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"id": "euclid-elements-book-xi",
"name": "Euclid's Elements, Book XI",
"subject": "solid_geometry",
"variant": "classical",
"description": "Solid geometry: planes, perpendiculars, parallelepipeds, prisms. 28 definitions, 39 propositions. Depends on Books I and VI. Source: David E. Joyce.",
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"citation": "Welz, G. (2026). Euclid's Elements Book XI Dependency Graph. Programming Framework.",
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{
"id": "joyce",
"type": "digital",
"authors": "Joyce, David E.",
"title": "Euclid's Elements, Book XI",
"year": "1996",
"url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html",
"notes": "Clark University"
}
],
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"shortLabel": "Book I",
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"id": "BookVI",
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{
"id": "Prop1",
"type": "proposition",
"label": "A part of a straight line cannot be in one plane and part in another elevated",
"shortLabel": "Prop. XI.1",
"short": "Line part in plane",
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{
"id": "Prop2",
"type": "proposition",
"label": "If two straight lines cut one another, they lie in one plane; every triangle in one plane",
"shortLabel": "Prop. XI.2",
"short": "Two lines cut: one plane",
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"number": 2,
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{
"id": "Prop3",
"type": "proposition",
"label": "If two planes cut one another, their intersection is a straight line",
"shortLabel": "Prop. XI.3",
"short": "Planes cut: line",
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{
"id": "Prop4",
"type": "proposition",
"label": "If line at right angles to two lines cutting at point, also perpendicular to plane through them",
"shortLabel": "Prop. XI.4",
"short": "Line perpendicular to plane",
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"number": 4,
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{
"id": "Prop5",
"type": "proposition",
"label": "If line at right angles to three lines meeting at point, the three lie in one plane",
"shortLabel": "Prop. XI.5",
"short": "Three lines from point",
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{
"id": "Prop6",
"type": "proposition",
"label": "If two lines at right angles to same plane, they are parallel",
"shortLabel": "Prop. XI.6",
"short": "Perpendicular to same plane: parallel",
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"colorClass": "proposition"
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{
"id": "Prop7",
"type": "proposition",
"label": "If two lines parallel, line joining points on each is in same plane",
"shortLabel": "Prop. XI.7",
"short": "Parallel lines: join in plane",
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{
"id": "Prop8",
"type": "proposition",
"label": "If two lines parallel, one perpendicular to plane, so is the other",
"shortLabel": "Prop. XI.8",
"short": "Parallel: one perpendicular",
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{
"id": "Prop9",
"type": "proposition",
"label": "Lines parallel to same line but not in same plane are parallel to each other",
"shortLabel": "Prop. XI.9",
"short": "Parallel to same: parallel",
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{
"id": "Prop10",
"type": "proposition",
"label": "Two lines meeting parallel to two meeting not in same plane: contain equal angles",
"shortLabel": "Prop. XI.10",
"short": "Skew lines: equal angles",
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"id": "Prop11",
"type": "proposition",
"label": "To draw line perpendicular to given plane from given elevated point",
"shortLabel": "Prop. XI.11",
"short": "Perpendicular from point to plane",
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"id": "Prop12",
"type": "proposition",
"label": "To set up line at right angles to plane from given point in it",
"shortLabel": "Prop. XI.12",
"short": "Perpendicular from point in plane",
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"id": "Prop13",
"type": "proposition",
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"short": "One perpendicular only",
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"id": "Prop14",
"type": "proposition",
"label": "Planes to which same line is perpendicular are parallel",
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"short": "Planes perpendicular to line: parallel",
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"id": "Prop15",
"type": "proposition",
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"short": "Skew lines: planes parallel",
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"id": "Prop16",
"type": "proposition",
"label": "If two parallel planes cut by any plane, intersections are parallel",
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"short": "Parallel planes cut: parallel",
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"id": "Prop17",
"type": "proposition",
"label": "If two lines cut by parallel planes, they are cut in same ratios",
"shortLabel": "Prop. XI.17",
"short": "Parallel planes: same ratio",
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"id": "Prop18",
"type": "proposition",
"label": "If line perpendicular to plane, all planes through it perpendicular to that plane",
"shortLabel": "Prop. XI.18",
"short": "Line perpendicular: planes through it",
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"number": 18,
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{
"id": "Prop19",
"type": "proposition",
"label": "If two planes cutting one another perpendicular to plane, intersection perpendicular",
"shortLabel": "Prop. XI.19",
"short": "Planes perpendicular: intersection",
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"number": 19,
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{
"id": "Prop20",
"type": "proposition",
"label": "Solid angle by three plane angles: sum of any two greater than third",
"shortLabel": "Prop. XI.20",
"short": "Solid angle: plane angles",
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"number": 20,
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{
"id": "Prop21",
"type": "proposition",
"label": "Any solid angle contained by plane angles summing to less than four right angles",
"shortLabel": "Prop. XI.21",
"short": "Solid angle: less than four right",
"book": 11,
"number": 21,
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{
"id": "Prop22",
"type": "proposition",
"label": "Three plane angles with sum of any two greater than third, equal sides: construct triangle",
"shortLabel": "Prop. XI.22",
"short": "Three plane angles: construct triangle",
"book": 11,
"number": 22,
"colorClass": "proposition"
},
{
"id": "Prop23",
"type": "proposition",
"label": "To construct solid angle from three plane angles (sum of any two greater than third)",
"shortLabel": "Prop. XI.23",
"short": "Construct solid angle",
"book": 11,
"number": 23,
"colorClass": "proposition"
},
{
"id": "Prop24",
"type": "proposition",
"label": "If solid contained by parallel planes, opposite planes equal and parallelogrammic",
"shortLabel": "Prop. XI.24",
"short": "Solid by parallel planes",
"book": 11,
"number": 24,
"colorClass": "proposition"
},
{
"id": "Prop25",
"type": "proposition",
"label": "Parallelepiped cut by plane parallel to opposite: base to base as solid to solid",
"shortLabel": "Prop. XI.25",
"short": "Parallelepiped cut: base ratio",
"book": 11,
"number": 25,
"colorClass": "proposition"
},
{
"id": "Prop26",
"type": "proposition",
"label": "To construct solid angle equal to given on given line at given point",
"shortLabel": "Prop. XI.26",
"short": "Construct equal solid angle",
"book": 11,
"number": 26,
"colorClass": "proposition"
},
{
"id": "Prop27",
"type": "proposition",
"label": "To describe parallelepiped similar to given on given straight line",
"shortLabel": "Prop. XI.27",
"short": "Similar parallelepiped on line",
"book": 11,
"number": 27,
"colorClass": "proposition"
},
{
"id": "Prop28",
"type": "proposition",
"label": "Parallelepiped cut by plane through diagonals of opposite planes: bisected",
"shortLabel": "Prop. XI.28",
"short": "Parallelepiped: diagonal plane bisects",
"book": 11,
"number": 28,
"colorClass": "proposition"
},
{
"id": "Prop29",
"type": "proposition",
"label": "Parallelepipeds same base, height, ends on same lines: equal",
"shortLabel": "Prop. XI.29",
"short": "Same base, height, same lines: equal",
"book": 11,
"number": 29,
"colorClass": "proposition"
},
{
"id": "Prop30",
"type": "proposition",
"label": "Parallelepipeds same base, height, ends not on same lines: equal",
"shortLabel": "Prop. XI.30",
"short": "Same base, height, different lines: equal",
"book": 11,
"number": 30,
"colorClass": "proposition"
},
{
"id": "Prop31",
"type": "proposition",
"label": "Parallelepipeds on equal bases, same height: equal",
"shortLabel": "Prop. XI.31",
"short": "Equal bases, same height: equal",
"book": 11,
"number": 31,
"colorClass": "proposition"
},
{
"id": "Prop32",
"type": "proposition",
"label": "Parallelepipeds same height: to one another as bases",
"shortLabel": "Prop. XI.32",
"short": "Same height: as bases",
"book": 11,
"number": 32,
"colorClass": "proposition"
},
{
"id": "Prop33",
"type": "proposition",
"label": "Similar parallelepipeds: to one another in triplicate ratio of corresponding sides",
"shortLabel": "Prop. XI.33",
"short": "Similar: triplicate ratio",
"book": 11,
"number": 33,
"colorClass": "proposition"
},
{
"id": "Prop34",
"type": "proposition",
"label": "Equal parallelepipeds: bases reciprocally proportional to heights",
"shortLabel": "Prop. XI.34",
"short": "Equal: bases reciprocally proportional",
"book": 11,
"number": 34,
"colorClass": "proposition"
},
{
"id": "Prop35",
"type": "proposition",
"label": "Equal plane angles, elevated lines with equal angles: perpendiculars, joins",
"shortLabel": "Prop. XI.35",
"short": "Equal plane angles: elevated lines",
"book": 11,
"number": 35,
"colorClass": "proposition"
},
{
"id": "Prop36",
"type": "proposition",
"label": "Three proportional lines: parallelepiped from three equals that on mean equilateral",
"shortLabel": "Prop. XI.36",
"short": "Three proportional: parallelepiped",
"book": 11,
"number": 36,
"colorClass": "proposition"
},
{
"id": "Prop37",
"type": "proposition",
"label": "Four proportional: similar parallelepipeds proportional; converse",
"shortLabel": "Prop. XI.37",
"short": "Four proportional: parallelepipeds",
"book": 11,
"number": 37,
"colorClass": "proposition"
},
{
"id": "Prop38",
"type": "proposition",
"label": "Cube opposite sides bisected, planes through: intersection and diameter bisect each other",
"shortLabel": "Prop. XI.38",
"short": "Cube: bisected by planes",
"book": 11,
"number": 38,
"colorClass": "proposition"
},
{
"id": "Prop39",
"type": "proposition",
"label": "Two prisms equal height, parallelogram and triangle bases, parallelogram double: equal",
"shortLabel": "Prop. XI.39",
"short": "Prisms: parallelogram, triangle",
"book": 11,
"number": 39,
"colorClass": "proposition"
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