{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "euclid-elements-book-vii",
    "name": "Euclid's Elements, Book VII",
    "subject": "number_theory",
    "variant": "classical",
    "description": "Number theory: GCD (Euclidean algorithm), proportions, primes, LCM. 22 definitions, 39 propositions. Does not depend on previous books. Source: David E. Joyce.",
    "structure": {
      "books": 7,
      "definitions": 22,
      "propositions": 39,
      "foundationTypes": [
        "definition"
      ]
    }
  },
  "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 VII Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book VII",
      "number theory",
      "GCD",
      "prime",
      "LCM"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book VII",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookVII/bookVII.html",
      "notes": "Clark University"
    }
  ],
  "nodes": [
    {
      "id": "Def1",
      "type": "definition",
      "label": "A unit is that by virtue of which each of the things that exist is called one",
      "shortLabel": "Def. VII.1",
      "short": "Unit",
      "book": 7,
      "number": 1,
      "colorClass": "definition"
    },
    {
      "id": "Def2",
      "type": "definition",
      "label": "A number is a multitude composed of units",
      "shortLabel": "Def. VII.2",
      "short": "Number",
      "book": 7,
      "number": 2,
      "colorClass": "definition"
    },
    {
      "id": "Def3",
      "type": "definition",
      "label": "A number is part of a number when it measures it",
      "shortLabel": "Def. VII.3",
      "short": "Part",
      "book": 7,
      "number": 3,
      "colorClass": "definition"
    },
    {
      "id": "Def4",
      "type": "definition",
      "label": "Parts when it does not measure it",
      "shortLabel": "Def. VII.4",
      "short": "Parts",
      "book": 7,
      "number": 4,
      "colorClass": "definition"
    },
    {
      "id": "Def5",
      "type": "definition",
      "label": "The greater is a multiple of the less when measured by the less",
      "shortLabel": "Def. VII.5",
      "short": "Multiple",
      "book": 7,
      "number": 5,
      "colorClass": "definition"
    },
    {
      "id": "Def6",
      "type": "definition",
      "label": "An even number is that which is divisible into two equal parts",
      "shortLabel": "Def. VII.6",
      "short": "Even",
      "book": 7,
      "number": 6,
      "colorClass": "definition"
    },
    {
      "id": "Def7",
      "type": "definition",
      "label": "An odd number is that which is not divisible into two equal parts",
      "shortLabel": "Def. VII.7",
      "short": "Odd",
      "book": 7,
      "number": 7,
      "colorClass": "definition"
    },
    {
      "id": "Def8",
      "type": "definition",
      "label": "Even-times even: measured by an even number an even number of times",
      "shortLabel": "Def. VII.8",
      "short": "Even-times even",
      "book": 7,
      "number": 8,
      "colorClass": "definition"
    },
    {
      "id": "Def9",
      "type": "definition",
      "label": "Even-times odd: measured by an even number an odd number of times",
      "shortLabel": "Def. VII.9",
      "short": "Even-times odd",
      "book": 7,
      "number": 9,
      "colorClass": "definition"
    },
    {
      "id": "Def10",
      "type": "definition",
      "label": "Odd-times odd: measured by an odd number an odd number of times",
      "shortLabel": "Def. VII.10",
      "short": "Odd-times odd",
      "book": 7,
      "number": 10,
      "colorClass": "definition"
    },
    {
      "id": "Def11",
      "type": "definition",
      "label": "A prime number is that which is measured by a unit alone",
      "shortLabel": "Def. VII.11",
      "short": "Prime",
      "book": 7,
      "number": 11,
      "colorClass": "definition"
    },
    {
      "id": "Def12",
      "type": "definition",
      "label": "Numbers relatively prime when only a unit measures both",
      "shortLabel": "Def. VII.12",
      "short": "Relatively prime",
      "book": 7,
      "number": 12,
      "colorClass": "definition"
    },
    {
      "id": "Def13",
      "type": "definition",
      "label": "A composite number is that measured by some number",
      "shortLabel": "Def. VII.13",
      "short": "Composite",
      "book": 7,
      "number": 13,
      "colorClass": "definition"
    },
    {
      "id": "Def14",
      "type": "definition",
      "label": "Numbers composite to one another when some number measures both",
      "shortLabel": "Def. VII.14",
      "short": "Composite to one another",
      "book": 7,
      "number": 14,
      "colorClass": "definition"
    },
    {
      "id": "Def15",
      "type": "definition",
      "label": "A number multiplies a number when the latter is added as many times as units in the former",
      "shortLabel": "Def. VII.15",
      "short": "Multiply",
      "book": 7,
      "number": 15,
      "colorClass": "definition"
    },
    {
      "id": "Def16",
      "type": "definition",
      "label": "When two numbers multiplied produce a number, the product is plane",
      "shortLabel": "Def. VII.16",
      "short": "Product",
      "book": 7,
      "number": 16,
      "colorClass": "definition"
    },
    {
      "id": "Def17",
      "type": "definition",
      "label": "Sides of the product are the numbers multiplied",
      "shortLabel": "Def. VII.17",
      "short": "Side",
      "book": 7,
      "number": 17,
      "colorClass": "definition"
    },
    {
      "id": "Def18",
      "type": "definition",
      "label": "A plane number is that produced by two numbers",
      "shortLabel": "Def. VII.18",
      "short": "Plane number",
      "book": 7,
      "number": 18,
      "colorClass": "definition"
    },
    {
      "id": "Def19",
      "type": "definition",
      "label": "A solid number is that produced by three numbers",
      "shortLabel": "Def. VII.19",
      "short": "Solid number",
      "book": 7,
      "number": 19,
      "colorClass": "definition"
    },
    {
      "id": "Def20",
      "type": "definition",
      "label": "Similar plane numbers have sides proportional",
      "shortLabel": "Def. VII.20",
      "short": "Similar plane",
      "book": 7,
      "number": 20,
      "colorClass": "definition"
    },
    {
      "id": "Def21",
      "type": "definition",
      "label": "Similar solid numbers have sides proportional",
      "shortLabel": "Def. VII.21",
      "short": "Similar solid",
      "book": 7,
      "number": 21,
      "colorClass": "definition"
    },
    {
      "id": "Def22",
      "type": "definition",
      "label": "A perfect number is that which equals its own parts",
      "shortLabel": "Def. VII.22",
      "short": "Perfect",
      "book": 7,
      "number": 22,
      "colorClass": "definition"
    },
    {
      "id": "Prop1",
      "type": "proposition",
      "label": "Unequal numbers: repeated subtraction; if unit left, relatively prime",
      "shortLabel": "Prop. VII.1",
      "short": "Antenaresis, relatively prime",
      "book": 7,
      "number": 1,
      "colorClass": "proposition"
    },
    {
      "id": "Prop2",
      "type": "proposition",
      "label": "To find greatest common measure of two numbers not relatively prime",
      "shortLabel": "Prop. VII.2",
      "short": "GCD of two numbers",
      "book": 7,
      "number": 2,
      "colorClass": "proposition"
    },
    {
      "id": "Prop3",
      "type": "proposition",
      "label": "To find greatest common measure of three numbers",
      "shortLabel": "Prop. VII.3",
      "short": "GCD of three numbers",
      "book": 7,
      "number": 3,
      "colorClass": "proposition"
    },
    {
      "id": "Prop4",
      "type": "proposition",
      "label": "Any number is part or parts of any number, less of greater",
      "shortLabel": "Prop. VII.4",
      "short": "Part or parts",
      "book": 7,
      "number": 4,
      "colorClass": "proposition"
    },
    {
      "id": "Prop5",
      "type": "proposition",
      "label": "If a is same part of b as c of d, then a+c same part of b+d",
      "shortLabel": "Prop. VII.5",
      "short": "Same part: sum",
      "book": 7,
      "number": 5,
      "colorClass": "proposition"
    },
    {
      "id": "Prop6",
      "type": "proposition",
      "label": "If a is same parts of b as c of d, then a+c same parts of b+d",
      "shortLabel": "Prop. VII.6",
      "short": "Same parts: sum",
      "book": 7,
      "number": 6,
      "colorClass": "proposition"
    },
    {
      "id": "Prop7",
      "type": "proposition",
      "label": "If a part of b as c of d, remainder same part of remainder",
      "shortLabel": "Prop. VII.7",
      "short": "Same part: remainder",
      "book": 7,
      "number": 7,
      "colorClass": "proposition"
    },
    {
      "id": "Prop8",
      "type": "proposition",
      "label": "If a parts of b as c of d, remainder same parts of remainder",
      "shortLabel": "Prop. VII.8",
      "short": "Same parts: remainder",
      "book": 7,
      "number": 8,
      "colorClass": "proposition"
    },
    {
      "id": "Prop9",
      "type": "proposition",
      "label": "If a part of b as c of d, alternately a part/parts of c as b of d",
      "shortLabel": "Prop. VII.9",
      "short": "Same part: alternately",
      "book": 7,
      "number": 9,
      "colorClass": "proposition"
    },
    {
      "id": "Prop10",
      "type": "proposition",
      "label": "If a parts of b as c of d, alternately a part/parts of c as b of d",
      "shortLabel": "Prop. VII.10",
      "short": "Same parts: alternately",
      "book": 7,
      "number": 10,
      "colorClass": "proposition"
    },
    {
      "id": "Prop11",
      "type": "proposition",
      "label": "If whole:whole as subtracted:subtracted, remainder:remainder as whole:whole",
      "shortLabel": "Prop. VII.11",
      "short": "Proportion: remainder",
      "book": 7,
      "number": 11,
      "colorClass": "proposition"
    },
    {
      "id": "Prop12",
      "type": "proposition",
      "label": "Proportional: one antecedent to consequent as sum antecedents to sum consequents",
      "shortLabel": "Prop. VII.12",
      "short": "Proportional: sum",
      "book": 7,
      "number": 12,
      "colorClass": "proposition"
    },
    {
      "id": "Prop13",
      "type": "proposition",
      "label": "If four numbers proportional, also proportional alternately",
      "shortLabel": "Prop. VII.13",
      "short": "Proportional: alternately",
      "book": 7,
      "number": 13,
      "colorClass": "proposition"
    },
    {
      "id": "Prop14",
      "type": "proposition",
      "label": "If a:b = d:e and b:c = e:f, then a:c = d:f",
      "shortLabel": "Prop. VII.14",
      "short": "Ex aequali",
      "book": 7,
      "number": 14,
      "colorClass": "proposition"
    },
    {
      "id": "Prop15",
      "type": "proposition",
      "label": "If unit measures a, b measures c same times, alternately unit:c as b:d",
      "shortLabel": "Prop. VII.15",
      "short": "Unit measures",
      "book": 7,
      "number": 15,
      "colorClass": "proposition"
    },
    {
      "id": "Prop16",
      "type": "proposition",
      "label": "If a×b and c×d, then a×b = c×d (commutativity)",
      "shortLabel": "Prop. VII.16",
      "short": "Commutativity of product",
      "book": 7,
      "number": 16,
      "colorClass": "proposition"
    },
    {
      "id": "Prop17",
      "type": "proposition",
      "label": "a:b = (a×c):(b×c)",
      "shortLabel": "Prop. VII.17",
      "short": "Ratio of products",
      "book": 7,
      "number": 17,
      "colorClass": "proposition"
    },
    {
      "id": "Prop18",
      "type": "proposition",
      "label": "a×c : b×c = a:b",
      "shortLabel": "Prop. VII.18",
      "short": "Ratio: multipliers",
      "book": 7,
      "number": 18,
      "colorClass": "proposition"
    },
    {
      "id": "Prop19",
      "type": "proposition",
      "label": "a:b = c:d iff a×d = b×c",
      "shortLabel": "Prop. VII.19",
      "short": "Proportional iff product",
      "book": 7,
      "number": 19,
      "colorClass": "proposition"
    },
    {
      "id": "Prop20",
      "type": "proposition",
      "label": "Least numbers in ratio measure others same number of times",
      "shortLabel": "Prop. VII.20",
      "short": "Least in ratio",
      "book": 7,
      "number": 20,
      "colorClass": "proposition"
    },
    {
      "id": "Prop21",
      "type": "proposition",
      "label": "Relatively prime numbers are least in their ratio",
      "shortLabel": "Prop. VII.21",
      "short": "Relatively prime: least",
      "book": 7,
      "number": 21,
      "colorClass": "proposition"
    },
    {
      "id": "Prop22",
      "type": "proposition",
      "label": "Least numbers in ratio are relatively prime",
      "shortLabel": "Prop. VII.22",
      "short": "Least: relatively prime",
      "book": 7,
      "number": 22,
      "colorClass": "proposition"
    },
    {
      "id": "Prop23",
      "type": "proposition",
      "label": "If a,b relatively prime, divisor of a relatively prime to b",
      "shortLabel": "Prop. VII.23",
      "short": "Relatively prime: divisor",
      "book": 7,
      "number": 23,
      "colorClass": "proposition"
    },
    {
      "id": "Prop24",
      "type": "proposition",
      "label": "If a,b relatively prime to c, then a×b relatively prime to c",
      "shortLabel": "Prop. VII.24",
      "short": "Product relatively prime",
      "book": 7,
      "number": 24,
      "colorClass": "proposition"
    },
    {
      "id": "Prop25",
      "type": "proposition",
      "label": "If a,b relatively prime, a² relatively prime to b",
      "shortLabel": "Prop. VII.25",
      "short": "Square relatively prime",
      "book": 7,
      "number": 25,
      "colorClass": "proposition"
    },
    {
      "id": "Prop26",
      "type": "proposition",
      "label": "If a,c and b,d relatively prime, a×b, c×d relatively prime",
      "shortLabel": "Prop. VII.26",
      "short": "Products relatively prime",
      "book": 7,
      "number": 26,
      "colorClass": "proposition"
    },
    {
      "id": "Prop27",
      "type": "proposition",
      "label": "If a,b relatively prime, a²,b² relatively prime; a×a², b×b²",
      "shortLabel": "Prop. VII.27",
      "short": "Squares relatively prime",
      "book": 7,
      "number": 27,
      "colorClass": "proposition"
    },
    {
      "id": "Prop28",
      "type": "proposition",
      "label": "If a,b relatively prime, a+b prime to each; converse",
      "shortLabel": "Prop. VII.28",
      "short": "Sum relatively prime",
      "book": 7,
      "number": 28,
      "colorClass": "proposition"
    },
    {
      "id": "Prop29",
      "type": "proposition",
      "label": "Prime relatively prime to any number it does not measure",
      "shortLabel": "Prop. VII.29",
      "short": "Prime to non-multiple",
      "book": 7,
      "number": 29,
      "colorClass": "proposition"
    },
    {
      "id": "Prop30",
      "type": "proposition",
      "label": "If prime measures product, it measures one factor",
      "shortLabel": "Prop. VII.30",
      "short": "Prime divides product",
      "book": 7,
      "number": 30,
      "colorClass": "proposition"
    },
    {
      "id": "Prop31",
      "type": "proposition",
      "label": "Any composite measured by some prime",
      "shortLabel": "Prop. VII.31",
      "short": "Composite has prime factor",
      "book": 7,
      "number": 31,
      "colorClass": "proposition"
    },
    {
      "id": "Prop32",
      "type": "proposition",
      "label": "Any number is prime or measured by some prime",
      "shortLabel": "Prop. VII.32",
      "short": "Prime or has prime factor",
      "book": 7,
      "number": 32,
      "colorClass": "proposition"
    },
    {
      "id": "Prop33",
      "type": "proposition",
      "label": "Given numbers, find least in same ratio",
      "shortLabel": "Prop. VII.33",
      "short": "Least in ratio",
      "book": 7,
      "number": 33,
      "colorClass": "proposition"
    },
    {
      "id": "Prop34",
      "type": "proposition",
      "label": "To find least number that two given numbers measure",
      "shortLabel": "Prop. VII.34",
      "short": "LCM of two",
      "book": 7,
      "number": 34,
      "colorClass": "proposition"
    },
    {
      "id": "Prop35",
      "type": "proposition",
      "label": "If two numbers measure some number, LCM also measures it",
      "shortLabel": "Prop. VII.35",
      "short": "LCM divides common multiple",
      "book": 7,
      "number": 35,
      "colorClass": "proposition"
    },
    {
      "id": "Prop36",
      "type": "proposition",
      "label": "To find least number that three given numbers measure",
      "shortLabel": "Prop. VII.36",
      "short": "LCM of three",
      "book": 7,
      "number": 36,
      "colorClass": "proposition"
    },
    {
      "id": "Prop37",
      "type": "proposition",
      "label": "If a measures b, b has part named by a",
      "shortLabel": "Prop. VII.37",
      "short": "Measured has part",
      "book": 7,
      "number": 37,
      "colorClass": "proposition"
    },
    {
      "id": "Prop38",
      "type": "proposition",
      "label": "If b has part named by a, a measures b",
      "shortLabel": "Prop. VII.38",
      "short": "Part implies measured",
      "book": 7,
      "number": 38,
      "colorClass": "proposition"
    },
    {
      "id": "Prop39",
      "type": "proposition",
      "label": "To find least number with given parts",
      "shortLabel": "Prop. VII.39",
      "short": "Least with given parts",
      "book": 7,
      "number": 39,
      "colorClass": "proposition"
    }
  ],
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      "from": "Prop1",
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      "from": "Prop2",
      "to": "Prop3"
    },
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      "from": "Prop19",
      "to": "Prop20"
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      "from": "Prop20",
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      "from": "Prop21",
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      "from": "Prop24",
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    {
      "from": "Prop23",
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      "from": "Prop31",
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      "from": "Prop20",
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    },
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      "from": "Prop22",
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    },
    {
      "from": "Prop33",
      "to": "Prop34"
    },
    {
      "from": "Prop34",
      "to": "Prop35"
    },
    {
      "from": "Prop34",
      "to": "Prop36"
    },
    {
      "from": "Prop37",
      "to": "Prop38"
    },
    {
      "from": "Prop38",
      "to": "Prop39"
    }
  ],
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    "definition": {
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