The Works of Euclid, Vol. 1

...to be able to fall either within the segment ΑΖΔ, or outside, or partly within and partly outside; these three cases are demonstrated in manuscript 190 and in the Parisian edition.

But in all other manuscripts, and in all other Greek editions, it is only demonstrated that the segment ΑΕΒ cannot fall partly within the segment ΓΖΔ, and partly outside. Commandinus gives a demonstration of the other cases. But Robert Simson removes from proposition 24 the part which he attaches to proposition 23.

In proposition 26 of the sixth book, a certain passage could not be understood at all; the third variant reading dispelled all obscurity from it.

Gregorius, speaking of the corollary to proposition 19 of the fifth book, speaks thus: This passage is most corrupt; nor can it be restored by the aid of ancient copies: therefore we have changed the version, so that the sense might be consistent. Clavius substituted another in place of this corollary. Robert Simson says: "This corollary clearly shows that the fifth book was corrupted by those ignorant of geometry, and this corollary does not depend in any way on proposition 19." In this, Robert Simson errs, and I shall show in my observations that he errs very frequently.

Gregorius' version is very difficult to understand.

I have suppressed the third word of the corollary, ἐδείχθη. In place of the proportion: ὡς τὸ ΑΒ πρὸς τὸ ΓΔ οὕτως τὸ ΕΒ πρὸς τὸ ΖΔ, I have written this proportion: ὡς τὸ ΑΒ πρὸς τὸ ΓΔ οὕτως τὸ ΑΕ πρὸς τὸ ΓΖ; finally, in place of the proportion: ὡς τὸ ΑΒ πρὸς τὸ ΑΕ οὕτως τὸ ΓΔ πρὸς τὸ ΓΖ, I have written this proportion: ὡς τὸ ΑΒ πρὸς τὸ ΕΒ οὕτως τὸ ΔΓ πρὸς τὸ ΖΔ. With the aid of these slight corrections, the corollary has become unassailable.

In my edition, the phrase ἐδείχθη δὲ ὡς τὸ ΑΒ πρὸς τὸ ΕΒ οὕτως τὸ ΔΓ πρὸς τὸ ΖΔ, but it is shown as AB to EB so ΔΓ to ΖΔ (19. 5), clearly takes the place of these two phrases: ἐδείχθη δὲ ὡς τὸ ΑΒ πρὸς τὸ ΓΔ οὕτως τὸ ΕΒ πρὸς τὸ ΖΔ, ἐναλλὰξ ἄρα ὡς τὸ ΑΒ πρὸς τὸ ΕΒ οὕτως τὸ ΓΔ πρὸς τὸ ΖΔ, it is shown autem as AB to ΓΔ so EB to ZΔ (19. 5); alternately therefore as AB to EB so ΓΔ to ZΔ (16. 5).