{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "euclid-elements-book-x",
    "name": "Euclid's Elements, Book X",
    "subject": "incommensurables",
    "variant": "classical",
    "description": "Incommensurables, binomials, apotomes. 16 definitions, 115 propositions. Depends on Books I, V, VI. Source: David E. Joyce.",
    "structure": {
      "books": 10,
      "definitions": 16,
      "propositions": 115,
      "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 X Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book X",
      "incommensurable",
      "binomial",
      "apotome",
      "medial"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book X",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookX/bookX.html",
      "notes": "Clark University"
    }
  ],
  "nodes": [
    {
      "id": "BookI",
      "type": "foundation",
      "label": "Book I — Plane geometry",
      "shortLabel": "Book I",
      "short": "Foundation",
      "book": 1,
      "colorClass": "foundation"
    },
    {
      "id": "BookV",
      "type": "foundation",
      "label": "Book V — Proportions",
      "shortLabel": "Book V",
      "short": "Foundation",
      "book": 5,
      "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": "Two unequal magnitudes: subtract more than half repeatedly; remainder less than lesser",
      "shortLabel": "Prop. X.1",
      "short": "Antenaresis for magnitudes",
      "book": 10,
      "number": 1,
      "colorClass": "proposition"
    },
    {
      "id": "Prop2",
      "type": "proposition",
      "label": "If less subtracted from greater repeatedly and remainder never measures prior: incommensurable",
      "shortLabel": "Prop. X.2",
      "short": "Incommensurable criterion",
      "book": 10,
      "number": 2,
      "colorClass": "proposition"
    },
    {
      "id": "Prop3",
      "type": "proposition",
      "label": "To find greatest common measure of two commensurable magnitudes",
      "shortLabel": "Prop. X.3",
      "short": "GCM of two commensurable",
      "book": 10,
      "number": 3,
      "colorClass": "proposition"
    },
    {
      "id": "Prop4",
      "type": "proposition",
      "label": "To find greatest common measure of three commensurable magnitudes",
      "shortLabel": "Prop. X.4",
      "short": "GCM of three commensurable",
      "book": 10,
      "number": 4,
      "colorClass": "proposition"
    },
    {
      "id": "Prop5",
      "type": "proposition",
      "label": "Commensurable magnitudes have ratio which a number has to a number",
      "shortLabel": "Prop. X.5",
      "short": "Commensurable: number ratio",
      "book": 10,
      "number": 5,
      "colorClass": "proposition"
    },
    {
      "id": "Prop6",
      "type": "proposition",
      "label": "If two magnitudes have number-to-number ratio, they are commensurable",
      "shortLabel": "Prop. X.6",
      "short": "Number ratio: commensurable",
      "book": 10,
      "number": 6,
      "colorClass": "proposition"
    },
    {
      "id": "Prop7",
      "type": "proposition",
      "label": "Incommensurable magnitudes do not have number-to-number ratio",
      "shortLabel": "Prop. X.7",
      "short": "Incommensurable: no number ratio",
      "book": 10,
      "number": 7,
      "colorClass": "proposition"
    },
    {
      "id": "Prop8",
      "type": "proposition",
      "label": "If magnitudes lack number-to-number ratio, they are incommensurable",
      "shortLabel": "Prop. X.8",
      "short": "No number ratio: incommensurable",
      "book": 10,
      "number": 8,
      "colorClass": "proposition"
    },
    {
      "id": "Prop9",
      "type": "proposition",
      "label": "Squares on commensurable lines have square-to-square ratio; converse",
      "shortLabel": "Prop. X.9",
      "short": "Squares: commensurable iff square ratio",
      "book": 10,
      "number": 9,
      "colorClass": "proposition"
    },
    {
      "id": "Prop10",
      "type": "proposition",
      "label": "To find two lines incommensurable with assigned: one in length, one in square",
      "shortLabel": "Prop. X.10",
      "short": "Find incommensurable lines",
      "book": 10,
      "number": 10,
      "colorClass": "proposition"
    },
    {
      "id": "Prop11",
      "type": "proposition",
      "label": "Four proportional: first commensurable with second iff third with fourth",
      "shortLabel": "Prop. X.11",
      "short": "Proportion preserves commensurability",
      "book": 10,
      "number": 11,
      "colorClass": "proposition"
    },
    {
      "id": "Prop12",
      "type": "proposition",
      "label": "Magnitudes commensurable with same magnitude are commensurable",
      "shortLabel": "Prop. X.12",
      "short": "Commensurable with same",
      "book": 10,
      "number": 12,
      "colorClass": "proposition"
    },
    {
      "id": "Prop13",
      "type": "proposition",
      "label": "If two commensurable, one incommensurable with third, so is the other",
      "shortLabel": "Prop. X.13",
      "short": "Commensurable: incommensurable partner",
      "book": 10,
      "number": 13,
      "colorClass": "proposition"
    },
    {
      "id": "Prop14",
      "type": "proposition",
      "label": "Given two lines: find square by which greater square exceeds less; find line whose square equals sum",
      "shortLabel": "Prop. X.14",
      "short": "Square difference, sum of squares",
      "book": 10,
      "number": 14,
      "colorClass": "proposition"
    },
    {
      "id": "Prop15",
      "type": "proposition",
      "label": "Two commensurable added: whole commensurable with each",
      "shortLabel": "Prop. X.15",
      "short": "Commensurable sum",
      "book": 10,
      "number": 15,
      "colorClass": "proposition"
    },
    {
      "id": "Prop16",
      "type": "proposition",
      "label": "Two incommensurable added: sum incommensurable with each",
      "shortLabel": "Prop. X.16",
      "short": "Incommensurable sum",
      "book": 10,
      "number": 16,
      "colorClass": "proposition"
    },
    {
      "id": "Prop17",
      "type": "proposition",
      "label": "Applied parallelogram falling short by square: parts commensurable iff square difference commensurable",
      "shortLabel": "Prop. X.17",
      "short": "Parallelogram falling short",
      "book": 10,
      "number": 17,
      "colorClass": "proposition"
    },
    {
      "id": "Prop18",
      "type": "proposition",
      "label": "Applied parallelogram: parts incommensurable iff square difference incommensurable",
      "shortLabel": "Prop. X.18",
      "short": "Parallelogram: incommensurable parts",
      "book": 10,
      "number": 18,
      "colorClass": "proposition"
    },
    {
      "id": "Prop19",
      "type": "proposition",
      "label": "Rectangle by rational lines commensurable in length is rational",
      "shortLabel": "Prop. X.19",
      "short": "Rational rectangle from rational",
      "book": 10,
      "number": 19,
      "colorClass": "proposition"
    },
    {
      "id": "Prop20",
      "type": "proposition",
      "label": "Rational area applied to rational line: breadth rational and commensurable",
      "shortLabel": "Prop. X.20",
      "short": "Rational area on rational",
      "book": 10,
      "number": 20,
      "colorClass": "proposition"
    },
    {
      "id": "Prop21",
      "type": "proposition",
      "label": "Rectangle by rational in square only is irrational; side called medial",
      "shortLabel": "Prop. X.21",
      "short": "Medial: rational in square only",
      "book": 10,
      "number": 21,
      "colorClass": "proposition"
    },
    {
      "id": "Prop22",
      "type": "proposition",
      "label": "Square on medial applied to rational: breadth rational, incommensurable in length",
      "shortLabel": "Prop. X.22",
      "short": "Medial applied to rational",
      "book": 10,
      "number": 22,
      "colorClass": "proposition"
    },
    {
      "id": "Prop23",
      "type": "proposition",
      "label": "Line commensurable with medial is medial",
      "shortLabel": "Prop. X.23",
      "short": "Commensurable with medial",
      "book": 10,
      "number": 23,
      "colorClass": "proposition"
    },
    {
      "id": "Prop24",
      "type": "proposition",
      "label": "Rectangle by medial lines commensurable in length is medial",
      "shortLabel": "Prop. X.24",
      "short": "Medial rectangle: commensurable length",
      "book": 10,
      "number": 24,
      "colorClass": "proposition"
    },
    {
      "id": "Prop25",
      "type": "proposition",
      "label": "Rectangle by medial in square only: rational or medial",
      "shortLabel": "Prop. X.25",
      "short": "Medial: square only",
      "book": 10,
      "number": 25,
      "colorClass": "proposition"
    },
    {
      "id": "Prop26",
      "type": "proposition",
      "label": "Medial area does not exceed medial by rational area",
      "shortLabel": "Prop. X.26",
      "short": "Medial minus medial",
      "book": 10,
      "number": 26,
      "colorClass": "proposition"
    },
    {
      "id": "Prop27",
      "type": "proposition",
      "label": "To find medial lines commensurable in square only containing rational rectangle",
      "shortLabel": "Prop. X.27",
      "short": "Find medial in square only, rational rect",
      "book": 10,
      "number": 27,
      "colorClass": "proposition"
    },
    {
      "id": "Prop28",
      "type": "proposition",
      "label": "To find medial lines commensurable in square only containing medial rectangle",
      "shortLabel": "Prop. X.28",
      "short": "Find medial in square only, medial rect",
      "book": 10,
      "number": 28,
      "colorClass": "proposition"
    },
    {
      "id": "Prop29",
      "type": "proposition",
      "label": "To find two rational in square only: square diff commensurable with greater",
      "shortLabel": "Prop. X.29",
      "short": "Find rational in square only, commensurable diff",
      "book": 10,
      "number": 29,
      "colorClass": "proposition"
    },
    {
      "id": "Prop30",
      "type": "proposition",
      "label": "To find two rational in square only: square diff incommensurable with greater",
      "shortLabel": "Prop. X.30",
      "short": "Find rational in square only, incommensurable diff",
      "book": 10,
      "number": 30,
      "colorClass": "proposition"
    },
    {
      "id": "Prop31",
      "type": "proposition",
      "label": "To find two medial in square only, rational rect: diff commensurable with greater",
      "shortLabel": "Prop. X.31",
      "short": "Find medial, rational rect, commensurable diff",
      "book": 10,
      "number": 31,
      "colorClass": "proposition"
    },
    {
      "id": "Prop32",
      "type": "proposition",
      "label": "To find two medial in square only, medial rect: diff commensurable with greater",
      "shortLabel": "Prop. X.32",
      "short": "Find medial, medial rect, commensurable diff",
      "book": 10,
      "number": 32,
      "colorClass": "proposition"
    },
    {
      "id": "Prop33",
      "type": "proposition",
      "label": "To find two incommensurable in square: sum of squares rational, rectangle medial",
      "shortLabel": "Prop. X.33",
      "short": "Find incommensurable: rational sum, medial rect",
      "book": 10,
      "number": 33,
      "colorClass": "proposition"
    },
    {
      "id": "Prop34",
      "type": "proposition",
      "label": "To find two incommensurable in square: sum medial, rectangle rational",
      "shortLabel": "Prop. X.34",
      "short": "Find incommensurable: medial sum, rational rect",
      "book": 10,
      "number": 34,
      "colorClass": "proposition"
    },
    {
      "id": "Prop35",
      "type": "proposition",
      "label": "To find two incommensurable in square: sum and rect medial, incommensurable",
      "shortLabel": "Prop. X.35",
      "short": "Find incommensurable: both medial",
      "book": 10,
      "number": 35,
      "colorClass": "proposition"
    },
    {
      "id": "Prop36",
      "type": "proposition",
      "label": "Two rational in square only added: whole irrational, called binomial",
      "shortLabel": "Prop. X.36",
      "short": "Binomial defined",
      "book": 10,
      "number": 36,
      "colorClass": "proposition"
    },
    {
      "id": "Prop37",
      "type": "proposition",
      "label": "Two medial in square only, rational rect added: first bimedial",
      "shortLabel": "Prop. X.37",
      "short": "First bimedial defined",
      "book": 10,
      "number": 37,
      "colorClass": "proposition"
    },
    {
      "id": "Prop38",
      "type": "proposition",
      "label": "Two medial in square only, medial rect added: second bimedial",
      "shortLabel": "Prop. X.38",
      "short": "Second bimedial defined",
      "book": 10,
      "number": 38,
      "colorClass": "proposition"
    },
    {
      "id": "Prop39",
      "type": "proposition",
      "label": "Two incommensurable in square, rational sum, medial rect added: major",
      "shortLabel": "Prop. X.39",
      "short": "Major defined",
      "book": 10,
      "number": 39,
      "colorClass": "proposition"
    },
    {
      "id": "Prop40",
      "type": "proposition",
      "label": "Two incommensurable in square, medial sum, rational rect added: side of rational plus medial",
      "shortLabel": "Prop. X.40",
      "short": "Side of rational plus medial",
      "book": 10,
      "number": 40,
      "colorClass": "proposition"
    },
    {
      "id": "Prop41",
      "type": "proposition",
      "label": "Two incommensurable in square, both medial added: side of sum of two medial",
      "shortLabel": "Prop. X.41",
      "short": "Side of sum of two medial",
      "book": 10,
      "number": 41,
      "colorClass": "proposition"
    },
    {
      "id": "Prop42",
      "type": "proposition",
      "label": "Binomial divided into terms at one point only",
      "shortLabel": "Prop. X.42",
      "short": "Binomial: unique division",
      "book": 10,
      "number": 42,
      "colorClass": "proposition"
    },
    {
      "id": "Prop43",
      "type": "proposition",
      "label": "First bimedial divided at one point only",
      "shortLabel": "Prop. X.43",
      "short": "First bimedial: unique division",
      "book": 10,
      "number": 43,
      "colorClass": "proposition"
    },
    {
      "id": "Prop44",
      "type": "proposition",
      "label": "Second bimedial divided at one point only",
      "shortLabel": "Prop. X.44",
      "short": "Second bimedial: unique division",
      "book": 10,
      "number": 44,
      "colorClass": "proposition"
    },
    {
      "id": "Prop45",
      "type": "proposition",
      "label": "Major divided at one point only",
      "shortLabel": "Prop. X.45",
      "short": "Major: unique division",
      "book": 10,
      "number": 45,
      "colorClass": "proposition"
    },
    {
      "id": "Prop46",
      "type": "proposition",
      "label": "Side of rational plus medial divided at one point only",
      "shortLabel": "Prop. X.46",
      "short": "Side rational+medial: unique division",
      "book": 10,
      "number": 46,
      "colorClass": "proposition"
    },
    {
      "id": "Prop47",
      "type": "proposition",
      "label": "Side of sum of two medial divided at one point only",
      "shortLabel": "Prop. X.47",
      "short": "Side two medial: unique division",
      "book": 10,
      "number": 47,
      "colorClass": "proposition"
    },
    {
      "id": "Prop48",
      "type": "proposition",
      "label": "To find the first binomial line",
      "shortLabel": "Prop. X.48",
      "short": "Find first binomial",
      "book": 10,
      "number": 48,
      "colorClass": "proposition"
    },
    {
      "id": "Prop49",
      "type": "proposition",
      "label": "To find the second binomial line",
      "shortLabel": "Prop. X.49",
      "short": "Find second binomial",
      "book": 10,
      "number": 49,
      "colorClass": "proposition"
    },
    {
      "id": "Prop50",
      "type": "proposition",
      "label": "To find the third binomial line",
      "shortLabel": "Prop. X.50",
      "short": "Find third binomial",
      "book": 10,
      "number": 50,
      "colorClass": "proposition"
    },
    {
      "id": "Prop51",
      "type": "proposition",
      "label": "To find the fourth binomial line",
      "shortLabel": "Prop. X.51",
      "short": "Find fourth binomial",
      "book": 10,
      "number": 51,
      "colorClass": "proposition"
    },
    {
      "id": "Prop52",
      "type": "proposition",
      "label": "To find the fifth binomial line",
      "shortLabel": "Prop. X.52",
      "short": "Find fifth binomial",
      "book": 10,
      "number": 52,
      "colorClass": "proposition"
    },
    {
      "id": "Prop53",
      "type": "proposition",
      "label": "To find the sixth binomial line",
      "shortLabel": "Prop. X.53",
      "short": "Find sixth binomial",
      "book": 10,
      "number": 53,
      "colorClass": "proposition"
    },
    {
      "id": "Prop54",
      "type": "proposition",
      "label": "Area by rational and first binomial: side is binomial",
      "shortLabel": "Prop. X.54",
      "short": "Rational × first binomial",
      "book": 10,
      "number": 54,
      "colorClass": "proposition"
    },
    {
      "id": "Prop55",
      "type": "proposition",
      "label": "Area by rational and second binomial: side is first bimedial",
      "shortLabel": "Prop. X.55",
      "short": "Rational × second binomial",
      "book": 10,
      "number": 55,
      "colorClass": "proposition"
    },
    {
      "id": "Prop56",
      "type": "proposition",
      "label": "Area by rational and third binomial: side is second bimedial",
      "shortLabel": "Prop. X.56",
      "short": "Rational × third binomial",
      "book": 10,
      "number": 56,
      "colorClass": "proposition"
    },
    {
      "id": "Prop57",
      "type": "proposition",
      "label": "Area by rational and fourth binomial: side is major",
      "shortLabel": "Prop. X.57",
      "short": "Rational × fourth binomial",
      "book": 10,
      "number": 57,
      "colorClass": "proposition"
    },
    {
      "id": "Prop58",
      "type": "proposition",
      "label": "Area by rational and fifth binomial: side is rational plus medial",
      "shortLabel": "Prop. X.58",
      "short": "Rational × fifth binomial",
      "book": 10,
      "number": 58,
      "colorClass": "proposition"
    },
    {
      "id": "Prop59",
      "type": "proposition",
      "label": "Area by rational and sixth binomial: side is sum of two medial",
      "shortLabel": "Prop. X.59",
      "short": "Rational × sixth binomial",
      "book": 10,
      "number": 59,
      "colorClass": "proposition"
    },
    {
      "id": "Prop60",
      "type": "proposition",
      "label": "Square on binomial applied to rational: breadth first binomial",
      "shortLabel": "Prop. X.60",
      "short": "Square on binomial",
      "book": 10,
      "number": 60,
      "colorClass": "proposition"
    },
    {
      "id": "Prop61",
      "type": "proposition",
      "label": "Square on first bimedial applied to rational: breadth second binomial",
      "shortLabel": "Prop. X.61",
      "short": "Square on first bimedial",
      "book": 10,
      "number": 61,
      "colorClass": "proposition"
    },
    {
      "id": "Prop62",
      "type": "proposition",
      "label": "Square on second bimedial applied to rational: breadth third binomial",
      "shortLabel": "Prop. X.62",
      "short": "Square on second bimedial",
      "book": 10,
      "number": 62,
      "colorClass": "proposition"
    },
    {
      "id": "Prop63",
      "type": "proposition",
      "label": "Square on major applied to rational: breadth fourth binomial",
      "shortLabel": "Prop. X.63",
      "short": "Square on major",
      "book": 10,
      "number": 63,
      "colorClass": "proposition"
    },
    {
      "id": "Prop64",
      "type": "proposition",
      "label": "Square on rational+medial applied to rational: breadth fifth binomial",
      "shortLabel": "Prop. X.64",
      "short": "Square on rational+medial",
      "book": 10,
      "number": 64,
      "colorClass": "proposition"
    },
    {
      "id": "Prop65",
      "type": "proposition",
      "label": "Square on sum of two medial applied to rational: breadth sixth binomial",
      "shortLabel": "Prop. X.65",
      "short": "Square on two medial",
      "book": 10,
      "number": 65,
      "colorClass": "proposition"
    },
    {
      "id": "Prop66",
      "type": "proposition",
      "label": "Line commensurable with binomial is binomial, same order",
      "shortLabel": "Prop. X.66",
      "short": "Commensurable with binomial",
      "book": 10,
      "number": 66,
      "colorClass": "proposition"
    },
    {
      "id": "Prop67",
      "type": "proposition",
      "label": "Line commensurable with bimedial is bimedial, same order",
      "shortLabel": "Prop. X.67",
      "short": "Commensurable with bimedial",
      "book": 10,
      "number": 67,
      "colorClass": "proposition"
    },
    {
      "id": "Prop68",
      "type": "proposition",
      "label": "Line commensurable with major is major",
      "shortLabel": "Prop. X.68",
      "short": "Commensurable with major",
      "book": 10,
      "number": 68,
      "colorClass": "proposition"
    },
    {
      "id": "Prop69",
      "type": "proposition",
      "label": "Line commensurable with rational+medial is rational+medial",
      "shortLabel": "Prop. X.69",
      "short": "Commensurable with rational+medial",
      "book": 10,
      "number": 69,
      "colorClass": "proposition"
    },
    {
      "id": "Prop70",
      "type": "proposition",
      "label": "Line commensurable with sum of two medial is sum of two medial",
      "shortLabel": "Prop. X.70",
      "short": "Commensurable with two medial",
      "book": 10,
      "number": 70,
      "colorClass": "proposition"
    },
    {
      "id": "Prop71",
      "type": "proposition",
      "label": "Rational and medial added: four irrationals arise",
      "shortLabel": "Prop. X.71",
      "short": "Rational + medial: four irrationals",
      "book": 10,
      "number": 71,
      "colorClass": "proposition"
    },
    {
      "id": "Prop72",
      "type": "proposition",
      "label": "Two medial incommensurable added: two irrationals arise",
      "shortLabel": "Prop. X.72",
      "short": "Two medial: two irrationals",
      "book": 10,
      "number": 72,
      "colorClass": "proposition"
    },
    {
      "id": "Prop73",
      "type": "proposition",
      "label": "Rational minus rational in square only: remainder irrational, apotome",
      "shortLabel": "Prop. X.73",
      "short": "Apotome defined",
      "book": 10,
      "number": 73,
      "colorClass": "proposition"
    },
    {
      "id": "Prop74",
      "type": "proposition",
      "label": "Medial minus medial, rational rect: first apotome of medial",
      "shortLabel": "Prop. X.74",
      "short": "First apotome of medial",
      "book": 10,
      "number": 74,
      "colorClass": "proposition"
    },
    {
      "id": "Prop75",
      "type": "proposition",
      "label": "Medial minus medial, medial rect: second apotome of medial",
      "shortLabel": "Prop. X.75",
      "short": "Second apotome of medial",
      "book": 10,
      "number": 75,
      "colorClass": "proposition"
    },
    {
      "id": "Prop76",
      "type": "proposition",
      "label": "Line minus incommensurable: sum rational, rect medial: remainder minor",
      "shortLabel": "Prop. X.76",
      "short": "Minor defined",
      "book": 10,
      "number": 76,
      "colorClass": "proposition"
    },
    {
      "id": "Prop77",
      "type": "proposition",
      "label": "Line minus incommensurable: sum medial, rect rational: produces rational+medial",
      "shortLabel": "Prop. X.77",
      "short": "Produces rational+medial",
      "book": 10,
      "number": 77,
      "colorClass": "proposition"
    },
    {
      "id": "Prop78",
      "type": "proposition",
      "label": "Line minus incommensurable: both medial, incommensurable: produces medial+medial",
      "shortLabel": "Prop. X.78",
      "short": "Produces medial+medial",
      "book": 10,
      "number": 78,
      "colorClass": "proposition"
    },
    {
      "id": "Prop79",
      "type": "proposition",
      "label": "To apotome only one rational can be annexed in square only",
      "shortLabel": "Prop. X.79",
      "short": "Apotome: unique annex",
      "book": 10,
      "number": 79,
      "colorClass": "proposition"
    },
    {
      "id": "Prop80",
      "type": "proposition",
      "label": "To first apotome of medial: unique medial annex, rational rect",
      "shortLabel": "Prop. X.80",
      "short": "First apotome medial: unique annex",
      "book": 10,
      "number": 80,
      "colorClass": "proposition"
    },
    {
      "id": "Prop81",
      "type": "proposition",
      "label": "To second apotome of medial: unique medial annex, medial rect",
      "shortLabel": "Prop. X.81",
      "short": "Second apotome medial: unique annex",
      "book": 10,
      "number": 81,
      "colorClass": "proposition"
    },
    {
      "id": "Prop82",
      "type": "proposition",
      "label": "To minor: unique annex incommensurable in square",
      "shortLabel": "Prop. X.82",
      "short": "Minor: unique annex",
      "book": 10,
      "number": 82,
      "colorClass": "proposition"
    },
    {
      "id": "Prop83",
      "type": "proposition",
      "label": "To produces rational+medial: unique annex",
      "shortLabel": "Prop. X.83",
      "short": "Rational+medial: unique annex",
      "book": 10,
      "number": 83,
      "colorClass": "proposition"
    },
    {
      "id": "Prop84",
      "type": "proposition",
      "label": "To produces medial+medial: unique annex",
      "shortLabel": "Prop. X.84",
      "short": "Medial+medial: unique annex",
      "book": 10,
      "number": 84,
      "colorClass": "proposition"
    },
    {
      "id": "Prop85",
      "type": "proposition",
      "label": "To find the first apotome",
      "shortLabel": "Prop. X.85",
      "short": "Find first apotome",
      "book": 10,
      "number": 85,
      "colorClass": "proposition"
    },
    {
      "id": "Prop86",
      "type": "proposition",
      "label": "To find the second apotome",
      "shortLabel": "Prop. X.86",
      "short": "Find second apotome",
      "book": 10,
      "number": 86,
      "colorClass": "proposition"
    },
    {
      "id": "Prop87",
      "type": "proposition",
      "label": "To find the third apotome",
      "shortLabel": "Prop. X.87",
      "short": "Find third apotome",
      "book": 10,
      "number": 87,
      "colorClass": "proposition"
    },
    {
      "id": "Prop88",
      "type": "proposition",
      "label": "To find the fourth apotome",
      "shortLabel": "Prop. X.88",
      "short": "Find fourth apotome",
      "book": 10,
      "number": 88,
      "colorClass": "proposition"
    },
    {
      "id": "Prop89",
      "type": "proposition",
      "label": "To find the fifth apotome",
      "shortLabel": "Prop. X.89",
      "short": "Find fifth apotome",
      "book": 10,
      "number": 89,
      "colorClass": "proposition"
    },
    {
      "id": "Prop90",
      "type": "proposition",
      "label": "To find the sixth apotome",
      "shortLabel": "Prop. X.90",
      "short": "Find sixth apotome",
      "book": 10,
      "number": 90,
      "colorClass": "proposition"
    },
    {
      "id": "Prop91",
      "type": "proposition",
      "label": "Area by rational and first apotome: side is apotome",
      "shortLabel": "Prop. X.91",
      "short": "Rational × first apotome",
      "book": 10,
      "number": 91,
      "colorClass": "proposition"
    },
    {
      "id": "Prop92",
      "type": "proposition",
      "label": "Area by rational and second apotome: side is first apotome of medial",
      "shortLabel": "Prop. X.92",
      "short": "Rational × second apotome",
      "book": 10,
      "number": 92,
      "colorClass": "proposition"
    },
    {
      "id": "Prop93",
      "type": "proposition",
      "label": "Area by rational and third apotome: side is second apotome of medial",
      "shortLabel": "Prop. X.93",
      "short": "Rational × third apotome",
      "book": 10,
      "number": 93,
      "colorClass": "proposition"
    },
    {
      "id": "Prop94",
      "type": "proposition",
      "label": "Area by rational and fourth apotome: side is minor",
      "shortLabel": "Prop. X.94",
      "short": "Rational × fourth apotome",
      "book": 10,
      "number": 94,
      "colorClass": "proposition"
    },
    {
      "id": "Prop95",
      "type": "proposition",
      "label": "Area by rational and fifth apotome: side produces rational+medial",
      "shortLabel": "Prop. X.95",
      "short": "Rational × fifth apotome",
      "book": 10,
      "number": 95,
      "colorClass": "proposition"
    },
    {
      "id": "Prop96",
      "type": "proposition",
      "label": "Area by rational and sixth apotome: side produces medial+medial",
      "shortLabel": "Prop. X.96",
      "short": "Rational × sixth apotome",
      "book": 10,
      "number": 96,
      "colorClass": "proposition"
    },
    {
      "id": "Prop97",
      "type": "proposition",
      "label": "Square on apotome of medial applied to rational: breadth first apotome",
      "shortLabel": "Prop. X.97",
      "short": "Square on apotome medial",
      "book": 10,
      "number": 97,
      "colorClass": "proposition"
    },
    {
      "id": "Prop98",
      "type": "proposition",
      "label": "Square on first apotome of medial: breadth second apotome",
      "shortLabel": "Prop. X.98",
      "short": "Square on first apotome medial",
      "book": 10,
      "number": 98,
      "colorClass": "proposition"
    },
    {
      "id": "Prop99",
      "type": "proposition",
      "label": "Square on second apotome of medial: breadth third apotome",
      "shortLabel": "Prop. X.99",
      "short": "Square on second apotome medial",
      "book": 10,
      "number": 99,
      "colorClass": "proposition"
    },
    {
      "id": "Prop100",
      "type": "proposition",
      "label": "Square on minor applied to rational: breadth fourth apotome",
      "shortLabel": "Prop. X.100",
      "short": "Square on minor",
      "book": 10,
      "number": 100,
      "colorClass": "proposition"
    },
    {
      "id": "Prop101",
      "type": "proposition",
      "label": "Square on produces rational+medial: breadth fifth apotome",
      "shortLabel": "Prop. X.101",
      "short": "Square on rational+medial",
      "book": 10,
      "number": 101,
      "colorClass": "proposition"
    },
    {
      "id": "Prop102",
      "type": "proposition",
      "label": "Square on produces medial+medial: breadth sixth apotome",
      "shortLabel": "Prop. X.102",
      "short": "Square on medial+medial",
      "book": 10,
      "number": 102,
      "colorClass": "proposition"
    },
    {
      "id": "Prop103",
      "type": "proposition",
      "label": "Line commensurable with apotome is apotome, same order",
      "shortLabel": "Prop. X.103",
      "short": "Commensurable with apotome",
      "book": 10,
      "number": 103,
      "colorClass": "proposition"
    },
    {
      "id": "Prop104",
      "type": "proposition",
      "label": "Line commensurable with apotome of medial is apotome of medial",
      "shortLabel": "Prop. X.104",
      "short": "Commensurable with apotome medial",
      "book": 10,
      "number": 104,
      "colorClass": "proposition"
    },
    {
      "id": "Prop105",
      "type": "proposition",
      "label": "Line commensurable with minor is minor",
      "shortLabel": "Prop. X.105",
      "short": "Commensurable with minor",
      "book": 10,
      "number": 105,
      "colorClass": "proposition"
    },
    {
      "id": "Prop106",
      "type": "proposition",
      "label": "Line commensurable with produces rational+medial is same",
      "shortLabel": "Prop. X.106",
      "short": "Commensurable with rational+medial",
      "book": 10,
      "number": 106,
      "colorClass": "proposition"
    },
    {
      "id": "Prop107",
      "type": "proposition",
      "label": "Line commensurable with produces medial+medial is same",
      "shortLabel": "Prop. X.107",
      "short": "Commensurable with medial+medial",
      "book": 10,
      "number": 107,
      "colorClass": "proposition"
    },
    {
      "id": "Prop108",
      "type": "proposition",
      "label": "From rational area subtract medial: side is apotome or minor",
      "shortLabel": "Prop. X.108",
      "short": "Rational minus medial",
      "book": 10,
      "number": 108,
      "colorClass": "proposition"
    },
    {
      "id": "Prop109",
      "type": "proposition",
      "label": "From medial subtract rational: first apotome of medial or rational+medial",
      "shortLabel": "Prop. X.109",
      "short": "Medial minus rational",
      "book": 10,
      "number": 109,
      "colorClass": "proposition"
    },
    {
      "id": "Prop110",
      "type": "proposition",
      "label": "From medial subtract medial incommensurable: second apotome or medial+medial",
      "shortLabel": "Prop. X.110",
      "short": "Medial minus medial",
      "book": 10,
      "number": 110,
      "colorClass": "proposition"
    },
    {
      "id": "Prop111",
      "type": "proposition",
      "label": "Apotome is not the same as binomial",
      "shortLabel": "Prop. X.111",
      "short": "Apotome ≠ binomial",
      "book": 10,
      "number": 111,
      "colorClass": "proposition"
    },
    {
      "id": "Prop112",
      "type": "proposition",
      "label": "Square on rational applied to binomial: breadth apotome, same order",
      "shortLabel": "Prop. X.112",
      "short": "Rational on binomial",
      "book": 10,
      "number": 112,
      "colorClass": "proposition"
    },
    {
      "id": "Prop113",
      "type": "proposition",
      "label": "Square on rational applied to apotome: breadth binomial, same order",
      "shortLabel": "Prop. X.113",
      "short": "Rational on apotome",
      "book": 10,
      "number": 113,
      "colorClass": "proposition"
    },
    {
      "id": "Prop114",
      "type": "proposition",
      "label": "Area by apotome and binomial (commensurable terms): side is rational",
      "shortLabel": "Prop. X.114",
      "short": "Apotome × binomial: rational",
      "book": 10,
      "number": 114,
      "colorClass": "proposition"
    },
    {
      "id": "Prop115",
      "type": "proposition",
      "label": "From medial arise irrationals infinite in number, none same as preceding",
      "shortLabel": "Prop. X.115",
      "short": "Medial: infinite irrationals",
      "book": 10,
      "number": 115,
      "colorClass": "proposition"
    }
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    },
    {
      "from": "BookVI",
      "to": "Prop44"
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      "from": "BookI",
      "to": "Prop45"
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    {
      "from": "BookV",
      "to": "Prop45"
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    {
      "from": "BookVI",
      "to": "Prop45"
    },
    {
      "from": "BookI",
      "to": "Prop46"
    },
    {
      "from": "BookV",
      "to": "Prop46"
    },
    {
      "from": "BookVI",
      "to": "Prop46"
    },
    {
      "from": "BookI",
      "to": "Prop47"
    },
    {
      "from": "BookV",
      "to": "Prop47"
    },
    {
      "from": "BookVI",
      "to": "Prop47"
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    {
      "from": "BookI",
      "to": "Prop48"
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    {
      "from": "BookV",
      "to": "Prop48"
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    {
      "from": "BookVI",
      "to": "Prop48"
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    {
      "from": "BookI",
      "to": "Prop49"
    },
    {
      "from": "BookV",
      "to": "Prop49"
    },
    {
      "from": "BookVI",
      "to": "Prop49"
    },
    {
      "from": "BookI",
      "to": "Prop50"
    },
    {
      "from": "BookV",
      "to": "Prop50"
    },
    {
      "from": "BookVI",
      "to": "Prop50"
    },
    {
      "from": "BookI",
      "to": "Prop51"
    },
    {
      "from": "BookV",
      "to": "Prop51"
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    {
      "from": "BookVI",
      "to": "Prop51"
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    {
      "from": "BookI",
      "to": "Prop52"
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    {
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      "to": "Prop52"
    },
    {
      "from": "BookVI",
      "to": "Prop52"
    },
    {
      "from": "BookI",
      "to": "Prop53"
    },
    {
      "from": "BookV",
      "to": "Prop53"
    },
    {
      "from": "BookVI",
      "to": "Prop53"
    },
    {
      "from": "BookI",
      "to": "Prop54"
    },
    {
      "from": "BookV",
      "to": "Prop54"
    },
    {
      "from": "BookVI",
      "to": "Prop54"
    },
    {
      "from": "BookI",
      "to": "Prop55"
    },
    {
      "from": "BookV",
      "to": "Prop55"
    },
    {
      "from": "BookVI",
      "to": "Prop55"
    },
    {
      "from": "BookI",
      "to": "Prop56"
    },
    {
      "from": "BookV",
      "to": "Prop56"
    },
    {
      "from": "BookVI",
      "to": "Prop56"
    },
    {
      "from": "BookI",
      "to": "Prop57"
    },
    {
      "from": "BookV",
      "to": "Prop57"
    },
    {
      "from": "BookVI",
      "to": "Prop57"
    },
    {
      "from": "BookI",
      "to": "Prop58"
    },
    {
      "from": "BookV",
      "to": "Prop58"
    },
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      "from": "BookVI",
      "to": "Prop58"
    },
    {
      "from": "BookI",
      "to": "Prop59"
    },
    {
      "from": "BookV",
      "to": "Prop59"
    },
    {
      "from": "BookVI",
      "to": "Prop59"
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    {
      "from": "BookI",
      "to": "Prop60"
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      "from": "BookV",
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      "from": "BookVI",
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      "from": "BookI",
      "to": "Prop61"
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      "from": "BookVI",
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    {
      "from": "BookI",
      "to": "Prop62"
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      "from": "BookVI",
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      "from": "BookI",
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      "to": "Prop64"
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      "to": "Prop65"
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      "from": "BookVI",
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    },
    {
      "from": "BookVI",
      "to": "Prop68"
    },
    {
      "from": "BookI",
      "to": "Prop69"
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      "from": "BookV",
      "to": "Prop69"
    },
    {
      "from": "BookVI",
      "to": "Prop69"
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      "from": "BookI",
      "to": "Prop70"
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    {
      "from": "BookVI",
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    {
      "from": "BookI",
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    {
      "from": "BookVI",
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    },
    {
      "from": "BookI",
      "to": "Prop72"
    },
    {
      "from": "BookV",
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    },
    {
      "from": "BookVI",
      "to": "Prop72"
    },
    {
      "from": "BookI",
      "to": "Prop73"
    },
    {
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    {
      "from": "BookVI",
      "to": "Prop73"
    },
    {
      "from": "BookI",
      "to": "Prop74"
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    {
      "from": "BookV",
      "to": "Prop74"
    },
    {
      "from": "BookVI",
      "to": "Prop74"
    },
    {
      "from": "BookI",
      "to": "Prop75"
    },
    {
      "from": "BookV",
      "to": "Prop75"
    },
    {
      "from": "BookVI",
      "to": "Prop75"
    },
    {
      "from": "BookI",
      "to": "Prop76"
    },
    {
      "from": "BookV",
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    },
    {
      "from": "BookVI",
      "to": "Prop76"
    },
    {
      "from": "BookI",
      "to": "Prop77"
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    {
      "from": "BookV",
      "to": "Prop77"
    },
    {
      "from": "BookVI",
      "to": "Prop77"
    },
    {
      "from": "BookI",
      "to": "Prop78"
    },
    {
      "from": "BookV",
      "to": "Prop78"
    },
    {
      "from": "BookVI",
      "to": "Prop78"
    },
    {
      "from": "BookI",
      "to": "Prop79"
    },
    {
      "from": "BookV",
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    },
    {
      "from": "BookVI",
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    {
      "from": "BookI",
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      "from": "BookI",
      "to": "Prop81"
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      "to": "Prop81"
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    {
      "from": "BookVI",
      "to": "Prop81"
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    {
      "from": "BookI",
      "to": "Prop82"
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      "from": "BookV",
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      "from": "BookVI",
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      "from": "BookI",
      "to": "Prop83"
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      "to": "Prop83"
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    {
      "from": "BookVI",
      "to": "Prop83"
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    {
      "from": "BookI",
      "to": "Prop84"
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    {
      "from": "BookV",
      "to": "Prop84"
    },
    {
      "from": "BookVI",
      "to": "Prop84"
    },
    {
      "from": "BookI",
      "to": "Prop85"
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    {
      "from": "BookV",
      "to": "Prop85"
    },
    {
      "from": "BookVI",
      "to": "Prop85"
    },
    {
      "from": "BookI",
      "to": "Prop86"
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      "from": "BookV",
      "to": "Prop86"
    },
    {
      "from": "BookVI",
      "to": "Prop86"
    },
    {
      "from": "BookI",
      "to": "Prop87"
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    {
      "from": "BookV",
      "to": "Prop87"
    },
    {
      "from": "BookVI",
      "to": "Prop87"
    },
    {
      "from": "BookI",
      "to": "Prop88"
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    {
      "from": "BookV",
      "to": "Prop88"
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    {
      "from": "BookVI",
      "to": "Prop88"
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    {
      "from": "BookI",
      "to": "Prop89"
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    {
      "from": "BookV",
      "to": "Prop89"
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    {
      "from": "BookVI",
      "to": "Prop89"
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    {
      "from": "BookI",
      "to": "Prop90"
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    {
      "from": "BookV",
      "to": "Prop90"
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    {
      "from": "BookVI",
      "to": "Prop90"
    },
    {
      "from": "BookI",
      "to": "Prop91"
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    {
      "from": "BookV",
      "to": "Prop91"
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    {
      "from": "BookVI",
      "to": "Prop91"
    },
    {
      "from": "BookI",
      "to": "Prop92"
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      "from": "BookV",
      "to": "Prop92"
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    {
      "from": "BookVI",
      "to": "Prop92"
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    {
      "from": "BookI",
      "to": "Prop93"
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      "to": "Prop93"
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    {
      "from": "BookVI",
      "to": "Prop93"
    },
    {
      "from": "BookI",
      "to": "Prop94"
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    {
      "from": "BookV",
      "to": "Prop94"
    },
    {
      "from": "BookVI",
      "to": "Prop94"
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    {
      "from": "BookI",
      "to": "Prop95"
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      "from": "BookV",
      "to": "Prop95"
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    {
      "from": "BookVI",
      "to": "Prop95"
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      "from": "BookI",
      "to": "Prop96"
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    {
      "from": "BookV",
      "to": "Prop96"
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    {
      "from": "BookVI",
      "to": "Prop96"
    },
    {
      "from": "BookI",
      "to": "Prop97"
    },
    {
      "from": "BookV",
      "to": "Prop97"
    },
    {
      "from": "BookVI",
      "to": "Prop97"
    },
    {
      "from": "BookI",
      "to": "Prop98"
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    {
      "from": "BookV",
      "to": "Prop98"
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    {
      "from": "BookVI",
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      "from": "BookI",
      "to": "Prop99"
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      "from": "BookVI",
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      "from": "BookI",
      "to": "Prop100"
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      "from": "BookV",
      "to": "Prop100"
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    {
      "from": "BookVI",
      "to": "Prop100"
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    {
      "from": "BookI",
      "to": "Prop101"
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    {
      "from": "BookV",
      "to": "Prop101"
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    {
      "from": "BookVI",
      "to": "Prop101"
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    {
      "from": "BookI",
      "to": "Prop102"
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      "from": "BookV",
      "to": "Prop102"
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    {
      "from": "BookVI",
      "to": "Prop102"
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    {
      "from": "BookI",
      "to": "Prop103"
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      "from": "BookV",
      "to": "Prop103"
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    {
      "from": "BookVI",
      "to": "Prop103"
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    {
      "from": "BookI",
      "to": "Prop104"
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      "from": "BookV",
      "to": "Prop104"
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    {
      "from": "BookVI",
      "to": "Prop104"
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      "from": "BookI",
      "to": "Prop105"
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      "from": "BookV",
      "to": "Prop105"
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    {
      "from": "BookVI",
      "to": "Prop105"
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      "from": "BookI",
      "to": "Prop106"
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      "from": "BookVI",
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    {
      "from": "BookI",
      "to": "Prop107"
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      "from": "BookV",
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      "from": "BookVI",
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    {
      "from": "BookI",
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      "from": "BookVI",
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    {
      "from": "BookI",
      "to": "Prop109"
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      "from": "BookV",
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      "from": "BookVI",
      "to": "Prop109"
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    {
      "from": "BookI",
      "to": "Prop110"
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      "from": "BookV",
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      "from": "BookVI",
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    {
      "from": "BookI",
      "to": "Prop111"
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      "from": "BookV",
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      "from": "BookVI",
      "to": "Prop111"
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      "from": "BookI",
      "to": "Prop112"
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      "from": "BookV",
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      "from": "BookVI",
      "to": "Prop112"
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      "from": "BookI",
      "to": "Prop113"
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      "from": "BookV",
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      "from": "BookVI",
      "to": "Prop113"
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    {
      "from": "BookI",
      "to": "Prop114"
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      "from": "BookV",
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      "from": "BookVI",
      "to": "Prop114"
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    {
      "from": "BookI",
      "to": "Prop115"
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      "from": "BookVI",
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