The Works of Euclid, Vol. 1

Euclid could have changed this corollary into a theorem, in this way:
If magnitudes are proportional, they will be proportional by conversion.

A geometric diagram consisting of two parallel horizontal lines. The top line is labeled with points A, E, and B. The bottom line is labeled with Greek letters Γ, Ζ, and Δ.

Let the proportional magnitudes be AB, AE, ΓΔ, ΓΖ, and let AB be to AE as ΓΔ is to ΓΖ; I say that by conversion, as AB is to EB, so is ΓΔ to ZΔ.

For since as AB is to AE, so is ΓΔ to ΓΖ, then alternately as AB is to ΓΔ, so is AE to ΓΖ (16. 5); but it has been shown that as AB is to ΓΔ, so is EB to ZΔ (19. 5); therefore, alternately, as AB is to EB, so is ΓΔ to ZΔ, that is, as AB is to AB—AE, so is ΓΔ to ΓΔ—ΓΖ (16. 5); which is by conversion. Which was to be demonstrated.

In the Greek text of manuscript 190, there is no mention of sectors of circles in the final proposition of the sixth book. An alien hand inscribed between the lines and in the margin of the manuscript everything pertaining to sectors, which is present in the Greek text of all other manuscripts and in the Basel and Oxford editions. This addition, which I ought not to have admitted into my edition, was made to the text by Theon. Thus speaks Theon in his commentaries on the Almagest, p. 50, l. 7, Basel edition, 1538: « ὅτι δὲ οἱ ἐπ' ἴσων κύκλων τομεῖς πρὸς ἀλλήλους εἰσὶν ὡς αἱ γωνίαι ἐφ' ὧν βεβήκασι δέδεικται ἡμῖν ἐν τῇ ἐκδόσει τῶν στοιχείων πρὸς τῷ τέλει τοῦ ἕκτου βιβλίου. » That sectors in equal circles are to one another as the angles upon which they stand, has been shown by us in the edition of the Elements at the end of the sixth book.

This addition of Theon, which has no utility in the subsequent material, brings a delay to Euclid’s speed. Especially in books 10, 14, and 15, as well as in the Data, one might find a great many superfluities, none of which are in the text of manuscript 190. For this reason, especially, have people admired Euclid: because he tends directly to the objective, never declining from his path for the sake of demonstrating things that are by no means necessary for proceeding. But this can only apply to manuscript 190; and thus I would not absurdly conjecture that an emended Euclid

(*) Four magnitudes are said to be proportional, when the second is a certain fraction of the first, and the fourth a certain fraction of the third.