Spaces:
Running
Running
| { | |
| "schemaVersion": "1.0", | |
| "discourse": { | |
| "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.", | |
| "structure": { | |
| "books": 11, | |
| "definitions": 28, | |
| "propositions": 39, | |
| "foundationTypes": [ | |
| "foundation" | |
| ] | |
| } | |
| }, | |
| "metadata": { | |
| "created": "2026-03-18", | |
| "lastUpdated": "2026-03-18", | |
| "version": "1.0.0", | |
| "license": "CC BY 4.0", | |
| "authors": [ | |
| "Welz, G." | |
| ], | |
| "methodology": "Programming Framework", | |
| "citation": "Welz, G. (2026). Euclid's Elements Book XI Dependency Graph. Programming Framework.", | |
| "keywords": [ | |
| "Euclid", | |
| "Elements", | |
| "Book XI", | |
| "solid geometry", | |
| "plane", | |
| "parallelepiped", | |
| "prism" | |
| ] | |
| }, | |
| "sources": [ | |
| { | |
| "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" | |
| } | |
| ], | |
| "nodes": [ | |
| { | |
| "id": "BookI", | |
| "type": "foundation", | |
| "label": "Book I — Plane geometry", | |
| "shortLabel": "Book I", | |
| "short": "Foundation", | |
| "book": 1, | |
| "colorClass": "foundation" | |
| }, | |
| { | |
| "id": "BookVI", | |
| "type": "foundation", | |
| "label": "Book VI — Similar figures", | |
| "shortLabel": "Book VI", | |
| "short": "Foundation", | |
| "book": 6, | |
| "colorClass": "foundation" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 1, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 2, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 3, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 4, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 5, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 6, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 7, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 8, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 9, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 10, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 11, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 12, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "id": "Prop13", | |
| "type": "proposition", | |
| "label": "From same point two lines cannot be perpendicular to same plane on same side", | |
| "shortLabel": "Prop. XI.13", | |
| "short": "One perpendicular only", | |
| "book": 11, | |
| "number": 13, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "id": "Prop14", | |
| "type": "proposition", | |
| "label": "Planes to which same line is perpendicular are parallel", | |
| "shortLabel": "Prop. XI.14", | |
| "short": "Planes perpendicular to line: parallel", | |
| "book": 11, | |
| "number": 14, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "id": "Prop15", | |
| "type": "proposition", | |
| "label": "Two lines meeting parallel to two meeting not in same plane: planes through them parallel", | |
| "shortLabel": "Prop. XI.15", | |
| "short": "Skew lines: planes parallel", | |
| "book": 11, | |
| "number": 15, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "id": "Prop16", | |
| "type": "proposition", | |
| "label": "If two parallel planes cut by any plane, intersections are parallel", | |
| "shortLabel": "Prop. XI.16", | |
| "short": "Parallel planes cut: parallel", | |
| "book": 11, | |
| "number": 16, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 17, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 18, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "id": "Prop19", | |
| "type": "proposition", | |
| "label": "If two planes cutting one another perpendicular to plane, intersection perpendicular", | |
| "shortLabel": "Prop. XI.19", | |
| "short": "Planes perpendicular: intersection", | |
| "book": 11, | |
| "number": 19, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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", | |
| "book": 11, | |
| "number": 20, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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, | |
| "colorClass": "proposition" | |
| }, | |
| { | |
| "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" | |
| } | |
| ], | |
| "edges": [ | |
| { | |
| "from": "BookI", | |
| "to": "Prop1" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop1" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop2" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop2" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop3" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop3" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop4" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop4" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop5" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop5" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop6" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop6" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop7" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop7" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop8" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop8" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop9" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop9" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop10" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop10" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop11" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop11" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop12" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop12" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop13" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop13" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop14" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop14" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop15" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop15" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop16" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop16" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop17" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop17" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop18" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop18" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop19" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop19" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop20" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop20" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop21" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop21" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop22" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop22" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop23" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop23" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop24" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop24" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop25" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop25" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop26" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop26" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop27" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop27" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop28" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop28" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop29" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop29" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop30" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop30" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop31" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop31" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop32" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop32" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop33" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop33" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop34" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop34" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop35" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop35" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop36" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop36" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop37" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop37" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop38" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop38" | |
| }, | |
| { | |
| "from": "BookI", | |
| "to": "Prop39" | |
| }, | |
| { | |
| "from": "BookVI", | |
| "to": "Prop39" | |
| } | |
| ], | |
| "colorScheme": { | |
| "foundation": { | |
| "fill": "#95a5a6", | |
| "stroke": "#7f8c8d" | |
| }, | |
| "proposition": { | |
| "fill": "#1abc9c", | |
| "stroke": "#16a085" | |
| } | |
| } | |
| } |