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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", | |
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| "colorScheme": { | |
| "postulate": { | |
| "fill": "#e74c3c", | |
| "stroke": "#c0392b" | |
| }, | |
| "commonNotion": { | |
| "fill": "#9b59b6", | |
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| }, | |
| "proposition": { | |
| "fill": "#1abc9c", | |
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| } | |
| } |