Euclid's Elements as software

Overview

Every proposition is a small program. It draws its own figure with a straightedge and compass, checks its own conclusion, and says what it depended on. Nothing is measured or approximated.

465propositions

7605steps checked

549dependency edges

2573unproved assumptions

The idea

A straightedge and compass can only produce certain numbers: the ones you reach from whole numbers by adding, subtracting, multiplying, dividing, and taking square roots. This project stores those numbers exactly, as square roots piled on square roots, never as decimals. Two lengths are either equal or they are not; there is no rounding, no tolerance to set, and no near-miss that might be mistaken for a theorem. Everything else on this site follows from that. See the findings page for the measurements and the commands that reproduce them.

The propositions

Book I: Rectilinear figures

I.1 I.2 I.3 I.4 I.5 I.6 I.7 I.8 I.9 I.10 I.11 I.12 I.13 I.14 I.15 I.16 I.17 I.18 I.19 I.20 I.21 I.22 I.23 I.24 I.25 I.26 I.27 I.28 I.29 I.30 I.31 I.32 I.33 I.34 I.35 I.36 I.37 I.38 I.39 I.40 I.41 I.42 I.43 I.44 I.45 I.46 I.47 I.48

Book II: Geometric algebra

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

Book III: Circles

Book IV: Inscribed and circumscribed figures

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

Book V: The theory of proportion

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

Book VI: Similar figures

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

Book VII: Elementary number theory

Book VIII: Continued proportions

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

Book IX: Numbers, primes and perfect numbers

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

Book X: Incommensurable magnitudes

Book XI: Solid geometry

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

Book XII: Method of exhaustion

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

Book XIII: The regular solids

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