{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "euclid-elements-book-i",
    "name": "Euclid's Elements, Book I",
    "subject": "geometry",
    "variant": "classical",
    "description": "The 48 propositions of Book I with dependencies on postulates (P1–P5), common notions (CN1–CN5), and prior propositions. Source: David E. Joyce, Clark University.",
    "structure": {
      "books": 1,
      "propositions": 48,
      "foundationTypes": [
        "postulate",
        "commonNotion"
      ]
    }
  },
  "metadata": {
    "created": "2026-03-15",
    "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 I Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book I",
      "plane geometry",
      "constructions",
      "Pythagorean theorem"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book I",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html",
      "notes": "Clark University; dependency table from Guide"
    },
    {
      "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": "P1",
      "type": "postulate",
      "label": "Draw a straight line from any point to any point",
      "shortLabel": "Post. 1",
      "book": 1,
      "number": 1,
      "colorClass": "postulate"
    },
    {
      "id": "P2",
      "type": "postulate",
      "label": "Produce a finite straight line continuously in a straight line",
      "shortLabel": "Post. 2",
      "book": 1,
      "number": 2,
      "colorClass": "postulate"
    },
    {
      "id": "P3",
      "type": "postulate",
      "label": "Describe a circle with any center and radius",
      "shortLabel": "Post. 3",
      "book": 1,
      "number": 3,
      "colorClass": "postulate"
    },
    {
      "id": "P4",
      "type": "postulate",
      "label": "All right angles equal one another",
      "shortLabel": "Post. 4",
      "book": 1,
      "number": 4,
      "colorClass": "postulate"
    },
    {
      "id": "P5",
      "type": "postulate",
      "label": "Parallel postulate: if interior angles < two right, lines meet",
      "shortLabel": "Post. 5",
      "book": 1,
      "number": 5,
      "colorClass": "postulate"
    },
    {
      "id": "CN1",
      "type": "commonNotion",
      "label": "Things equal to the same thing are equal to each other",
      "shortLabel": "CN 1",
      "book": 1,
      "number": 1,
      "colorClass": "commonNotion"
    },
    {
      "id": "CN2",
      "type": "commonNotion",
      "label": "If equals are added to equals, the wholes are equal",
      "shortLabel": "CN 2",
      "book": 1,
      "number": 2,
      "colorClass": "commonNotion"
    },
    {
      "id": "CN3",
      "type": "commonNotion",
      "label": "If equals are subtracted from equals, the remainders are equal",
      "shortLabel": "CN 3",
      "book": 1,
      "number": 3,
      "colorClass": "commonNotion"
    },
    {
      "id": "CN4",
      "type": "commonNotion",
      "label": "Things coinciding with one another are equal",
      "shortLabel": "CN 4",
      "book": 1,
      "number": 4,
      "colorClass": "commonNotion"
    },
    {
      "id": "CN5",
      "type": "commonNotion",
      "label": "The whole is greater than the part",
      "shortLabel": "CN 5",
      "book": 1,
      "number": 5,
      "colorClass": "commonNotion"
    },
    {
      "id": "Prop1",
      "type": "proposition",
      "label": "To construct an equilateral triangle on a given finite straight line",
      "shortLabel": "Prop. I.1",
      "short": "Equilateral triangle on given line",
      "book": 1,
      "number": 1,
      "colorClass": "proposition"
    },
    {
      "id": "Prop2",
      "type": "proposition",
      "label": "To place a straight line equal to a given straight line with one end at a given point",
      "shortLabel": "Prop. I.2",
      "short": "Place line equal to given at point",
      "book": 1,
      "number": 2,
      "colorClass": "proposition"
    },
    {
      "id": "Prop3",
      "type": "proposition",
      "label": "To cut off from the greater of two given unequal straight lines a straight line equal to the less",
      "shortLabel": "Prop. I.3",
      "short": "Cut off from greater segment equal to less",
      "book": 1,
      "number": 3,
      "colorClass": "proposition"
    },
    {
      "id": "Prop4",
      "type": "proposition",
      "label": "If two triangles have two sides equal to two sides respectively, and the angles contained equal, then bases and remaining angles equal",
      "shortLabel": "Prop. I.4",
      "short": "SAS congruence",
      "book": 1,
      "number": 4,
      "colorClass": "proposition"
    },
    {
      "id": "Prop5",
      "type": "proposition",
      "label": "In isosceles triangles the angles at the base equal one another",
      "shortLabel": "Prop. I.5",
      "short": "Base angles of isosceles equal",
      "book": 1,
      "number": 5,
      "colorClass": "proposition"
    },
    {
      "id": "Prop6",
      "type": "proposition",
      "label": "If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another",
      "shortLabel": "Prop. I.6",
      "short": "Sides opposite equal angles equal",
      "book": 1,
      "number": 6,
      "colorClass": "proposition"
    },
    {
      "id": "Prop7",
      "type": "proposition",
      "label": "Given two lines from ends of a line meeting at a point, no other such pair from same ends on same side",
      "shortLabel": "Prop. I.7",
      "short": "Uniqueness of triangle from ends",
      "book": 1,
      "number": 7,
      "colorClass": "proposition"
    },
    {
      "id": "Prop8",
      "type": "proposition",
      "label": "If two triangles have two sides equal to two sides respectively, and the base equal to the base, then the angles contained are equal",
      "shortLabel": "Prop. I.8",
      "short": "SSS congruence",
      "book": 1,
      "number": 8,
      "colorClass": "proposition"
    },
    {
      "id": "Prop9",
      "type": "proposition",
      "label": "To bisect a given rectilinear angle",
      "shortLabel": "Prop. I.9",
      "short": "Bisect angle",
      "book": 1,
      "number": 9,
      "colorClass": "proposition"
    },
    {
      "id": "Prop10",
      "type": "proposition",
      "label": "To bisect a given finite straight line",
      "shortLabel": "Prop. I.10",
      "short": "Bisect line",
      "book": 1,
      "number": 10,
      "colorClass": "proposition"
    },
    {
      "id": "Prop11",
      "type": "proposition",
      "label": "To draw a straight line at right angles to a given straight line from a given point on it",
      "shortLabel": "Prop. I.11",
      "short": "Perpendicular from point on line",
      "book": 1,
      "number": 11,
      "colorClass": "proposition"
    },
    {
      "id": "Prop12",
      "type": "proposition",
      "label": "To draw a straight line perpendicular to a given infinite straight line from a given point not on it",
      "shortLabel": "Prop. I.12",
      "short": "Perpendicular from point not on line",
      "book": 1,
      "number": 12,
      "colorClass": "proposition"
    },
    {
      "id": "Prop13",
      "type": "proposition",
      "label": "If a straight line stands on a straight line, it makes either two right angles or angles whose sum equals two right angles",
      "shortLabel": "Prop. I.13",
      "short": "Angles on line sum to two right",
      "book": 1,
      "number": 13,
      "colorClass": "proposition"
    },
    {
      "id": "Prop14",
      "type": "proposition",
      "label": "If with any straight line, at a point, two lines not on same side make adjacent angles equal to two right, they are in a straight line",
      "shortLabel": "Prop. I.14",
      "short": "If angles sum to two right, straight line",
      "book": 1,
      "number": 14,
      "colorClass": "proposition"
    },
    {
      "id": "Prop15",
      "type": "proposition",
      "label": "If two straight lines cut one another, they make the vertical angles equal to one another",
      "shortLabel": "Prop. I.15",
      "short": "Vertical angles equal",
      "book": 1,
      "number": 15,
      "colorClass": "proposition"
    },
    {
      "id": "Prop16",
      "type": "proposition",
      "label": "In any triangle, if one side is produced, the exterior angle is greater than either interior opposite angle",
      "shortLabel": "Prop. I.16",
      "short": "Exterior angle > interior opposite",
      "book": 1,
      "number": 16,
      "colorClass": "proposition"
    },
    {
      "id": "Prop17",
      "type": "proposition",
      "label": "In any triangle the sum of any two angles is less than two right angles",
      "shortLabel": "Prop. I.17",
      "short": "Sum of two angles < two right",
      "book": 1,
      "number": 17,
      "colorClass": "proposition"
    },
    {
      "id": "Prop18",
      "type": "proposition",
      "label": "In any triangle the angle opposite the greater side is greater",
      "shortLabel": "Prop. I.18",
      "short": "Angle opposite greater side greater",
      "book": 1,
      "number": 18,
      "colorClass": "proposition"
    },
    {
      "id": "Prop19",
      "type": "proposition",
      "label": "In any triangle the side opposite the greater angle is greater",
      "shortLabel": "Prop. I.19",
      "short": "Side opposite greater angle greater",
      "book": 1,
      "number": 19,
      "colorClass": "proposition"
    },
    {
      "id": "Prop20",
      "type": "proposition",
      "label": "In any triangle the sum of any two sides is greater than the remaining one",
      "shortLabel": "Prop. I.20",
      "short": "Triangle inequality",
      "book": 1,
      "number": 20,
      "colorClass": "proposition"
    },
    {
      "id": "Prop21",
      "type": "proposition",
      "label": "If from ends of one side two lines meet within the triangle, their sum < sum of other two sides",
      "shortLabel": "Prop. I.21",
      "short": "Lines from ends within triangle",
      "book": 1,
      "number": 21,
      "colorClass": "proposition"
    },
    {
      "id": "Prop22",
      "type": "proposition",
      "label": "To construct a triangle out of three straight lines which equal three given straight lines",
      "shortLabel": "Prop. I.22",
      "short": "Construct triangle from three lines",
      "book": 1,
      "number": 22,
      "colorClass": "proposition"
    },
    {
      "id": "Prop23",
      "type": "proposition",
      "label": "To construct a rectilinear angle equal to a given rectilinear angle on a given straight line",
      "shortLabel": "Prop. I.23",
      "short": "Construct angle equal to given",
      "book": 1,
      "number": 23,
      "colorClass": "proposition"
    },
    {
      "id": "Prop24",
      "type": "proposition",
      "label": "If two triangles have two sides equal but one contained angle greater, the base is greater",
      "shortLabel": "Prop. I.24",
      "short": "SAS for greater angle => greater base",
      "book": 1,
      "number": 24,
      "colorClass": "proposition"
    },
    {
      "id": "Prop25",
      "type": "proposition",
      "label": "If two triangles have two sides equal but base greater, the contained angle is greater",
      "shortLabel": "Prop. I.25",
      "short": "SAS for greater base => greater angle",
      "book": 1,
      "number": 25,
      "colorClass": "proposition"
    },
    {
      "id": "Prop26",
      "type": "proposition",
      "label": "If two triangles have two angles equal and one side equal, the remaining sides and angle equal",
      "shortLabel": "Prop. I.26",
      "short": "AAS congruence",
      "book": 1,
      "number": 26,
      "colorClass": "proposition"
    },
    {
      "id": "Prop27",
      "type": "proposition",
      "label": "If a line falling on two lines makes alternate angles equal, the lines are parallel",
      "shortLabel": "Prop. I.27",
      "short": "Alternate angles equal => parallel",
      "book": 1,
      "number": 27,
      "colorClass": "proposition"
    },
    {
      "id": "Prop28",
      "type": "proposition",
      "label": "If exterior angle equals interior opposite, or interior same-side sum to two right, lines parallel",
      "shortLabel": "Prop. I.28",
      "short": "Exterior = interior opposite => parallel",
      "book": 1,
      "number": 28,
      "colorClass": "proposition"
    },
    {
      "id": "Prop29",
      "type": "proposition",
      "label": "A line falling on parallel lines makes alternate angles equal, exterior = interior opposite",
      "shortLabel": "Prop. I.29",
      "short": "Parallel => alternate angles equal",
      "book": 1,
      "number": 29,
      "colorClass": "proposition"
    },
    {
      "id": "Prop30",
      "type": "proposition",
      "label": "Straight lines parallel to the same straight line are also parallel to one another",
      "shortLabel": "Prop. I.30",
      "short": "Transitivity of parallel",
      "book": 1,
      "number": 30,
      "colorClass": "proposition"
    },
    {
      "id": "Prop31",
      "type": "proposition",
      "label": "To draw a straight line through a given point parallel to a given straight line",
      "shortLabel": "Prop. I.31",
      "short": "Draw parallel through point",
      "book": 1,
      "number": 31,
      "colorClass": "proposition"
    },
    {
      "id": "Prop32",
      "type": "proposition",
      "label": "In any triangle, exterior angle equals sum of two interior opposite; three angles = two right",
      "shortLabel": "Prop. I.32",
      "short": "Exterior angle = sum interior opposite",
      "book": 1,
      "number": 32,
      "colorClass": "proposition"
    },
    {
      "id": "Prop33",
      "type": "proposition",
      "label": "Straight lines which join the ends of equal and parallel straight lines in same directions are equal and parallel",
      "shortLabel": "Prop. I.33",
      "short": "Joining ends of equal parallel lines",
      "book": 1,
      "number": 33,
      "colorClass": "proposition"
    },
    {
      "id": "Prop34",
      "type": "proposition",
      "label": "In parallelogrammic areas the opposite sides and angles equal one another, diameter bisects",
      "shortLabel": "Prop. I.34",
      "short": "Parallelogram properties",
      "book": 1,
      "number": 34,
      "colorClass": "proposition"
    },
    {
      "id": "Prop35",
      "type": "proposition",
      "label": "Parallelograms which are on the same base and in the same parallels equal one another",
      "shortLabel": "Prop. I.35",
      "short": "Parallelograms same base equal",
      "book": 1,
      "number": 35,
      "colorClass": "proposition"
    },
    {
      "id": "Prop36",
      "type": "proposition",
      "label": "Parallelograms which are on equal bases and in the same parallels equal one another",
      "shortLabel": "Prop. I.36",
      "short": "Parallelograms equal bases equal",
      "book": 1,
      "number": 36,
      "colorClass": "proposition"
    },
    {
      "id": "Prop37",
      "type": "proposition",
      "label": "Triangles which are on the same base and in the same parallels equal one another",
      "shortLabel": "Prop. I.37",
      "short": "Triangles same base equal",
      "book": 1,
      "number": 37,
      "colorClass": "proposition"
    },
    {
      "id": "Prop38",
      "type": "proposition",
      "label": "Triangles which are on equal bases and in the same parallels equal one another",
      "shortLabel": "Prop. I.38",
      "short": "Triangles equal bases equal",
      "book": 1,
      "number": 38,
      "colorClass": "proposition"
    },
    {
      "id": "Prop39",
      "type": "proposition",
      "label": "Equal triangles on same base and same side are in the same parallels",
      "shortLabel": "Prop. I.39",
      "short": "Equal triangles same base same side",
      "book": 1,
      "number": 39,
      "colorClass": "proposition"
    },
    {
      "id": "Prop40",
      "type": "proposition",
      "label": "Equal triangles on equal bases and same side are in the same parallels",
      "shortLabel": "Prop. I.40",
      "short": "Equal triangles equal bases same side",
      "book": 1,
      "number": 40,
      "colorClass": "proposition"
    },
    {
      "id": "Prop41",
      "type": "proposition",
      "label": "If a parallelogram has same base with triangle and same parallels, parallelogram is double the triangle",
      "shortLabel": "Prop. I.41",
      "short": "Parallelogram = 2× triangle",
      "book": 1,
      "number": 41,
      "colorClass": "proposition"
    },
    {
      "id": "Prop42",
      "type": "proposition",
      "label": "To construct a parallelogram equal to a given triangle in a given rectilinear angle",
      "shortLabel": "Prop. I.42",
      "short": "Construct parallelogram = triangle",
      "book": 1,
      "number": 42,
      "colorClass": "proposition"
    },
    {
      "id": "Prop43",
      "type": "proposition",
      "label": "In any parallelogram the complements of the parallelograms about the diameter equal one another",
      "shortLabel": "Prop. I.43",
      "short": "Complements of parallelogram",
      "book": 1,
      "number": 43,
      "colorClass": "proposition"
    },
    {
      "id": "Prop44",
      "type": "proposition",
      "label": "To a given straight line in a given angle, to apply a parallelogram equal to a given triangle",
      "shortLabel": "Prop. I.44",
      "short": "Apply parallelogram to line",
      "book": 1,
      "number": 44,
      "colorClass": "proposition"
    },
    {
      "id": "Prop45",
      "type": "proposition",
      "label": "To construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle",
      "shortLabel": "Prop. I.45",
      "short": "Construct parallelogram = rectilinear figure",
      "book": 1,
      "number": 45,
      "colorClass": "proposition"
    },
    {
      "id": "Prop46",
      "type": "proposition",
      "label": "To describe a square on a given straight line",
      "shortLabel": "Prop. I.46",
      "short": "Construct square on line",
      "book": 1,
      "number": 46,
      "colorClass": "proposition"
    },
    {
      "id": "Prop47",
      "type": "proposition",
      "label": "In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle",
      "shortLabel": "Prop. I.47",
      "short": "Pythagorean theorem",
      "book": 1,
      "number": 47,
      "colorClass": "proposition"
    },
    {
      "id": "Prop48",
      "type": "proposition",
      "label": "If in a triangle the square on one side equals the sum of squares on the other two, the angle contained by those sides is right",
      "shortLabel": "Prop. I.48",
      "short": "Converse Pythagorean",
      "book": 1,
      "number": 48,
      "colorClass": "proposition"
    }
  ],
  "edges": [
    {
      "from": "P1",
      "to": "Prop1"
    },
    {
      "from": "P3",
      "to": "Prop1"
    },
    {
      "from": "Prop1",
      "to": "Prop2"
    },
    {
      "from": "P1",
      "to": "Prop2"
    },
    {
      "from": "P2",
      "to": "Prop2"
    },
    {
      "from": "P3",
      "to": "Prop2"
    },
    {
      "from": "Prop2",
      "to": "Prop3"
    },
    {
      "from": "P3",
      "to": "Prop3"
    },
    {
      "from": "CN4",
      "to": "Prop4"
    },
    {
      "from": "CN5",
      "to": "Prop4"
    },
    {
      "from": "Prop3",
      "to": "Prop5"
    },
    {
      "from": "Prop4",
      "to": "Prop5"
    },
    {
      "from": "Prop3",
      "to": "Prop6"
    },
    {
      "from": "Prop4",
      "to": "Prop6"
    },
    {
      "from": "Prop5",
      "to": "Prop7"
    },
    {
      "from": "Prop7",
      "to": "Prop8"
    },
    {
      "from": "Prop1",
      "to": "Prop9"
    },
    {
      "from": "Prop3",
      "to": "Prop9"
    },
    {
      "from": "Prop8",
      "to": "Prop9"
    },
    {
      "from": "Prop1",
      "to": "Prop10"
    },
    {
      "from": "Prop4",
      "to": "Prop10"
    },
    {
      "from": "Prop9",
      "to": "Prop10"
    },
    {
      "from": "Prop1",
      "to": "Prop11"
    },
    {
      "from": "Prop3",
      "to": "Prop11"
    },
    {
      "from": "Prop8",
      "to": "Prop11"
    },
    {
      "from": "Prop8",
      "to": "Prop12"
    },
    {
      "from": "Prop10",
      "to": "Prop12"
    },
    {
      "from": "Prop11",
      "to": "Prop13"
    },
    {
      "from": "Prop13",
      "to": "Prop14"
    },
    {
      "from": "Prop13",
      "to": "Prop15"
    },
    {
      "from": "Prop3",
      "to": "Prop16"
    },
    {
      "from": "Prop4",
      "to": "Prop16"
    },
    {
      "from": "Prop10",
      "to": "Prop16"
    },
    {
      "from": "Prop15",
      "to": "Prop16"
    },
    {
      "from": "Prop13",
      "to": "Prop17"
    },
    {
      "from": "Prop16",
      "to": "Prop17"
    },
    {
      "from": "Prop3",
      "to": "Prop18"
    },
    {
      "from": "Prop5",
      "to": "Prop18"
    },
    {
      "from": "Prop16",
      "to": "Prop18"
    },
    {
      "from": "Prop5",
      "to": "Prop19"
    },
    {
      "from": "Prop18",
      "to": "Prop19"
    },
    {
      "from": "Prop3",
      "to": "Prop20"
    },
    {
      "from": "Prop5",
      "to": "Prop20"
    },
    {
      "from": "Prop19",
      "to": "Prop20"
    },
    {
      "from": "Prop16",
      "to": "Prop21"
    },
    {
      "from": "Prop20",
      "to": "Prop21"
    },
    {
      "from": "Prop3",
      "to": "Prop22"
    },
    {
      "from": "Prop20",
      "to": "Prop22"
    },
    {
      "from": "Prop8",
      "to": "Prop23"
    },
    {
      "from": "Prop22",
      "to": "Prop23"
    },
    {
      "from": "Prop3",
      "to": "Prop24"
    },
    {
      "from": "Prop4",
      "to": "Prop24"
    },
    {
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