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"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"
},
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