The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...

PROPOSITION XXIII. THEOREM.

THe Reason of equality without order, if Two Ranks of terms are in the same Reason ill Ranked the first and the last of the one and of the other shall be propor∣tional.

A,	B,	C.	D,	E,	F,	G.
12,	6,	3.	8	4,	2,	1.

If the Magni∣tudes A, B, C, and the others D, E, F, in like number be in the same Reason ill Ranked, that is to say, that there be the same Reason of A to B, as of E to F; and the same Reason of B to C, as of D to E: there will be the same Reason of A to C, as of D to F. Let there be the same Reason of B to C, as of F to G.

Demonstration. Seeing there is the same Reason of A to B, as of E to F; and of B to C, as of F to G: there shall be also the same Reason of A to C, as of E to G (by the 22d) Moreover, seeing there is the same Reason of B to C as of D to E, and of F to G; there shall be (by the 11th.) the same Reason of D to F, as of E to G: Now as E is to

G, so is A to C, as we have already pro∣ved; thence as A is to C, so is D to F.