{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "euclid-elements-book-ix",
    "name": "Euclid's Elements, Book IX",
    "subject": "number_theory",
    "variant": "classical",
    "description": "Primes, perfect numbers, odd/even. 36 propositions. Depends on Books VII and VIII. Source: David E. Joyce.",
    "structure": {
      "books": 9,
      "propositions": 36,
      "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 IX Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book IX",
      "prime",
      "perfect",
      "odd",
      "even"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book IX",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookIX/bookIX.html",
      "notes": "Clark University; IX.20 infinitude of primes"
    }
  ],
  "nodes": [
    {
      "id": "BookVII",
      "type": "foundation",
      "label": "Book VII — Number theory",
      "shortLabel": "Book VII",
      "short": "Foundation",
      "book": 7,
      "colorClass": "foundation"
    },
    {
      "id": "BookVIII",
      "type": "foundation",
      "label": "Book VIII — Continued proportions",
      "shortLabel": "Book VIII",
      "short": "Foundation",
      "book": 8,
      "colorClass": "foundation"
    },
    {
      "id": "Prop1",
      "type": "proposition",
      "label": "Two similar plane numbers multiplied: product is square",
      "shortLabel": "Prop. IX.1",
      "short": "Similar plane product square",
      "book": 9,
      "number": 1,
      "colorClass": "proposition"
    },
    {
      "id": "Prop2",
      "type": "proposition",
      "label": "Two numbers product square: they are similar plane",
      "shortLabel": "Prop. IX.2",
      "short": "Product square: similar plane",
      "book": 9,
      "number": 2,
      "colorClass": "proposition"
    },
    {
      "id": "Prop3",
      "type": "proposition",
      "label": "Cubic number multiplied by itself: product is cube",
      "shortLabel": "Prop. IX.3",
      "short": "Cube times itself",
      "book": 9,
      "number": 3,
      "colorClass": "proposition"
    },
    {
      "id": "Prop4",
      "type": "proposition",
      "label": "Cubic times cubic: product is cube",
      "shortLabel": "Prop. IX.4",
      "short": "Cube times cube",
      "book": 9,
      "number": 4,
      "colorClass": "proposition"
    },
    {
      "id": "Prop5",
      "type": "proposition",
      "label": "If cube times any makes cube, the multiplied is cubic",
      "shortLabel": "Prop. IX.5",
      "short": "Cube times any makes cube",
      "book": 9,
      "number": 5,
      "colorClass": "proposition"
    },
    {
      "id": "Prop6",
      "type": "proposition",
      "label": "If number times itself makes cubic, it is cubic",
      "shortLabel": "Prop. IX.6",
      "short": "Number times itself cubic",
      "book": 9,
      "number": 6,
      "colorClass": "proposition"
    },
    {
      "id": "Prop7",
      "type": "proposition",
      "label": "Composite times any: product is solid",
      "shortLabel": "Prop. IX.7",
      "short": "Composite times any: solid",
      "book": 9,
      "number": 7,
      "colorClass": "proposition"
    },
    {
      "id": "Prop8",
      "type": "proposition",
      "label": "Numbers from unit in continued proportion: 3rd square, 4th cube, 7th both",
      "shortLabel": "Prop. IX.8",
      "short": "Continued proportion from unit",
      "book": 9,
      "number": 8,
      "colorClass": "proposition"
    },
    {
      "id": "Prop9",
      "type": "proposition",
      "label": "If second from unit square, all square; if cubic, all cubic",
      "shortLabel": "Prop. IX.9",
      "short": "Second square: all square",
      "book": 9,
      "number": 9,
      "colorClass": "proposition"
    },
    {
      "id": "Prop10",
      "type": "proposition",
      "label": "If second not square, only 3rd and every other square; similar for cube",
      "shortLabel": "Prop. IX.10",
      "short": "Second not square",
      "book": 9,
      "number": 10,
      "colorClass": "proposition"
    },
    {
      "id": "Prop11",
      "type": "proposition",
      "label": "Continued proportion from unit: less measures greater",
      "shortLabel": "Prop. IX.11",
      "short": "Less measures greater",
      "book": 9,
      "number": 11,
      "colorClass": "proposition"
    },
    {
      "id": "Prop12",
      "type": "proposition",
      "label": "If prime measures last, it measures second from unit",
      "shortLabel": "Prop. IX.12",
      "short": "Prime measures last",
      "book": 9,
      "number": 12,
      "colorClass": "proposition"
    },
    {
      "id": "Prop13",
      "type": "proposition",
      "label": "Continued proportion, second prime: only proportional numbers measure last",
      "shortLabel": "Prop. IX.13",
      "short": "Second prime: only those measure",
      "book": 9,
      "number": 13,
      "colorClass": "proposition"
    },
    {
      "id": "Prop14",
      "type": "proposition",
      "label": "Least measured by given primes: not measured by any other prime",
      "shortLabel": "Prop. IX.14",
      "short": "Least measured by primes",
      "book": 9,
      "number": 14,
      "colorClass": "proposition"
    },
    {
      "id": "Prop15",
      "type": "proposition",
      "label": "Three least in ratio: sum of any two prime to remainder",
      "shortLabel": "Prop. IX.15",
      "short": "Three in proportion",
      "book": 9,
      "number": 15,
      "colorClass": "proposition"
    },
    {
      "id": "Prop16",
      "type": "proposition",
      "label": "Two relatively prime: no third as first to second",
      "shortLabel": "Prop. IX.16",
      "short": "Relatively prime: no third",
      "book": 9,
      "number": 16,
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    },
    {
      "id": "Prop17",
      "type": "proposition",
      "label": "Continued proportion, extremes prime: last not to any as first to second",
      "shortLabel": "Prop. IX.17",
      "short": "Extremes prime: no extension",
      "book": 9,
      "number": 17,
      "colorClass": "proposition"
    },
    {
      "id": "Prop18",
      "type": "proposition",
      "label": "Given two numbers, investigate if third proportional exists",
      "shortLabel": "Prop. IX.18",
      "short": "Third proportional",
      "book": 9,
      "number": 18,
      "colorClass": "proposition"
    },
    {
      "id": "Prop19",
      "type": "proposition",
      "label": "Given three numbers, investigate when fourth proportional exists",
      "shortLabel": "Prop. IX.19",
      "short": "Fourth proportional",
      "book": 9,
      "number": 19,
      "colorClass": "proposition"
    },
    {
      "id": "Prop20",
      "type": "proposition",
      "label": "Prime numbers are more than any assigned multitude",
      "shortLabel": "Prop. IX.20",
      "short": "Infinitude of primes",
      "book": 9,
      "number": 20,
      "colorClass": "proposition"
    },
    {
      "id": "Prop21",
      "type": "proposition",
      "label": "Sum of even numbers is even",
      "shortLabel": "Prop. IX.21",
      "short": "Sum of evens even",
      "book": 9,
      "number": 21,
      "colorClass": "proposition"
    },
    {
      "id": "Prop22",
      "type": "proposition",
      "label": "Sum of odd numbers, even multitude: sum even",
      "shortLabel": "Prop. IX.22",
      "short": "Sum of odds (even count) even",
      "book": 9,
      "number": 22,
      "colorClass": "proposition"
    },
    {
      "id": "Prop23",
      "type": "proposition",
      "label": "Sum of odd numbers, odd multitude: sum odd",
      "shortLabel": "Prop. IX.23",
      "short": "Sum of odds (odd count) odd",
      "book": 9,
      "number": 23,
      "colorClass": "proposition"
    },
    {
      "id": "Prop24",
      "type": "proposition",
      "label": "Even minus even: remainder even",
      "shortLabel": "Prop. IX.24",
      "short": "Even minus even",
      "book": 9,
      "number": 24,
      "colorClass": "proposition"
    },
    {
      "id": "Prop25",
      "type": "proposition",
      "label": "Even minus odd: remainder odd",
      "shortLabel": "Prop. IX.25",
      "short": "Even minus odd",
      "book": 9,
      "number": 25,
      "colorClass": "proposition"
    },
    {
      "id": "Prop26",
      "type": "proposition",
      "label": "Odd minus odd: remainder even",
      "shortLabel": "Prop. IX.26",
      "short": "Odd minus odd",
      "book": 9,
      "number": 26,
      "colorClass": "proposition"
    },
    {
      "id": "Prop27",
      "type": "proposition",
      "label": "Odd minus even: remainder odd",
      "shortLabel": "Prop. IX.27",
      "short": "Odd minus even",
      "book": 9,
      "number": 27,
      "colorClass": "proposition"
    },
    {
      "id": "Prop28",
      "type": "proposition",
      "label": "Odd times even: product even",
      "shortLabel": "Prop. IX.28",
      "short": "Odd times even",
      "book": 9,
      "number": 28,
      "colorClass": "proposition"
    },
    {
      "id": "Prop29",
      "type": "proposition",
      "label": "Odd times odd: product odd",
      "shortLabel": "Prop. IX.29",
      "short": "Odd times odd",
      "book": 9,
      "number": 29,
      "colorClass": "proposition"
    },
    {
      "id": "Prop30",
      "type": "proposition",
      "label": "Odd measuring even: measures half",
      "shortLabel": "Prop. IX.30",
      "short": "Odd measures even",
      "book": 9,
      "number": 30,
      "colorClass": "proposition"
    },
    {
      "id": "Prop31",
      "type": "proposition",
      "label": "Odd relatively prime to any: also prime to its double",
      "shortLabel": "Prop. IX.31",
      "short": "Odd prime to double",
      "book": 9,
      "number": 31,
      "colorClass": "proposition"
    },
    {
      "id": "Prop32",
      "type": "proposition",
      "label": "Numbers doubled from 2: even-times even only",
      "shortLabel": "Prop. IX.32",
      "short": "Powers of 2",
      "book": 9,
      "number": 32,
      "colorClass": "proposition"
    },
    {
      "id": "Prop33",
      "type": "proposition",
      "label": "Number with half odd: even-times odd only",
      "shortLabel": "Prop. IX.33",
      "short": "Half odd",
      "book": 9,
      "number": 33,
      "colorClass": "proposition"
    },
    {
      "id": "Prop34",
      "type": "proposition",
      "label": "Number neither: both even-times even and even-times odd",
      "shortLabel": "Prop. IX.34",
      "short": "Neither",
      "book": 9,
      "number": 34,
      "colorClass": "proposition"
    },
    {
      "id": "Prop35",
      "type": "proposition",
      "label": "Continued proportion: (second−first):first = (last−first):sum of rest",
      "shortLabel": "Prop. IX.35",
      "short": "Geometric series",
      "book": 9,
      "number": 35,
      "colorClass": "proposition"
    },
    {
      "id": "Prop36",
      "type": "proposition",
      "label": "If sum of powers of 2 is prime, product with last is perfect",
      "shortLabel": "Prop. IX.36",
      "short": "Perfect numbers",
      "book": 9,
      "number": 36,
      "colorClass": "proposition"
    }
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