that a claim about uniform velocity depends on which clock is used, since the same motion can look uniform or not.", "use": [ "lesson", "website" ], "concepts": [ "concept/measure-of-time", "quantity/velocity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-169e98a85b", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "168", "location": "Periodicity in Nature", "latex": "Thus the question as to whether the train has been running uniformly or not entirely depends on the standard of time which we adopt.", "markdown": "Thus the question as to whether the train has been running uniformly or not entirely depends on the standard of time which we adopt.", "why": "It states the chapter's point in one sentence: whether motion is uniform depends on the chosen measure of time.", "use": [ "lesson" ], "concepts": [ "concept/measure-of-time", "quantity/velocity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-dcd7f88142", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "166", "location": "Periodicity in Nature", "latex": "It has been one of the first tasks of science among civilized or semi-civilized nations, to fuse them into one coherent measure.", "markdown": "It has been one of the first tasks of science among civilized or semi-civilized nations, to fuse them into one coherent measure.", "why": "It gives the historical background to the problem of combining days, months and years into a single measure of time.", "use": [ "history" ], "concepts": [ "concept/measure-of-time" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-31fd18280c", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "170", "location": "Periodicity in Nature", "latex": "Any one wanting to upset a rocking stone will push ``in tune'' with the oscillations of the stone, so as always to secure a favourable moment for a push.", "markdown": "Any one wanting to upset a rocking stone will push “in tune” with the oscillations of the stone, so as always to secure a favourable moment for a push.", "why": "It gives a concrete picture of resonance, where repeated pushes in step with a body's own rhythm build up its motion.", "use": [ "lesson", "website" ], "concepts": [ "concept/resonance" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-97e361698d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "165", "location": "Periodicity in Nature", "latex": "The presupposition of periodicity is indeed fundamental to our very conception of life.", "markdown": "The presupposition of periodicity is indeed fundamental to our very conception of life.", "why": "It links the abstract idea of periodicity to everyday experience, such as the heartbeat and breathing.", "use": [ "website", "lesson" ], "concepts": [ "concept/periodicity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-ea0fb9a934", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "170", "location": "Periodicity in Nature", "latex": "Thus a pendulum has but one period of vibration, while a suspension bridge will have many.", "markdown": "Thus a pendulum has but one period of vibration, while a suspension bridge will have many.", "why": "It contrasts a body with one free period against a body with many, which prepares the learner for resonance.", "use": [ "lesson" ], "concepts": [ "quantity/free-period-of-vibration" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-9b1c34f8d4", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "174", "location": "Trigonometry", "latex": "The origin of trigonometry was practical; it was invented because it was necessary for astronomical research.", "markdown": "The origin of trigonometry was practical; it was invented because it was necessary for astronomical research.", "why": "It gives the learner the reason trigonometry exists: it was built to solve practical astronomical measurement.", "use": [ "lesson", "history" ], "concepts": [ "concept/trigonometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-dfa1ceb749", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "177", "location": "Trigonometry", "latex": "Thus if the scale of a plan be an inch to a yard, a length of three inches in the plan means a length of three yards in the original.", "markdown": "Thus if the scale of a plan be an inch to a yard, a length of three inches in the plan means a length of three yards in the original.", "why": "A concrete plan-and-scale example makes the idea of similarity and scale easy to picture before any formula appears.", "use": [ "lesson" ], "concepts": [ "concept/scale-of-a-map", "concept/similarity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-162220dcaa", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "179", "location": "Trigonometry", "latex": "This peculiar property of the triangle, which is not shared by other rectilinear figures, makes it the fundamental figure in the theory of similarity.", "markdown": "This peculiar property of the triangle, which is not shared by other rectilinear figures, makes it the fundamental figure in the theory of similarity.", "why": "It explains why triangles, rather than other polygons, are the natural starting point for trigonometry.", "use": [ "lesson" ], "concepts": [ "concept/rectilinear-figure", "concept/similarity", "concept/triangle" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-65c9c73fec", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "183", "location": "Trigonometry", "latex": "We have called $v$ the \\emph{sine} of~$u$, and $w$ the \\emph{cosine} of~$u$.", "markdown": "We have called $v$ the *sine* of $u$, and $w$ the *cosine* of $u$.", "why": "It states plainly what sine and cosine are, as functions correlating an angle with a pair of ratios.", "use": [ "lesson" ], "concepts": [ "concept/argument-of-a-function", "concept/cosine", "concept/sine" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-fc6f4eac7d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "186", "location": "Trigonometry", "latex": "It can be proved that $\\pi$~is an incommensurable number, and that therefore its value cannot be expressed by any fraction, or by any terminating or recurring decimal.", "markdown": "It can be proved that $\\pi$ is an incommensurable number, and that therefore its value cannot be expressed by any fraction, or by any terminating or recurring decimal.", "why": "It shows learners why pi cannot be written exactly, which motivates approximation and infinite series.", "use": [ "lesson" ], "concepts": [ "concept/approximation", "concept/incommensurable-magnitudes", "concept/pi" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-b511d1f679", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "193", "location": "Trigonometry", "latex": "We are here in the presence of one of the fundamental processes of mathematical physics---namely, nothing less than its general method of dealing with the great natural fact of Periodicity.", "markdown": "We are here in the presence of one of the fundamental processes of mathematical physics---namely, nothing less than its general method of dealing with the great natural fact of Periodicity.", "why": "It connects the trigonometric functions to the wider physical idea of periodicity in tides, strings and light.", "use": [ "lesson", "website" ], "concepts": [ "concept/mathematical-physics", "concept/periodicity", "method/harmonic-analysis" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-fb6279194e", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "174", "location": "Trigonometry", "latex": "Characteristically enough conic sections were invented about $150$~years earlier than trigonometry, during the very best period of Greek thought.", "markdown": "Characteristically enough conic sections were invented about $150$ years earlier than trigonometry, during the very best period of Greek thought.", "why": "A vivid historical remark that places trigonometry in the context of Greek mathematics.", "use": [ "history", "website" ], "concepts": [ "concept/trigonometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-8ae0b01c36", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "194", "location": "Series", "latex": "The general mathematical idea of a series is that of a set of things ranged in order, that is, in sequence; This meaning is accurately represented in the common use of the term.", "markdown": "The general mathematical idea of a series is that of a set of things ranged in order, that is, in sequence; This meaning is accurately represented in the common use of the term.", "why": "It gives the learner the plain meaning of a series before any formulas, so the later technical sense has something concrete to attach to.", "use": [ "lesson", "website" ], "concepts": [ "concept/infinite-sequence" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-288e0af93f", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "202", "location": "Series", "latex": "It is evident that nothing that has been said gives the slightest idea as to how the ``sum to infinity'' of a series is to be found.", "markdown": "It is evident that nothing that has been said gives the slightest idea as to how the “sum to infinity” of a series is to be found.", "why": "It warns that defining a sum to infinity does not give a method for computing it.", "use": [ "lesson" ], "concepts": [ "concept/infinite-sequence", "concept/sum-to-infinity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-faba375778", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "197", "location": "Series", "latex": "But, if the series has an infinite number of terms, this process of successively forming the sums of the terms never terminates; and in this sense there is no such thing as the sum of an infinite series.", "markdown": "But, if the series has an infinite number of terms, this process of successively forming the sums of the terms never terminates; and in this sense there is no such thing as the sum of an infinite series.", "why": "It explains why an infinite series has no sum in the everyday sense, which is the reason the limit idea is needed.", "use": [ "lesson" ], "concepts": [ "concept/infinite-sequence", "concept/sum-to-infinity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-d3a408d718", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "195", "location": "Series", "latex": "When the number of things considered is finite, the number of ways of arranging them in order is called the number of their permutations.", "markdown": "When the number of things considered is finite, the number of ways of arranging them in order is called the number of their permutations.", "why": "It defines a permutation in plain words, so the learner can count arrangements before meeting the formula n!.", "use": [ "lesson" ], "concepts": [ "concept/permutation" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-981a1254c4", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "202", "location": "Series", "latex": "This decimal is merely a way of symbolizing the ``sum to infinity'' of the series", "markdown": "This decimal is merely a way of symbolizing the “sum to infinity” of the series", "why": "It shows a familiar object, the recurring decimal, as the sum to infinity of a series, with the worked check that follows in the text.", "use": [ "lesson", "website" ], "concepts": [ "concept/recurring-decimal", "concept/sum-to-infinity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-68b09adc7a", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "200", "location": "Series", "latex": "The summation of a series approximates to a limit when the sum of any number of its terms, provided the number be large enough, is as nearly equal to the limit as you care to approach.", "markdown": "The summation of a series approximates to a limit when the sum of any number of its terms, provided the number be large enough, is as nearly equal to the limit as you care to approach.", "why": "It gives a learner the plain intuition of a limit of partial sums before the book makes it precise.", "use": [ "lesson" ], "concepts": [ "concept/approximation", "concept/infinite-sequence", "concept/limit" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-2177f55248", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "200", "location": "Series", "latex": "But this description of the meaning of approximating to a limit evidently will not stand the vigorous scrutiny of modern mathematics.", "markdown": "But this description of the meaning of approximating to a limit evidently will not stand the vigorous scrutiny of modern mathematics.", "why": "It shows why vague phrases such as nearly equal must be replaced by exact definitions, a useful point for historical context.", "use": [ "history" ], "concepts": [ "concept/approximation", "concept/limit" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-462fbb605a", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "203", "location": "Series", "latex": "Mathematics would be a much easier science than it is, if this were the case. Unfortunately the supposition is not true.", "markdown": "Mathematics would be a much easier science than it is, if this were the case. Unfortunately the supposition is not true.", "why": "It warns learners against the common belief that terms shrinking to zero guarantees a sum to infinity.", "use": [ "lesson" ], "concepts": [ "concept/convergent-series", "concept/divergent-series" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-d66abb8814", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "198", "location": "Series", "latex": "The statesman in framing his speech puts the dominating issues first and lets the details fall naturally into their subordinate places.", "markdown": "The statesman in framing his speech puts the dominating issues first and lets the details fall naturally into their subordinate places.", "why": "It gives a memorable everyday analogy for why adding the largest terms first gives successive approximations.", "use": [ "lesson", "website" ], "concepts": [ "concept/approximation" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-29affacd83", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "196", "location": "Series", "latex": "It is easy to verify in the case of small values of~$n$ that $n!$ is the number of ways of arranging $n$~things in order.", "markdown": "It is easy to verify in the case of small values of $n$ that $n!$ is the number of ways of arranging $n$ things in order.", "why": "It links the factorial to counting permutations, with small cases a learner can check by hand.", "use": [ "lesson" ], "concepts": [ "concept/factorial", "concept/permutation" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-cf9e730982", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "213", "location": "Series", "latex": "The importance of the exponential function is that it represents any changing physical quantity whose rate of increase at any instant is a uniform percentage of its value at that instant.", "markdown": "The importance of the exponential function is that it represents any changing physical quantity whose rate of increase at any instant is a uniform percentage of its value at that instant.", "why": "It explains in plain terms what kind of growth the exponential function models, such as populations.", "use": [ "lesson", "website" ], "concepts": [ "concept/exponential-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-10ea9056ea", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "214", "location": "Series", "latex": "The curve, which is something like a cocked hat, is called the curve of normal error.", "markdown": "The curve, which is something like a cocked hat, is called the curve of normal error.", "why": "It gives a vivid picture of the normal curve that learners can recognise from their own statistics work.", "use": [ "website" ], "concepts": [ "concept/curve-of-normal-error" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-9227aa7012", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "220", "location": "The Differential Calculus", "latex": "The really inspiring reflection suggested by the history of mathematics is the unity of thought and interest among men of so many epochs, so many nations, and so many races.", "markdown": "The really inspiring reflection suggested by the history of mathematics is the unity of thought and interest among men of so many epochs, so many nations, and so many races.", "why": "It gives learners a historical frame: the subject was built by many peoples, not one nation.", "use": [ "history", "website" ], "concepts": [ "concept/mathematics" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-29bd387c8d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "220", "location": "The Differential Calculus", "latex": "This idea is immediately presented to us by the study of nature; velocity is the rate of increase of the distance travelled, and acceleration is the rate of increase of velocity.", "markdown": "This idea is immediately presented to us by the study of nature; velocity is the rate of increase of the distance travelled, and acceleration is the rate of increase of velocity.", "why": "It ties the abstract idea of rate of change to velocity and acceleration, which learners already know from physics.", "use": [ "lesson" ], "concepts": [ "concept/rate-of-change", "quantity/acceleration", "quantity/distance", "quantity/velocity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-e8718671da", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "225", "location": "The Differential Calculus", "latex": "When $x$~increases to $x + h$, the function~$x^{2}$ increases to $(x + h)^{2}$; so that the total increase has been $(x + h)^{2} - x^{2}$, due to an increase~$h$ in the argument. Hence throughout the interval $x$~to $(x + h)$ the average increase of the function per unit increase of the argument is $\\dfrac{(x + h)^{2} - x^{2}}{h}$.", "markdown": "When $x$ increases to $x + h$, the function $x^{2}$ increases to $(x + h)^{2}$; so that the total increase has been $(x + h)^{2} - x^{2}$, due to an increase $h$ in the argument. Hence throughout the interval $x$ to $(x + h)$ the average increase of the function per unit increase of the argument is $\\dfrac{(x + h)^{2} - x^{2}}{h}$.", "why": "It shows the average increase over an interval as a worked calculation before any limit is taken.", "use": [ "lesson" ], "concepts": [ "concept/argument-of-a-function", "concept/function", "concept/interval", "concept/rate-of-change" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-f56181d402", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "227", "location": "The Differential Calculus", "latex": "In reading over the Newtonian method of statement, it is tempting to seek simplicity by saying that $2x + h$ is~$2x$, when $h$~is zero. But this will not do; for it thereby abolishes the interval from $x$ to~$x + h$, over which the average increase was calculated.", "markdown": "In reading over the Newtonian method of statement, it is tempting to seek simplicity by saying that $2x + h$ is $2x$, when $h$ is zero. But this will not do; for it thereby abolishes the interval from $x$ to $x + h$, over which the average increase was calculated.", "why": "It warns against the common mistake of setting h to zero too early, which removes the interval the average was taken over.", "use": [ "lesson" ], "concepts": [ "concept/interval", "concept/limit", "concept/rate-of-change" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-40ca4ae07b", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "229", "location": "The Differential Calculus", "latex": "A function~$f(x)$ has the limit~$l$ at a value~$a$ of its argument~$x$, when in the neighbourhood of~$a$ its values approximate to~$l$ within \\emph{every} standard of approximation.", "markdown": "A function $f(x)$ has the limit $l$ at a value $a$ of its argument $x$, when in the neighbourhood of $a$ its values approximate to $l$ within *every* standard of approximation.", "why": "It gives the precise definition of a limit that replaces the idea of infinitely small quantities.", "use": [ "lesson" ], "concepts": [ "concept/approximation", "concept/limit", "concept/neighbourhood", "concept/standard-of-approximation" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-b50a92e976", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "232", "location": "The Differential Calculus", "latex": "Thus the limit of~$\\dfrac{2x}{x}$ at $x = 0$ is~$2$, and it has no value at $x = 0$.", "markdown": "Thus the limit of $\\dfrac{2x}{x}$ at $x = 0$ is $2$, and it has no value at $x = 0$.", "why": "It shows a limit that exists where the function has no value, which is the key to the derivative definition.", "use": [ "lesson" ], "concepts": [ "concept/limit", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-cc425bc87f", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "223", "location": "The Differential Calculus", "latex": "It is a well-founded historical generalization, that the last thing to be discovered in any science is what the science is really about.", "markdown": "It is a well-founded historical generalization, that the last thing to be discovered in any science is what the science is really about.", "why": "A memorable line on how a science's purpose often becomes clear only late, useful for opening a discussion of why limits matter.", "use": [ "website", "history" ], "concepts": [ "concept/mathematics" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-9ba46eba54", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "236", "location": "Geometry", "latex": "But we do not need to consider who is observing the things, or whether he becomes acquainted with them by sight or touch or hearing. In short, we ignore all particular sensations.", "markdown": "But we do not need to consider who is observing the things, or whether he becomes acquainted with them by sight or touch or hearing. In short, we ignore all particular sensations.", "why": "It states plainly what kind of abstraction geometry performs, so a learner sees that a theorem does not depend on who observes the figure.", "use": [ "lesson" ], "concepts": [ "concept/abstractness", "concept/geometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-9ac1b0cb5e", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "236", "location": "Geometry", "latex": "Furthermore, particular things such as the Houses of Parliament, or the terrestrial globe are ignored. Every proposition refers to any things with such and such geometrical properties.", "markdown": "Furthermore, particular things such as the Houses of Parliament, or the terrestrial globe are ignored. Every proposition refers to any things with such and such geometrical properties.", "why": "It shows that a geometrical proposition is about any figure of a kind, not the one drawn on the page.", "use": [ "lesson", "website" ], "concepts": [ "concept/abstractness", "concept/geometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-6e7dee6b77", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "237", "location": "Geometry", "latex": "The answer is that the triangles are in all respects equal, if:---", "markdown": "The answer is that the triangles are in all respects equal, if:---", "why": "It opens the classic study of which partial correspondences between two triangles force full equality.", "use": [ "lesson" ], "concepts": [ "concept/congruent-figures", "concept/triangle" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-5d223a0d9e", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "238", "location": "Geometry", "latex": "Also there is the still simpler correlation between the angles of the triangle, namely, that their sum is equal to two right angles;", "markdown": "Also there is the still simpler correlation between the angles of the triangle, namely, that their sum is equal to two right angles;", "why": "It gives a short, memorable statement of a basic triangle fact that a learner can check against the angle sum.", "use": [ "lesson", "website" ], "concepts": [ "concept/triangle", "theorem/sum-of-the-angles-of-a-triangle" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-ebbee8947f", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "240", "location": "Geometry", "latex": "It is as though the great science of Anthropology were named the Study of Noses, owing to the fact that noses are a prominent part of the human body.", "markdown": "It is as though the great science of Anthropology were named the Study of Noses, owing to the fact that noses are a prominent part of the human body.", "why": "A vivid analogy that makes the point that a field should not be named after one of its subdivisions.", "use": [ "website", "history" ], "concepts": [ "concept/conic-section", "concept/coordinate-geometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-ad678ce0a3", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "242", "location": "Geometry", "latex": "The peculiarity of geometry is the fixity and overwhelming importance of the one particular example which occurs to our minds.", "markdown": "The peculiarity of geometry is the fixity and overwhelming importance of the one particular example which occurs to our minds.", "why": "It explains why a single drawn figure dominates a learner's thinking while the proof itself rests on abstract properties.", "use": [ "lesson" ], "concepts": [ "concept/abstractness", "concept/geometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-2bd7053c9d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "243", "location": "Geometry", "latex": "The abstract logical form of the propositions when fully stated is, ``If any collections of things have such and such abstract properties, they also have such and such other abstract properties.''", "markdown": "The abstract logical form of the propositions when fully stated is, “If any collections of things have such and such abstract properties, they also have such and such other abstract properties.”", "why": "It gives the full logical shape of a geometrical proposition in one sentence, a clear model for stating any theorem.", "use": [ "lesson" ], "concepts": [ "concept/abstractness", "concept/geometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-515371b2fa", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "241", "location": "Geometry", "latex": "The number of the archangels can be counted just because they are things.", "markdown": "The number of the archangels can be counted just because they are things.", "why": "A memorable contrast showing why counting needs only things, while the location of things is a further matter.", "use": [ "website", "history" ], "concepts": [ "concept/abstractness" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-946352486e", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "248", "location": "Quantity", "latex": "For example, astronomers tell us that the earth's rotation is slowing down, so that each day gains in length by some inconceivably minute fraction of a second.", "markdown": "For example, astronomers tell us that the earth’s rotation is slowing down, so that each day gains in length by some inconceivably minute fraction of a second.", "why": "It is a historical example of a scientist adjusting the measure of time to keep the laws of motion simple, and it is worth reading with the author's date in mind.", "use": [ "history" ], "concepts": [ "concept/measure-of-time", "law/laws-of-motion" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-7bf0fc5e88", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "245", "location": "Quantity", "latex": "When we have a set of things such as lengths which are measurable in terms of any one of them, we say that they are quantities of the same kind.", "markdown": "When we have a set of things such as lengths which are measurable in terms of any one of them, we say that they are quantities of the same kind.", "why": "It gives the learner the chapter's own definition of a family of quantities that can be compared and measured against one another.", "use": [ "lesson" ], "concepts": [ "concept/quantities-of-the-same-kind" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-672dcfcbf7", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "245", "location": "Quantity", "latex": "Lengths are measured by the foot-rule. By transporting the foot-rule from place to place we judge of the equality of lengths.", "markdown": "Lengths are measured by the foot-rule. By transporting the foot-rule from place to place we judge of the equality of lengths.", "why": "It shows a concrete measuring act, with equality tested by moving a rule, that a learner can picture before the abstraction.", "use": [ "lesson", "website" ], "concepts": [ "concept/equality", "instrument/foot-rule", "quantity/length" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-19fbb8d81d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "246", "location": "Quantity", "latex": "These preconceived conditions when accurately formulated may be called axioms of quantity.", "markdown": "These preconceived conditions when accurately formulated may be called axioms of quantity.", "why": "It explains that the rules for equality and addition are chosen conditions that measurement must satisfy, not facts found by test.", "use": [ "lesson" ], "concepts": [ "concept/axiom", "concept/equality", "method/addition" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-9c66e36fc4", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "248", "location": "Quantity", "latex": "A rule which made days of violently different lengths, and which made the speeds of apparently similar operations vary utterly out of proportion to the apparent minuteness of their differences, would never do.", "markdown": "A rule which made days of violently different lengths, and which made the speeds of apparently similar operations vary utterly out of proportion to the apparent minuteness of their differences, would never do.", "why": "It warns that a measuring rule must agree with common sense about small differences, which is a useful check on any proposed standard.", "use": [ "lesson" ], "concepts": [ "concept/measure-of-time" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-08b76209a5", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "249", "location": "Quantity", "latex": "A sense of the flux of time accompanies all our sensations and perceptions, and practically all that interests us in regard to time can be paralleled by the abstract mathematical properties which we ascribe to it.", "markdown": "A sense of the flux of time accompanies all our sensations and perceptions, and practically all that interests us in regard to time can be paralleled by the abstract mathematical properties which we ascribe to it.", "why": "It presents time as a quantity whose everyday experience is matched by abstract mathematical properties, which invites learners to see the link between intuition and formal measure.", "use": [ "website" ], "concepts": [ "quantity/time" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-d2e66bcecd", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "250", "location": "Quantity", "latex": "The mathematical sciences associated with them do not form the whole of mathematics, but they are the substratum of mathematical physics as at present existing.", "markdown": "The mathematical sciences associated with them do not form the whole of mathematics, but they are the substratum of mathematical physics as at present existing.", "why": "It places the study of number, quantity, space and time at the foundation of mathematical physics, giving the learner a sense of why these ideas matter.", "use": [ "website", "history" ], "concepts": [ "concept/mathematical-physics", "concept/number" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-17e4c4ec6a", "chapter": "whitehead-introduction-to-mathematics-1911/ch-notes", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "250", "location": "Notes", "latex": "In reading these equations it must be noted that a bracket is used in mathematical symbolism to mean that the operations within it are to be performed first. Thus $(1 + 3) + 2$ directs us first to add $3$ to~$1$, and then to add~$2$ to the result; and $1 + (3 + 2)$ directs us first to add $2$ to~$3$, and then to add the result to~$1$.", "markdown": "In reading these equations it must be noted that a bracket is used in mathematical symbolism to mean that the operations within it are to be performed first. Thus $(1 + 3) + 2$ directs us first to add $3$ to $1$, and then to add $2$ to the result; and $1 + (3 + 2)$ directs us first to add $2$ to $3$, and then to add the result to $1$.", "why": "It shows a learner, with concrete examples, that brackets change the order in which operations are done.", "use": [ "lesson" ], "concepts": [ "concept/parenthesis", "method/addition", "method/order-of-operations" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-6b64ff05e2", "chapter": "whitehead-introduction-to-mathematics-1911/ch-notes", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "250", "location": "Notes", "latex": "We perform first the operations in brackets and obtain \\[ 2 × 7 = 6 + 8 \\] which is obviously true.", "markdown": "We perform first the operations in brackets and obtain 2 × 7 = 6 + 8 which is obviously true.", "why": "It works a small example step by step, showing the bracket rule applied in order.", "use": [ "lesson" ], "concepts": [ "concept/parenthesis", "method/multiplication", "method/order-of-operations" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-5681a760cf", "chapter": "whitehead-introduction-to-mathematics-1911/ch-notes", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "250", "location": "Notes", "latex": "This fundamental ratio~$\\dfrac{SP}{PN}$ is called the eccentricity of the curve.", "markdown": "This fundamental ratio $\\dfrac{SP}{PN}$ is called the eccentricity of the curve.", "why": "It gives a clear definition of eccentricity as the ratio that fixes a conic's shape.", "use": [ "website" ], "concepts": [ "quantity/eccentricity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-5c560cfcb8", "chapter": "whitehead-introduction-to-mathematics-1911/ch-notes", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "250", "location": "Notes", "latex": "An ellipse with small eccentricity is very nearly a circle, and an ellipse of eccentricity only slightly less than unity is a long flat oval.", "markdown": "An ellipse with small eccentricity is very nearly a circle, and an ellipse of eccentricity only slightly less than unity is a long flat oval.", "why": "It gives a vivid picture of how changing eccentricity changes an ellipse from circle-like to flat.", "use": [ "website" ], "concepts": [ "concept/ellipse", "quantity/eccentricity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-a5a3942699", "chapter": "whitehead-introduction-to-mathematics-1911/ch-notes", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "251", "location": "Notes", "latex": "But it is possible for a series with terms partly positive and partly negative to be convergent, although the corresponding series with all its terms positive is divergent.", "markdown": "But it is possible for a series with terms partly positive and partly negative to be convergent, although the corresponding series with all its terms positive is divergent.", "why": "It warns learners that a series can converge even when its all-positive version diverges, a common surprise.", "use": [ "lesson" ], "concepts": [ "concept/convergent-series", "concept/divergent-series" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-035cad72ec", "chapter": "whitehead-introduction-to-mathematics-1911/ch-notes", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "251", "location": "Notes", "latex": "Such convergent series, which are not absolutely convergent, are much more difficult to deal with than absolutely convergent series.", "markdown": "Such convergent series, which are not absolutely convergent, are much more difficult to deal with than absolutely convergent series.", "why": "It tells learners why the distinction between absolute and ordinary convergence matters in practice.", "use": [ "lesson" ], "concepts": [ "concept/absolute-convergence", "concept/convergent-series" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-71ba536915", "chapter": "whitehead-introduction-to-mathematics-1911/ch-bibliography", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "251", "location": "Bibliography", "latex": "the algebra should be studied graphically, so that in practice the ideas of elementary coordinate geometry are also being assimilated.", "markdown": "the algebra should be studied graphically, so that in practice the ideas of elementary coordinate geometry are also being assimilated.", "why": "It shows a learner that algebra and coordinate geometry are meant to be learned together through graphs.", "use": [ "lesson" ], "concepts": [ "concept/algebra", "concept/coordinate-geometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-81c9e14c58", "chapter": "whitehead-introduction-to-mathematics-1911/ch-bibliography", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "251", "location": "Bibliography", "latex": "But in all these courses great care should be taken not to overload the mind with more detail than is necessary for the exemplification of the fundamental ideas.", "markdown": "But in all these courses great care should be taken not to overload the mind with more detail than is necessary for the exemplification of the fundamental ideas.", "why": "It gives a teacher a principle for pacing: keep to the fundamental ideas and leave out surplus detail.", "use": [ "lesson", "website" ], "concepts": [ "concept/mathematics" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-9c4b0f1da9", "chapter": "whitehead-introduction-to-mathematics-1911/ch-bibliography", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "252", "location": "Bibliography", "latex": "The science has grown to such vast proportions that probably no living mathematician can claim to have achieved this.", "markdown": "The science has grown to such vast proportions that probably no living mathematician can claim to have achieved this.", "why": "It is a historical remark on how far the subject has grown, which shows learners that no one masters all of it.", "use": [ "history" ], "concepts": [ "concept/mathematics" ] }, { "id": "whitehead-introduction-to-mathematics-1911/x-faf95e8c62", "chapter": "whitehead-introduction-to-mathematics-1911/ch-bibliography", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "252", "location": "Bibliography", "latex": "This elementary course of mathematics is sufficient for some types of professional career.", "markdown": "This elementary course of mathematics is sufficient for some types of professional career.", "why": "It tells a learner what the elementary course is for, and that further study is optional for many careers.", "use": [ "website" ], "concepts": [ "concept/mathematics" ] } ], "equations": [ { "id": "whitehead-introduction-to-mathematics-1911/eq-47a7093dac", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "16", "location": "Variables", "latex": "x + 2 = 2 + x", "name": null, "statement": "For any number x, adding 2 gives the same result as adding x to 2, so the order of addition does not matter.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "any number" } ], "sympy": "Eq(x + 2, 2 + x)", "physics": false, "states": [], "concepts": [ "concept/arithmetic", "concept/real-number", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-4918b8bdf4", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "15", "location": "Variables", "latex": "x + y = y + x", "name": null, "statement": "For any two numbers x and y, x + y equals y + x (the commutative law of addition, stated for any pair).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "any number" }, { "unit": null, "symbol": "y", "meaning": "any number" } ], "sympy": "Eq(x + y, y + x)", "physics": false, "states": [ "law/commutative-law" ], "concepts": [ "concept/algebra", "concept/arithmetic", "concept/real-number", "concept/variable", "method/addition" ], "pages": [ "15", "60" ], "chapters": [ "whitehead-introduction-to-mathematics-1911/ch-ii", "whitehead-introduction-to-mathematics-1911/ch-v" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-f9ea264b61", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "16", "location": "Variables", "latex": "x + 2 = 3", "name": null, "statement": "For some number x, x + 2 equals 3; the book notes that the only such number is 1, so this is an equation whose unknown is determined.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown number" } ], "sympy": "Eq(x + 2, 3)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/solution", "concept/unknown", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-30eea3d4c4", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "16", "location": "Variables", "latex": "x + 2 > 3", "name": null, "statement": "For some number x, x + 2 is greater than 3; every number greater than 1 satisfies this, so the set of such x is infinite.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a number satisfying the inequality" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/infinite-set", "concept/solution", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-ec53b9efae", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "15", "location": "Variables", "latex": "y > x", "name": null, "statement": "For any number x there exists some number y greater than x; the book identifies this assumption as the source of the notion of infinity.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "x", "meaning": "any number" }, { "unit": null, "symbol": "y", "meaning": "some number greater than x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/infinity", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-cb7bdbb482", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "18", "location": "Variables", "latex": "x + y = 1", "name": null, "statement": "A fixed relation between two correlated variables x and y: the pairs (x, y) satisfying it form the aggregate studied as a relation between variables.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a variable number" }, { "unit": null, "symbol": "y", "meaning": "a variable number correlated with x" } ], "sympy": "Eq(x + y, 1)", "physics": false, "states": [], "concepts": [ "concept/coordinate-geometry", "concept/equation", "concept/relation-between-variables", "concept/variable" ], "pages": [ "18", "117" ], "chapters": [ "whitehead-introduction-to-mathematics-1911/ch-ii", "whitehead-introduction-to-mathematics-1911/ch-ix" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-e1db74d023", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "19", "location": "Variables", "latex": "y + x = 1", "name": null, "statement": "An equivalent form of the relation x + y = 1, obtained by reordering the terms.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a variable number" }, { "unit": null, "symbol": "y", "meaning": "a variable number" } ], "sympy": "Eq(y + x, 1)", "physics": false, "states": [], "concepts": [ "concept/coordinate-geometry", "concept/equivalent-solids", "concept/line", "concept/locus", "concept/parallel-lines", "concept/relation-between-variables" ], "pages": [ "19", "124" ], "chapters": [ "whitehead-introduction-to-mathematics-1911/ch-ii", "whitehead-introduction-to-mathematics-1911/ch-ix" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-0be5ba2e17", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "19", "location": "Variables", "latex": "6x + 6y = 6", "name": null, "statement": "An equivalent form of the relation x + y = 1, obtained by multiplying both sides by 6.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a variable number" }, { "unit": null, "symbol": "y", "meaning": "a variable number" } ], "sympy": "Eq(6*x + 6*y, 6)", "physics": false, "states": [], "concepts": [ "concept/equivalent-solids", "concept/relation-between-variables" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-bc66d0e9ba", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "19", "location": "Variables", "latex": "y^{2} = x", "name": null, "statement": "A relation between x and y in which x is determined by y as its square; for x = 4, y can be plus or minus 2, so y is not uniquely determined by x.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a variable number" }, { "unit": null, "symbol": "y", "meaning": "a variable number whose square is x" } ], "sympy": "Eq(y**2, x)", "physics": false, "states": [], "concepts": [ "concept/field-of-a-variable", "concept/relation-between-variables", "concept/square" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-44c574dd98", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "19", "location": "Variables", "latex": "x + y > 1", "name": null, "statement": "An inequality relating x and y; when either variable is given, an indefinite number of values remain open for the other.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a variable number" }, { "unit": null, "symbol": "y", "meaning": "a variable number" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/relation-between-variables" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-f48010edab", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "22", "location": "Variables", "latex": "pv = 1", "name": "Boyle's law", "statement": "For a fixed mass of gas at constant temperature, the product of pressure and volume is constant; the book takes the constant to be 1 for illustration.", "kind": "law", "symbols": [ { "unit": "lb. weight per square inch", "symbol": "p", "meaning": "pressure of the gas" }, { "unit": "cubic feet", "symbol": "v", "meaning": "volume of the gas" } ], "sympy": "Eq(p*v, 1)", "physics": true, "states": [ "law/boyle-s-law" ], "concepts": [ "concept/constant", "concept/pressure", "concept/relation-between-variables", "concept/temperature", "quantity/volume" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-558ca1186d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "25", "location": "Methods of Application", "latex": "20y = x", "name": null, "statement": "If the cost of a house is y pounds and its volume x cubic feet, the assumed building-cost law makes x equal to 20 times y; the chapter uses this only to illustrate how a mathematical relation is applied to a fact.", "kind": "law", "symbols": [ { "unit": "cubic foot", "symbol": "x", "meaning": "number of cubic feet (cubic content) of the house" }, { "unit": "pound sterling", "symbol": "y", "meaning": "cost of the house to the owner" } ], "sympy": "Eq(20*y, x)", "physics": false, "states": [], "concepts": [ "concept/relation-between-variables", "concept/rule", "concept/variable", "quantity/volume" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-31b65eeaa1", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "29", "location": "Methods of Application", "latex": "F = k\\dfrac{mM}{d^{2}}", "name": "law of gravity", "statement": "The attraction between two bodies is a constant k times the product of their masses divided by the square of the distance between them, so the force F varies with the masses and inversely as the square of the distance.", "kind": "law", "symbols": [ { "unit": null, "symbol": "F", "meaning": "force of attraction on either body, due to the other and directed towards it" }, { "unit": null, "symbol": "k", "meaning": "a definite number depending on the absolute magnitude of the attraction and on the scale chosen for measuring forces" }, { "unit": "lb.", "symbol": "m", "meaning": "mass of one of the two bodies" }, { "unit": "lb.", "symbol": "M", "meaning": "mass of the other body" }, { "unit": "mile", "symbol": "d", "meaning": "distance between the two bodies" } ], "sympy": "Eq(F, k*m*M/d**2)", "physics": true, "states": [ "law/law-of-gravity" ], "concepts": [ "concept/constant", "concept/proportion", "concept/relation-between-variables", "quantity/distance", "quantity/force", "quantity/mass" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-1e679a11d7", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "39", "location": "Methods of Application", "latex": "F = W - w", "name": null, "statement": "The apparent weight of the crown in water, F, equals its weight in air W minus the weight w of the water it displaces, which is the upward force of the water.", "kind": "result", "symbols": [ { "unit": "lb.", "symbol": "F", "meaning": "apparent weight of the crown when hung in water (the scale reading)" }, { "unit": "lb.", "symbol": "W", "meaning": "weight of the crown as weighed in air" }, { "unit": "lb.", "symbol": "w", "meaning": "weight of the water displaced by the crown when completely immersed" } ], "sympy": "Eq(F, W - w)", "physics": true, "states": [], "concepts": [ "concept/buoyancy", "concept/specific-gravity", "concept/weight", "law/law-of-gravity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-6d143eb486", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "39", "location": "Methods of Application", "latex": "w = W - F", "name": null, "statement": "The weight of the displaced water w equals the weight of the crown in air W minus its apparent weight F in water; this is the same relation as F = W - w, rearranged.", "kind": "result", "symbols": [ { "unit": "lb.", "symbol": "w", "meaning": "weight of the water displaced by the crown when completely immersed" }, { "unit": "lb.", "symbol": "W", "meaning": "weight of the crown as weighed in air" }, { "unit": "lb.", "symbol": "F", "meaning": "apparent weight of the crown when hung in water" } ], "sympy": "Eq(w, W - F)", "physics": true, "states": [], "concepts": [ "concept/buoyancy", "concept/specific-gravity", "concept/weight" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-d22e01c187", "chapter": "whitehead-introduction-to-mathematics-1911/ch-iii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "39", "location": "Methods of Application", "latex": "\\frac{W}{w} = \\frac{W}{W - F}", "name": null, "statement": "The ratio of the weight of the crown to the weight of an equal volume of water equals W divided by W minus F, so it can be computed from two weighings; this ratio is the specific gravity of the metal.", "kind": "result", "symbols": [ { "unit": "lb.", "symbol": "W", "meaning": "weight of the crown as weighed in air" }, { "unit": "lb.", "symbol": "w", "meaning": "weight of the water displaced by the crown (weight of an equal volume of water)" }, { "unit": "lb.", "symbol": "F", "meaning": "apparent weight of the crown when hung in water" } ], "sympy": "Eq(W/w, W/(W - F))", "physics": true, "states": [], "concepts": [ "concept/buoyancy", "concept/common-ratio", "concept/specific-gravity", "concept/weight" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-0b01b5e8b1", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "60", "location": "The Symbolism of Mathematics", "latex": "(x + y) + z = x + (y + z)", "name": "associative law of addition", "statement": "When adding three numbers, the grouping of the two additions does not change the sum.", "kind": "law", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number" }, { "unit": null, "symbol": "y", "meaning": "variable number" }, { "unit": null, "symbol": "z", "meaning": "variable number" } ], "sympy": "Eq((x + y) + z, x + (y + z))", "physics": false, "states": [ "law/associative-law-of-addition" ], "concepts": [ "concept/algebra", "concept/real-number", "concept/variable", "method/addition" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-875b6ddbdb", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "60", "location": "The Symbolism of Mathematics", "latex": "x × y = y × x", "name": "commutative law of multiplication", "statement": "Multiplying x by y gives the same result as multiplying y by x.", "kind": "law", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number" }, { "unit": null, "symbol": "y", "meaning": "variable number" } ], "sympy": "Eq(x*y, y*x)", "physics": false, "states": [ "law/commutative-law-of-multiplication" ], "concepts": [ "concept/algebra", "concept/real-number", "concept/variable", "method/multiplication" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-e78d91f228", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "60", "location": "The Symbolism of Mathematics", "latex": "(x × y) × z = x × (y × z)", "name": "associative law of multiplication", "statement": "When multiplying three numbers, the grouping of the two multiplications does not change the product.", "kind": "law", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number" }, { "unit": null, "symbol": "y", "meaning": "variable number" }, { "unit": null, "symbol": "z", "meaning": "variable number" } ], "sympy": "Eq((x*y)*z, x*(y*z))", "physics": false, "states": [ "law/associative-law" ], "concepts": [ "concept/algebra", "concept/real-number", "concept/variable", "method/multiplication" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-bd4a33de69", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "60", "location": "The Symbolism of Mathematics", "latex": "x × (y + z) = (x × y) + (x × z)", "name": "distributive law", "statement": "Multiplying a number by a sum equals the sum of the two products with that number.", "kind": "law", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number" }, { "unit": null, "symbol": "y", "meaning": "variable number" }, { "unit": null, "symbol": "z", "meaning": "variable number" } ], "sympy": "Eq(x*(y + z), x*y + x*z)", "physics": false, "states": [ "law/distributive-law" ], "concepts": [ "concept/algebra", "concept/variable", "method/addition", "method/multiplication" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-2199efc8dd", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "65", "location": "The Symbolism of Mathematics", "latex": "x + y - 1 = 0", "name": null, "statement": "The standard form of the correlation x + y = 1, with everything moved to the left so the right side is zero.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number" }, { "unit": null, "symbol": "y", "meaning": "variable number" } ], "sympy": "Eq(x + y - 1, 0)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/equation", "concept/relation-between-variables", "concept/variable", "concept/zero-displacement" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-6b2ba8fcf5", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "69", "location": "The Symbolism of Mathematics", "latex": "ax + by - c = 0", "name": null, "statement": "The general linear relation between variables x and y, with a, b, c as parameters (constants for a given relation) that can be varied to give any linear correlation.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "parameter (constant for a given correlation)" }, { "unit": null, "symbol": "b", "meaning": "parameter (constant for a given correlation)" }, { "unit": null, "symbol": "c", "meaning": "parameter (constant for a given correlation)" }, { "unit": null, "symbol": "x", "meaning": "variable number" }, { "unit": null, "symbol": "y", "meaning": "variable number" } ], "sympy": "Eq(a*x + b*y - c, 0)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/constant", "concept/equation", "concept/linear", "concept/linear-form", "concept/parameter", "concept/relation-between-variables", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-b89e2577df", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "67", "location": "The Symbolism of Mathematics", "latex": "0 × x = 0", "name": null, "statement": "Multiplying any number by zero gives zero.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "any number" } ], "sympy": "Eq(0*x, 0)", "physics": false, "states": [], "concepts": [ "concept/real-number", "concept/zero-displacement", "method/multiplication" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-f5aa07468e", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "67", "location": "The Symbolism of Mathematics", "latex": "x + 0 = x", "name": null, "statement": "Adding zero to any number leaves the number unchanged.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "any number" } ], "sympy": "Eq(x + 0, x)", "physics": false, "states": [], "concepts": [ "concept/real-number", "concept/zero-displacement", "method/addition" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-df1e46abc5", "chapter": "whitehead-introduction-to-mathematics-1911/ch-v", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "68", "location": "The Symbolism of Mathematics", "latex": "x^{2} + (0 × x) - 4 = 0", "name": null, "statement": "The equation x^2 - 4 = 0 written with the zero term 0 × x inserted, so that it has the same algebraic form as the other quadratic equations.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown" } ], "sympy": "Eq(x**2 + (0*x) - 4, 0)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/equation", "concept/quadratic-equation", "concept/unknown", "concept/zero-displacement" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-658b227017", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "83", "location": "Generalizations of Number", "latex": "x + a = b", "name": null, "statement": "The general equation of the form x + a = b, with a and b any constants, which the book uses to show why operations remove the need for limits on the constants.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown, a variable" }, { "unit": null, "symbol": "a", "meaning": "a constant" }, { "unit": null, "symbol": "b", "meaning": "a constant" } ], "sympy": "Eq(x + a, b)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/constant", "concept/equation", "concept/generality", "concept/operation", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-3295306d5b", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "83", "location": "Generalizations of Number", "latex": "x = b - a", "name": null, "statement": "The solution of the general equation x + a = b is x = b - a, an operation of addition or subtraction as the case may be.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown, a variable" }, { "unit": null, "symbol": "a", "meaning": "a constant" }, { "unit": null, "symbol": "b", "meaning": "a constant" } ], "sympy": "Eq(x, b - a)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/equation", "concept/generality", "concept/operation", "concept/solution", "method/subtraction" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-a35a7d4dbe", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "98", "location": "Imaginary Numbers", "latex": "(x, y) - (u, v) = (x, y) + (-u, -v)", "name": null, "statement": "Subtracting an ordered couple is the same as adding the couple with both numbers negated.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first number of the ordered couple" }, { "unit": null, "symbol": "y", "meaning": "second number of the ordered couple" }, { "unit": null, "symbol": "u", "meaning": "first number of the subtracted ordered couple" }, { "unit": null, "symbol": "v", "meaning": "second number of the subtracted ordered couple" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/ordered-pair", "method/adding-ordered-pairs", "method/subtraction" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-9002e32a61", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "89", "location": "Imaginary Numbers", "latex": "x = ±\\sqrt{(b - a)}", "name": null, "statement": "The two solutions of x² + a = b are ±√(b − a), and they exist only when b is not less than a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown number in the equation x² + a = b" }, { "unit": null, "symbol": "a", "meaning": "constant term in the general equation x² + a = b" }, { "unit": null, "symbol": "b", "meaning": "constant right-hand side in the general equation x² + a = b" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant", "concept/imaginary-root", "concept/solution", "concept/square", "concept/unknown" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-2d110c8238", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "90", "location": "Imaginary Numbers", "latex": "\\sqrt{(-1)} \\sqrt{c^{2}} = c\\sqrt{(-1)}", "name": null, "statement": "For a positive c, the square root of −c² equals c times √(−1), so the imaginary unit √(−1) carries the root of any negative number.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "c", "meaning": "a positive number, so that c² is positive and a = −c²" } ], "sympy": "Eq(sqrt(-1)*sqrt(c**2), c*sqrt(-1))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/imaginary-root", "concept/root", "concept/square" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-0a9bbcdb1f", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "97", "location": "Imaginary Numbers", "latex": "(x, y) + (x', y') = (x + x', y + y')", "name": null, "statement": "By definition, the sum of two ordered couples is the ordered couple whose first number is the sum of the first numbers and whose second number is the sum of the second numbers.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first number of the first ordered couple" }, { "unit": null, "symbol": "y", "meaning": "second number of the first ordered couple" }, { "unit": null, "symbol": "x'", "meaning": "first number of the second ordered couple" }, { "unit": null, "symbol": "y'", "meaning": "second number of the second ordered couple" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/ordered-pair", "method/adding-ordered-pairs", "method/addition" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-34e9a2a67e", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "96", "location": "Imaginary Numbers", "latex": "(x, y) = (c, d) - (a, b)", "name": null, "statement": "The unknown ordered couple (x, y) satisfying (x, y) + (a, b) = (c, d) is unique and equals (c, d) minus (a, b).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first number of the unknown ordered couple" }, { "unit": null, "symbol": "y", "meaning": "second number of the unknown ordered couple" }, { "unit": null, "symbol": "a", "meaning": "first number of the given ordered couple" }, { "unit": null, "symbol": "b", "meaning": "second number of the given ordered couple" }, { "unit": null, "symbol": "c", "meaning": "first number of the ordered couple on the right-hand side" }, { "unit": null, "symbol": "d", "meaning": "second number of the ordered couple on the right-hand side" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ordered-pair", "concept/solution", "concept/uniqueness", "concept/unknown", "method/subtraction" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-d8eb32fa8a", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "97", "location": "Imaginary Numbers", "latex": "(x, y) - (u, v) = (x - u, y - v)", "name": null, "statement": "Subtraction of ordered couples is defined componentwise: the first numbers are subtracted and the second numbers are subtracted.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first number of the minuend ordered couple" }, { "unit": null, "symbol": "y", "meaning": "second number of the minuend ordered couple" }, { "unit": null, "symbol": "u", "meaning": "first number of the subtrahend ordered couple" }, { "unit": null, "symbol": "v", "meaning": "second number of the subtrahend ordered couple" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/ordered-pair", "method/subtraction" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-21081019cf", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "98", "location": "Imaginary Numbers", "latex": "(x, y) - (x, y) = (0, 0)", "name": null, "statement": "Any ordered couple minus itself gives the couple (0, 0), which is therefore the zero ordered couple.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first number of the ordered couple" }, { "unit": null, "symbol": "y", "meaning": "second number of the ordered couple" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ordered-pair", "concept/zero-displacement", "method/subtraction" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-5db9163105", "chapter": "whitehead-introduction-to-mathematics-1911/ch-vii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "98", "location": "Imaginary Numbers", "latex": "(x, y) + (0, 0) = (x, y)", "name": null, "statement": "Adding the zero ordered couple (0, 0) to any ordered couple leaves it unchanged.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first number of the ordered couple" }, { "unit": null, "symbol": "y", "meaning": "second number of the ordered couple" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ordered-pair", "concept/zero-displacement", "method/adding-ordered-pairs", "method/addition" ], "pages": [ "98", "103" ], "chapters": [ "whitehead-introduction-to-mathematics-1911/ch-vii", "whitehead-introduction-to-mathematics-1911/ch-viii" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-bf6ed0c745", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "102", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(x, y) × (x', y') = \\{(xx' - yy'), (xy' + x'y)\\}", "name": null, "statement": "By definition, the product of two ordered couples is the couple whose first term is xx' - yy' and whose second term is xy' + x'y.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first term of the couple (x, y)" }, { "unit": null, "symbol": "y", "meaning": "second term of the couple (x, y)" }, { "unit": null, "symbol": "x'", "meaning": "first term of the couple (x', y')" }, { "unit": null, "symbol": "y'", "meaning": "second term of the couple (x', y')" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ordered-pair", "method/multiplication", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-063e8496ee", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "101", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(x, y) × (x', y') = (x', y') × (x, y)", "name": "commutative law", "statement": "The product of two ordered couples does not depend on the order of the factors.", "kind": "law", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first term of the couple (x, y)" }, { "unit": null, "symbol": "y", "meaning": "second term of the couple (x, y)" }, { "unit": null, "symbol": "x'", "meaning": "first term of the couple (x', y')" }, { "unit": null, "symbol": "y'", "meaning": "second term of the couple (x', y')" } ], "sympy": null, "physics": false, "states": [ "law/commutative-law" ], "concepts": [ "concept/ordered-pair", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-76b2ae3912", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "101", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "\\{(x, y) × (x', y')\\} × (u, v) = (x, y) × \\{(x', y') × (u, v)\\}", "name": "associative law", "statement": "Multiplying ordered couples gives the same result whichever pair of adjacent factors is multiplied first.", "kind": "law", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first term of the couple (x, y)" }, { "unit": null, "symbol": "y", "meaning": "second term of the couple (x, y)" }, { "unit": null, "symbol": "x'", "meaning": "first term of the couple (x', y')" }, { "unit": null, "symbol": "y'", "meaning": "second term of the couple (x', y')" }, { "unit": null, "symbol": "u", "meaning": "first term of the couple (u, v)" }, { "unit": null, "symbol": "v", "meaning": "second term of the couple (u, v)" } ], "sympy": null, "physics": false, "states": [ "law/associative-law" ], "concepts": [ "concept/ordered-pair", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-6697d12633", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "101", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(x, y) × (a, b) = (c, d)", "name": null, "statement": "The unknown couple (x, y) that multiplies (a, b) to give (c, d) is required to be unique (except when (a, b) is the zero couple).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown first term of the couple to be found" }, { "unit": null, "symbol": "y", "meaning": "unknown second term of the couple to be found" }, { "unit": null, "symbol": "a", "meaning": "first term of the given divisor couple" }, { "unit": null, "symbol": "b", "meaning": "second term of the given divisor couple" }, { "unit": null, "symbol": "c", "meaning": "first term of the given product couple" }, { "unit": null, "symbol": "d", "meaning": "second term of the given product couple" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ordered-pair", "concept/unknown", "method/division", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-6713816e59", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "101", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(x, y) = \\frac{(c, d)}{(a, b)}", "name": null, "statement": "The quotient of ordered couples (c, d) by (a, b) is the unique couple (x, y) satisfying (x, y) × (a, b) = (c, d).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first term of the quotient couple" }, { "unit": null, "symbol": "y", "meaning": "second term of the quotient couple" }, { "unit": null, "symbol": "a", "meaning": "first term of the divisor couple" }, { "unit": null, "symbol": "b", "meaning": "second term of the divisor couple" }, { "unit": null, "symbol": "c", "meaning": "first term of the dividend couple" }, { "unit": null, "symbol": "d", "meaning": "second term of the dividend couple" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ordered-pair", "method/division" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-5745fa9041", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "102", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(x,y) × \\{(a, b) + (c, d)\\} = \\{(x, y) × (a, b)\\} + \\{(x, y) × (c, d)\\}", "name": "distributive law", "statement": "Multiplying an ordered couple by a sum of couples equals the sum of the separate products.", "kind": "law", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first term of the couple (x, y)" }, { "unit": null, "symbol": "y", "meaning": "second term of the couple (x, y)" }, { "unit": null, "symbol": "a", "meaning": "first term of the couple (a, b)" }, { "unit": null, "symbol": "b", "meaning": "second term of the couple (a, b)" }, { "unit": null, "symbol": "c", "meaning": "first term of the couple (c, d)" }, { "unit": null, "symbol": "d", "meaning": "second term of the couple (c, d)" } ], "sympy": null, "physics": false, "states": [ "law/distributive-law" ], "concepts": [ "method/addition", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-80b23b3091", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "103", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(x, y) × (0, 0) = (0, 0)", "name": null, "statement": "Multiplying any ordered couple by the zero couple (0, 0) gives the zero couple.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first term of the couple (x, y)" }, { "unit": null, "symbol": "y", "meaning": "second term of the couple (x, y)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ordered-pair", "concept/zero-displacement", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-4e291b291c", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "103", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "x × 1 = x", "name": null, "statement": "Multiplying any value of x by 1 leaves x unchanged, the characteristic property of 1.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "any value" } ], "sympy": "Eq(x*1, x)", "physics": false, "states": [], "concepts": [ "method/multiplication", "unit/unit" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-0f02213b66", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "103", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(x, y) × (1, 0) = \\{(x - 0), (y + 0)\\} = (x, y)", "name": null, "statement": "The couple (1, 0) leaves every ordered couple unchanged under multiplication, so it is the unit couple.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first term of the couple (x, y)" }, { "unit": null, "symbol": "y", "meaning": "second term of the couple (x, y)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ordered-pair", "method/multiplying-ordered-pairs", "unit/unit" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-e33102fc05", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "103", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "\\sqrt{(-1)} × \\sqrt{(-1)} = -1", "name": null, "statement": "The symbol sqrt(-1) must be defined so that its square is -1; this is the property the ordered couple (0, 1) is required to have.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\sqrt{(-1)}", "meaning": "the imaginary unit, a square root of -1" } ], "sympy": "Eq(sqrt(-1)*sqrt(-1), -1)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/imaginary-root", "concept/root" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-9ceeac8775", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "104", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(0, 1) × (0, 1) = \\{(0 - 1), (0 + 0)\\} = (-1, 0)", "name": null, "statement": "The couple (0, 1) squares to the negative unit couple (-1, 0), so it interprets the square root of -1.", "kind": "result", "symbols": [], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/imaginary-root", "method/multiplying-ordered-pairs", "unit/unit" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-af5d7d8f30", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "104", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(0, -1) × (0, -1) = (-1, 0)", "name": null, "statement": "The couple (0, -1) also squares to (-1, 0), so it interprets the other square root of -1.", "kind": "result", "symbols": [], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/imaginary-root", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-8dd0dfa263", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "105", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(a, 0) × (x, y) = (ax, ay)", "name": null, "statement": "Multiplying a complex couple by a real couple multiplies each term by the real number a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real number multiplier" }, { "unit": null, "symbol": "x", "meaning": "first term of the couple (x, y)" }, { "unit": null, "symbol": "y", "meaning": "second term of the couple (x, y)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/real-number", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-2b904d9b2c", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "105", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(0, b) × (x, y) = (-by, bx)", "name": null, "statement": "Multiplying a complex couple by a pure imaginary couple (0, b) turns it through a right angle and scales it.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b", "meaning": "real number coefficient of the pure imaginary couple" }, { "unit": null, "symbol": "x", "meaning": "first term of the couple (x, y)" }, { "unit": null, "symbol": "y", "meaning": "second term of the couple (x, y)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/imaginary-root", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-cffdca1d92", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "105", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(a, 0) × (0, b) = (0, ab)", "name": null, "statement": "Multiplying a real couple by a pure imaginary couple gives a pure imaginary couple.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real number" }, { "unit": null, "symbol": "b", "meaning": "real number coefficient of the pure imaginary couple" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/imaginary-root", "concept/real-number", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-983c9f3c65", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "105", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(a, 0) × (a', 0) =( aa', 0)", "name": null, "statement": "Multiplying two real couples gives the real couple whose term is the product of the two real numbers.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "first real number" }, { "unit": null, "symbol": "a'", "meaning": "second real number" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/real-number", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-a5aa93f352", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "105", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(0, b) × (0, b') = (-bb', 0)", "name": null, "statement": "Multiplying two pure imaginary couples gives a real couple equal to the negative of the product bb'.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b", "meaning": "real coefficient of the first pure imaginary couple" }, { "unit": null, "symbol": "b'", "meaning": "real coefficient of the second pure imaginary couple" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/imaginary-root", "concept/real-number", "method/multiplying-ordered-pairs" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-0451322401", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "108", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "\\text{the angle } QOR = \\text{the angle } XOP", "name": null, "statement": "Multiplying OQ by OP rotates OQ through the angle XOP, so the angle QOR equals the angle XOP.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "O", "meaning": "origin of the vectors in the diagram" }, { "unit": "angle", "symbol": "XOP", "meaning": "plane angle from the axis OX to the vector OP" }, { "unit": "angle", "symbol": "QOR", "meaning": "plane angle from the vector OQ to the product vector OR" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/rotation", "method/multiplying-vectors-in-a-plane" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-59ac801585", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "110", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "(u, v) + (3, 0) = (2, 0)", "name": null, "statement": "The equation x + 3 = 2 becomes this equation for the unknown couple (u, v), where x is represented by (u, v).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "u", "meaning": "first term of the unknown couple, representing x" }, { "unit": null, "symbol": "v", "meaning": "second term of the unknown couple, representing x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/equation", "concept/ordered-pair", "concept/unknown", "method/addition" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-581898a0dd", "chapter": "whitehead-introduction-to-mathematics-1911/ch-viii", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "110", "location": "Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})", "latex": "\\{(u, v) + (3, 0)\\}^{2} = (-2, 0)", "name": null, "statement": "The equation (x + 3)^2 = -2 becomes this equation for the unknown couple (u, v).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "u", "meaning": "first term of the unknown couple, representing x" }, { "unit": null, "symbol": "v", "meaning": "second term of the unknown couple, representing x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/equation", "concept/ordered-pair", "concept/quadratic-equation", "concept/square", "concept/unknown" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-f74fcbc030", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "117", "location": "Coordinate Geometry", "latex": "x - y = 1", "name": null, "statement": "A second example of a correlation between the variable numbers x and y.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number (abscissa of a point, the length OM)" }, { "unit": null, "symbol": "y", "meaning": "variable number (ordinate of a point, the length PM)" } ], "sympy": "Eq(x - y, 1)", "physics": false, "states": [], "concepts": [ "concept/coordinate-geometry", "concept/equation", "concept/relation-between-variables", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-99c1c95f75", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "117", "location": "Coordinate Geometry", "latex": "ax + by = c", "name": null, "statement": "The general linear correlation between x and y, in which a, b and c are constants (parameters) determining the correlation; it stands for the general algebraic form of a linear relation.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number (abscissa of a point)" }, { "unit": null, "symbol": "y", "meaning": "variable number (ordinate of a point)" }, { "unit": null, "symbol": "a", "meaning": "constant (parameter) determining the correlation; a letter standing for any number" }, { "unit": null, "symbol": "b", "meaning": "constant (parameter) determining the correlation; a letter standing for any number" }, { "unit": null, "symbol": "c", "meaning": "constant (parameter) determining the correlation; a letter standing for any number" } ], "sympy": "Eq(a*x + b*y, c)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/constant", "concept/coordinate-geometry", "concept/equation", "concept/line", "concept/locus", "concept/parameter", "concept/relation-between-variables" ], "pages": [ "117", "141" ], "chapters": [ "whitehead-introduction-to-mathematics-1911/ch-ix", "whitehead-introduction-to-mathematics-1911/ch-x" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-e5210717a3", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "118", "location": "Coordinate Geometry", "latex": "x^{2} + y^{2} = 1", "name": null, "statement": "A correlation between x and y of the form of a circle centred at the origin with unit radius, the starting point for generalizing to quadratic algebraic forms.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number (abscissa of a point)" }, { "unit": null, "symbol": "y", "meaning": "variable number (ordinate of a point)" } ], "sympy": "Eq(x**2 + y**2, 1)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/circle", "concept/coordinate-geometry", "concept/equation", "concept/relation-between-variables" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-6bed1edc72", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "118", "location": "Coordinate Geometry", "latex": "ax^{2} + by^{2} = c", "name": null, "statement": "The generalization of x^2 + y^2 = 1 with constant coefficients a, b and right-hand side c, a further algebraic form of correlation between x and y.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number (abscissa of a point)" }, { "unit": null, "symbol": "y", "meaning": "variable number (ordinate of a point)" }, { "unit": null, "symbol": "a", "meaning": "constant (parameter) coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "constant (parameter) coefficient of y^2" }, { "unit": null, "symbol": "c", "meaning": "constant (parameter) on the right-hand side" } ], "sympy": "Eq(a*x**2 + b*y**2, c)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/equation", "concept/parameter", "concept/relation-between-variables" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-e1e2c22c01", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "118", "location": "Coordinate Geometry", "latex": "ax^{2} + 2hxy + by^{2} = c", "name": null, "statement": "A further generalization of the quadratic correlation, adding a cross term in xy with constant coefficient 2h.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number (abscissa of a point)" }, { "unit": null, "symbol": "y", "meaning": "variable number (ordinate of a point)" }, { "unit": null, "symbol": "a", "meaning": "constant (parameter) coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "constant (parameter) coefficient of y^2" }, { "unit": null, "symbol": "h", "meaning": "constant (parameter) coefficient of the cross term, written as 2h" }, { "unit": null, "symbol": "c", "meaning": "constant (parameter) on the right-hand side" } ], "sympy": "Eq(a*x**2 + 2*h*x*y + b*y**2, c)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/equation", "concept/parameter", "concept/relation-between-variables" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-c1cb2f2c8f", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "118", "location": "Coordinate Geometry", "latex": "ax^{2} + 2hxy + by^{2} + 2gx + 2fy = c", "name": null, "statement": "The most general of the quadratic algebraic forms listed, with linear terms added to the quadratic terms, a correlation between x and y indicating a variable correlation of a given algebraic form.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable number (abscissa of a point)" }, { "unit": null, "symbol": "y", "meaning": "variable number (ordinate of a point)" }, { "unit": null, "symbol": "a", "meaning": "constant (parameter) coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "constant (parameter) coefficient of y^2" }, { "unit": null, "symbol": "h", "meaning": "constant (parameter) coefficient of the cross term, written as 2h" }, { "unit": null, "symbol": "g", "meaning": "constant (parameter) coefficient of x, written as 2g" }, { "unit": null, "symbol": "f", "meaning": "constant (parameter) coefficient of y, written as 2f" }, { "unit": null, "symbol": "c", "meaning": "constant (parameter) on the right-hand side" } ], "sympy": "Eq(a*x**2 + 2*h*x*y + b*y**2 + 2*g*x + 2*f*y, c)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/equation", "concept/locus", "concept/parameter", "concept/relation-between-variables" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-d45bc5789e", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "124", "location": "Coordinate Geometry", "latex": "y - x = 0", "name": null, "statement": "The equation of a straight line through the origin O that bisects the angle XOY; obtained from ax + by = c with a = -1, b = 1, c = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of a point on the locus (length OM)" }, { "unit": null, "symbol": "y", "meaning": "ordinate of a point on the locus (length PM)" } ], "sympy": "Eq(y - x, 0)", "physics": false, "states": [], "concepts": [ "concept/bisector", "concept/cartesian-coordinates", "concept/coordinate-geometry", "concept/line", "concept/locus", "concept/origin" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-40afa4cdfd", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "124", "location": "Coordinate Geometry", "latex": "y + x = 0", "name": null, "statement": "The equation of a straight line through the origin that bisects the angle X'OY, the line L1OL1' of the diagram.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of a point on the locus (length OM)" }, { "unit": null, "symbol": "y", "meaning": "ordinate of a point on the locus (length PM)" } ], "sympy": "Eq(y + x, 0)", "physics": false, "states": [], "concepts": [ "concept/coordinate-geometry", "concept/line", "concept/locus", "concept/origin" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-7b8ab7c0b7", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "124", "location": "Coordinate Geometry", "latex": "ax + by = 0", "name": null, "statement": "The general form of the equation of any straight line through the origin.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of a point on the locus (length OM)" }, { "unit": null, "symbol": "y", "meaning": "ordinate of a point on the locus (length PM)" }, { "unit": null, "symbol": "a", "meaning": "constant (parameter) determining the line" }, { "unit": null, "symbol": "b", "meaning": "constant (parameter) determining the line" } ], "sympy": "Eq(a*x + b*y, 0)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/line", "concept/locus", "concept/origin", "concept/parameter" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-f3e2110c91", "chapter": "whitehead-introduction-to-mathematics-1911/ch-ix", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "124", "location": "Coordinate Geometry", "latex": "y - x = 1", "name": null, "statement": "The equation of a straight line not passing through the origin; it meets the axis OX at (1, 0) and the axis OY at (0, 1), and is parallel to LOL'.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of a point on the locus (length OM)" }, { "unit": null, "symbol": "y", "meaning": "ordinate of a point on the locus (length PM)" } ], "sympy": "Eq(y - x, 1)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/coordinate-geometry", "concept/line", "concept/locus", "concept/origin", "concept/parallel-lines" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-60fcddf9e0", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "136", "location": "Conic Sections", "latex": "\\dfrac{SP}{PN}", "name": "focus-directrix property of conic sections", "statement": "For a focus S and its corresponding directrix XN, the ratio of the distance SP to the perpendicular PN from any point P on the curve to the directrix is constant, for ellipses, parabolas and hyperbolas alike.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "S", "meaning": "focus of the curve" }, { "unit": null, "symbol": "P", "meaning": "any point on the curve" }, { "unit": null, "symbol": "N", "meaning": "foot of the perpendicular from P on the directrix" }, { "unit": null, "symbol": "PN", "meaning": "perpendicular on the directrix from P" }, { "unit": null, "symbol": "SP", "meaning": "distance from the focus S to the point P" } ], "sympy": null, "physics": false, "states": [ "theorem/focus-directrix-property-of-conic-sections" ], "concepts": [ "concept/common-ratio", "concept/conic-section", "concept/constant", "concept/curve", "concept/directrix", "concept/ellipse", "concept/focus", "concept/hyperbola", "concept/parabola", "concept/perpendicular", "quantity/eccentricity" ], "pages": [ "136", "250" ], "chapters": [ "whitehead-introduction-to-mathematics-1911/ch-x", "whitehead-introduction-to-mathematics-1911/ch-notes" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-b07cecfdaa", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "141", "location": "Conic Sections", "latex": "ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0", "name": null, "statement": "The general algebraic equation of the second degree, which when it represents any locus always represents a conic section, and to which the equation of every conic section can be brought.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of a point" }, { "unit": null, "symbol": "y", "meaning": "coordinate of a point" }, { "unit": null, "symbol": "a", "meaning": "constant coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "constant coefficient of y^2" }, { "unit": null, "symbol": "h", "meaning": "constant coefficient of xy (half of the xy term)" }, { "unit": null, "symbol": "g", "meaning": "constant coefficient of x" }, { "unit": null, "symbol": "f", "meaning": "constant coefficient of y" }, { "unit": null, "symbol": "c", "meaning": "constant term" } ], "sympy": "Eq(a*x**2 + 2*h*x*y + b*y**2 + 2*g*x + 2*f*y + c, 0)", "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/constant", "concept/coordinate-geometry", "concept/equation", "concept/locus", "concept/variable", "unit/degree-of-angle" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-8de156b5fc", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "142", "location": "Conic Sections", "latex": "ab - h^{2} = 0", "name": null, "statement": "When ab - h^2 is zero, the conic given by the general second-degree equation is a parabola.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant coefficient of x^2 in the general equation" }, { "unit": null, "symbol": "b", "meaning": "constant coefficient of y^2 in the general equation" }, { "unit": null, "symbol": "h", "meaning": "constant coefficient of xy in the general equation (half the xy term)" } ], "sympy": "Eq(a*b - h**2, 0)", "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/constant", "concept/equation-of-the-second-degree", "concept/parabola", "theorem/discriminant-of-a-conic" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-c315b5ac1f", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "142", "location": "Conic Sections", "latex": "a(x^{2} + y^{2}) + 2gx + 2fy + c = 0", "name": null, "statement": "The equation of any circle can be written in this form, with the coefficients of x^2 and y^2 equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of a point" }, { "unit": null, "symbol": "y", "meaning": "coordinate of a point" }, { "unit": null, "symbol": "a", "meaning": "constant common coefficient of x^2 and y^2" }, { "unit": null, "symbol": "g", "meaning": "constant coefficient of x" }, { "unit": null, "symbol": "f", "meaning": "constant coefficient of y" }, { "unit": null, "symbol": "c", "meaning": "constant term" } ], "sympy": "Eq(a*(x**2 + y**2) + 2*g*x + 2*f*y + c, 0)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/conic-section", "concept/constant", "concept/coordinate-geometry", "concept/ellipse", "concept/equation" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-4e9953e6f0", "chapter": "whitehead-introduction-to-mathematics-1911/ch-x", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "142", "location": "Conic Sections", "latex": "(dx + ey)^{2} + 2gx + 2fy + c = 0", "name": null, "statement": "The general form of the equation of a parabola, in which the second-degree terms form a perfect square.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of a point" }, { "unit": null, "symbol": "y", "meaning": "coordinate of a point" }, { "unit": null, "symbol": "d", "meaning": "constant coefficient in the perfect square" }, { "unit": null, "symbol": "e", "meaning": "constant coefficient in the perfect square" }, { "unit": null, "symbol": "g", "meaning": "constant coefficient of x" }, { "unit": null, "symbol": "f", "meaning": "constant coefficient of y" }, { "unit": null, "symbol": "c", "meaning": "constant term" } ], "sympy": "Eq((d*x + e*y)**2 + 2*g*x + 2*f*y + c, 0)", "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/constant", "concept/coordinate-geometry", "concept/parabola", "concept/perfect-square", "unit/degree-of-angle" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-98e73ff08d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "145", "location": "Functions", "latex": "s = 20 × t", "name": null, "statement": "A train travelling at 20 miles per hour has gone s miles after t hours, so s is a function of t.", "kind": "formula", "symbols": [ { "unit": "mile", "symbol": "s", "meaning": "distance gone" }, { "unit": "hour", "symbol": "t", "meaning": "number of hours travelled (argument)" } ], "sympy": "Eq(s, 20*t)", "physics": true, "states": [], "concepts": [ "concept/argument-of-a-function", "concept/function", "concept/rate-of-change", "concept/value-of-a-function", "quantity/distance", "quantity/time", "quantity/velocity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-77e6d1fe89", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "145", "location": "Functions", "latex": "y = x + 1", "name": null, "statement": "John's age y is one year more than Thomas's age x, so y is the function x + 1 of x.", "kind": "formula", "symbols": [ { "unit": "year", "symbol": "y", "meaning": "John's age (value of the function)" }, { "unit": "year", "symbol": "x", "meaning": "Thomas's age (argument)" } ], "sympy": "Eq(y, x + 1)", "physics": false, "states": [], "concepts": [ "concept/argument-of-a-function", "concept/function", "concept/relation-between-variables", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-2f78e0fd4d", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "146", "location": "Functions", "latex": "y = x^{2}", "name": null, "statement": "y is the square of the argument x, an example of a function of x.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "argument of the function" } ], "sympy": "Eq(y, x**2)", "physics": false, "states": [], "concepts": [ "concept/argument-of-a-function", "concept/function", "concept/square", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-481c31abbb", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "146", "location": "Functions", "latex": "y = 2x^{2} + 3x + 1", "name": null, "statement": "y is the quadratic polynomial 2x^2 + 3x + 1 of the argument x, an example of a function.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "argument of the function" } ], "sympy": "Eq(y, 2*x**2 + 3*x + 1)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/argument-of-a-function", "concept/function", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-93048ab716", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "145", "location": "Functions", "latex": "y = x", "name": null, "statement": "y equals the argument x itself, the simplest example of a function.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "argument of the function" } ], "sympy": "Eq(y, x)", "physics": false, "states": [], "concepts": [ "concept/argument-of-a-function", "concept/function", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-5112129174", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "146", "location": "Functions", "latex": "y = \\log x", "name": null, "statement": "y is the logarithm of x, quoted as an example of a function of x.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "argument of the function" } ], "sympy": "Eq(y, log(x))", "physics": false, "states": [], "concepts": [ "concept/argument-of-a-function", "concept/function", "concept/logarithm", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-2230f48d01", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "147", "location": "Functions", "latex": "y = f(x)", "name": null, "statement": "y is the value of some undetermined function f of the argument x; once x is given, y is definitely determined.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "value of the function" }, { "unit": null, "symbol": "f", "meaning": "the undetermined function (functional letter)" }, { "unit": null, "symbol": "x", "meaning": "argument of the function" } ], "sympy": "Eq(y, f(x))", "physics": false, "states": [], "concepts": [ "concept/argument-of-a-function", "concept/function", "concept/function-notation", "concept/relation-between-variables", "concept/value-of-a-function", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-c9972e28a3", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "148", "location": "Functions", "latex": "f(1) = 0", "name": null, "statement": "With f defined to be 0 at integers and 1 otherwise, the value of f at the argument 1 is 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the function defined to be 0 for integral x and 1 otherwise" } ], "sympy": "Eq(f(1), 0)", "physics": false, "states": [], "concepts": [ "concept/discontinuous-function", "concept/function", "concept/integer", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-44daafd179", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "155", "location": "Functions", "latex": "f(x) = 1", "name": null, "statement": "For the example function, f(x) takes the value 1 when x is a fractional (non-integral) number.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "the example function, value at a fractional argument x" }, { "unit": null, "symbol": "x", "meaning": "argument (a fractional number)" } ], "sympy": "Eq(f(x), 1)", "physics": false, "states": [], "concepts": [ "concept/discontinuous-function", "concept/function", "concept/rational-number", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-c4759de494", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "155", "location": "Functions", "latex": "f(x) = 2", "name": null, "statement": "For the example function, f(x) takes the value 2 when x is an incommensurable number; this makes f discontinuous at every point.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "the example function, value at an incommensurable argument x" }, { "unit": null, "symbol": "x", "meaning": "argument (an incommensurable number)" } ], "sympy": "Eq(f(x), 2)", "physics": false, "states": [], "concepts": [ "concept/discontinuous-function", "concept/function", "concept/incommensurable-magnitudes", "concept/value-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-495dcc2ca8", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "153", "location": "Functions", "latex": "p = \\dfrac{1}{v}", "name": null, "statement": "The value p of the function is the reciprocal of the argument v (graphed in Chapter II for positive v).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "value of the function" }, { "unit": null, "symbol": "v", "meaning": "argument of the function (positive values in the physical application)" } ], "sympy": "Eq(p, 1/v)", "physics": true, "states": [], "concepts": [ "concept/continuous-function", "concept/function", "concept/graph-of-a-function", "concept/relation-between-variables" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-cb7891b045", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "153", "location": "Functions", "latex": "y = \\dfrac{1}{x}", "name": null, "statement": "y is the reciprocal of x for any value of x except 0; it is discontinuous at x = 0 and continuous on positive or on negative values only.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "argument of the function" } ], "sympy": "Eq(y, 1/x)", "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/discontinuous-function", "concept/function", "concept/graph-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-0ce27a921e", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "195", "location": "Series", "latex": "n × (n - 1) × (n - 2) × (n - 3) × \\dots × 4 × 3 × 2 × 1\\Add{,}", "name": null, "statement": "The number of permutations of n things is the product of the first n integers, which is written n!.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "n", "meaning": "finite integer, the number of things arranged" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/arrangement", "concept/finite-set", "concept/integer", "concept/permutation", "concept/product" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-f25ee20249", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "200", "location": "Series", "latex": "s_{n} = u_{1} + u_{2} + u_{3} + \\dots + u_{n}", "name": null, "statement": "The sum s_n of the first n terms of a series is the sum of the terms u_1 through u_n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s_{n}", "meaning": "sum of the first n terms of the series" }, { "unit": null, "symbol": "u_{n}", "meaning": "nth term of the series" }, { "unit": null, "symbol": "n", "meaning": "number of terms summed" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/partial-sum", "concept/sum", "concept/term" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-6d9d81f711", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "202", "location": "Series", "latex": "\\tfrac{1}{9} = .1 + \\tfrac{1}{90}", "name": null, "statement": "The recurring decimal .1111... equals 1/9, shown by writing 1/9 as the first term .1 plus the remainder 1/90.", "kind": "result", "symbols": [], "sympy": "Eq(Rational(1, 9), Rational(1, 10) + Rational(1, 90))", "physics": false, "states": [], "concepts": [ "concept/decimal-fraction", "concept/limit", "concept/rational-number", "concept/recurring-decimal", "concept/sum-to-infinity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-02b51eb846", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "206", "location": "Series", "latex": "s_{n} = 1 + x + x^{2} + x^{3} + \\dots + x^{n}", "name": null, "statement": "The sum of the first n terms of the geometrical series in x is s_n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s_{n}", "meaning": "sum of the first n terms of the geometric series" }, { "unit": null, "symbol": "x", "meaning": "variable (common ratio of the geometric series)" }, { "unit": null, "symbol": "n", "meaning": "number of terms" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/geometrical-progression", "concept/sum", "concept/term", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-38b8fe1c89", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "206", "location": "Series", "latex": "s_{n} = \\frac{1 - x^{n+1}}{1 - x}", "name": null, "statement": "For x not equal to 1, the sum of n terms of the geometric series equals (1 - x^(n+1))/(1 - x).", "kind": "result", "symbols": [ { "unit": null, "symbol": "s_{n}", "meaning": "sum of the first n terms of the geometric series" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "n", "meaning": "number of terms" } ], "sympy": "Eq(s_n, (1 - x**(n + 1))/(1 - x))", "physics": false, "states": [], "concepts": [ "concept/geometrical-progression", "concept/sum", "concept/variable" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-b6e6e99deb", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "207", "location": "Series", "latex": "\\frac{1}{1 - x} = 1 + x + x^{2} + \\dots + x^{n} + \\dots", "name": "sum of the geometric series", "statement": "For |x| < 1 the geometric series 1 + x + x^2 + ... converges, with sum 1/(1 - x).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable, with -1 < x < 1" } ], "sympy": null, "physics": false, "states": [ "theorem/sum-of-the-geometric-series" ], "concepts": [ "concept/convergent-series", "concept/function", "concept/geometrical-progression", "concept/interval", "concept/limit", "concept/sum-to-infinity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-e5c7020535", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "211", "location": "Series", "latex": "\\exp x = 1 + x + \\frac{x^{2}}{2!} + \\frac{x^{3}}{3!} + \\dots + \\frac{x^{n}}{n!} + \\dots", "name": "exponential series", "statement": "The exponential function exp x is defined as the sum to infinity of the exponential series.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "n", "meaning": "integer index of the term" } ], "sympy": null, "physics": false, "states": [ "theorem/exponential-series" ], "concepts": [ "concept/convergent-series", "concept/exponential-function", "concept/factorial", "concept/function", "concept/sum-to-infinity" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-f517f9b38a", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "211", "location": "Series", "latex": "(\\exp x) × (\\exp y) = \\exp(x + y)", "name": null, "statement": "The product of exp x and exp y equals exp(x + y); this is the addition-theorem of the exponential function.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" } ], "sympy": "Eq(exp(x)*exp(y), exp(x + y))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/function", "theorem/addition-theorem" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-5718dd3c31", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "212", "location": "Series", "latex": "\\sin(x + y) = \\sin x \\cos y + \\cos x \\sin y", "name": "addition theorem for the sine", "statement": "The sine of a sum equals sin x cos y plus cos x sin y.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle or variable" }, { "unit": null, "symbol": "y", "meaning": "angle or variable" } ], "sympy": "Eq(sin(x + y), sin(x)*cos(y) + cos(x)*sin(y))", "physics": false, "states": [ "theorem/addition-theorem-for-the-sine" ], "concepts": [ "concept/cosine", "concept/sine", "theorem/addition-theorem" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-5c5cb37b34", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "212", "location": "Series", "latex": "\\cos(x + y) = \\cos x \\cos y - \\sin x \\sin y", "name": "addition theorem for the cosine", "statement": "The cosine of a sum equals cos x cos y minus sin x sin y.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle or variable" }, { "unit": null, "symbol": "y", "meaning": "angle or variable" } ], "sympy": "Eq(cos(x + y), cos(x)*cos(y) - sin(x)*sin(y))", "physics": false, "states": [ "theorem/addition-theorem-for-the-cosine" ], "concepts": [ "concept/cosine", "concept/sine", "theorem/addition-theorem" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-7fe52db00b", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "212", "location": "Series", "latex": "\\sin x = x - \\frac{x^{3}}{3!} + \\frac{x^{5}}{5!} - \\frac{x^{7}}{7!} + \\text{etc.} \\dots", "name": null, "statement": "The sine is defined as the limit of the alternating power series in x.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (angle)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/factorial", "concept/limit", "concept/sine" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-f283bbb29c", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "212", "location": "Series", "latex": "\\cos x = 1 - \\frac{x^{2}}{2!} + \\frac{x^{4}}{4!} - \\frac{x^{6}}{6!} + \\text{etc.} \\dots", "name": null, "statement": "The cosine is defined as the limit of the alternating power series in x.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (angle)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/cosine", "concept/factorial", "concept/limit" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-eefb986697", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "214", "location": "Series", "latex": "y = \\exp(-x^{2})", "name": "curve of normal error", "statement": "The curve of normal error is the graph of exp(-x^2), whose function is central to statistics.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "value of the function at x" }, { "unit": null, "symbol": "x", "meaning": "argument of the function" } ], "sympy": "Eq(y, exp(-x**2))", "physics": false, "states": [ "concept/curve-of-normal-error" ], "concepts": [ "concept/argument-of-a-function", "concept/exponential-function", "concept/graph-of-a-function" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-c7d8305ef1", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xiv", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "215", "location": "Series", "latex": "y = \\exp(-cx) × \\sin \\frac{2\\pi x}{p}", "name": null, "statement": "A damped vibration: a sine of period p multiplied by an exponential decay exp(-cx) with constant percentage damping.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "displacement of the vibration at x" }, { "unit": null, "symbol": "x", "meaning": "time" }, { "unit": null, "symbol": "c", "meaning": "damping constant" }, { "unit": null, "symbol": "p", "meaning": "period of the vibration apart from friction" } ], "sympy": "Eq(y, exp(-c*x)*sin(2*pi*x/p))", "physics": true, "states": [], "concepts": [ "concept/damped-vibration", "concept/exponential-function", "concept/periodic-function", "concept/periodicity", "concept/sine", "quantity/pi" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-7aec5d73b3", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "238", "location": "Geometry", "latex": "\\frac{\\sin A}{a} = \\frac{\\sin B}{b} = \\frac{\\sin C}{c}", "name": "law of sines", "statement": "In a triangle, the sine of each angle divided by the side opposite it has the same value for all three angles.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A", "meaning": "plane angle of the triangle ABC at A" }, { "unit": null, "symbol": "B", "meaning": "plane angle of the triangle ABC at B" }, { "unit": null, "symbol": "C", "meaning": "plane angle of the triangle ABC at C" }, { "unit": null, "symbol": "a", "meaning": "side of the triangle opposite angle A" }, { "unit": null, "symbol": "b", "meaning": "side of the triangle opposite angle B" }, { "unit": null, "symbol": "c", "meaning": "side of the triangle opposite angle C" } ], "sympy": "And(Eq(sin(A)/a, sin(B)/b), Eq(sin(B)/b, sin(C)/c))", "physics": false, "states": [ "theorem/law-of-sines" ], "concepts": [ "concept/equality", "concept/plane-angle", "concept/side", "concept/sine", "concept/triangle", "concept/trigonometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-f4fb62a4cd", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "238", "location": "Geometry", "latex": "a^{2} = b^{2} + c^{2} - 2bc \\cos A", "name": "law of cosines", "statement": "The square of one side of a triangle equals the sum of the squares of the other two sides minus twice their product times the cosine of the angle between them.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side of the triangle opposite angle A" }, { "unit": null, "symbol": "b", "meaning": "side of the triangle opposite angle B" }, { "unit": null, "symbol": "c", "meaning": "side of the triangle opposite angle C" }, { "unit": null, "symbol": "A", "meaning": "plane angle of the triangle ABC at A" } ], "sympy": "Eq(a**2, b**2 + c**2 - 2*b*c*cos(A))", "physics": false, "states": [ "theorem/law-of-cosines" ], "concepts": [ "concept/cosine", "concept/equality", "concept/plane-angle", "concept/side", "concept/triangle", "concept/trigonometry" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-8f3934de24", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "239", "location": "Geometry", "latex": "A + B + C = 180°", "name": null, "statement": "The three angles of a triangle, counted in degrees, add up to 180 degrees (two right angles).", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "A", "meaning": "number of degrees in the angle of triangle ABC at A" }, { "unit": "degree", "symbol": "B", "meaning": "number of degrees in the angle of triangle ABC at B" }, { "unit": "degree", "symbol": "C", "meaning": "number of degrees in the angle of triangle ABC at C" } ], "sympy": "Eq(A + B + C, 180)", "physics": false, "states": [], "concepts": [ "concept/equality", "concept/plane-angle", "concept/sum", "concept/triangle", "quantity/right-angle", "unit/degree-of-angle" ] }, { "id": "whitehead-introduction-to-mathematics-1911/eq-b006ad1e88", "chapter": "whitehead-introduction-to-mathematics-1911/ch-xvi", "book": "whitehead-introduction-to-mathematics-1911", "edition": "Williams and Norgate (Home University Library), \"New and Revised Edition\", no year printed (first published 1911; edition to be confirmed from the copy)", "page": "239", "location": "Geometry", "latex": "a < b + c", "name": "triangle inequality", "statement": "Each side of a triangle is shorter than the sum of the other two sides (the same holds for b and c in the book's companion inequalities b < c + a and c < a + b).", "kind": "rule", "symbols": [ { "unit": "foot", "symbol": "a", "meaning": "length of the side of the triangle opposite angle A" }, { "unit": "foot", "symbol": "b", "meaning": "length of the side of the triangle opposite angle B" }, { "unit": "foot", "symbol": "c", "meaning": "length of the side of the triangle opposite angle C" } ], "sympy": "Lt(a, b + c)", "physics": false, "states": [ "theorem/sum-of-two-sides-of-a-triangle-exceeds-the-third" ], "concepts": [ "concept/inequality", "concept/real-number", "concept/side", "concept/sum", "concept/triangle", "quantity/length" ] } ], "exercise_sets": [], "problems": [], "errata_statuses": [ "transcriber_marked", "candidate", "probable", "confirmed", "dismissed", "note" ], "errata": [] }