Min. You mean, by 'conclude,' that we may prove our conclusion?
Euc. Yes, wherever proof is needed. Conclusions (1) and (4) need none, and are usually stated as Axioms.
Min. In subject (4), instead of 'identical directions,' why not say 'the same direction'?
Euc. Because I want to keep clearly in view that there are two Lines.
Min. In predicate (2), you speak of 'any transversal': a little while ago, you spoke of 'every exterior angle.' Do you make any distinction between 'any' and 'every'?
Euc. Where the things spoken of are limited in number, I use 'every'; where infinite, I use 'any' in order to bring the idea within the grasp of our finite intellects. For instance, you may talk of 'every grain of sand in the world': there are, no doubt, what country-folk would call 'a good few' of them, but still the number is limited, and the mind can just grasp the idea. But if you tell me that 'every cubic inch of Space contains eight cubic half-inches,' my mind is unable to form a distinct conception of the subject of your Proposition: you would convey the same truth, and in a form I could grasp, by saying 'any cubic inch.'
Min. The angles made with the transversal are a little bewildering when the Pair of Lines shrinks, as it does in this case, into one Line. For instance, what becomes of the pair of interior angles on the same side of the transversal?
Euc. A diagram will make it clear.
