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BOOK V: The Sphere", "pages": [ "142", "160" ], "concepts": [ "concept/axis-of-a-circle-on-a-sphere", "concept/birectangular-spherical-triangle", "concept/circle", "concept/circumscribed-polyhedron", "concept/diameter", "concept/great-circle", "concept/isosceles-spherical-triangle", "concept/lune", "concept/perpendicular", "concept/polar-distance-of-a-circle", "concept/polar-triangle", "concept/pole-of-a-circle", "concept/polyhedral-angle", "concept/polyhedron-inscribed-in-a-sphere", "concept/quadrant", "concept/right-spherical-triangle", "concept/small-circle", "concept/sphere", "concept/sphere-inscribed-in-a-polyhedron", "concept/spherical-angle", "concept/spherical-cone", "concept/spherical-distance", "concept/spherical-polygon", "concept/spherical-sector", "concept/spherical-segment", "concept/spherical-triangle", "concept/spherical-zone", "concept/tangent-plane-to-a-sphere", "concept/tetrahedron", "method/approximating-the-volume-of-a-sphere-by-pyramids", "method/constructing-a-spherical-triangle-from-its-angles", "method/constructing-a-spherical-triangle-from-its-sides", "method/finding-the-diameter-of-a-sphere", "method/inscribing-a-sphere-in-a-tetrahedron", "person/adrien-marie-legendre", "person/cavalieri", "quantity/altitude-of-a-spherical-zone", "quantity/radius", "quantity/spherical-excess", "theorem/angle-and-corresponding-arc-of-polar-triangle-sum-to-180-degrees", "theorem/angles-opposite-equal-sides-of-a-spherical-triangle-are-equal", "theorem/area-of-a-sphere", "theorem/area-of-a-spherical-polygon", "theorem/area-of-a-spherical-triangle", "theorem/area-of-a-spherical-zone", "theorem/assumption-on-the-area-of-a-sphere", "theorem/equal-circles-on-a-sphere-have-equidistant-planes", "theorem/intersection-of-two-spherical-surfaces-is-a-circle", "theorem/plane-section-of-a-sphere-is-a-circle", "theorem/points-of-a-circle-on-a-sphere-are-equidistant-from-its-pole", "theorem/polar-triangle-relation-is-symmetric", 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"slaught-lennes-solid-geometry-1919/eq-255f3085ef", "slaught-lennes-solid-geometry-1919/eq-e1301e6d39", "slaught-lennes-solid-geometry-1919/eq-869308b42c", "slaught-lennes-solid-geometry-1919/eq-f9c89fb8ad", "slaught-lennes-solid-geometry-1919/eq-c125141b9f", "slaught-lennes-solid-geometry-1919/eq-fb261286ae", "slaught-lennes-solid-geometry-1919/eq-4507496565", "slaught-lennes-solid-geometry-1919/eq-6f6e7b3580", "slaught-lennes-solid-geometry-1919/eq-52f362833c", "slaught-lennes-solid-geometry-1919/eq-2a99cddeb3", "slaught-lennes-solid-geometry-1919/eq-d90f6ceff7", "slaught-lennes-solid-geometry-1919/eq-fd4248d33c", "slaught-lennes-solid-geometry-1919/eq-95329cfabd", "slaught-lennes-solid-geometry-1919/eq-dc738616b3", "slaught-lennes-solid-geometry-1919/eq-b40f3341f6", "slaught-lennes-solid-geometry-1919/eq-53cd9dc361", "slaught-lennes-solid-geometry-1919/eq-3ede131a14", "slaught-lennes-solid-geometry-1919/eq-4ccd124ffd", "slaught-lennes-solid-geometry-1919/eq-7e50e6461e", "slaught-lennes-solid-geometry-1919/eq-0a2fa5b1db", "slaught-lennes-solid-geometry-1919/eq-67f555ee50", "slaught-lennes-solid-geometry-1919/eq-0529a7d720", "slaught-lennes-solid-geometry-1919/eq-97a9521634", "slaught-lennes-solid-geometry-1919/eq-38413ca8c5", "slaught-lennes-solid-geometry-1919/eq-7ff4b23bbc", "slaught-lennes-solid-geometry-1919/eq-d73b772dd5", "slaught-lennes-solid-geometry-1919/eq-e3efb21594", "slaught-lennes-solid-geometry-1919/eq-bd680b3170", "slaught-lennes-solid-geometry-1919/eq-a30389bbc7" ], "exercise_sets": [] }, { "id": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "number": "Appendix I", "title": "SIMILAR SOLIDS", "name": "Slaught & Lennes 1919, ch. Appendix I: SIMILAR SOLIDS", "pages": [ "160", "176" ], "concepts": [ "concept/center-of-similitude", "concept/corresponding-linear-dimensions", "concept/corresponding-parts", "concept/ratio-of-similitude", "concept/similar-cones", "concept/similar-cylinders", "concept/similar-polyhedron", "concept/similar-solids", "instrument/pantograph", "theorem/area-and-volume-ratios-of-similar-solids", "theorem/figures-with-a-center-of-similitude-are-similar", "theorem/ratios-of-similar-cones", "theorem/ratios-of-similar-cylinders", "theorem/two-similar-triangles-may-be-placed-with-a-center-of-similitude", "theorem/two-tetrahedrons-are-similar-if-three-faces-are-similar-and-similarly-placed", "theorem/volumes-of-similar-polyhedrons", "theorem/volumes-of-similar-solids", "theorem/volumes-of-similar-tetrahedrons", "theorem/volumes-of-tetrahedrons-with-equal-trihedral-angles" ], "excerpts": [ "slaught-lennes-solid-geometry-1919/x-adcc3e958f", "slaught-lennes-solid-geometry-1919/x-f80a05cef5", "slaught-lennes-solid-geometry-1919/x-6b3219a64c", "slaught-lennes-solid-geometry-1919/x-6f4a53fc18", "slaught-lennes-solid-geometry-1919/x-3d5b568d5a", "slaught-lennes-solid-geometry-1919/x-72a4d0b1b5" ], "equations": [ "slaught-lennes-solid-geometry-1919/eq-2a99cddeb3", "slaught-lennes-solid-geometry-1919/eq-8802f8e67d", "slaught-lennes-solid-geometry-1919/eq-aebb1a1c99", "slaught-lennes-solid-geometry-1919/eq-d540dbf5a7", "slaught-lennes-solid-geometry-1919/eq-d4cfef6e4b", "slaught-lennes-solid-geometry-1919/eq-4b7ffe76a7", "slaught-lennes-solid-geometry-1919/eq-346882baa5", "slaught-lennes-solid-geometry-1919/eq-96e1a345b5", "slaught-lennes-solid-geometry-1919/eq-6aaab052f2", "slaught-lennes-solid-geometry-1919/eq-5aa83460ab", "slaught-lennes-solid-geometry-1919/eq-dc1d03ed3b", "slaught-lennes-solid-geometry-1919/eq-ccda6ac7b8", "slaught-lennes-solid-geometry-1919/eq-5430f639bf", "slaught-lennes-solid-geometry-1919/eq-d0175894ee" ], "exercise_sets": [] }, { "id": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "number": "Appendix II", "title": "PROJECTION OF LINE-SEGMENTS", "name": "Slaught & Lennes 1919, ch. Appendix II: PROJECTION OF LINE-SEGMENTS", "pages": [ "176", "188" ], "concepts": [ "concept/angle-between-line-and-plane", "concept/cosine", "concept/ellipse", "concept/oblique-prism", "concept/projection-of-a-line", "concept/right-section", "concept/right-triangle", "concept/sine", "concept/tangent-function", "method/interpolating-in-logarithmic-tables", "theorem/altitude-of-an-oblique-prism-or-cylinder", "theorem/area-of-an-ellipse", "theorem/area-of-the-projection-of-a-plane-segment", "theorem/dihedral-angle-of-an-oblique-prism" ], "excerpts": [ "slaught-lennes-solid-geometry-1919/x-456a72a179", "slaught-lennes-solid-geometry-1919/x-daf88d8f44", "slaught-lennes-solid-geometry-1919/x-0b217327b2", "slaught-lennes-solid-geometry-1919/x-89b21a1191", "slaught-lennes-solid-geometry-1919/x-03d29ae0d6", "slaught-lennes-solid-geometry-1919/x-8c87fb6de6", "slaught-lennes-solid-geometry-1919/x-239be3bc6b" ], "equations": [ "slaught-lennes-solid-geometry-1919/eq-cdcc0d802d", "slaught-lennes-solid-geometry-1919/eq-3f9f666c54", "slaught-lennes-solid-geometry-1919/eq-8994ca1c43", "slaught-lennes-solid-geometry-1919/eq-642039cf78", "slaught-lennes-solid-geometry-1919/eq-b1d96f5e7f", "slaught-lennes-solid-geometry-1919/eq-d55b598482", "slaught-lennes-solid-geometry-1919/eq-a2416ba81b", "slaught-lennes-solid-geometry-1919/eq-9c25319856" ], "exercise_sets": [] }, { "id": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "number": "Appendix III", "title": "VARIABLES. LIMITS", "name": "Slaught & Lennes 1919, ch. Appendix III: VARIABLES. LIMITS", "pages": [ "202", "210" ], "concepts": [ "concept/approximating-sequence-of-polygons", "concept/circumscribed-prism", "concept/constant", "concept/convex-plane-curve", "concept/element-of-a-cylinder", "concept/function", "concept/hemisphere", "concept/incommensurable-magnitudes", "concept/infinite-sequence", "concept/inscribed-prism", "concept/irrational-number", "concept/least-upper-bound", "concept/limit", "concept/perimeter", "concept/rational-number", "concept/rectangular-parallelepiped", "concept/right-section", "concept/variable", "person/cavalieri", "quantity/area", "quantity/circumference", "quantity/lateral-area", "quantity/volume", "theorem/area-of-a-rectangle", "theorem/area-of-a-sphere", "theorem/bounded-monotone-sequence-has-a-limit", "theorem/circumferences-and-areas-of-circles-vary-as-radii-and-squares-of-radii", "theorem/equal-cross-sections-imply-equal-volumes", "theorem/lateral-area-formula-for-a-cylinder", "theorem/limits-of-equal-variables-are-equal", "theorem/mutually-bounded-decreasing-sequences-have-the-same-limit", "theorem/mutually-bounded-increasing-sequences-have-the-same-limit", "theorem/pyramids-with-equal-altitudes-and-equal-base-areas-have-equal-volumes", "theorem/volume-of-a-cylinder", "theorem/volume-of-a-rectangular-parallelepiped", "theorem/volume-of-a-sphere" ], "excerpts": [ "slaught-lennes-solid-geometry-1919/x-ea4e45500d", "slaught-lennes-solid-geometry-1919/x-0c5cdcc596", "slaught-lennes-solid-geometry-1919/x-c7eb4f04a9", "slaught-lennes-solid-geometry-1919/x-bc0ff198d8", "slaught-lennes-solid-geometry-1919/x-2309e7ddda", "slaught-lennes-solid-geometry-1919/x-cbda4c3b61", "slaught-lennes-solid-geometry-1919/x-e58899fe20", "slaught-lennes-solid-geometry-1919/x-4df6f1895b", "slaught-lennes-solid-geometry-1919/x-29d93f112a", "slaught-lennes-solid-geometry-1919/x-c8f0038d6a", "slaught-lennes-solid-geometry-1919/x-d007cf4a46" ], "equations": [ "slaught-lennes-solid-geometry-1919/eq-7aa76b6a93", "slaught-lennes-solid-geometry-1919/eq-e3873c2f19", "slaught-lennes-solid-geometry-1919/eq-19bdc9073b", "slaught-lennes-solid-geometry-1919/eq-3e79b28097", "slaught-lennes-solid-geometry-1919/eq-b0ea85892c", "slaught-lennes-solid-geometry-1919/eq-fa944052f1", "slaught-lennes-solid-geometry-1919/eq-6f4168ebad", "slaught-lennes-solid-geometry-1919/eq-bd67296c88", "slaught-lennes-solid-geometry-1919/eq-db5918bc12", "slaught-lennes-solid-geometry-1919/eq-021b0ad7eb" ], "exercise_sets": [] } ], "excerpts": [ { "id": "slaught-lennes-solid-geometry-1919/x-a08831d69a", "chapter": "slaught-lennes-solid-geometry-1919/ch-introduction", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "1", "location": "", "latex": "In plane geometry each figure is restricted so that all of its parts lie in the same plane. Such figures are called \\emph{two-dimensional figures}.", "markdown": "In plane geometry each figure is restricted so that all of its parts lie in the same plane. Such figures are called *two-dimensional figures*.", "why": "Gives the learner the defining restriction that separates plane figures from solid ones.", "use": [ "lesson" ], "concepts": [ "concept/two-dimensional-figure" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-635024d72b", "chapter": "slaught-lennes-solid-geometry-1919/ch-introduction", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "1", "location": "", "latex": "A figure, all parts of which lie in one straight line, is a \\emph{one-dimensional figure}, while a point is of zero dimensions.", "markdown": "A figure, all parts of which lie in one straight line, is a *one-dimensional figure*, while a point is of zero dimensions.", "why": "Places the line and the point on the same ladder of dimensions as plane and solid figures.", "use": [ "lesson" ], "concepts": [ "concept/one-dimensional-figure", "concept/point" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-07981ac5b3", "chapter": "slaught-lennes-solid-geometry-1919/ch-introduction", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "3", "location": "", "latex": "In plane geometry, two lines which do not meet are parallel, while in solid geometry, two lines which do not meet need not be parallel. That is, they may not be in the same plane. Lines which are not parallel and do not meet are called \\emph{skew} lines.", "markdown": "In plane geometry, two lines which do not meet are parallel, while in solid geometry, two lines which do not meet need not be parallel. That is, they may not be in the same plane. Lines which are not parallel and do not meet are called *skew* lines.", "why": "Shows a learner why a fact that is true in the plane fails in space, and names the new case.", "use": [ "lesson", "website" ], "concepts": [ "concept/parallel-lines", "concept/skew-lines", "concept/solid-geometry" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-a6ef59e1f1", "chapter": "slaught-lennes-solid-geometry-1919/ch-introduction", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "2", "location": "", "latex": "In plane geometry, ``the locus of all points at a given distance from a given point'' is a circle, while in solid geometry this locus is a sphere.", "markdown": "In plane geometry, “the locus of all points at a given distance from a given point” is a circle, while in solid geometry this locus is a sphere.", "why": "A short contrast that lets a learner see how the same definition of locus changes its shape when a dimension is added.", "use": [ "lesson", "website" ], "concepts": [ "concept/locus", "concept/sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-dc448934d2", "chapter": "slaught-lennes-solid-geometry-1919/ch-introduction", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "2", "location": "", "latex": "If all parts of the figure are not required to lie in one plane, the theorem just quoted is far from true.", "markdown": "If all parts of the figure are not required to lie in one plane, the theorem just quoted is far from true.", "why": "Warns the learner that a plane theorem cannot be carried into space without checking its hypothesis.", "use": [ "lesson" ], "concepts": [ "concept/perpendicular", "concept/plane" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-325976aead", "chapter": "slaught-lennes-solid-geometry-1919/ch-introduction", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "1", "location": "", "latex": "A plane is designated by a single letter in it, by two letters at opposite corners of the parallelogram representing it, or by any three letters in it but not in the same straight line.", "markdown": "A plane is designated by a single letter in it, by two letters at opposite corners of the parallelogram representing it, or by any three letters in it but not in the same straight line.", "why": "Gives the three standard ways to name a plane, which the learner will use throughout the book.", "use": [ "lesson" ], "concepts": [ "concept/plane" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-84f36c2072", "chapter": "slaught-lennes-solid-geometry-1919/ch-introduction", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "8", "location": "", "latex": "The area of a circle is one half the circumference times the radius, or in symbols:", "markdown": "The area of a circle is one half the circumference times the radius, or in symbols:", "why": "Presents the area of a circle as a consequence of circumference and radius, a link a learner can check against the formula that follows.", "use": [ "lesson", "website" ], "concepts": [ "quantity/circumference", "quantity/radius", "theorem/area-of-a-circle" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-954cc854e9", "chapter": "slaught-lennes-solid-geometry-1919/ch-introduction", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "5", "location": "", "latex": "Through a point not on a given line only one straight line can be drawn parallel to that line.", "markdown": "Through a point not on a given line only one straight line can be drawn parallel to that line.", "why": "States the parallel postulate that plane geometry rests on, the premise the space results later qualify.", "use": [ "history" ], "concepts": [ "concept/parallel-lines", "theorem/parallel-postulate" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-beb86089f3", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "9", "location": "Properties of the Plane", "latex": "If a line or a plane contains a point, the point is said to be \\emph{on} the line or \\emph{in} the plane and the line or plane is said to pass \\emph{through} the point.", "markdown": "If a line or a plane contains a point, the point is said to be *on* the line or *in* the plane and the line or plane is said to pass *through* the point.", "why": "It fixes the basic vocabulary of 'on', 'in' and 'through' that every later figure relies on.", "use": [ "lesson" ], "concepts": [ "concept/line", "concept/plane", "concept/point" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-4d6f7b6163", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "9", "location": "Properties of the Plane", "latex": "While two points determine a straight line it is obvious that two points do not determine a plane.", "markdown": "While two points determine a straight line it is obvious that two points do not determine a plane.", "why": "It shows a learner why three points, not two, are needed to fix a plane.", "use": [ "lesson" ], "concepts": [ "concept/determination-of-a-plane", "concept/line", "concept/point" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-9a1a9d95a3", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "9", "location": "Properties of the Plane", "latex": "We, therefore, say that three non-collinear points determine a plane, while any number of collinear points fail to determine a plane.", "markdown": "We, therefore, say that three non-collinear points determine a plane, while any number of collinear points fail to determine a plane.", "why": "It states the rule for when points fix a plane, in the form a student can apply directly.", "use": [ "lesson" ], "concepts": [ "concept/collinear-points", "concept/determination-of-a-plane" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-955bbba1bc", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "10", "location": "Properties of the Plane", "latex": "In the figure, $PA$ is perpendicular to the plane~$M$ and $QA$ is oblique to it.", "markdown": "In the figure, $PA$ is perpendicular to the plane $M$ and $QA$ is oblique to it.", "why": "It sets a perpendicular line beside an oblique one from the same foot, so the contrast is visible.", "use": [ "lesson" ], "concepts": [ "concept/oblique-line", "concept/perpendicular" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-175d5645cb", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "17", "location": "Properties of the Plane", "latex": "The perpendicular is the shortest distance from a point to a plane.", "markdown": "The perpendicular is the shortest distance from a point to a plane.", "why": "It explains why the distance from a point to a plane is measured along the perpendicular.", "use": [ "lesson" ], "concepts": [ "concept/distance-from-a-point-to-a-plane", "concept/perpendicular" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-91e91c5753", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "24", "location": "Properties of the Plane", "latex": "Note that the four vertices of a quadrilateral in space do not necessarily all lie in the same plane.", "markdown": "Note that the four vertices of a quadrilateral in space do not necessarily all lie in the same plane.", "why": "It warns learners not to assume that four points in space are coplanar.", "use": [ "lesson" ], "concepts": [ "concept/plane" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-60f72a7a4e", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "20", "location": "Properties of the Plane", "latex": "Any line, as $l_1$, in either of two parallel planes, $M$ and $N$, is parallel to the other plane.", "markdown": "Any line, as $l_1$, in either of two parallel planes, $M$ and $N$, is parallel to the other plane.", "why": "It links parallel planes to lines parallel to a plane, which is how the parallel theorems are later used.", "use": [ "lesson" ], "concepts": [ "concept/line-parallel-to-a-plane", "concept/parallel-planes" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-96f111d37b", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "26", "location": "Properties of the Plane", "latex": "If two straight lines are cut by three parallel planes, the intercepted segments on one line are proportional to the corresponding segments on the other.", "markdown": "If two straight lines are cut by three parallel planes, the intercepted segments on one line are proportional to the corresponding segments on the other.", "why": "States the central result of the section in a form a learner can apply to a slanting rod cut by parallel shelves.", "use": [ "lesson" ], "concepts": [ "concept/parallel-planes", "theorem/parallel-planes-intercept-proportional-segments" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-2a940e24b4", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "28", "location": "Properties of the Plane", "latex": "Two half-planes meeting in a common edge form a \\emph{dihedral angle}.", "markdown": "Two half-planes meeting in a common edge form a *dihedral angle*.", "why": "Gives the defining sentence of a dihedral angle, the solid-geometry analogue of a plane angle.", "use": [ "lesson" ], "concepts": [ "concept/dihedral-angle", "concept/half-plane" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-9917bf5de1", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "28", "location": "Properties of the Plane", "latex": "angle may be thought of as \\emph{generated} by the rotation of a half-plane about its edge. The magnitude of the angle depends solely upon the \\emph{amount of rotation}.", "markdown": "angle may be thought of as *generated* by the rotation of a half-plane about its edge. The magnitude of the angle depends solely upon the *amount of rotation*.", "why": "A concrete picture of a dihedral angle as a turning half-plane, which makes its size easy to grasp.", "use": [ "lesson" ], "concepts": [ "concept/dihedral-angle", "quantity/angle" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-5588179ddd", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "32", "location": "Properties of the Plane", "latex": "point on a plane is the foot of the perpendicular from the point to the plane.", "markdown": "point on a plane is the foot of the perpendicular from the point to the plane.", "why": "Defines the projection of a point, the basis for every later projection argument in the chapter.", "use": [ "lesson" ], "concepts": [ "concept/foot-of-a-line", "concept/projection" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-7f11d71368", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "33", "location": "Properties of the Plane", "latex": "The acute angle formed by a straight line with its own projection on a plane is the least angle which it makes with any line in that plane.", "markdown": "The acute angle formed by a straight line with its own projection on a plane is the least angle which it makes with any line in that plane.", "why": "A vivid minimum property that explains why the angle between a line and a plane is defined through its projection.", "use": [ "lesson", "website" ], "concepts": [ "concept/angle-between-line-and-plane", "theorem/acute-angle-with-projection-is-the-least-angle-with-lines-in-the-plane" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-8a33e53a3f", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "41", "location": "Properties of the Plane", "latex": "The trihedral angles $O$ and $O'$ cannot be made to coincide, even though their corresponding parts are equal. This can be illustrated by attempting to put a left glove on the right hand.", "markdown": "The trihedral angles $O$ and $O'$ cannot be made to coincide, even though their corresponding parts are equal. This can be illustrated by attempting to put a left glove on the right hand.", "why": "The glove image makes the idea of mirror-image order of parts memorable for a learner.", "use": [ "website" ], "concepts": [ "concept/order-of-parts", "concept/trihedral-angle" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-f5687e2e63", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "27", "location": "Properties of the Plane", "latex": "Show that line-segments included between parallel planes and perpendicular to them are equal, and hence that parallel planes are everywhere equally distant.", "markdown": "Show that line-segments included between parallel planes and perpendicular to them are equal, and hence that parallel planes are everywhere equally distant.", "why": "A practical exercise that links parallel planes to the carpenter's task of placing two shelves level.", "use": [ "lesson" ], "concepts": [ "concept/parallel-planes", "concept/perpendicular" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-4b77751d59", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "50", "location": "Regular Polyhedrons", "latex": "A polyhedron is \\emph{convex} if every section of it made by a plane is a convex polygon.", "markdown": "A polyhedron is *convex* if every section of it made by a plane is a convex polygon.", "why": "It gives the test a learner can apply to any solid: slice it with a plane and check whether every section is convex.", "use": [ "lesson" ], "concepts": [ "concept/convex-polyhedron", "concept/polyhedron" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-e7b3b477fb", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "50", "location": "Regular Polyhedrons", "latex": "These names are all derived from the Greek and refer to the number of faces.", "markdown": "These names are all derived from the Greek and refer to the number of faces.", "why": "It shows the learner that the names of the polyhedrons are a code: the Greek root gives the number of faces.", "use": [ "lesson", "history" ], "concepts": [ "concept/dodecahedron", "concept/hexahedron", "concept/icosahedron", "concept/octahedron", "concept/tetrahedron" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-2df6f50a5a", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "50", "location": "Regular Polyhedrons", "latex": "The faces, edges, and vertices taken together form the surface of the polyhedron.", "markdown": "The faces, edges, and vertices taken together form the surface of the polyhedron.", "why": "It pins down what the surface of a solid is, so learners can separate the boundary from the solid it encloses.", "use": [ "lesson" ], "concepts": [ "concept/edge", "concept/face", "concept/surface-of-a-polyhedron", "concept/vertex" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-4dd8dca176", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "53", "location": "Regular Polyhedrons", "latex": "From these propositions it follows that each of the polyhedral angles of a regular polyhedron may be formed by three, four, or five (but not six) equilateral triangles; by three (but not four) squares; or by three (but not four) regular pentagons.", "markdown": "From these propositions it follows that each of the polyhedral angles of a regular polyhedron may be formed by three, four, or five (but not six) equilateral triangles; by three (but not four) squares; or by three (but not four) regular pentagons.", "why": "It turns the two face-angle facts into the concrete list of which regular polygons can meet at a vertex, which is the core reason there are only five.", "use": [ "lesson" ], "concepts": [ "concept/polyhedral-angle", "concept/regular-polyhedron", "theorem/polyhedral-angle-has-at-least-three-face-angles", "theorem/sum-of-face-angles-of-a-polyhedral-angle-is-less-than-360-degrees" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-f13cc37fe2", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "53", "location": "Regular Polyhedrons", "latex": "\\textit{The sum of the face angles of a polyhedral angle is less than~$360$°}.", "markdown": "*The sum of the face angles of a polyhedral angle is less than $360$°*.", "why": "It states the limit that stops a vertex from closing up flat, which the learner needs before testing any candidate polygon.", "use": [ "lesson" ], "concepts": [ "concept/face-angle", "concept/polyhedral-angle", "theorem/sum-of-face-angles-of-a-polyhedral-angle-is-less-than-360-degrees" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-fdeabe6260", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "53", "location": "Regular Polyhedrons", "latex": "At the center~$E$ of an equilateral triangle~$ABC$ erect a perpendicular to the plane of the triangle.", "markdown": "At the center $E$ of an equilateral triangle $ABC$ erect a perpendicular to the plane of the triangle.", "why": "It shows how a regular solid is built from one equilateral triangle by erecting a perpendicular at its centre, a step the learner can carry out with card and a ruler.", "use": [ "lesson" ], "concepts": [ "method/constructing-a-line-perpendicular-to-a-plane", "method/constructing-a-regular-tetrahedron" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-face78773c", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "64", "location": "Prisms and Cylinders", "latex": "\\textit{Two polyhedrons have the same volume if they can be made to coincide, or if they can be divided into parts which can be made to coincide in pairs}.", "markdown": "*Two polyhedrons have the same volume if they can be made to coincide, or if they can be divided into parts which can be made to coincide in pairs*.", "why": "It states the principle that lets volumes of general prisms be found by cutting and rearranging solids, without new measurement.", "use": [ "lesson" ], "concepts": [ "concept/equivalent-solids", "quantity/volume", "theorem/volume-of-a-prism" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-1db6a72c74", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "68", "location": "Prisms and Cylinders", "latex": "Any prism, can be divided into triangular prisms by planes passing through one edge and each of the other non-adjacent edges.", "markdown": "Any prism, can be divided into triangular prisms by planes passing through one edge and each of the other non-adjacent edges.", "why": "This is the method for reducing any prism to triangular prisms, whose volume is already known.", "use": [ "lesson" ], "concepts": [ "concept/prism", "theorem/volume-of-a-prism", "theorem/volume-of-a-triangular-prism" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-3133683442", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "62", "location": "Prisms and Cylinders", "latex": "but no attempt has been made to \\emph{measure the space occupied} by such a solid. For this purpose we consider first a rectangular parallelopiped.", "markdown": "but no attempt has been made to *measure the space occupied* by such a solid. For this purpose we consider first a rectangular parallelopiped.", "why": "It frames volume as a new question about the space a solid occupies, and says why the book starts with the simplest box.", "use": [ "lesson" ], "concepts": [ "concept/rectangular-parallelepiped", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-0e18348065", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "62", "location": "Prisms and Cylinders", "latex": "Since the edge~$BE$ is $5$~units long, $5$ such tiers will exactly fill the space within the solid. That is, $5\\cdot 3\\cdot 4 = 60$ is the number of cubic units in the solid.", "markdown": "Since the edge $BE$ is $5$ units long, $5$ such tiers will exactly fill the space within the solid. That is, $5\\cdot 3\\cdot 4 = 60$ is the number of cubic units in the solid.", "why": "A small whole-number example shows, by counting layers of unit cubes, why volume is the product of the three edges.", "use": [ "lesson" ], "concepts": [ "concept/rectangular-parallelepiped", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-4b826000d5", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "63", "location": "Prisms and Cylinders", "latex": "E.g.,~if the length is $5$~inches and the width is $\\sqrt{3}$~inches, then the base cannot be exactly covered with equal cubes, however small.", "markdown": "E.g., if the length is $5$ inches and the width is $\\sqrt{3}$ inches, then the base cannot be exactly covered with equal cubes, however small.", "why": "It shows learners that some lengths cannot be measured by any unit cube exactly, which is why the book uses approximation.", "use": [ "lesson", "history" ], "concepts": [ "concept/rectangular-parallelepiped", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-8256f36273", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "57", "location": "Prisms and Cylinders", "latex": "The form of statement in this theorem is the usual abbreviation for the more \\emph{precise} form:", "markdown": "The form of statement in this theorem is the usual abbreviation for the more *precise* form:", "why": "It warns learners that the statement 'lateral area equals edge times perimeter' means the numerical measures of those lengths, a common source of confusion.", "use": [ "lesson" ], "concepts": [ "concept/lateral-surface", "theorem/lateral-area-formula-for-a-prism" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-fde93cb217", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "74", "location": "Prisms and Cylinders", "latex": "Indeed, in these cases approximate measurement only is possible, since no square unit exists in terms of which such areas can be \\emph{exactly} measured.", "markdown": "Indeed, in these cases approximate measurement only is possible, since no square unit exists in terms of which such areas can be *exactly* measured.", "why": "It explains why curved surfaces need approximation, by contrast with plane figures, which can be measured exactly.", "use": [ "lesson", "history" ], "concepts": [ "concept/curved-surface", "quantity/area" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-e7a92791bb", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "75", "location": "Prisms and Cylinders", "latex": "\\textit{A cylinder has a definite lateral area and a definite volume which may be approximated as nearly as we please by taking the lateral areas and the volumes of successive inscribed or circumscribed prisms}.", "markdown": "*A cylinder has a definite lateral area and a definite volume which may be approximated as nearly as we please by taking the lateral areas and the volumes of successive inscribed or circumscribed prisms*.", "why": "It states the fundamental assumption that lets a learner treat a cylinder's area and volume as limits of prism values.", "use": [ "lesson" ], "concepts": [ "method/approximating-the-area-and-volume-of-a-cylinder-by-prisms", "quantity/lateral-area", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-55a9bd14b8", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "76", "location": "Prisms and Cylinders", "latex": "The lateral area of a cylinder can be computed only in case the perimeter of a right section can be found, and this is possible by elementary methods only in case the right section is a circle.", "markdown": "The lateral area of a cylinder can be computed only in case the perimeter of a right section can be found, and this is possible by elementary methods only in case the right section is a circle.", "why": "It shows why circular cylinders are the case that elementary geometry can handle, which motivates the formulas that follow.", "use": [ "lesson" ], "concepts": [ "concept/circular-cylinder", "quantity/lateral-area", "quantity/perimeter" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-76d6cdfa3a", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "70", "location": "Prisms and Cylinders", "latex": "The moving line is the \\emph{generator}, and the generator in any one of its positions is an \\emph{element} of the surface.", "markdown": "The moving line is the *generator*, and the generator in any one of its positions is an *element* of the surface.", "why": "It gives the learner a clear mental picture of how a cylinder's surface is built by a sliding line.", "use": [ "lesson", "website" ], "concepts": [ "concept/cylindrical-surface", "concept/element-of-a-cylinder", "concept/generator" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-625bba13fc", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "76", "location": "Prisms and Cylinders", "latex": "In this case $r = r_1 = r_2$, and $e = h$, and the formulas of corollaries~1 and~2 become identical.", "markdown": "In this case $r = r_1 = r_2$, and $e = h$, and the formulas of corollaries 1 and 2 become identical.", "why": "It shows the learner that the general volume formula reduces to the familiar right circular cylinder formula, tying the cases together.", "use": [ "lesson" ], "concepts": [ "concept/circular-cylinder", "theorem/volume-of-a-cylinder" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-a08d3297d2", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "80", "location": "Pyramids and Cones", "latex": "If a line through the fixed point moves so as to touch the boundary of the polygon and is made to traverse it completely, the line is said to generate a convex \\emph{pyramidal surface}.", "markdown": "If a line through the fixed point moves so as to touch the boundary of the polygon and is made to traverse it completely, the line is said to generate a convex *pyramidal surface*.", "why": "It gives the learner the motion-based definition of a pyramidal surface, which makes the generator, element and vertex intelligible as parts of one picture.", "use": [ "lesson" ], "concepts": [ "concept/generator", "concept/pyramidal-surface", "concept/vertex" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-48340aac51", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "82", "location": "Pyramids and Cones", "latex": "The lateral area of a regular pyramid is equal to one half the product of its slant height and the perimeter of the base.", "markdown": "The lateral area of a regular pyramid is equal to one half the product of its slant height and the perimeter of the base.", "why": "It states the lateral area rule in words a learner can apply before reading the proof, linking slant height to surface area.", "use": [ "lesson" ], "concepts": [ "concept/regular-pyramid", "concept/slant-height", "quantity/lateral-area", "theorem/lateral-area-of-a-regular-pyramid" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-5133043da9", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "88", "location": "Pyramids and Cones", "latex": "The volume of any pyramid is one third of the product of its base and altitude.", "markdown": "The volume of any pyramid is one third of the product of its base and altitude.", "why": "It is the central volume result for pyramids and the sentence a learner should be able to state and use.", "use": [ "lesson", "website" ], "concepts": [ "concept/altitude-of-a-pyramid", "concept/base-of-a-pyramid", "quantity/volume", "theorem/volume-of-a-pyramid" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-07e3683e62", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "85", "location": "Pyramids and Cones", "latex": "\\textit{A pyramid has a definite volume which is less than the combined volume of any set of circumscribed prisms and greater than that of any set of inscribed prisms}.", "markdown": "*A pyramid has a definite volume which is less than the combined volume of any set of circumscribed prisms and greater than that of any set of inscribed prisms*.", "why": "It shows the book openly adopting an assumption about volume rather than proving it, so learners see where the argument's foundation lies.", "use": [ "lesson", "history" ], "concepts": [ "concept/circumscribed-prism", "concept/inscribed-prism", "theorem/volume-of-a-pyramid-lies-between-inscribed-and-circumscribed-prisms" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-085bb9303a", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "89", "location": "Pyramids and Cones", "latex": "The volume of a frustum of a pyramid is equal to the combined volumes of three pyramids whose common altitude is the same as that of the frustum, and the areas of whose bases are those of the upper and lower bases of the frustum and the mean proportional between these areas.", "markdown": "The volume of a frustum of a pyramid is equal to the combined volumes of three pyramids whose common altitude is the same as that of the frustum, and the areas of whose bases are those of the upper and lower bases of the frustum and the mean proportional between these areas.", "why": "It states the frustum volume result as a sum of pyramid volumes, a clear bridge from pyramids to frustums for a learner.", "use": [ "lesson" ], "concepts": [ "concept/frustum-of-a-pyramid", "theorem/volume-of-a-frustum-of-a-pyramid", "theorem/volume-of-a-pyramid" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-450ae84868", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "85", "location": "Pyramids and Cones", "latex": "This process may be repeated by doubling the number of planes drawn parallel to the base and thus doubling the number of inscribed or circumscribed prisms. By continuing in this way either set of prisms may be made to coincide as nearly as we please with the pyramid.", "markdown": "This process may be repeated by doubling the number of planes drawn parallel to the base and thus doubling the number of inscribed or circumscribed prisms. By continuing in this way either set of prisms may be made to coincide as nearly as we please with the pyramid.", "why": "It explains the limiting idea behind approximating a pyramid's volume by prisms, in the book's own old style of exhaustion.", "use": [ "history" ], "concepts": [ "concept/circumscribed-prism", "concept/inscribed-prism", "theorem/volume-of-a-pyramid-lies-between-inscribed-and-circumscribed-prisms" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-7b9e0bdb4e", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "81", "location": "Pyramids and Cones", "latex": "In this case every face is a triangle, and any one of them may be taken as the base.", "markdown": "In this case every face is a triangle, and any one of them may be taken as the base.", "why": "It warns learners that a tetrahedron has no privileged base, which avoids a common confusion when choosing the altitude.", "use": [ "lesson" ], "concepts": [ "concept/altitude-of-a-pyramid", "concept/base-of-a-pyramid", "concept/tetrahedron" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-3641077eee", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "94", "location": "Pyramids and Cones", "latex": "by rotating a right triangle $PMB$ about one of its legs, $PM$, as an axis.", "markdown": "by rotating a right triangle $PMB$ about one of its legs, $PM$, as an axis.", "why": "It gives a concrete rotation picture for generating a right circular cone, which a learner can visualise or model.", "use": [ "website" ], "concepts": [ "concept/cone", "concept/generator", "concept/right-circular-cone" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-e08370fde1", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "97", "location": "Pyramids and Cones", "latex": "The circumscribed pyramids all have the same slant height as that of the cone, and in case of the inscribed pyramids, the slant height may be made to differ by as little as we please from that of the cone by making the number of faces great enough.", "markdown": "The circumscribed pyramids all have the same slant height as that of the cone, and in case of the inscribed pyramids, the slant height may be made to differ by as little as we please from that of the cone by making the number of faces great enough.", "why": "It shows a learner why the slant height of an approximating pyramid settles to that of the cone, the idea behind the lateral area argument.", "use": [ "lesson" ], "concepts": [ "concept/circumscribed-pyramid", "concept/inscribed-pyramid", "concept/slant-height", "method/approximating-the-lateral-area-of-a-cone-by-inscribed-and-circumscribed-pyramids" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-adb53ded8c", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "97", "location": "Pyramids and Cones", "latex": "Evidently either of these processes may be repeated indefinitely and the surfaces of the inscribed or circumscribed pyramids may be made to lie as close to the surface of the cone as we please.", "markdown": "Evidently either of these processes may be repeated indefinitely and the surfaces of the inscribed or circumscribed pyramids may be made to lie as close to the surface of the cone as we please.", "why": "It states the limiting idea behind the approximation method in plain words a learner can follow.", "use": [ "lesson", "website" ], "concepts": [ "concept/cone", "method/approximating-the-lateral-area-of-a-cone-by-inscribed-and-circumscribed-pyramids" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-03e9b17713", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "98", "location": "Pyramids and Cones", "latex": "In the case of a cone which is not a right circular cone the slant height varies from point to point and the process of computation of §~\\Sref{266} fails.", "markdown": "In the case of a cone which is not a right circular cone the slant height varies from point to point and the process of computation of § [unit:266.]266 fails.", "why": "It warns that the right-cone lateral area formula does not carry over to oblique cones, a common overreach.", "use": [ "lesson" ], "concepts": [ "concept/right-circular-cone", "theorem/lateral-area-of-a-right-circular-cone" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-a14ee2911e", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "102", "location": "Pyramids and Cones", "latex": "The theorem of §~\\Sref{272} holds for any cone whatever, whether right or oblique.", "markdown": "The theorem of § [unit:272.]272 holds for any cone whatever, whether right or oblique.", "why": "It tells a learner that the volume rule is general while the elementary computation of the base area is not.", "use": [ "lesson" ], "concepts": [ "concept/cone", "theorem/volume-of-a-cone" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-bac58dce38", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "106", "location": "Pyramids and Cones", "latex": "His work on the circle, cone, cylinder, and sphere is reflected in our treatment of surfaces and volumes at the present day.", "markdown": "His work on the circle, cone, cylinder, and sphere is reflected in our treatment of surfaces and volumes at the present day.", "why": "It places the cone measurements of this book in the history of Archimedes' work.", "use": [ "history", "website" ], "concepts": [ "person/archimedes" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-cfa0595a1b", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "136", "location": "The Sphere", "latex": "The spherical degree differs fundamentally from the units of measure hitherto used. This unit is a certain fraction of the surface of the sphere and hence its actual size depends upon the size of the sphere.", "markdown": "The spherical degree differs fundamentally from the units of measure hitherto used. This unit is a certain fraction of the surface of the sphere and hence its actual size depends upon the size of the sphere.", "why": "Warns that a spherical degree is a fraction of the sphere's surface, so its size changes with the sphere, unlike plane units.", "use": [ "lesson" ], "concepts": [ "unit/spherical-degree" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-0539f9fbee", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "113", "location": "The Sphere", "latex": "A plane tangent to a sphere is perpendicular to the radius from the point of tangency; and conversely, a plane perpendicular to a radius at its extremity is tangent to the sphere.", "markdown": "A plane tangent to a sphere is perpendicular to the radius from the point of tangency; and conversely, a plane perpendicular to a radius at its extremity is tangent to the sphere.", "why": "States the tangent-plane criterion in both directions, which a learner needs to test tangency in problems.", "use": [ "lesson" ], "concepts": [ "concept/perpendicular", "concept/tangent-plane-to-a-sphere", "quantity/radius" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-2c01e4d9ce", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "122", "location": "The Sphere", "latex": "The shortest distance on a sphere between two of its points is measured along the minor arc of a great circle passing through these points.", "markdown": "The shortest distance on a sphere between two of its points is measured along the minor arc of a great circle passing through these points.", "why": "Shows why great circles are the natural 'straight lines' of spherical geometry, linking distance to great-circle arcs.", "use": [ "lesson" ], "concepts": [ "concept/great-circle", "concept/spherical-distance" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-18d3c962f5", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "111", "location": "The Sphere", "latex": "It follows from the preceding theorem and corollaries that, if a \\emph{spherical} blackboard is at hand, circles may be constructed on it by means of crayon and string the same as on a \\emph{plane} blackboard. Likewise, curve-legged compasses may be used.", "markdown": "It follows from the preceding theorem and corollaries that, if a *spherical* blackboard is at hand, circles may be constructed on it by means of crayon and string the same as on a *plane* blackboard. Likewise, curve-legged compasses may be used.", "why": "A concrete picture of how sphere geometry relates to ordinary plane drawing, from a book written for practical surveyors and engineers.", "use": [ "website", "history" ], "concepts": [ "concept/circle", "concept/sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-cf99d5fa51", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "116", "location": "The Sphere", "latex": "$PB$ and $BD$ being known, we may now \\emph{compute} $PD$ from the right triangle $PBD$, and then compute $PP'$ from the similar triangles $PBD$ and $PP'B$, for the latter using the relation $PD : PB = PB : PP'$ or $PD × PP' = \\overline{PB}^2$.", "markdown": "$PB$ and $BD$ being known, we may now *compute* $PD$ from the right triangle $PBD$, and then compute $PP'$ from the similar triangles $PBD$ and $PP'B$, for the latter using the relation $PD : PB = PB : PP'$ or $PD × PP' = \\overline{PB}^2$.", "why": "Shows a worked method for recovering a sphere's diameter from a measured circle by similar right triangles.", "use": [ "lesson" ], "concepts": [ "concept/diameter", "method/finding-the-diameter-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-4c0bf344db", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "135", "location": "The Sphere", "latex": "The number of degrees by which the sum of the angles of a spherical triangle exceeds $180$° is called the \\emph{spherical excess} of the triangle.", "markdown": "The number of degrees by which the sum of the angles of a spherical triangle exceeds $180$° is called the *spherical excess* of the triangle.", "why": "Defines the spherical excess before the area theorem uses it, so the learner meets the quantity first.", "use": [ "lesson" ], "concepts": [ "quantity/spherical-excess" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-aeb5293f79", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "139", "location": "The Sphere", "latex": "A sphere has a definite area which is less than the surface of any circumscribed figure and greater than the surface of any inscribed convex figure", "markdown": "A sphere has a definite area which is less than the surface of any circumscribed figure and greater than the surface of any inscribed convex figure", "why": "States the assumption on which the sphere's area rests, given as an assumption rather than proved.", "use": [ "lesson" ], "concepts": [ "theorem/assumption-on-the-area-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-0a2d3fdd25", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "139", "location": "The Sphere", "latex": "The student should note that while the statement just preceding is obviously true, it is not capable of proof by pure deduction.", "markdown": "The student should note that while the statement just preceding is obviously true, it is not capable of proof by pure deduction.", "why": "Tells the learner that this assumption is a stated axiom, not a gap in the argument.", "use": [ "lesson" ], "concepts": [ "theorem/assumption-on-the-area-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-f66b2ee7ac", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "126", "location": "The Sphere", "latex": "The triangle $A'B'C'$ as thus described is \\emph{the polar triangle} of $ABC$.", "markdown": "The triangle $A'B'C'$ as thus described is *the polar triangle* of $ABC$.", "why": "Shows exactly which triangle counts as the polar triangle, through the side condition on its vertices.", "use": [ "lesson" ], "concepts": [ "concept/polar-triangle" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-296601393a", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "136", "location": "The Sphere", "latex": "This theorem was discovered by Cavalieri.", "markdown": "This theorem was discovered by Cavalieri.", "why": "Credits the discovery of the spherical-triangle area theorem to Cavalieri, a historical remark for a history page.", "use": [ "history" ], "concepts": [ "theorem/area-of-a-spherical-triangle" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-3b118a1c5f", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "140", "location": "The Sphere", "latex": "The area of a sphere whose radius is $r$ is $4\\pi r^2$.", "markdown": "The area of a sphere whose radius is $r$ is $4\\pi r^2$.", "why": "The headline result, stated plainly so a learner can compare it with the area of a circle of the same radius.", "use": [ "website", "lesson" ], "concepts": [ "theorem/area-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-c2d0e9e24e", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "144", "location": "The Sphere", "latex": "The volume of a sphere whose radius is $r$ is $\\tfrac{4}{3} \\pi r^3$.", "markdown": "The volume of a sphere whose radius is $r$ is $\\tfrac{4}{3} \\pi r^3$.", "why": "States the sphere-volume result as a named theorem a learner can cite and apply.", "use": [ "lesson" ], "concepts": [ "theorem/volume-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-44e906cf71", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "143", "location": "The Sphere", "latex": "The sphere is covered with a network of spherical quadrilaterals. If these are taken small enough, they may be regarded as approximately \\emph{plane surfaces}.", "markdown": "The sphere is covered with a network of spherical quadrilaterals. If these are taken small enough, they may be regarded as approximately *plane surfaces*.", "why": "Gives the picture behind the volume formula: small patches of a sphere look flat, so pyramids can stand in for it.", "use": [ "lesson" ], "concepts": [ "method/approximating-the-volume-of-a-sphere-by-pyramids", "theorem/volume-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-0288b4429d", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "143", "location": "The Sphere", "latex": "Hence, their combined volume is $\\frac{1}{3} r × (\\text{area of sphere})$ or $\\tfrac{1}{3} r \\cdot 4\\pi r^2$. That is, the volume is $\\tfrac{4}{3} \\pi r^3$.", "markdown": "Hence, their combined volume is $\\frac{1}{3} r × (\\text{area of sphere})$ or $\\tfrac{1}{3} r \\cdot 4\\pi r^2$. That is, the volume is $\\tfrac{4}{3} \\pi r^3$.", "why": "Shows the one-third base-times-height idea at work, reaching the sphere's volume in a single line.", "use": [ "lesson" ], "concepts": [ "theorem/volume-of-a-pyramid", "theorem/volume-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-978643a2b7", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "143", "location": "The Sphere", "latex": "This is of course obvious at a glance, though a formal deductive proof is very difficult.", "markdown": "This is of course obvious at a glance, though a formal deductive proof is very difficult.", "why": "An honest remark that a result can be plainly evident yet hard to prove formally, which helps learners see the gap between intuition and proof.", "use": [ "history", "lesson" ], "concepts": [ "theorem/assumption-on-the-area-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-d5c5b6f3dd", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "142", "location": "The Sphere", "latex": "The total area of the sphere is $4\\pi × 6^2 = 452.3904$~sq.~in., and one spherical degree is $\\tfrac{1}{720}$ of $452.3904 = .62832$~sq.~in.", "markdown": "The total area of the sphere is $4\\pi × 6^2 = 452.3904$ sq. in., and one spherical degree is $\\tfrac{1}{720}$ of $452.3904 = .62832$ sq. in.", "why": "Shows how a spherical degree turns a spherical excess into an area in square inches.", "use": [ "lesson" ], "concepts": [ "quantity/spherical-excess", "theorem/area-of-a-spherical-polygon", "unit/spherical-degree" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-4b7ec6ba4d", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "159", "location": "The Sphere", "latex": "He was one of three commissioners who introduced the metric system in France, having also been a member of the commission for determining the length of the meter.", "markdown": "He was one of three commissioners who introduced the metric system in France, having also been a member of the commission for determining the length of the meter.", "why": "A short historical note tying the geometry to the metric system.", "use": [ "history" ], "concepts": [ "person/adrien-marie-legendre" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-a9b3b090d7", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "146", "location": "The Sphere", "latex": "The portion of a sphere included between two parallel planes cutting it is called a \\emph{spherical segment}, and the two circular sections made by the parallel planes are its \\emph{bases}.", "markdown": "The portion of a sphere included between two parallel planes cutting it is called a *spherical segment*, and the two circular sections made by the parallel planes are its *bases*.", "why": "Defines the spherical segment and its bases in the words a learner will meet again.", "use": [ "lesson" ], "concepts": [ "concept/spherical-segment" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-f00b67f93e", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "146", "location": "The Sphere", "latex": "The perpendicular distance between the planes is the \\emph{altitude} of the zone and of the corresponding segment.", "markdown": "The perpendicular distance between the planes is the *altitude* of the zone and of the corresponding segment.", "why": "Pins down what altitude means for a zone or segment, a term easily confused with a pyramid's height.", "use": [ "lesson" ], "concepts": [ "concept/spherical-segment", "quantity/altitude-of-a-spherical-zone" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-adcc3e958f", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "172", "location": "SIMILAR SOLIDS", "latex": "The fact that the ratio of the areas of corresponding surfaces of similar solids is equal to the \\emph{square} of their ratio of similitude, while the ratio of their volumes equals the \\emph{cube} of this ratio is one of the most important and far-reaching conclusions of geometry.", "markdown": "The fact that the ratio of the areas of corresponding surfaces of similar solids is equal to the *square* of their ratio of similitude, while the ratio of their volumes equals the *cube* of this ratio is one of the most important and far-reaching conclusions of geometry.", "why": "It states in one sentence the square-and-cube pattern that lets a learner scale areas and volumes from a linear ratio.", "use": [ "history", "lesson" ], "concepts": [ "concept/ratio-of-similitude", "theorem/area-and-volume-ratios-of-similar-solids" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-f80a05cef5", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "165", "location": "SIMILAR SOLIDS", "latex": "Any two figures which have a center of similitude are similar.", "markdown": "Any two figures which have a center of similitude are similar.", "why": "It gives the chapter's central idea, that a shared center of similitude forces similarity, which a learner can test on any pair of figures.", "use": [ "lesson" ], "concepts": [ "concept/center-of-similitude", "theorem/figures-with-a-center-of-similitude-are-similar" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-6b3219a64c", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "170", "location": "SIMILAR SOLIDS", "latex": "This proposition may be rendered evident by noticing that any two similar three-dimensional figures may be built up to any degree of approximation by means of pairs of similar tetrahedrons similarly placed.", "markdown": "This proposition may be rendered evident by noticing that any two similar three-dimensional figures may be built up to any degree of approximation by means of pairs of similar tetrahedrons similarly placed.", "why": "It gives a picture of why volumes of similar solids scale as cubes: each solid is built from similar tetrahedrons.", "use": [ "lesson" ], "concepts": [ "theorem/volumes-of-similar-polyhedrons", "theorem/volumes-of-similar-solids" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-6f4a53fc18", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "171", "location": "SIMILAR SOLIDS", "latex": "The essential property of all such contrivances is that one point $O$ is kept fixed, while two points $A$ and $B$ are allowed to move so that $O$, $A$, and $B$ always remain in a straight line, and so that the ratio $OA : OB$ remains the same.", "markdown": "The essential property of all such contrivances is that one point $O$ is kept fixed, while two points $A$ and $B$ are allowed to move so that $O$, $A$, and $B$ always remain in a straight line, and so that the ratio $OA : OB$ remains the same.", "why": "It explains in plain terms how a pantograph copies a figure at a fixed scale, using a fixed centre and collinear points.", "use": [ "website", "lesson" ], "concepts": [ "concept/center-of-similitude", "instrument/pantograph" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-3d5b568d5a", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "170", "location": "SIMILAR SOLIDS", "latex": "Note that the ratio of similitude of two similar figures may be obtained from the ratio of any pair of their corresponding linear dimensions.", "markdown": "Note that the ratio of similitude of two similar figures may be obtained from the ratio of any pair of their corresponding linear dimensions.", "why": "It warns learners that one matching pair of lengths is enough to find the ratio of similitude for the whole figure.", "use": [ "lesson" ], "concepts": [ "concept/corresponding-linear-dimensions", "concept/ratio-of-similitude" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-72a4d0b1b5", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "172", "location": "SIMILAR SOLIDS", "latex": "Thus the ratio of the weights of two similar shells used in gunnery, or the ratio of the weights of two men of similar build, may be found when their ratio of similitude is known.", "markdown": "Thus the ratio of the weights of two similar shells used in gunnery, or the ratio of the weights of two men of similar build, may be found when their ratio of similitude is known.", "why": "It gives a concrete everyday use of cube scaling, so the learner can see why a linear ratio fixes a weight ratio.", "use": [ "website", "history" ], "concepts": [ "concept/ratio-of-similitude", "concept/similar-solids" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-456a72a179", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "179", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "The length of the projection of a line-segment upon a given line is equal to the length of the line-segment multiplied by the cosine of the projection angle.", "markdown": "The length of the projection of a line-segment upon a given line is equal to the length of the line-segment multiplied by the cosine of the projection angle.", "why": "States the central result of the chapter in one sentence, so a learner can see that projection length is a cosine times the segment.", "use": [ "lesson" ], "concepts": [ "concept/angle-between-line-and-plane", "concept/cosine", "concept/projection-of-a-line" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-daf88d8f44", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "176", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "Likewise we define the \\emph{sine} of an acute angle of a right triangle as the \\emph{ratio of the opposite side to the hypotenuse}, and the \\emph{tangent} of an acute angle of a right triangle as the \\emph{ratio of the opposite side to the adjacent side}.", "markdown": "Likewise we define the *sine* of an acute angle of a right triangle as the *ratio of the opposite side to the hypotenuse*, and the *tangent* of an acute angle of a right triangle as the *ratio of the opposite side to the adjacent side*.", "why": "Gives the sine and tangent as side ratios in a right triangle, the definitions a learner needs before any table work.", "use": [ "lesson" ], "concepts": [ "concept/right-triangle", "concept/sine", "concept/tangent-function" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-0b217327b2", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "183", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "The area of the projection of a plane-segment on a plane is equal to the area of the plane-segment multiplied by the cosine of the projection angle.", "markdown": "The area of the projection of a plane-segment on a plane is equal to the area of the plane-segment multiplied by the cosine of the projection angle.", "why": "Shows that projection scales area by the cosine of the angle between the planes, which carries the idea from lengths to areas.", "use": [ "lesson" ], "concepts": [ "concept/cosine", "theorem/area-of-the-projection-of-a-plane-segment" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-89b21a1191", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "181", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "The altitude of an oblique prism or cylinder is equal to an element multiplied by the cosine of the angle between the plane of the base and that of a right section.", "markdown": "The altitude of an oblique prism or cylinder is equal to an element multiplied by the cosine of the angle between the plane of the base and that of a right section.", "why": "Connects the slant length of an oblique solid to its height through one cosine, a usable check for the height of a tilted prism.", "use": [ "lesson" ], "concepts": [ "concept/oblique-prism", "concept/right-section", "theorem/altitude-of-an-oblique-prism-or-cylinder" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-03d29ae0d6", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "185", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "Note that when $a$ and $b$ are equal, the ellipse becomes a circle, and this formula reduces to $\\pi a^2$ as it should.", "markdown": "Note that when $a$ and $b$ are equal, the ellipse becomes a circle, and this formula reduces to $\\pi a^2$ as it should.", "why": "Lets a learner check the ellipse formula against the circle they already know, a good sanity check to show with the result.", "use": [ "website" ], "concepts": [ "concept/ellipse", "theorem/area-of-an-ellipse" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-8c87fb6de6", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "186", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "The comparatively low temperature of the earth's surface near the pole, even in summer, when the sun does not set for months, is due partly to the \\emph{obliqueness} with which the sun's rays strike the earth.", "markdown": "The comparatively low temperature of the earth’s surface near the pole, even in summer, when the sun does not set for months, is due partly to the *obliqueness* with which the sun’s rays strike the earth.", "why": "Gives a concrete physical reason why projection matters, as a vivid and readable opening for a general audience.", "use": [ "website", "history" ], "concepts": [ "theorem/area-of-the-projection-of-a-plane-segment" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-239be3bc6b", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "179", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "We assume that it lies halfway between these numbers. This assumption, while not quite correct, is very nearly so for small differences of angles, as in this case, where the total difference is only one degree.", "markdown": "We assume that it lies halfway between these numbers. This assumption, while not quite correct, is very nearly so for small differences of angles, as in this case, where the total difference is only one degree.", "why": "Warns learners that linear interpolation in a trigonometric table is an approximation, and says when the error is small.", "use": [ "lesson" ], "concepts": [ "method/interpolating-in-logarithmic-tables" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-ea4e45500d", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "188", "location": "VARIABLES. LIMITS", "latex": "Now the greater the number of sides the more nearly does the apothem equal the radius in length.", "markdown": "Now the greater the number of sides the more nearly does the apothem equal the radius in length.", "why": "It shows a learner the idea of a limit through the inscribed polygon, where the apothem approaches the radius as the sides multiply.", "use": [ "lesson", "website" ], "concepts": [ "concept/approximating-sequence-of-polygons", "concept/limit" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-0c5cdcc596", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "191", "location": "VARIABLES. LIMITS", "latex": "For \\emph{practical purposes} the lengths of such segments are approximated to any desired degree of accuracy, and their ratios are understood to be the ratios of these approximate numerical measures.", "markdown": "For *practical purposes* the lengths of such segments are approximated to any desired degree of accuracy, and their ratios are understood to be the ratios of these approximate numerical measures.", "why": "It separates the practical approximation of incommensurable lengths from the theoretical meaning, so the learner knows why limits are needed.", "use": [ "lesson" ], "concepts": [ "concept/incommensurable-magnitudes" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-c7eb4f04a9", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "193", "location": "VARIABLES. LIMITS", "latex": "Not every infinite sequence serves to single out a definite point in the manner shown above.", "markdown": "Not every infinite sequence serves to single out a definite point in the manner shown above.", "why": "It warns that an endless list of numbers does not always approach a limit, which corrects a common assumption.", "use": [ "lesson" ], "concepts": [ "concept/infinite-sequence", "concept/limit" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-bc0ff198d8", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "193", "location": "VARIABLES. LIMITS", "latex": "That is, $1$ is the \\emph{smallest number} beyond which the sequence does not go.", "markdown": "That is, $1$ is the *smallest number* beyond which the sequence does not go.", "why": "It gives an intuitive picture of the least upper bound of an increasing sequence that a learner can hold onto.", "use": [ "lesson", "website" ], "concepts": [ "concept/infinite-sequence", "concept/least-upper-bound" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-2309e7ddda", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "194", "location": "VARIABLES. LIMITS", "latex": "The Greeks dealt with the incommensurable cases in an interesting manner. Thus, to prove that two incommensurable ratios are equal they showed that neither can be less than the other.", "markdown": "The Greeks dealt with the incommensurable cases in an interesting manner. Thus, to prove that two incommensurable ratios are equal they showed that neither can be less than the other.", "why": "It places the Greek method of proving incommensurable ratios equal in historical context and shows the origin of the limit approach.", "use": [ "history" ], "concepts": [ "concept/incommensurable-magnitudes", "concept/limit" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-cbda4c3b61", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "202", "location": "VARIABLES. LIMITS", "latex": "Let $a_1$, $a_2$, $a_3$, $\\dotsc$ be a sequence of rational numbers whose limit is the side~$a$.", "markdown": "Let $a_1$, $a_2$, $a_3$, $\\dotsc$ be a sequence of rational numbers whose limit is the side $a$.", "why": "It shows how a volume with incommensurable edges is approached by a sequence of rational approximations, the core idea of the section.", "use": [ "lesson" ], "concepts": [ "concept/incommensurable-magnitudes", "concept/infinite-sequence", "concept/limit", "concept/rectangular-parallelepiped" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-e58899fe20", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "202", "location": "VARIABLES. LIMITS", "latex": "But the limit of this sequence is by definition the product of the limits of the three sequences $a_1$, $a_2$, $a_3$, $\\dotsc$, $b_1$, $b_2$, $b_3$, $\\dotsc$, $c_1$, $c_2$, $c_3$, $\\dotsc$, or $abc$.", "markdown": "But the limit of this sequence is by definition the product of the limits of the three sequences $a_1$, $a_2$, $a_3$, $\\dotsc$, $b_1$, $b_2$, $b_3$, $\\dotsc$, $c_1$, $c_2$, $c_3$, $\\dotsc$, or $abc$.", "why": "It shows the proof step that turns a limit of products into the product of limits, which is what makes the volume formula work.", "use": [ "lesson" ], "concepts": [ "concept/limit", "theorem/volume-of-a-rectangular-parallelepiped" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-4df6f1895b", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "203", "location": "VARIABLES. LIMITS", "latex": "That $U = V$ follows from the fact that the sum of the volumes of the circumscribed prisms exceeds that of the inscribed prisms by the volume of the lowest circumscribed prism, and this may be made as small as we please.", "markdown": "That $U = V$ follows from the fact that the sum of the volumes of the circumscribed prisms exceeds that of the inscribed prisms by the volume of the lowest circumscribed prism, and this may be made as small as we please.", "why": "It gives a clear geometric reason why inscribed and circumscribed approximations converge to the same volume.", "use": [ "lesson", "website" ], "concepts": [ "concept/circumscribed-prism", "concept/inscribed-prism", "concept/limit" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-29d93f112a", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "207", "location": "VARIABLES. LIMITS", "latex": "The proof of this general theorem is more difficult than any thus far given, inasmuch as it involves sequences which \\emph{oscillate}; that is, which are neither constantly increasing nor constantly decreasing.", "markdown": "The proof of this general theorem is more difficult than any thus far given, inasmuch as it involves sequences which *oscillate*; that is, which are neither constantly increasing nor constantly decreasing.", "why": "It warns the learner that some limit arguments need sequences that go up and down, not only monotone ones.", "use": [ "lesson" ], "concepts": [ "concept/infinite-sequence", "theorem/equal-cross-sections-imply-equal-volumes" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-c8f0038d6a", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "207", "location": "VARIABLES. LIMITS", "latex": "Indeed, this theorem and also that of §~\\Sref{437} are special cases of what is known as \\textbf{Cavalieri's Theorem}.", "markdown": "Indeed, this theorem and also that of § [unit:437.]437 are special cases of what is known as **Cavalieri’s Theorem**.", "why": "It shows that several volume results are instances of one principle, which is a memorable point for learners and a good historical note.", "use": [ "history", "website" ], "concepts": [ "person/cavalieri", "theorem/equal-cross-sections-imply-equal-volumes", "theorem/volume-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/x-d007cf4a46", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "207", "location": "VARIABLES. LIMITS", "latex": "Hence, by §~\\Sref{428}, $V = \\tfrac{1}{3}rS$. But by §~\\Sref{443}, $V = \\tfrac{4}{3} \\pi r^3$.", "markdown": "Hence, by § [unit:428.]428, $V = \\tfrac{1}{3}rS$. But by § [unit:443.]443, $V = \\tfrac{4}{3} \\pi r^3$.", "why": "It shows how the area of a sphere is obtained from its volume by using the volume-surface relation, a clear model of deriving one result from another.", "use": [ "lesson" ], "concepts": [ "theorem/area-of-a-sphere", "theorem/volume-of-a-sphere" ] } ], "equations": [ { "id": "slaught-lennes-solid-geometry-1919/eq-5df8500792", "chapter": "slaught-lennes-solid-geometry-1919/ch-introduction", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "8", "location": "", "latex": "ah = bk", "name": null, "statement": "In any triangle, the product of one side and its altitude equals the product of another side and its altitude.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a side of a triangle" }, { "unit": null, "symbol": "h", "meaning": "the altitude on side a" }, { "unit": null, "symbol": "b", "meaning": "another side of the triangle" }, { "unit": null, "symbol": "k", "meaning": "the altitude on side b" } ], "sympy": "Eq(a*h, b*k)", "physics": false, "states": [], "concepts": [ "concept/angle-of-elevation", "concept/triangle", "quantity/area" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-fa0de5ceaa", "chapter": "slaught-lennes-solid-geometry-1919/ch-introduction", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "8", "location": "", "latex": "a = \\tfrac{1}{2} \\cdot 2\\pi r \\cdot r = \\pi r^2.", "name": null, "statement": "The area of a circle is one half the circumference times the radius, which equals pi times the square of the radius.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "area of the circle" }, { "unit": null, "symbol": "r", "meaning": "radius of the circle" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of the circumference to the diameter of a circle" } ], "sympy": "Eq(a, Rational(1, 2)*2*pi*r*r)", "physics": false, "states": [], "concepts": [ "quantity/circumference", "quantity/pi", "quantity/radius", "theorem/area-of-a-circle" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-3fff19f93b", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "26", "location": "Properties of the Plane", "latex": "\\dfrac{AE}{EB} = \\dfrac{CG}{GD}", "name": null, "statement": "Two lines cut by three parallel planes are divided in the same ratio: the ratio of the segments on one line equals the ratio of the corresponding segments on the other.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AE", "meaning": "segment of line AB between planes M and N" }, { "unit": null, "symbol": "EB", "meaning": "segment of line AB between planes N and P" }, { "unit": null, "symbol": "CG", "meaning": "segment of line CD between planes M and N" }, { "unit": null, "symbol": "GD", "meaning": "segment of line CD between planes N and P" } ], "sympy": "Eq(AE/EB, CG/GD)", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/line", "concept/parallel-planes", "concept/theorem-parallel-planes-intercept-proportional-segments" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-8276d077d9", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "34", "location": "Properties of the Plane", "latex": "EC = ED", "name": null, "statement": "Any point in the bisecting half-plane of a dihedral angle is equally distant from the two faces, measured along the perpendiculars to each face.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "point in the bisecting half-plane P" }, { "unit": null, "symbol": "C", "meaning": "foot of the perpendicular from E to face M" }, { "unit": null, "symbol": "D", "meaning": "foot of the perpendicular from E to face N" } ], "sympy": "Eq(EC, ED)", "physics": false, "states": [], "concepts": [ "concept/bisector-of-a-dihedral-angle", "concept/dihedral-angle", "concept/distance-from-a-point-to-a-plane", "theorem/locus-of-points-equidistant-from-the-faces-of-a-dihedral-angle" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-c8876d264c", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "29", "location": "Properties of the Plane", "latex": "\\angle M-AB-N = \\angle M'-A'B'-N'", "name": null, "statement": "Two dihedral angles are equal when their plane angles are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "M-AB-N", "meaning": "dihedral angle formed by half-planes M and N along edge AB" }, { "unit": null, "symbol": "M'-A'B'-N'", "meaning": "dihedral angle formed by half-planes M' and N' along edge A'B'" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/dihedral-angle", "concept/equal-dihedral-angles", "concept/plane-angle", "concept/theorem-dihedral-angles-equal-when-plane-angles-are-equal" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-526fca03e6", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "29", "location": "Properties of the Plane", "latex": "\\angle CDE = \\angle C'D'E'", "name": null, "statement": "Two plane angles are equal when the dihedral angles that they measure are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "CDE", "meaning": "plane angle of the dihedral angle M-AB-N" }, { "unit": null, "symbol": "C'D'E'", "meaning": "plane angle of the dihedral angle M'-A'B'-N'" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/dihedral-angle", "concept/plane-angle", "concept/theorem-dihedral-angles-equal-when-plane-angles-are-equal" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-f030a1f6e7", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "33", "location": "Properties of the Plane", "latex": "\\angle ABD > \\angle ABC", "name": null, "statement": "The acute angle a slanting line makes with its own projection on a plane is smaller than its angle with any other line in that plane through the same point.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "ABD", "meaning": "angle between oblique line AB and another line BD in plane M" }, { "unit": null, "symbol": "ABC", "meaning": "angle between oblique line AB and its projection BC on plane M" } ], "sympy": "Gt(ABD, ABC)", "physics": false, "states": [], "concepts": [ "concept/angle-between-line-and-plane", "concept/projection-of-a-line", "concept/theorem-acute-angle-with-projection-is-the-least-angle-with-lines-in-the-plane", "quantity/angle" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-55c2dea1fc", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "42", "location": "Properties of the Plane", "latex": "\\angle DFE = \\angle D'F'E'", "name": null, "statement": "If two trihedral angles have their three face angles equal respectively, the dihedral angles opposite the equal face angles are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "DFE", "meaning": "dihedral angle of trihedral angle P opposite face angle a" }, { "unit": null, "symbol": "D'F'E'", "meaning": "dihedral angle of trihedral angle P' opposite face angle a'" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/dihedral-angle", "concept/face-angle", "concept/theorem-equal-face-angles-give-equal-opposite-dihedral-angles", "concept/trihedral-angle" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-8d0dfcd620", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "53", "location": "Regular Polyhedrons", "latex": "AD = AC", "name": null, "statement": "In the construction of the regular tetrahedron, the fourth vertex D is placed on the perpendicular at E so that its distance from A equals the side AC.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "length of the segment from vertex A to the new vertex D (the fourth vertex of the tetrahedron)" }, { "unit": null, "symbol": "AC", "meaning": "length of the side of the equilateral triangle ABC" } ], "sympy": "Eq(AD, AC)", "physics": false, "states": [], "concepts": [ "concept/edge", "concept/method-constructing-a-regular-tetrahedron", "concept/regular-polygon", "concept/regular-polyhedron" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-8fe14bf5f3", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "53", "location": "Regular Polyhedrons", "latex": "AF = AE = AB", "name": null, "statement": "In the construction of the regular octahedron, the two apex points E and F are placed on the perpendicular through the center O so that each is at distance AB, the side of the square, from A.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "AF", "meaning": "length of the segment from vertex A to apex F" }, { "unit": null, "symbol": "AE", "meaning": "length of the segment from vertex A to apex E" }, { "unit": null, "symbol": "AB", "meaning": "length of the side of the square ABCD" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/edge", "concept/method-constructing-a-regular-octahedron", "concept/regular-polyhedron" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-508d936ec1", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "62", "location": "Prisms and Cylinders", "latex": "\\textit{Volume} = \\textit{Length}\\/ × \\textit{Width}\\/ × \\textit{Height}.", "name": null, "statement": "The volume of a rectangular parallelopiped with commensurable edges is the product of its length, width and height.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "Volume", "meaning": "volume of the rectangular parallelopiped (number of unit cubes it contains)" }, { "unit": null, "symbol": "Length", "meaning": "length of the rectangular parallelopiped" }, { "unit": null, "symbol": "Width", "meaning": "width of the rectangular parallelopiped" }, { "unit": null, "symbol": "Height", "meaning": "height of the rectangular parallelopiped" } ], "sympy": "Eq(Volume, Length*Width*Height)", "physics": false, "states": [], "concepts": [ "concept/rectangular-parallelepiped", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-7462ffcf12", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "62", "location": "Prisms and Cylinders", "latex": "\\textbf{Volume} = \\textbf{Length} × \\textbf{Width} × \\textbf{Height}.", "name": null, "statement": "The volume of a rectangular parallelopiped with commensurable dimensions is given by the product of its length, width and height.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "Volume", "meaning": "volume of the rectangular parallelopiped" }, { "unit": null, "symbol": "Length", "meaning": "length of the rectangular parallelopiped" }, { "unit": null, "symbol": "Width", "meaning": "width of the rectangular parallelopiped" }, { "unit": null, "symbol": "Height", "meaning": "height of the rectangular parallelopiped" } ], "sympy": "Eq(Volume, Length*Width*Height)", "physics": false, "states": [], "concepts": [ "concept/formula-for-volume-of-a-rectangular-solid", "concept/rectangular-parallelepiped", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-76989eef52", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "63", "location": "Prisms and Cylinders", "latex": "V = l\\cdot w\\cdot h", "name": null, "statement": "The volume of a rectangular parallelopiped is the product of its length, width and height.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the rectangular parallelopiped" }, { "unit": null, "symbol": "l", "meaning": "length of the rectangular parallelopiped" }, { "unit": null, "symbol": "w", "meaning": "width of the rectangular parallelopiped" }, { "unit": null, "symbol": "h", "meaning": "height of the rectangular parallelopiped" } ], "sympy": "Eq(V, l*w*h)", "physics": false, "states": [], "concepts": [ "concept/rectangular-parallelepiped", "concept/theorem-volume-of-a-rectangular-parallelopiped", "concept/volume-of-a-rectangular-parallelopiped", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-36543c5642", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "64", "location": "Prisms and Cylinders", "latex": "\\dfrac{V}{V'} = \\dfrac{a\\cdot b\\cdot c} {a'\\cdot b'\\cdot c'} = \\dfrac{a}{a'}", "name": null, "statement": "The volumes of two rectangular parallelopipeds are in the ratio of the products of their corresponding dimensions, and so in the ratio of a pair of dimensions when the other two are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the first rectangular parallelopiped" }, { "unit": null, "symbol": "V'", "meaning": "volume of the second rectangular parallelopiped" }, { "unit": null, "symbol": "a", "meaning": "first dimension of the first rectangular parallelopiped" }, { "unit": null, "symbol": "b", "meaning": "second dimension of the first rectangular parallelopiped" }, { "unit": null, "symbol": "c", "meaning": "third dimension of the first rectangular parallelopiped" }, { "unit": null, "symbol": "a'", "meaning": "first dimension of the second rectangular parallelopiped" }, { "unit": null, "symbol": "b'", "meaning": "second dimension of the second rectangular parallelopiped" }, { "unit": null, "symbol": "c'", "meaning": "third dimension of the second rectangular parallelopiped" } ], "sympy": "Eq(V/Vp, a*b*c/(ap*bp*cp))", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/corresponding-linear-dimensions", "concept/rectangular-parallelepiped", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-b58cf178b8", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "75", "location": "Prisms and Cylinders", "latex": "S = 2 \\pi re", "name": null, "statement": "The lateral surface of a circular cylinder equals 2π times the radius of its right section times the length of an element.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "S", "meaning": "lateral surface of the cylinder" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of the circumference to the diameter" }, { "unit": null, "symbol": "r", "meaning": "radius of a right section of the circular cylinder" }, { "unit": null, "symbol": "e", "meaning": "element of the cylinder" } ], "sympy": "Eq(S, 2*pi*r*e)", "physics": false, "states": [], "concepts": [ "concept/circular-cylinder", "concept/element-of-a-cylinder", "concept/right-section", "quantity/lateral-area", "quantity/pi", "quantity/radius" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-e3efb21594", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "75", "location": "Prisms and Cylinders", "latex": "S = 2 \\pi rh", "name": null, "statement": "The lateral surface of a right circular cylinder equals 2π times its radius times its altitude.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "S", "meaning": "lateral surface of the right circular cylinder" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of the circumference to the diameter" }, { "unit": null, "symbol": "r", "meaning": "radius of the right circular cylinder" }, { "unit": null, "symbol": "h", "meaning": "altitude of the right circular cylinder" } ], "sympy": "Eq(S, 2*pi*r*h)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-cylinder", "concept/lune", "concept/right-circular-cylinder", "quantity/area", "quantity/lateral-area", "quantity/pi", "quantity/radius" ], "pages": [ "75", "152" ], "chapters": [ "slaught-lennes-solid-geometry-1919/ch-book-iii", "slaught-lennes-solid-geometry-1919/ch-book-v" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-9330181188", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "76", "location": "Prisms and Cylinders", "latex": "V = \\pi r_1^2 e", "name": null, "statement": "The volume of a circular cylinder equals π times the square of the radius of its right section times the length of an element.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the cylinder" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of the circumference to the diameter" }, { "unit": null, "symbol": "r_1", "meaning": "radius of the right section of the circular cylinder" }, { "unit": null, "symbol": "e", "meaning": "element of the cylinder" } ], "sympy": "Eq(V, pi*r_1**2*e)", "physics": false, "states": [], "concepts": [ "concept/circular-cylinder", "concept/element-of-a-cylinder", "concept/right-section", "quantity/pi", "quantity/radius", "quantity/volume", "theorem/volume-of-a-cylinder" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-53576c3e5b", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "76", "location": "Prisms and Cylinders", "latex": "V = \\pi r_2^2 h", "name": null, "statement": "The volume of a cylinder with a circular base equals π times the square of the base radius times the altitude.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the cylinder" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of the circumference to the diameter" }, { "unit": null, "symbol": "r_2", "meaning": "radius of the circular base of the cylinder" }, { "unit": null, "symbol": "h", "meaning": "altitude of the cylinder" } ], "sympy": "Eq(V, pi*r_2**2*h)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-cylinder", "quantity/pi", "quantity/radius", "quantity/volume", "theorem/volume-of-a-cylinder" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-aebb1a1c99", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "76", "location": "Prisms and Cylinders", "latex": "V = \\pi r^2h", "name": null, "statement": "The volume of a right circular cylinder equals π times the square of its radius times its altitude.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the right circular cylinder" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of the circumference to the diameter" }, { "unit": null, "symbol": "r", "meaning": "radius of the base of the right circular cylinder" }, { "unit": null, "symbol": "h", "meaning": "altitude of the right circular cylinder" } ], "sympy": "Eq(V, pi*r**2*h)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-cylinder", "concept/cylinder", "concept/right-circular-cylinder", "quantity/pi", "quantity/radius", "quantity/volume", "theorem/volume-of-a-cylinder" ], "pages": [ "76", "160" ], "chapters": [ "slaught-lennes-solid-geometry-1919/ch-book-iii", "slaught-lennes-solid-geometry-1919/ch-appendix-i" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-cf2f6763cf", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "82", "location": "Pyramids and Cones", "latex": "S = \\tfrac{1}{2} l\\cdot p", "name": null, "statement": "The lateral area of a regular pyramid equals one half the product of its slant height and the perimeter of its base.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "lateral area of the regular pyramid" }, { "unit": null, "symbol": "l", "meaning": "slant height (PK) of the pyramid" }, { "unit": null, "symbol": "p", "meaning": "perimeter of the base" } ], "sympy": "Eq(S, l*p/2)", "physics": false, "states": [], "concepts": [ "concept/regular-pyramid", "concept/slant-height", "concept/theorem-lateral-area-of-a-regular-pyramid", "quantity/lateral-area", "quantity/perimeter" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-a3f9b2e582", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "83", "location": "Pyramids and Cones", "latex": "\\dfrac{b'}{b} = \\dfrac{\\overline{PK'}^2}{\\overline{PK}^2}", "name": null, "statement": "When a pyramid is cut by a plane parallel to its base, the ratio of the section's area to the base's area equals the ratio of the squares of their distances from the vertex.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b'", "meaning": "area of the polygonal section cut parallel to the base" }, { "unit": null, "symbol": "b", "meaning": "area of the base" }, { "unit": null, "symbol": "PK'", "meaning": "perpendicular distance from the vertex P to the cutting plane" }, { "unit": null, "symbol": "PK", "meaning": "altitude of the pyramid (distance from the vertex P to the base plane)" } ], "sympy": "Eq(bp/b, PKp**2/PK**2)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-pyramid", "concept/similar-solids", "concept/theorem-section-of-a-pyramid-parallel-to-its-base", "concept/vertex", "quantity/area", "theorem/section-of-a-pyramid-parallel-to-its-base" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-97a9521634", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "87", "location": "Pyramids and Cones", "latex": "V = \\tfrac{1}{3} bh", "name": null, "statement": "The volume of any pyramid is one third of the product of its base area and its altitude.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the pyramid" }, { "unit": null, "symbol": "b", "meaning": "area of the base" }, { "unit": null, "symbol": "h", "meaning": "altitude of the pyramid" } ], "sympy": "Eq(V, b*h/3)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-pyramid", "concept/base-of-a-pyramid", "concept/theorem-volume-of-a-pyramid", "quantity/volume", "theorem/volume-of-a-cone", "theorem/volume-of-a-pyramid" ], "pages": [ "87", "152" ], "chapters": [ "slaught-lennes-solid-geometry-1919/ch-book-iv", "slaught-lennes-solid-geometry-1919/ch-book-v" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-43931c622a", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "89", "location": "Pyramids and Cones", "latex": "V = \\tfrac{1}{3} hb + \\tfrac{1}{3} hb' + \\tfrac{1}{3} h \\sqrt{bb'} = \\tfrac{1}{3} h [b + b' + \\sqrt{bb'}]", "name": null, "statement": "The volume of a frustum of a pyramid equals one third of its altitude times the sum of the two base areas and their mean proportional.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the frustum of the pyramid" }, { "unit": null, "symbol": "h", "meaning": "altitude of the frustum" }, { "unit": null, "symbol": "b", "meaning": "area of the lower base of the frustum" }, { "unit": null, "symbol": "b'", "meaning": "area of the upper base of the frustum" } ], "sympy": "Eq(V, h*(b + bp + sqrt(b*bp))/3)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-pyramid", "concept/frustum-of-a-pyramid", "concept/geometrical-mean", "concept/theorem-volume-of-a-frustum-of-a-pyramid", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-4f6895c1d4", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "96", "location": "Pyramids and Cones", "latex": "\\dfrac{OF}{MB} = \\dfrac{OP}{MP} = \\dfrac{OG}{MC}", "name": null, "statement": "In a plane section parallel to the base of a circular cone, the ratio of a perimeter point's distance from the section's center O to the base radius is the same as the ratio OP/MP, so equal base radii give equal distances from O.", "kind": "result", "symbols": [ { "unit": null, "symbol": "OF", "meaning": "distance from the center O of the parallel section to the point F on its perimeter" }, { "unit": null, "symbol": "MB", "meaning": "radius of the cone's circular base, along the plane through PM and F" }, { "unit": null, "symbol": "OP", "meaning": "distance from O (center of the section) to the vertex P" }, { "unit": null, "symbol": "MP", "meaning": "distance from the center M of the base to the vertex P (the axis length)" }, { "unit": null, "symbol": "OG", "meaning": "distance from the center O of the parallel section to the point G on its perimeter" }, { "unit": null, "symbol": "MC", "meaning": "radius of the cone's circular base, along the plane through PM and G" } ], "sympy": "Eq(OF/MB, OP/MP)", "physics": false, "states": [], "concepts": [ "concept/base-of-a-logarithm-system", "concept/center-of-similitude", "concept/circle", "concept/cone", "concept/perpendicular" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-208805e749", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "98", "location": "Pyramids and Cones", "latex": "S = \\tfrac{1}{2}\\cdot 2 \\pi r\\cdot l = \\pi r l", "name": null, "statement": "The lateral area of a right circular cone equals pi times the base radius times the slant height.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "lateral area of the cone" }, { "unit": null, "symbol": "r", "meaning": "radius of the base of the cone" }, { "unit": null, "symbol": "l", "meaning": "slant height of the cone" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter of a circle" } ], "sympy": "Eq(S, pi*r*l)", "physics": false, "states": [], "concepts": [ "concept/right-circular-cone", "concept/slant-height", "quantity/lateral-area", "quantity/pi", "quantity/radius" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-ffc7c3fb8c", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "99", "location": "Pyramids and Cones", "latex": "S = \\tfrac{1}{2} (2\\pi r + 2\\pi r')l = \\pi l(r + r')", "name": null, "statement": "The lateral area of a frustum of a right circular cone equals pi times the slant height times the sum of the radii of its two bases.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "lateral area of the frustum" }, { "unit": null, "symbol": "r", "meaning": "radius of one base of the frustum" }, { "unit": null, "symbol": "r'", "meaning": "radius of the other base of the frustum" }, { "unit": null, "symbol": "l", "meaning": "slant height of the frustum" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter of a circle" } ], "sympy": "Eq(S, pi*l*(r + rp))", "physics": false, "states": [], "concepts": [ "concept/frustum-of-a-cone", "concept/right-circular-cone", "concept/slant-height", "quantity/lateral-area", "quantity/pi", "quantity/radius" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-ebde28790a", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "101", "location": "Pyramids and Cones", "latex": "V = \\tfrac{1}{3}\\cdot \\pi r^2\\cdot h = \\tfrac{1}{3} \\pi r^2 h", "name": null, "statement": "The volume of a cone with a circular base is one third of pi times the square of the base radius times the altitude.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the cone" }, { "unit": null, "symbol": "r", "meaning": "radius of the circular base" }, { "unit": null, "symbol": "h", "meaning": "altitude of the cone" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter of a circle" } ], "sympy": "Eq(V, Rational(1,3)*pi*r**2*h)", "physics": false, "states": [], "concepts": [ "concept/angle-of-elevation", "concept/circle", "concept/cone", "quantity/pi", "quantity/radius", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-e2cdc63f3d", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "102", "location": "Pyramids and Cones", "latex": "V = \\tfrac{1}{3} h(b + b' + \\sqrt{bb'})", "name": null, "statement": "The volume of a frustum of a cone equals one third of its altitude times the sum of the two base areas and the square root of their product.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the frustum of the cone" }, { "unit": null, "symbol": "h", "meaning": "altitude of the frustum" }, { "unit": null, "symbol": "b", "meaning": "area of one base of the frustum" }, { "unit": null, "symbol": "b'", "meaning": "area of the other base of the frustum" } ], "sympy": "Eq(V, Rational(1,3)*h*(b + bp + sqrt(b*bp)))", "physics": false, "states": [], "concepts": [ "concept/angle-of-elevation", "concept/base-of-a-logarithm-system", "concept/cone", "concept/frustum-of-a-cone", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-d8b2b9c0af", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-iv", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "102", "location": "Pyramids and Cones", "latex": "V = \\tfrac{1}{3} \\pi h(r^2 + {r'}^2 + rr')", "name": null, "statement": "The volume of a frustum of a right circular cone equals one third of pi times its altitude times the sum of the squares of the two base radii and their product.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the frustum of the right circular cone" }, { "unit": null, "symbol": "h", "meaning": "altitude of the frustum" }, { "unit": null, "symbol": "r", "meaning": "radius of one base of the frustum" }, { "unit": null, "symbol": "r'", "meaning": "radius of the other base of the frustum" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter of a circle" } ], "sympy": "Eq(V, Rational(1,3)*pi*h*(r**2 + rp**2 + r*rp))", "physics": false, "states": [], "concepts": [ "concept/angle-of-elevation", "concept/frustum-of-a-cone", "concept/right-circular-cone", "quantity/pi", "quantity/radius", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-90264bf7ec", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "116", "location": "The Sphere", "latex": "PD : PB = PB : PP'", "name": null, "statement": "On the sphere, the ratio of PD to PB equals the ratio of PB to PP', a relation the book uses to find the diameter from a point P on a circle and its pole-to-point distance.", "kind": "result", "symbols": [ { "unit": null, "symbol": "PD", "meaning": "segment from the pole P to the center D of the circle on the sphere" }, { "unit": null, "symbol": "PB", "meaning": "segment from the pole P to a point B on the circle" }, { "unit": null, "symbol": "PP'", "meaning": "diameter of the sphere through the two poles P and P' of the circle (the axis of the circle)" } ], "sympy": "Eq(PD/PB, PB/PP')", "physics": false, "states": [], "concepts": [ "concept/axis-of-a-circle-on-a-sphere", "concept/diameter", "concept/method-finding-the-diameter-of-a-sphere", "concept/pole-of-a-circle", "concept/sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-8f3ff69009", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "116", "location": "The Sphere", "latex": "PD × PP' = \\overline{PB}^2", "name": null, "statement": "The product of PD and the sphere's diameter PP' equals the square of the segment PB, which lets the diameter be computed from measured segments.", "kind": "result", "symbols": [ { "unit": null, "symbol": "PD", "meaning": "segment from the pole P to the center D of the circle on the sphere" }, { "unit": null, "symbol": "PP'", "meaning": "diameter of the sphere through the two poles P and P' of the circle (the axis of the circle)" }, { "unit": null, "symbol": "PB", "meaning": "segment from the pole P to a point B on the circle" } ], "sympy": "Eq(PD*PPp, PB**2)", "physics": false, "states": [], "concepts": [ "concept/diameter", "concept/method-finding-the-diameter-of-a-sphere", "concept/pole-of-a-circle", "concept/sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-cfec2bda51", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "122", "location": "The Sphere", "latex": "\\wideparen{AD} + \\wideparen{DC} + \\wideparen{CB} > \\wideparen{AB}", "name": null, "statement": "A path made of great-circle arcs from A through D and C to B on a sphere is longer than the minor great-circle arc AB, so the shortest distance between two points is the arc of a great circle.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "great-circle arc from A to D on the sphere" }, { "unit": null, "symbol": "DC", "meaning": "great-circle arc from D to C on the sphere" }, { "unit": null, "symbol": "CB", "meaning": "great-circle arc from C to B on the sphere" }, { "unit": null, "symbol": "AB", "meaning": "minor great-circle arc from A to B on the sphere" } ], "sympy": "Gt(AD + DC + CB, AB)", "physics": false, "states": [], "concepts": [ "concept/spherical-distance", "concept/spherical-triangle", "concept/theorem-shortest-distance-on-a-sphere-follows-a-great-circle-arc", "concept/theorem-sum-of-two-sides-of-a-spherical-triangle-exceeds-the-third" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-255f3085ef", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "128", "location": "The Sphere", "latex": "A + a' &= 180\\text{°}", "name": null, "statement": "An angle of a spherical triangle and the corresponding side of its polar triangle together make 180 degrees.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "A", "meaning": "angle A of the spherical triangle ABC" }, { "unit": "degree", "symbol": "a'", "meaning": "side a' of the polar triangle A'B'C', corresponding to angle A" } ], "sympy": "Eq(A + a_prime, 180)", "physics": false, "states": [], "concepts": [ "concept/corresponding-parts", "concept/polar-triangle", "concept/spherical-triangle", "quantity/angle", "quantity/arc-of-a-circle" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-e1301e6d39", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "129", "location": "The Sphere", "latex": "\\angle A + \\angle B + \\angle C + a' + b' + c' = 6 \\text{ rt.\\ } \\Angles", "name": null, "statement": "The angles of a spherical triangle plus the sides of its polar triangle together total six right angles.", "kind": "result", "symbols": [ { "unit": "right angle", "symbol": "A, B, C", "meaning": "angles of the spherical triangle ABC" }, { "unit": "right angle", "symbol": "a', b', c'", "meaning": "sides of the polar triangle A'B'C'" } ], "sympy": "Eq(A + B + C + a_prime + b_prime + c_prime, 6)", "physics": false, "states": [], "concepts": [ "concept/polar-triangle", "quantity/angle", "quantity/arc-of-a-circle", "quantity/right-angle", "theorem/sum-of-angles-of-a-spherical-triangle-lies-between-two-and-six-right-angles" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-869308b42c", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "129", "location": "The Sphere", "latex": "\\angle A + \\angle B + \\angle C < 6 \\text{ rt.\\ } \\Angles", "name": null, "statement": "The sum of the angles of a spherical triangle is less than six right angles.", "kind": "rule", "symbols": [ { "unit": "right angle", "symbol": "A, B, C", "meaning": "angles of the spherical triangle ABC" } ], "sympy": "Lt(A + B + C, 6)", "physics": false, "states": [], "concepts": [ "concept/spherical-triangle", "quantity/angle", "quantity/right-angle", "theorem/sum-of-angles-of-a-spherical-triangle-lies-between-two-and-six-right-angles" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-f9c89fb8ad", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "129", "location": "The Sphere", "latex": "\\angle A + \\angle B + \\angle C > 2 \\text{ rt.\\ } \\Angles", "name": null, "statement": "The sum of the angles of a spherical triangle is greater than two right angles.", "kind": "rule", "symbols": [ { "unit": "right angle", "symbol": "A, B, C", "meaning": "angles of the spherical triangle ABC" } ], "sympy": "Gt(A + B + C, 2)", "physics": false, "states": [], "concepts": [ "concept/spherical-triangle", "quantity/angle", "quantity/right-angle", "theorem/sum-of-angles-of-a-spherical-triangle-lies-between-two-and-six-right-angles" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-c125141b9f", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "134", "location": "The Sphere", "latex": "\\text{area } \\triangle ABC = \\text{area } \\triangle A_1B_1C_1", "name": null, "statement": "Two symmetrical spherical triangles have equal areas.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ABC", "meaning": "spherical triangle with vertices A, B, C" }, { "unit": null, "symbol": "A_1B_1C_1", "meaning": "spherical triangle symmetrical to ABC" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "quantity/area", "theorem/area-of-a-spherical-triangle", "theorem/spherical-triangles-equal-by-two-angles-and-included-side" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-fb261286ae", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "136", "location": "The Sphere", "latex": "\\triangle ABC = \\angle A + \\angle B + \\angle C - 180", "name": null, "statement": "The area of a spherical triangle, in spherical degrees, equals its spherical excess: the sum of its angles minus 180 degrees.", "kind": "result", "symbols": [ { "unit": "spherical degree", "symbol": "ABC", "meaning": "area of the spherical triangle ABC" }, { "unit": "degree", "symbol": "A, B, C", "meaning": "angles of the spherical triangle ABC" } ], "sympy": "Eq(area_ABC, A + B + C - 180)", "physics": false, "states": [], "concepts": [ "quantity/angle", "quantity/area", "quantity/spherical-excess", "theorem/area-of-a-spherical-triangle", "unit/spherical-degree" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-4507496565", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "140", "location": "The Sphere", "latex": "2\\pi r × AB = 2\\pi r × 2r = 4\\pi r^2", "name": null, "statement": "The area of a sphere of radius r is 4 pi r squared, found as the limit of circumscribed surfaces with AB = 2r.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "AB", "meaning": "diameter of the sphere" }, { "unit": null, "symbol": "π", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(2*pi*r*AB, 4*pi*r**2)", "physics": false, "states": [], "concepts": [ "concept/sphere", "quantity/area", "quantity/pi", "quantity/radius", "theorem/area-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-6f6e7b3580", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "143", "location": "The Sphere", "latex": "\\tfrac{4}{3} \\pi r^3", "name": "volume of a sphere", "statement": "The volume of a sphere of radius r is four-thirds pi times r cubed.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the sphere" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "π", "meaning": "ratio of the circumference to the diameter" } ], "sympy": "Eq(V, 4*pi*r**3/3)", "physics": false, "states": [ "theorem/volume-of-a-sphere" ], "concepts": [ "concept/sphere", "quantity/pi", "quantity/radius", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-52f362833c", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "143", "location": "The Sphere", "latex": "4\\pi r^2", "name": "area of a sphere", "statement": "The surface of a sphere of radius r has area four pi r squared.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "π", "meaning": "ratio of the circumference to the diameter" } ], "sympy": "Eq(S, 4*pi*r**2)", "physics": false, "states": [ "theorem/area-of-a-sphere" ], "concepts": [ "concept/sphere", "quantity/area", "quantity/pi", "quantity/radius" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-2a99cddeb3", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "147", "location": "The Sphere", "latex": "s = 2\\pi rh", "name": null, "statement": "The area of a zone is two pi times the radius of the sphere times the altitude of the zone.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s", "meaning": "area of the zone" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "h", "meaning": "altitude of the zone" } ], "sympy": "Eq(s, 2*pi*r*h)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-cylinder", "concept/cylinder", "concept/spherical-zone", "quantity/altitude-of-a-spherical-zone", "quantity/lateral-area", "quantity/pi", "quantity/radius", "theorem/area-of-a-spherical-zone" ], "pages": [ "147", "160" ], "chapters": [ "slaught-lennes-solid-geometry-1919/ch-book-v", "slaught-lennes-solid-geometry-1919/ch-appendix-i" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-d90f6ceff7", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "147", "location": "The Sphere", "latex": "v = \\dfrac{r}{3} \\cdot s", "name": null, "statement": "The volume of a spherical cone is one third the radius times the area of the zone cut out of the sphere by the cone.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "v", "meaning": "volume of the spherical cone" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "s", "meaning": "area of the zone of one base cut out by the cone" } ], "sympy": "Eq(v, r*s/3)", "physics": false, "states": [], "concepts": [ "concept/spherical-cone", "quantity/radius", "theorem/area-of-a-spherical-zone", "theorem/volume-of-a-spherical-cone" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-fd4248d33c", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "147", "location": "The Sphere", "latex": "v = \\dfrac{r}{3} \\cdot 2\\pi rh = \\dfrac{2\\pi}{3} r^2h", "name": null, "statement": "Substituting the zone area into the spherical cone formula gives volume two pi r squared h over three.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "volume of the spherical cone" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "h", "meaning": "altitude of the zone cut out by the cone" } ], "sympy": "Eq(v, 2*pi*r**2*h/3)", "physics": false, "states": [], "concepts": [ "quantity/altitude-of-a-spherical-zone", "quantity/pi", "quantity/radius", "theorem/volume-of-a-spherical-cone" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-95329cfabd", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "147", "location": "The Sphere", "latex": "v = \\dfrac{2\\pi}{3} r^2h", "name": null, "statement": "The volume of a spherical sector is two pi over three times the square of the radius times the altitude of its zone.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "volume of the spherical sector" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "h", "meaning": "altitude of the zone of two bases cut out by the sector" } ], "sympy": "Eq(v, 2*pi*r**2*h/3)", "physics": false, "states": [], "concepts": [ "concept/spherical-sector", "quantity/altitude-of-a-spherical-zone", "quantity/pi", "quantity/radius", "theorem/volume-of-a-spherical-sector" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-dc738616b3", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "148", "location": "The Sphere", "latex": "v = \\dfrac{\\pi h}{2} (r_1^2 + r_2^2) + \\dfrac{\\pi}{6} h^3", "name": null, "statement": "The volume of a spherical segment is pi h over two times the sum of the squares of the base radii, plus pi h cubed over six.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "volume of the spherical segment" }, { "unit": null, "symbol": "r_1", "meaning": "radius of one base of the segment" }, { "unit": null, "symbol": "r_2", "meaning": "radius of the other base of the segment" }, { "unit": null, "symbol": "h", "meaning": "altitude of the segment" }, { "unit": null, "symbol": "π", "meaning": "ratio of the circumference to the diameter" } ], "sympy": "Eq(v, pi*h/2*(r1**2 + r2**2) + pi*h**3/6)", "physics": false, "states": [], "concepts": [ "concept/spherical-segment", "quantity/altitude-of-a-spherical-zone", "quantity/pi", "quantity/radius", "theorem/volume-of-a-spherical-segment" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-b40f3341f6", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "148", "location": "The Sphere", "latex": "v = \\pi h^2 \\left( r - \\dfrac{h}{3} \\right)", "name": null, "statement": "The volume of a spherical segment of one base is pi h squared times the quantity r minus h over three.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "volume of the spherical segment of one base" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "h", "meaning": "altitude of the segment" } ], "sympy": "Eq(v, pi*h**2*(r - h/3))", "physics": false, "states": [], "concepts": [ "concept/spherical-segment", "quantity/altitude-of-a-spherical-zone", "quantity/pi", "quantity/radius", "theorem/volume-of-a-spherical-segment" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-53cd9dc361", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "148", "location": "The Sphere", "latex": "d &= \\dfrac{r_2^2 - r_1^2 - h^2}{2h}", "name": null, "statement": "The distance d from the center of the sphere to the plane of the smaller base is expressed in terms of the base radii and the altitude h.", "kind": "result", "symbols": [ { "unit": null, "symbol": "d", "meaning": "distance from the center of the sphere to the base plane of radius r_2 (segment construction)" }, { "unit": null, "symbol": "r_1", "meaning": "radius of one base of the segment" }, { "unit": null, "symbol": "r_2", "meaning": "radius of the other base of the segment" }, { "unit": null, "symbol": "h", "meaning": "altitude of the segment" } ], "sympy": "Eq(d, (r2**2 - r1**2 - h**2)/(2*h))", "physics": false, "states": [], "concepts": [ "concept/spherical-segment", "quantity/altitude-of-a-spherical-zone", "quantity/radius" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-3ede131a14", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "148", "location": "The Sphere", "latex": "r^2 = \\dfrac{r_2^4 + r_1^4 + h^4 - 2r_1^2r_2^2 + 2h^2r_2^2 + 2h^2r_1^2}{4h^2}", "name": null, "statement": "The squared sphere radius r is written in terms of the base radii r_1, r_2 and the altitude h of the segment.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "r_1", "meaning": "radius of one base of the segment" }, { "unit": null, "symbol": "r_2", "meaning": "radius of the other base of the segment" }, { "unit": null, "symbol": "h", "meaning": "altitude of the segment" } ], "sympy": "Eq(r**2, (r2**4 + r1**4 + h**4 - 2*r1**2*r2**2 + 2*h**2*r2**2 + 2*h**2*r1**2)/(4*h**2))", "physics": false, "states": [], "concepts": [ "concept/spherical-segment", "quantity/radius" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-4ccd124ffd", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "148", "location": "The Sphere", "latex": "r^2 = r_2^2 + d^2", "name": null, "statement": "By the Pythagorean theorem, the squared sphere radius equals the squared base radius r_2 plus the squared distance d.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "r_2", "meaning": "radius of the base of the segment" }, { "unit": null, "symbol": "d", "meaning": "distance from the center of the sphere to the base plane of radius r_2" } ], "sympy": "Eq(r**2, r2**2 + d**2)", "physics": false, "states": [], "concepts": [ "concept/spherical-segment", "quantity/radius", "theorem/pythagorean-theorem" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-7e50e6461e", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "148", "location": "The Sphere", "latex": "r^2 = r_1^2 + (h+d)^2", "name": null, "statement": "By the Pythagorean theorem, the squared sphere radius equals the squared base radius r_1 plus the squared distance h plus d.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "r_1", "meaning": "radius of the other base of the segment" }, { "unit": null, "symbol": "h", "meaning": "altitude of the segment" }, { "unit": null, "symbol": "d", "meaning": "distance from the center of the sphere to the base plane of radius r_2" } ], "sympy": "Eq(r**2, r1**2 + (h + d)**2)", "physics": false, "states": [], "concepts": [ "concept/spherical-segment", "quantity/radius", "theorem/pythagorean-theorem" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-0a2fa5b1db", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "152", "location": "The Sphere", "latex": "V = abc", "name": null, "statement": "The volume of a rectangular parallelepiped with dimensions a, b, c is the product abc.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the rectangular parallelopiped" }, { "unit": null, "symbol": "a", "meaning": "dimension of the rectangular parallelopiped" }, { "unit": null, "symbol": "b", "meaning": "dimension of the rectangular parallelopiped" }, { "unit": null, "symbol": "c", "meaning": "dimension of the rectangular parallelopiped" } ], "sympy": "Eq(V, a*b*c)", "physics": false, "states": [], "concepts": [ "concept/rectangular-parallelepiped", "concept/volume-of-a-rectangular-parallelopiped", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-67f555ee50", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "152", "location": "The Sphere", "latex": "V = hb", "name": null, "statement": "The volume of a prism or cylinder is its base area times its altitude.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the prism or cylinder" }, { "unit": null, "symbol": "b", "meaning": "area of the base" }, { "unit": null, "symbol": "h", "meaning": "altitude of the prism or cylinder" } ], "sympy": "Eq(V, b*h)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-prism", "concept/base-of-a-prism", "concept/cylinder", "concept/prism", "quantity/area", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-0529a7d720", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "152", "location": "The Sphere", "latex": "S = pe", "name": null, "statement": "The lateral surface of a prism or cylinder is the perimeter of a right section times the lateral edge or element.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "S", "meaning": "lateral surface" }, { "unit": null, "symbol": "p", "meaning": "perimeter of a right section of the prism or cylinder" }, { "unit": null, "symbol": "e", "meaning": "lateral edge or element" } ], "sympy": "Eq(S, p*e)", "physics": false, "states": [], "concepts": [ "concept/cylinder", "concept/edge", "concept/prism", "concept/right-section", "quantity/lateral-area", "quantity/perimeter" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-38413ca8c5", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "152", "location": "The Sphere", "latex": "S = \\tfrac{1}{2} pl", "name": null, "statement": "The lateral area of a regular pyramid or cone is one half the base perimeter times the slant height.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "S", "meaning": "lateral area" }, { "unit": null, "symbol": "p", "meaning": "perimeter of the base" }, { "unit": null, "symbol": "l", "meaning": "slant height of the regular pyramid or cone" } ], "sympy": "Eq(S, p*l/2)", "physics": false, "states": [], "concepts": [ "concept/slant-height", "quantity/lateral-area", "quantity/perimeter", "theorem/lateral-area-of-a-regular-pyramid" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-7ff4b23bbc", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "152", "location": "The Sphere", "latex": "V = \\tfrac{1}{3} h (b + b' + \\sqrt{bb'})", "name": null, "statement": "The volume of a frustum of a pyramid or cone is one third its altitude times the sum of the two base areas plus the square root of their product.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the frustum" }, { "unit": null, "symbol": "b", "meaning": "area of the lower base" }, { "unit": null, "symbol": "b'", "meaning": "area of the upper base" }, { "unit": null, "symbol": "h", "meaning": "altitude of the frustum" } ], "sympy": "Eq(V, h*(b + bp + sqrt(b*bp))/3)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-pyramid", "concept/frustum-of-a-cone", "concept/frustum-of-a-pyramid", "theorem/volume-of-a-frustum-of-a-pyramid" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-d73b772dd5", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "152", "location": "The Sphere", "latex": "S = \\dfrac{a}{720} \\cdot 4 \\pi r^2", "name": null, "statement": "The area of a spherical polygon is its spherical excess in degrees divided by 720 times the total area of the sphere.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "S", "meaning": "area of the spherical polygon in square units" }, { "unit": "degree", "symbol": "a", "meaning": "spherical excess of the spherical polygon in degrees" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "π", "meaning": "ratio of the circumference to the diameter" } ], "sympy": "Eq(S, a/720*4*pi*r**2)", "physics": false, "states": [], "concepts": [ "quantity/pi", "quantity/radius", "quantity/spherical-excess", "theorem/area-of-a-sphere", "theorem/area-of-a-spherical-polygon", "unit/spherical-degree" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-bd680b3170", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "153", "location": "The Sphere", "latex": "V = \\dfrac{2\\pi}{3}r^2h", "name": null, "statement": "The volume of a spherical cone is two pi over three times the square of the sphere's radius times the altitude of the cone's zone.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the spherical cone" }, { "unit": null, "symbol": "h", "meaning": "altitude of the spherical zone cut out by the cone" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": "Eq(V, 2*pi*r**2*h/3)", "physics": false, "states": [], "concepts": [ "concept/spherical-cone", "quantity/altitude-of-a-spherical-zone", "quantity/pi", "quantity/radius", "theorem/volume-of-a-spherical-cone" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-a30389bbc7", "chapter": "slaught-lennes-solid-geometry-1919/ch-book-v", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "152", "location": "The Sphere", "latex": "S = 4 \\pi r^2", "name": null, "statement": "The surface of a sphere of radius r is four pi r squared.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "surface of the sphere" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "π", "meaning": "ratio of the circumference to the diameter" } ], "sympy": "Eq(S, 4*pi*r**2)", "physics": false, "states": [], "concepts": [ "concept/sphere", "quantity/pi", "quantity/radius", "theorem/area-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-8802f8e67d", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "160", "location": "SIMILAR SOLIDS", "latex": "S = 2\\pi r(r + h)", "name": null, "statement": "The total area of a right circular cylinder equals 2 pi times the radius times the sum of the radius and the altitude.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "S", "meaning": "total area of a cylinder" }, { "unit": null, "symbol": "r", "meaning": "radius of the cylinder" }, { "unit": null, "symbol": "h", "meaning": "altitude of the cylinder" } ], "sympy": "Eq(S, 2*pi*r*(r + h))", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-cylinder", "concept/cylinder", "quantity/pi", "quantity/radius", "quantity/total-area" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-d540dbf5a7", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "160", "location": "SIMILAR SOLIDS", "latex": "V = \\pi {r'}^2h'", "name": null, "statement": "The volume of the second right circular cylinder equals pi times the square of its radius times its altitude.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the second cylinder" }, { "unit": null, "symbol": "r'", "meaning": "radius of the second cylinder" }, { "unit": null, "symbol": "h'", "meaning": "altitude of the second cylinder" } ], "sympy": "Eq(V, pi*rp**2*hp)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-cylinder", "concept/cylinder", "quantity/pi", "quantity/radius", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-d4cfef6e4b", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "160", "location": "SIMILAR SOLIDS", "latex": "\\dfrac{r}{r'} = \\dfrac{h}{h'}", "name": null, "statement": "For two similar cylinders the ratio of radii equals the ratio of altitudes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the first cylinder" }, { "unit": null, "symbol": "r'", "meaning": "radius of the second cylinder" }, { "unit": null, "symbol": "h", "meaning": "altitude of the first cylinder" }, { "unit": null, "symbol": "h'", "meaning": "altitude of the second cylinder" } ], "sympy": "Eq(r/rp, h/hp)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-cylinder", "concept/ratio-of-similitude", "concept/similar-cylinders", "quantity/radius" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-4b7ffe76a7", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "160", "location": "SIMILAR SOLIDS", "latex": "\\dfrac{s}{s'} = \\dfrac{S}{S'} = \\dfrac{r^2}{r'^2} = \\dfrac{h^2}{h'^2}", "name": null, "statement": "For two similar right circular cones, the ratios of lateral areas and total areas equal the squares of the ratios of radii and altitudes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s", "meaning": "lateral area of the first cone" }, { "unit": null, "symbol": "s'", "meaning": "lateral area of the second cone" }, { "unit": null, "symbol": "S", "meaning": "total area of the first cone" }, { "unit": null, "symbol": "S'", "meaning": "total area of the second cone" }, { "unit": null, "symbol": "r", "meaning": "radius of the first cone" }, { "unit": null, "symbol": "r'", "meaning": "radius of the second cone" }, { "unit": null, "symbol": "h", "meaning": "altitude of the first cone" }, { "unit": null, "symbol": "h'", "meaning": "altitude of the second cone" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/angle-of-elevation", "concept/similar-cones", "quantity/lateral-area", "quantity/radius", "quantity/total-area" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-346882baa5", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "160", "location": "SIMILAR SOLIDS", "latex": "\\dfrac{s}{s'} = \\dfrac{S}{S'} = \\dfrac{r^2}{r'^2} = \\dfrac{h^2}{h'^2} \\text{ and } \\dfrac{V}{V'} = \\dfrac{r^3}{r'^3} = \\dfrac{h^3}{h'^3}", "name": null, "statement": "For two similar right circular cones, the area ratios equal the squares of the radius and altitude ratios, and the volume ratio equals their cubes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s", "meaning": "lateral area of the first cone" }, { "unit": null, "symbol": "s'", "meaning": "lateral area of the second cone" }, { "unit": null, "symbol": "S", "meaning": "total area of the first cone" }, { "unit": null, "symbol": "S'", "meaning": "total area of the second cone" }, { "unit": null, "symbol": "V", "meaning": "volume of the first cone" }, { "unit": null, "symbol": "V'", "meaning": "volume of the second cone" }, { "unit": null, "symbol": "r", "meaning": "radius of the first cone" }, { "unit": null, "symbol": "r'", "meaning": "radius of the second cone" }, { "unit": null, "symbol": "h", "meaning": "altitude of the first cone" }, { "unit": null, "symbol": "h'", "meaning": "altitude of the second cone" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-cylinder", "concept/angle-of-elevation", "concept/ratio-of-similitude", "concept/similar-cones", "concept/similar-cylinders", "quantity/lateral-area", "quantity/radius", "quantity/total-area", "quantity/volume", "theorem/ratios-of-similar-cones", "theorem/volume-of-a-cone" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-96e1a345b5", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "163", "location": "SIMILAR SOLIDS", "latex": "\\dfrac{V}{V'} = \\dfrac{PA \\cdot PB \\cdot PC}{P'A' \\cdot P'B' \\cdot P'C'}", "name": null, "statement": "Two tetrahedrons with an equal trihedral angle at P have volumes in the ratio of the products of the edges meeting at that vertex.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of tetrahedron P-ABC" }, { "unit": null, "symbol": "V'", "meaning": "volume of tetrahedron P'-A'B'C'" }, { "unit": null, "symbol": "PA", "meaning": "edge PA of the first tetrahedron" }, { "unit": null, "symbol": "PB", "meaning": "edge PB of the first tetrahedron" }, { "unit": null, "symbol": "PC", "meaning": "edge PC of the first tetrahedron" }, { "unit": null, "symbol": "P'A'", "meaning": "edge P'A' of the second tetrahedron" }, { "unit": null, "symbol": "P'B'", "meaning": "edge P'B' of the second tetrahedron" }, { "unit": null, "symbol": "P'C'", "meaning": "edge P'C' of the second tetrahedron" } ], "sympy": "Eq(V/Vp, PA*PB*PC/(PAp*PBp*PCp))", "physics": false, "states": [], "concepts": [ "concept/edge", "concept/equal-trihedral-angles", "concept/tetrahedron", "concept/trihedral-angle", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-6aaab052f2", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "164", "location": "SIMILAR SOLIDS", "latex": "\\dfrac{V}{V'} = \\dfrac{\\overline{PA}^3}{\\overline{P'A'}^3}", "name": null, "statement": "The volumes of two similar tetrahedrons are in the ratio of the cubes of corresponding edges.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of tetrahedron P-ABC" }, { "unit": null, "symbol": "V'", "meaning": "volume of tetrahedron P'-A'B'C'" }, { "unit": null, "symbol": "PA", "meaning": "edge PA of the first tetrahedron" }, { "unit": null, "symbol": "P'A'", "meaning": "corresponding edge of the second tetrahedron" } ], "sympy": "Eq(V/Vp, PA**3/PAp**3)", "physics": false, "states": [], "concepts": [ "concept/corresponding-parts", "concept/similar-polyhedron", "concept/tetrahedron", "quantity/volume", "theorem/volumes-of-similar-tetrahedrons" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-5aa83460ab", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "165", "location": "SIMILAR SOLIDS", "latex": "OA : OA' = OB : OB'", "name": null, "statement": "Two figures have a center of similitude O when the segments from O to corresponding points stand in a constant ratio.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "O", "meaning": "center of similitude" }, { "unit": null, "symbol": "A", "meaning": "a point of the first figure" }, { "unit": null, "symbol": "A'", "meaning": "the point of the second figure corresponding to A" }, { "unit": null, "symbol": "B", "meaning": "a second point of the first figure" }, { "unit": null, "symbol": "B'", "meaning": "the point corresponding to B" } ], "sympy": "Eq(OA/OAp, OB/OBp)", "physics": false, "states": [], "concepts": [ "concept/center-of-similitude", "concept/corresponding-parts", "concept/ratio-of-similitude", "concept/similar-solids" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-dc1d03ed3b", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "170", "location": "SIMILAR SOLIDS", "latex": "\\dfrac{V}{V'} = \\dfrac{\\overline{AB}^3}{\\overline{A'B'}^3}", "name": null, "statement": "The volumes of any two similar polyhedrons are proportional to the cubes of their corresponding edges.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the first polyhedron" }, { "unit": null, "symbol": "V'", "meaning": "volume of the second polyhedron" }, { "unit": null, "symbol": "AB", "meaning": "an edge of the first polyhedron" }, { "unit": null, "symbol": "A'B'", "meaning": "the corresponding edge of the second polyhedron" } ], "sympy": "Eq(V/Vp, AB**3/ApBp**3)", "physics": false, "states": [], "concepts": [ "concept/corresponding-linear-dimensions", "concept/edge", "concept/similar-polyhedron", "quantity/volume", "theorem/volumes-of-similar-polyhedrons" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-ccda6ac7b8", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "172", "location": "SIMILAR SOLIDS", "latex": "\\dfrac{\\text{area } ABC}{\\text{area } A'B'C'} = \\dfrac{m^2}{n^2}", "name": null, "statement": "Corresponding triangles of two similar figures have areas in the square of their ratio of similitude.", "kind": "result", "symbols": [ { "unit": null, "symbol": "area ABC", "meaning": "area of a triangle in the first figure" }, { "unit": null, "symbol": "area A'B'C'", "meaning": "area of the corresponding triangle in the second figure" }, { "unit": null, "symbol": "m", "meaning": "first term of the ratio of similitude m : n" }, { "unit": null, "symbol": "n", "meaning": "second term of the ratio of similitude m : n" } ], "sympy": "Eq(area_ABC/area_ApBpCp, m**2/n**2)", "physics": false, "states": [], "concepts": [ "concept/corresponding-cross-sections", "concept/corresponding-parts", "concept/ratio-of-similitude", "concept/similar-solids", "quantity/area" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-5430f639bf", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "172", "location": "SIMILAR SOLIDS", "latex": "\\dfrac{\\text{vol.\\ } ABCD}{\\text{vol.\\ } A'B'C'D'} = \\dfrac{m^3}{n^3}", "name": null, "statement": "Corresponding tetrahedrons of two similar figures have volumes in the cube of their ratio of similitude.", "kind": "result", "symbols": [ { "unit": null, "symbol": "vol. ABCD", "meaning": "volume of a tetrahedron in the first figure" }, { "unit": null, "symbol": "vol. A'B'C'D'", "meaning": "volume of the corresponding tetrahedron in the second figure" }, { "unit": null, "symbol": "m", "meaning": "first term of the ratio of similitude m : n" }, { "unit": null, "symbol": "n", "meaning": "second term of the ratio of similitude m : n" } ], "sympy": "Eq(vol_ABCD/vol_ApBpCpDp, m**3/n**3)", "physics": false, "states": [], "concepts": [ "concept/corresponding-parts", "concept/ratio-of-similitude", "concept/similar-solids", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-d0175894ee", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-i", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "174", "location": "SIMILAR SOLIDS", "latex": "w = kh^3", "name": null, "statement": "Assuming schoolboys' weights vary as the cube of their heights, weight equals a constant k times height cubed.", "kind": "formula", "symbols": [ { "unit": "pound", "symbol": "w", "meaning": "weight of a schoolboy" }, { "unit": "foot", "symbol": "h", "meaning": "height of a schoolboy" }, { "unit": null, "symbol": "k", "meaning": "constant of proportionality for the boys' weights" } ], "sympy": "Eq(w, k*h**3)", "physics": true, "states": [], "concepts": [ "concept/constant", "concept/proportion", "concept/similar-solids", "quantity/volume" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-cdcc0d802d", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "176", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "\\dfrac{p}{l} = \\text{cosine}\\, \\angle BAE", "name": null, "statement": "The cosine of the projection angle is the ratio of the length p of a segment's projection to the length l of the segment.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "p", "meaning": "length of the projection of a line-segment" }, { "unit": null, "symbol": "l", "meaning": "length of the line-segment" } ], "sympy": "Eq(p/l, cos(BAE))", "physics": false, "states": [], "concepts": [ "concept/angle-between-line-and-plane", "concept/cosine", "concept/projection-of-a-line" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-3f9f666c54", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "176", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "\\sin A = \\dfrac{a}{c},\\quad \\cos A = \\dfrac{b}{c},\\quad \\tan A = \\dfrac{a}{b}", "name": null, "statement": "In a right triangle, the sine of an acute angle A is opposite side over hypotenuse, its cosine is adjacent side over hypotenuse, and its tangent is opposite side over adjacent side.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "A", "meaning": "acute angle of the right triangle ABC" }, { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "b", "meaning": "side adjacent to angle A" }, { "unit": null, "symbol": "c", "meaning": "hypotenuse" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/hypotenuse", "concept/right-triangle", "concept/side", "concept/sine", "concept/tangent-function" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-8994ca1c43", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "179", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "p = l \\cos A", "name": null, "statement": "The length of the projection of a line-segment on a given line equals the length of the segment multiplied by the cosine of the projection angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "length of the projection of the line-segment on the given line" }, { "unit": null, "symbol": "l", "meaning": "length of the line-segment" }, { "unit": null, "symbol": "A", "meaning": "projection angle" } ], "sympy": "Eq(p, l*cos(A))", "physics": false, "states": [], "concepts": [ "concept/angle-between-line-and-plane", "concept/cosine", "concept/projection-of-a-line", "concept/theorem-projection-of-a-line-on-a-line" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-642039cf78", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "181", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "BE &= AB \\cdot \\cos D", "name": null, "statement": "The altitude of an oblique prism or cylinder equals an element multiplied by the cosine of the dihedral angle between the base plane and the right-section plane.", "kind": "result", "symbols": [ { "unit": null, "symbol": "BE", "meaning": "altitude of the oblique prism or cylinder (perpendicular between the planes of the bases)" }, { "unit": null, "symbol": "AB", "meaning": "element (lateral edge) of the oblique prism or cylinder" }, { "unit": null, "symbol": "D", "meaning": "dihedral angle between the plane of the base and the plane of a right section" } ], "sympy": "Eq(BE, AB*cos(D))", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-cylinder", "concept/altitude-of-a-prism", "concept/dihedral-angle", "concept/element-of-a-cylinder", "concept/theorem-altitude-of-an-oblique-prism-or-cylinder" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-b1d96f5e7f", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "184", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "c &= b \\cos \\angle 1", "name": null, "statement": "The area of the projection of a plane-segment equals the area of the plane-segment multiplied by the cosine of the projection angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "area of the projection of the plane-segment b upon the given plane" }, { "unit": null, "symbol": "b", "meaning": "area of the plane-segment being projected" }, { "unit": null, "symbol": "angle 1", "meaning": "projection angle between the two planes" } ], "sympy": "Eq(c, b*cos(angle1))", "physics": false, "states": [], "concepts": [ "concept/angle-between-line-and-plane", "concept/cosine", "concept/projection", "quantity/area", "theorem/area-of-the-projection-of-a-plane-segment" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-d55b598482", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "184", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "S' &= S \\cos \\angle 1", "name": null, "statement": "For a rectangle with one side parallel to the line of intersection of the planes, its projection has area equal to the rectangle's area times the cosine of the projection angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "area of the given rectangle" }, { "unit": null, "symbol": "S'", "meaning": "area of the projection of the rectangle" }, { "unit": null, "symbol": "angle 1", "meaning": "angle between the two planes" } ], "sympy": "Eq(Sprime, S*cos(angle1))", "physics": false, "states": [], "concepts": [ "concept/angle-between-line-and-plane", "concept/cosine", "concept/projection", "quantity/area", "theorem/area-of-the-projection-of-a-plane-segment" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-a2416ba81b", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "185", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "\\pi ab", "name": null, "statement": "The area of an ellipse, formed as the projection of a circle, equals pi times the semimajor axis times the semiminor axis.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A", "meaning": "area of the ellipse" }, { "unit": null, "symbol": "a", "meaning": "semimajor axis of the ellipse (O'B')" }, { "unit": null, "symbol": "b", "meaning": "semiminor axis of the ellipse (O'C')" } ], "sympy": "Eq(A, pi*a*b)", "physics": false, "states": [], "concepts": [ "concept/ellipse", "concept/theorem-area-of-an-ellipse", "quantity/area", "quantity/pi", "theorem/area-of-an-ellipse" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-9c25319856", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-ii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "186", "location": "PROJECTION OF LINE-SEGMENTS", "latex": "ABCD = A'BCD' \\cos \\angle 1", "name": null, "statement": "The right cross-section of a beam of sunlight equals the area it spreads over on the ground multiplied by the cosine of the angle between the rays and the ground.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ABCD", "meaning": "right cross-section of the beam of sunlight" }, { "unit": null, "symbol": "A'BCD'", "meaning": "patch of ground over which the beam is spread" }, { "unit": null, "symbol": "angle 1", "meaning": "projection angle between the sun's rays and the horizontal ground" } ], "sympy": "Eq(ABCD, Aprime_BCD*cos(angle1))", "physics": true, "states": [], "concepts": [ "concept/angle-between-line-and-plane", "concept/cosine", "quantity/area", "theorem/area-of-the-projection-of-a-plane-segment" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-7aa76b6a93", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "188", "location": "VARIABLES. LIMITS", "latex": "a = bh", "name": null, "statement": "The area of a rectangle equals its base times its altitude.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "area of the rectangle" }, { "unit": null, "symbol": "b", "meaning": "base of the rectangle" }, { "unit": null, "symbol": "h", "meaning": "altitude of the rectangle" } ], "sympy": "Eq(a, b*h)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-prism", "quantity/area", "theorem/area-of-a-rectangle" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-e3873c2f19", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "189", "location": "VARIABLES. LIMITS", "latex": "d = \\sqrt{2}", "name": null, "statement": "The diagonal of a square whose side is unity has length the square root of 2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "d", "meaning": "diagonal of a square whose side is unity" } ], "sympy": "Eq(d, sqrt(2))", "physics": false, "states": [], "concepts": [ "concept/incommensurable-magnitudes", "concept/irrational-number", "concept/surd" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-19bdc9073b", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "198", "location": "VARIABLES. LIMITS", "latex": "\\dfrac{AD}{AB} = \\dfrac{AE}{AC}", "name": null, "statement": "A line DE parallel to side BC of triangle ABC, cutting sides AB and AC, divides them in the same ratio.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "segment from A to D on side AB" }, { "unit": null, "symbol": "AB", "meaning": "side of the triangle from A to B" }, { "unit": null, "symbol": "AE", "meaning": "segment from A to E on side AC" }, { "unit": null, "symbol": "AC", "meaning": "side of the triangle from A to C" } ], "sympy": "Eq(AD/AB, AE/AC)", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/parallel-lines", "concept/proportion", "concept/triangle" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-3e79b28097", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "201", "location": "VARIABLES. LIMITS", "latex": "\\dfrac{k^2A}{A} = k^2 = \\dfrac{r'^2}{r^2}", "name": null, "statement": "For two circles whose radii are r and r' with r' = kr, the ratio of their areas equals the square of the ratio of their radii, k squared.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "ratio r'/r of the radii of the two circles" }, { "unit": null, "symbol": "A", "meaning": "area of the first circle" }, { "unit": null, "symbol": "r", "meaning": "radius of the first circle" }, { "unit": null, "symbol": "r'", "meaning": "radius of the second circle" } ], "sympy": "Eq(k**2*A/A, k**2)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/common-ratio", "concept/similar-solids", "quantity/area", "quantity/radius" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-b0ea85892c", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "207", "location": "VARIABLES. LIMITS", "latex": "V = \\tfrac{4}{3} \\pi r^3", "name": null, "statement": "The volume of a sphere of radius r is four thirds of pi times r cubed.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the sphere" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": "Eq(V, 4*pi*r**3/3)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/sphere", "quantity/pi", "quantity/radius", "theorem/volume-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-fa944052f1", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "207", "location": "VARIABLES. LIMITS", "latex": "V = \\tfrac{1}{3}rS", "name": null, "statement": "The volume of a sphere equals one third of its radius times its surface area, by the limit of the volumes of the circumscribed polyhedrons.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the sphere, defined as the limit of the volumes of circumscribed polyhedrons" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "S", "meaning": "surface of the sphere, defined as the limit of the total surfaces of circumscribed polyhedrons" } ], "sympy": "Eq(V, r*S/3)", "physics": false, "states": [], "concepts": [ "concept/circumscribed-polyhedron", "concept/limit", "concept/sphere", "theorem/area-of-a-sphere", "theorem/volume-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-6f4168ebad", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "207", "location": "VARIABLES. LIMITS", "latex": "S = \\tfrac{3}{r} \\cdot \\tfrac{4}{3} \\pi r^3 = 4 \\pi r^2", "name": null, "statement": "The surface of a sphere of radius r equals four pi times r squared, obtained by substituting the volume formula into the relation V = r S / 3.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "surface of the sphere" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": "Eq(S, 4*pi*r**2)", "physics": false, "states": [], "concepts": [ "concept/sphere", "quantity/pi", "quantity/radius", "theorem/area-of-a-sphere", "theorem/volume-of-a-sphere" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-bd67296c88", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "203", "location": "VARIABLES. LIMITS", "latex": "U = V", "name": null, "statement": "Defining the volume of a pyramid by circumscribed prisms gives the same limit as defining it by inscribed prisms.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U", "meaning": "limit of the volumes of the decreasing sequence of circumscribed prisms of a pyramid" }, { "unit": null, "symbol": "V", "meaning": "volume of the pyramid, defined as the limit of the volumes of inscribed prisms" } ], "sympy": "Eq(U, V)", "physics": false, "states": [], "concepts": [ "concept/circumscribed-prism", "concept/inscribed-prism", "concept/limit", "theorem/volume-of-a-pyramid" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-db5918bc12", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "203", "location": "VARIABLES. LIMITS", "latex": "p = p'", "name": null, "statement": "The perimeter of a convex closed curve is the same whether it is found as the limit of inscribed polygon perimeters or of circumscribed polygon perimeters.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "p", "meaning": "limit of the perimeters of the inscribed polygons, defined as the perimeter of the curve" }, { "unit": null, "symbol": "p'", "meaning": "limit of the perimeters of the circumscribed polygons" } ], "sympy": "Eq(p, pp)", "physics": false, "states": [], "concepts": [ "concept/convex-plane-curve", "concept/limit", "concept/regular-polygon", "quantity/perimeter" ] }, { "id": "slaught-lennes-solid-geometry-1919/eq-021b0ad7eb", "chapter": "slaught-lennes-solid-geometry-1919/ch-appendix-iii", "book": "slaught-lennes-solid-geometry-1919", "edition": "Allyn and Bacon, revised ed., 1919 (edition to be confirmed from the copy)", "page": "203", "location": "VARIABLES. LIMITS", "latex": "A = A'", "name": null, "statement": "The area of a convex closed curve is the same whether it is found as the limit of inscribed polygon areas or of circumscribed polygon areas.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "A", "meaning": "limit of the areas of the inscribed polygons, defined as the area of the curve" }, { "unit": null, "symbol": "A'", "meaning": "limit of the areas of the circumscribed polygons" } ], "sympy": "Eq(A, Ap)", "physics": false, "states": [], "concepts": [ "concept/convex-plane-curve", "concept/limit", "concept/regular-polygon", "quantity/area" ] } ], "exercise_sets": [], "problems": [], "errata_statuses": [ "transcriber_marked", "candidate", "probable", "confirmed", "dismissed", "note" ], "errata": [] }