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"discourse": {
"id": "euclid-elements-book-ii",
"name": "Euclid's Elements, Book II",
"subject": "geometry",
"variant": "classical",
"description": "The 14 propositions of Book II on geometric algebra (rectangles, squares). Props 1-10 are logically independent within Book II; 11-14 depend on 6, 4, 7, 5. All depend on Book I. Source: David E. Joyce, Clark University.",
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"metadata": {
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"lastUpdated": "2026-03-15",
"version": "1.0.0",
"license": "CC BY 4.0",
"authors": [
"Welz, G."
],
"methodology": "Programming Framework",
"citation": "Welz, G. (2026). Euclid's Elements Book II Dependency Graph. Programming Framework.",
"keywords": [
"Euclid",
"Elements",
"Book II",
"geometric algebra",
"rectangles",
"squares",
"golden section",
"quadrature"
]
},
"sources": [
{
"id": "joyce",
"type": "digital",
"authors": "Joyce, David E.",
"title": "Euclid's Elements, Book II",
"year": "1996",
"url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookII/bookII.html",
"notes": "Clark University; dependency table from Logical structure"
},
{
"id": "euclid-heath",
"type": "primary",
"authors": "Heath, T.L.",
"title": "The Thirteen Books of Euclid's Elements",
"year": "1908",
"edition": "2nd",
"publisher": "Cambridge University Press",
"url": "https://archive.org/details/euclidheath00heatiala",
"notes": "Standard English translation"
}
],
"nodes": [
{
"id": "BookI",
"type": "foundation",
"label": "Book I — Fundamentals of plane geometry",
"shortLabel": "Book I",
"short": "Foundation",
"book": 1,
"colorClass": "foundation"
},
{
"id": "Def1",
"type": "definition",
"label": "Rectangle contained by two straight lines containing the right angle",
"shortLabel": "Def. II.1",
"book": 2,
"number": 1,
"colorClass": "definition"
},
{
"id": "Def2",
"type": "definition",
"label": "Gnomon: parallelogram about diameter with two complements",
"shortLabel": "Def. II.2",
"book": 2,
"number": 2,
"colorClass": "definition"
},
{
"id": "Prop1",
"type": "proposition",
"label": "If one line is cut into segments, rectangle by whole equals sum of rectangles by each segment",
"shortLabel": "Prop. II.1",
"short": "Rectangle = sum of rectangles",
"book": 2,
"number": 1,
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{
"id": "Prop2",
"type": "proposition",
"label": "If a line is cut at random, sum of rectangles by whole and each segment equals square on whole",
"shortLabel": "Prop. II.2",
"short": "Sum of rectangles = square on whole",
"book": 2,
"number": 2,
"colorClass": "proposition"
},
{
"id": "Prop3",
"type": "proposition",
"label": "If a line is cut at random, rectangle by whole and one segment equals rectangle by segments plus square on that segment",
"shortLabel": "Prop. II.3",
"short": "Rectangle = rectangle + square",
"book": 2,
"number": 3,
"colorClass": "proposition"
},
{
"id": "Prop4",
"type": "proposition",
"label": "If a line is cut at random, square on whole equals squares on segments plus twice rectangle contained by segments",
"shortLabel": "Prop. II.4",
"short": "Square on whole = squares + 2×rectangle",
"book": 2,
"number": 4,
"colorClass": "proposition"
},
{
"id": "Prop5",
"type": "proposition",
"label": "If a line cut into equal and unequal segments, rectangle by unequal segments plus square on difference equals square on half",
"shortLabel": "Prop. II.5",
"short": "Unequal segments: rectangle + square = square on half",
"book": 2,
"number": 5,
"colorClass": "proposition"
},
{
"id": "Prop6",
"type": "proposition",
"label": "If a line bisected and added to, rectangle by whole-with-added and added plus square on half equals square on half-plus-added",
"shortLabel": "Prop. II.6",
"short": "Bisected + added: rectangle + square = square",
"book": 2,
"number": 6,
"colorClass": "proposition"
},
{
"id": "Prop7",
"type": "proposition",
"label": "If a line cut at random, square on whole plus square on one segment equals twice rectangle by whole and segment plus square on remainder",
"shortLabel": "Prop. II.7",
"short": "Square on whole + square on segment",
"book": 2,
"number": 7,
"colorClass": "proposition"
},
{
"id": "Prop8",
"type": "proposition",
"label": "If a line cut at random, four times rectangle by whole and one segment plus square on remainder equals square on whole-plus-segment",
"shortLabel": "Prop. II.8",
"short": "Four times rectangle + square",
"book": 2,
"number": 8,
"colorClass": "proposition"
},
{
"id": "Prop9",
"type": "proposition",
"label": "If a line cut into equal and unequal segments, sum of squares on unequal segments is double sum of square on half and square on difference",
"shortLabel": "Prop. II.9",
"short": "Unequal segments: sum of squares",
"book": 2,
"number": 9,
"colorClass": "proposition"
},
{
"id": "Prop10",
"type": "proposition",
"label": "If a line bisected and added to, square on whole-with-added plus square on added equals double sum of square on half and square on half-plus-added",
"shortLabel": "Prop. II.10",
"short": "Bisected + added: sum of squares",
"book": 2,
"number": 10,
"colorClass": "proposition"
},
{
"id": "Prop11",
"type": "proposition",
"label": "To cut a given line so that rectangle by whole and one segment equals square on remaining segment",
"shortLabel": "Prop. II.11",
"short": "Cut line: rectangle = square (golden section)",
"book": 2,
"number": 11,
"colorClass": "proposition"
},
{
"id": "Prop12",
"type": "proposition",
"label": "In obtuse-angled triangles, square on side opposite obtuse angle greater than sum of squares on sides containing it",
"shortLabel": "Prop. II.12",
"short": "Obtuse triangle: law of cosines",
"book": 2,
"number": 12,
"colorClass": "proposition"
},
{
"id": "Prop13",
"type": "proposition",
"label": "In acute-angled triangles, square on side opposite acute angle less than sum of squares on sides containing it",
"shortLabel": "Prop. II.13",
"short": "Acute triangle: law of cosines",
"book": 2,
"number": 13,
"colorClass": "proposition"
},
{
"id": "Prop14",
"type": "proposition",
"label": "To construct a square equal to a given rectilinear figure",
"shortLabel": "Prop. II.14",
"short": "Construct square = rectilinear figure",
"book": 2,
"number": 14,
"colorClass": "proposition"
}
],
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