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 XXIV. THEOREM.

IF there be the same Reason of the first Magnitude to the Second, as of the Third to the Fourth; and the same of the Fifth to the Second, as of the Sixth to the Fourth: there will be the same Reason of the First with the Fifth to the Second, as of the Third with the Sixth to the Fourth.

E,		F,
4.		6.
6.	2.	9.	3.
A,	B,	C,	D.

If there be the same Reason of A to B as of C to D, of E to B as of F to D; there shall be the same Rea∣son of AE to B, as of CF to D.

Demonstration. Seeing there is the same Reason of A to B as of C to D, A will contain any Aliquot part of B whatever, as many times as C contains a like Aliquot part of D (by the 5th. Def.) In like manner E will contain the same Aliquot part of B, as many times as F will contain a like Aliquout part of D: So then AE will contain any Aliquot part of B whatever, as many times as C and F will contain

a like Aliquot part of D. There shall thence be the same Reason of AE to B, as of CF to D.