Euclid, Elements I 14©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 14: If at some line and the point on it two straight-lines, not positioned on the same sides, make successive angles equal to two right-angles, the straight-lines will be on a straight-line with one another.

(general diagram)

Diagrams for book 1 proposition 14, following the display, construction, and demonstration(diagram 1) For at some line AB and the point on it, B, let two straight-lines, BG, BD, not positioned on the same sides, make successive angles, those by ABG, ABD, equal to two right-angles. I say that BD is on a straight-line with GB. (diagram 2) For if BD is not on a straight-line with BG, let BE be on a straight-line with GB. (diagram 3) And so, since a straight-line, AB, stands-on a straight-line, GBE, therefore, the angles by ABG, ABE are equal to two right-angles. But the angles by ABG, ABD are also equal to two right-angles. Therefore, the angles by GBA, ABE are equal to those by GBA, ABD. (diagram 4) Let a common be taken away, that by GBA. Therefore, a remainder, the angle by ABE, is equal to a remainder, that by ABD, the smaller to the larger, which is impossible. Therefore, BE is not on a straight-line with GB. Similarly, in fact, we will show that not any other straight-line except BD is either. Therefore, GB is on a straight-line with BD. Therefore, if at some line and the point on it two straight-lines, not positioned on the same sides, make successive angles equal to two right-angles, the straight-lines will be on a straight-line with one another, just what it was required to show.

Note the form of generalization, an intention to prove the case for every other line. This gesture is not uncommon in Euclid nor Greek mathematics (c.f., Autolycus, On moving spheres). Cf. I 39, I 40. In these cases, the context is a reductio where the reasoning is:

No figure that is not a has F. Some figure has F. Therefore, a has F.

The invitation to show in each case that no other figure has F is in actuality an infinite task. So far as I can tell, this explicit form of generalization to unlimited cases occurs in reductios and but very rarely in positive proofs (XI 18, XII 4 alternatve prrof, Optics 35, and possibly X 115, a later addition to the text). Is this significant?

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