Everything Pythagoras needs, and nothing else
Of the 465 propositions written out here, 25 are needed to reach I.47. Below they are in order, each one resting only on those above it.
This is computed over both kinds of dependency edge, and most of them are citations written by hand from Heath's margins. So read this as a tidy presentation of Euclid's own cross-references. The split is on the graph page.
I.47 is the one shown here because it is the traditional end of Book I and the obvious thing to aim at. Nothing about the calculation is special to it — any proposition can be tree-shaken the same way, and the command takes whichever you like:
euclid minimal III.35 euclid minimal X.115
executedcited
| Proposition | Statement |
|---|---|
| I.1 | On a given finite straight line to construct an equilateral triangle. |
| I.4 | If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend. |
| I.13 | If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles. |
| I.2 | To place at a given point (as an extremity) a straight line equal to a given straight line. |
| I.15 | If two straight lines cut one another, they make the vertical angles equal to one another. |
| I.3 | Given two unequal straight lines, to cut off from the greater a straight line equal to the less. |
| I.29 | A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles. |
| I.5 | In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another. |
| I.7 | Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it. |
| I.8 | If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines. |
| I.9 | To bisect a given rectilineal angle. |
| I.11 | To draw a straight line at right angles to a given straight line from a given point on it. |
| I.22 | Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one. |
| I.10 | To bisect a given finite straight line. |
| I.23 | On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle. |
| I.16 | In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles. |
| I.26 | If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle to the remaining angle. |
| I.27 | If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. |
| I.31 | Through a given point to draw a straight line parallel to a given straight line. |
| I.34 | In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas. |
| I.35 | Parallelograms which are on the same base and in the same parallels are equal to one another. |
| I.46 | On a given straight line to describe a square. |
| I.37 | Triangles which are on the same base and in the same parallels are equal to one another. |
| I.41 | If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle. |
| I.47 | In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. |
I.6 I.12 I.14 I.17 I.18 I.19 I.20 I.21 I.24 I.25 I.28 I.30 I.32 I.33 I.36 I.38 I.39 I.40 I.42 I.43 I.44 I.45 I.48 II.1 II.2 II.3 II.4 II.5 II.6 II.7 II.8 II.9 II.10 II.11 II.12 II.13 II.14 III.1 III.2 III.3 III.4 III.5 III.6 III.7 III.8 III.9 III.10 III.11 III.12 III.13 III.14 III.15 III.16 III.17 III.18 III.19 III.20 III.21 III.22 III.23 III.24 III.25 III.26 III.27 III.28 III.29 III.30 III.31 III.32 III.33 III.34 III.35 III.36 III.37 IV.1 IV.2 IV.3 IV.4 IV.5 IV.6 IV.7 IV.8 IV.9 IV.10 IV.11 IV.12 IV.13 IV.14 IV.15 IV.16 V.1 V.2 V.3 V.4 V.5 V.6 V.7 V.8 V.9 V.10 V.11 V.12 V.13 V.14 V.15 V.16 V.17 V.18 V.19 V.20 V.21 V.22 V.23 V.24 V.25 VI.1 VI.2 VI.3 VI.4 VI.5 VI.6 VI.7 VI.8 VI.9 VI.10 VI.11 VI.12 VI.13 VI.14 VI.15 VI.16 VI.17 VI.18 VI.19 VI.20 VI.21 VI.22 VI.23 VI.24 VI.25 VI.26 VI.27 VI.28 VI.29 VI.30 VI.31 VI.32 VI.33 VII.1 VII.2 VII.3 VII.4 VII.5 VII.6 VII.7 VII.8 VII.9 VII.10 VII.11 VII.12 VII.13 VII.14 VII.15 VII.16 VII.17 VII.18 VII.19 VII.20 VII.21 VII.22 VII.23 VII.24 VII.25 VII.26 VII.27 VII.28 VII.29 VII.30 VII.31 VII.32 VII.33 VII.34 VII.35 VII.36 VII.37 VII.38 VII.39 VIII.1 VIII.2 VIII.3 VIII.4 VIII.5 VIII.6 VIII.7 VIII.8 VIII.9 VIII.10 VIII.11 VIII.12 VIII.13 VIII.14 VIII.15 VIII.16 VIII.17 VIII.18 VIII.19 VIII.20 VIII.21 VIII.22 VIII.23 VIII.24 VIII.25 VIII.26 VIII.27 IX.1 IX.2 IX.3 IX.4 IX.5 IX.6 IX.7 IX.8 IX.9 IX.10 IX.11 IX.12 IX.13 IX.14 IX.15 IX.16 IX.17 IX.18 IX.19 IX.20 IX.21 IX.22 IX.23 IX.24 IX.25 IX.26 IX.27 IX.28 IX.29 IX.30 IX.31 IX.32 IX.33 IX.34 IX.35 IX.36 X.1 X.2 X.3 X.4 X.5 X.6 X.7 X.8 X.9 X.10 X.11 X.12 X.13 X.14 X.15 X.16 X.17 X.18 X.19 X.20 X.21 X.22 X.23 X.24 X.25 X.26 X.27 X.28 X.29 X.30 X.31 X.32 X.33 X.34 X.35 X.36 X.37 X.38 X.39 X.40 X.41 X.42 X.43 X.44 X.45 X.46 X.47 X.48 X.49 X.50 X.51 X.52 X.53 X.54 X.55 X.56 X.57 X.58 X.59 X.60 X.61 X.62 X.63 X.64 X.65 X.66 X.67 X.68 X.69 X.70 X.71 X.72 X.73 X.74 X.75 X.76 X.77 X.78 X.79 X.80 X.81 X.82 X.83 X.84 X.85 X.86 X.87 X.88 X.89 X.90 X.91 X.92 X.93 X.94 X.95 X.96 X.97 X.98 X.99 X.100 X.101 X.102 X.103 X.104 X.105 X.106 X.107 X.108 X.109 X.110 X.111 X.112 X.113 X.114 X.115 XI.1 XI.2 XI.3 XI.4 XI.5 XI.6 XI.7 XI.8 XI.9 XI.10 XI.11 XI.12 XI.13 XI.14 XI.15 XI.16 XI.17 XI.18 XI.19 XI.20 XI.21 XI.22 XI.23 XI.24 XI.25 XI.26 XI.27 XI.28 XI.29 XI.30 XI.31 XI.32 XI.33 XI.34 XI.35 XI.36 XI.37 XI.38 XI.39 XII.1 XII.2 XII.3 XII.4 XII.5 XII.6 XII.7 XII.8 XII.9 XII.10 XII.11 XII.12 XII.13 XII.14 XII.15 XII.16 XII.17 XII.18 XIII.1 XIII.2 XIII.3 XIII.4 XIII.5 XIII.6 XIII.7 XIII.8 XIII.9 XIII.10 XIII.11 XIII.12 XIII.13 XIII.14 XIII.15 XIII.16 XIII.17 XIII.18