The Works of Euclid, Vol. 1

that the segment AEB cannot fall either inside the segment AZA, or outside, or partly inside and partly outside. These three cases are demonstrated in manuscript 190 and in the Paris edition.

But in all the other manuscripts and in all the other Greek editions, it is only demonstrated that the segment AEB cannot fall partly inside the segment ΓΖΔ and partly outside. Commandinus provides the demonstration for the other two cases. Robert Simson removes a portion of proposition 24, which he adds to proposition 23.

In proposition 26 of book six, there was a passage completely unintelligible; variant 3 has caused its obscurity to disappear.

Gregorius, in speaking of the corollary to proposition 19 of book five, expresses himself thus: Corruptissimus est hic locus; nec ope veterum exemplarium restitui potest: versionem ideo mutavimus, ut sensus constaret. Clavius replaced this corollary with one of his own making. Robert Simson tells us that this corollary manifests clearly that the fifth book has been corrupted by those ignorant of geometry, and that this corollary does not depend in any way on proposition 19. Robert Simson is wrong here, as in a host of other occasions, as I shall show in my remarks.

The version of Gregorius is unintelligible.

I have caused the third word of the corollary, ἐδείχθη, to disappear. In place of ὡς τὸ ΑΒ πρὸς τὸ ΓΔ οὕτως τὸ ΕΒ πρὸς τὸ ΖΔ, I have placed ὡς τὸ ΑΒ πρὸς τὸ ΓΔ οὕτως τὸ ΑΕ πρὸς τὸ ΓΖ; and in place of ὡς τὸ ΑΒ πρὸς τὸ ΑΕ οὕτως τὸ ΓΔ πρὸς τὸ ΓΖ, I have written ὡς τὸ ΑΒ πρὸς τὸ ΕΒ οὕτως τὸ ΔΓ πρὸς τὸ ΖΔ. By means of these slight corrections, the corollary is found to be restored to all its purity.

In my edition, the phrase ἐδείχθη δὲ ὡς τὸ ΑΒ πρὸς τὸ ΕΒ οὕτως τὸ ΔΓ πρὸς τὸ ΖΔ, but it has been shown that AB is to EB as ΔΓ is to ΖΔ (19. 5), obviously takes the place of ἐδείχθη δὲ ὡς τὸ ΑΒ πρὸς τὸ ΓΔ οὕτως τὸ ΕΒ πρὸς τὸ ΖΔ, ἐναλλάξ ἄρα ὡς τὸ ΑΒ πρὸς τὸ ΕΒ οὕτως τὸ ΓΔ πρὸς τὸ ΖΔ, but it has been shown that AB is to ΓΔ as EB is to ΖΔ (19. 5); therefore by permutation AB is to EB as ΓΔ is to ΖΔ (16. 5).