Found by running the proofs
His rules let you draw a straight line and a circle. None of them says two circles ever cross.
Points assumed to exist. Every time the construction uses a point where a line meets a circle, or two circles meet. Euclid's rules never promise those points are there. This is read straight off the record of what was drawn, so nothing is missed.
Facts read off the picture. Claims that one point lies between two others, or that two points are on the same side of a line — things the diagram shows but the argument never establishes.
Steps that vary by figure. Each proposition is run on many different figures and the runs compared. A step that holds in one figure but not another appears here, and 43 do — in III.2, III.14, III.23, III.24, III.29, III.32, III.33, III.34, VI.9, VI.11, XII.17. Three of them evaluate a different number of facts depending on where the points fall, which means the proof is taking more than one route through the diagram; III.32 has a step about two points lying on the same side of a line that is simply true in some figures and false in others. These are the places where a case analysis is doing work that the argument does not set out.
| Proposition | Points assumed | Read off | Varies | The first of them |
|---|---|---|---|---|
| I.1 | 1 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.2 | 3 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.3 | 4 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.5 | 7 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.6 | 1 | — | — | a line-circle intersection is used, though no postulate asserts that a line meets a circle |
| I.7 | 7 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.9 | 6 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.10 | 8 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.11 | 6 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.12 | 9 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.13 | — | 1 | — | the argument reads a 'between' fact off the figure |
| I.14 | — | 1 | — | the argument reads a 'between' fact off the figure |
| I.15 | — | 1 | — | the argument reads a 'between' fact off the figure |
| I.16 | 8 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.17 | 24 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.18 | 19 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.19 | 4 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.20 | 24 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.21 | 14 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.22 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.23 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.24 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.25 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.26 | 8 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.31 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.32 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.38 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.39 | 2 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.40 | 4 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.42 | 16 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.43 | 4 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.44 | 30 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.45 | 46 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.46 | 11 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.47 | 35 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| I.48 | 42 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.1 | — | 1 | — | the argument reads a 'between' fact off the figure |
| II.2 | — | 1 | — | the argument reads a 'between' fact off the figure |
| II.3 | — | 1 | — | the argument reads a 'between' fact off the figure |
| II.4 | 26 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.5 | 23 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.6 | 27 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.7 | 11 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.8 | 4 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.9 | 119 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.10 | 119 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.11 | 64 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.12 | 79 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.13 | 79 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| II.14 | 69 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.1 | 22 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.2 | 12 | — | 1 | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.3 | 8 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.4 | 16 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.7 | 18 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.8 | 18 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.9 | 22 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.10 | 8 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.14 | 94 | — | 2 | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.15 | 89 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.17 | 1 | — | — | a line-circle intersection is used, though no postulate asserts that a line meets a circle |
| III.20 | 9 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.21 | — | 1 | — | the argument reads a 'same_side' fact off the figure |
| III.23 | — | 1 | 9 | the argument reads a 'same_side' fact off the figure |
| III.24 | — | 1 | 9 | the argument reads a 'same_side' fact off the figure |
| III.25 | 8 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.27 | — | 1 | — | the argument reads a 'same_side' fact off the figure |
| III.29 | — | 1 | 5 | the argument reads a 'same_side' fact off the figure |
| III.30 | 9 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.32 | — | 1 | 6 | the argument reads a 'same_side' fact off the figure |
| III.33 | 9 | 1 | 5 | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.34 | — | 1 | 3 | the argument reads a 'same_side' fact off the figure |
| III.35 | 102 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.36 | 53 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| III.37 | 106 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.1 | 1 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.4 | 15 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.5 | 22 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.6 | 6 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.8 | 49 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.9 | 17 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.10 | 69 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.11 | 25 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.12 | 60 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.13 | 74 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.14 | 26 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.15 | 5 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| IV.16 | 58 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.1 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.2 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.3 | 8 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.4 | 4 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.6 | 4 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.8 | 11 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.9 | 6 | — | 1 | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.10 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.11 | 2 | — | 1 | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.12 | 2 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.13 | 15 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.20 | 4 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.24 | 4 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.27 | 16 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.28 | 24 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.29 | 16 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.30 | 21 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| VI.31 | 35 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| X.17 | 8 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XI.1 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.2 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.3 | 1 | — | — | a plane-plane intersection is used, though no postulate asserts that two planes meet |
| XI.4 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.5 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.6 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.7 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.8 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.9 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.11 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.12 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.13 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.14 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.15 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.16 | 2 | — | — | a plane-plane intersection is used, though no postulate asserts that two planes meet |
| XI.17 | 2 | — | — | a line-plane intersection is used, though no postulate asserts that a line meets a plane |
| XI.18 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.19 | 1 | — | — | a plane-plane intersection is used, though no postulate asserts that two planes meet |
| XI.25 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.28 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.35 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XI.38 | 3 | — | — | a line-plane intersection is used, though no postulate asserts that a line meets a plane |
| XI.39 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XII.4 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XII.5 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XII.9 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XII.13 | — | — | — | a plane is described, and XI Def. 1-2; no postulate describes a plane |
| XII.17 | — | — | 1 | a sphere is described, and XI Def. 14; no postulate describes a sphere |
| XII.18 | — | — | — | a sphere is described, and XI Def. 14; no postulate describes a sphere |
| XIII.1 | 37 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.2 | 21 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.3 | 37 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.4 | 21 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.5 | 21 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.6 | 21 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.7 | 7 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.8 | 25 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.9 | 40 | 1 | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.10 | 75 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.11 | 25 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.12 | 40 | — | — | a circle-circle intersection is used, though no postulate asserts that two circles meet |
| XIII.13 | — | — | — | a sphere is described, and XI Def. 14; no postulate describes a sphere |
| XIII.14 | — | — | — | a sphere is described, and XI Def. 14; no postulate describes a sphere |
| XIII.15 | — | — | — | a sphere is described, and XI Def. 14; no postulate describes a sphere |
| XIII.16 | — | — | — | a sphere is described, and XI Def. 14; no postulate describes a sphere |
| XIII.17 | — | — | — | a sphere is described, and XI Def. 14; no postulate describes a sphere |