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 XIX. THEOREM.

IF the Wholes are in the same reason as the Parts taken away, those which re∣main shall be in the same reason.

A,	C,
12.	6.
B,	D,
4.	2.

If there be the same reason of the Magnitude AB to the Magnitude CD, as of the part B to the part D; I de∣monstrate, that there shall be the same reason of A to C as of AB to CD.

Demonstration. It is supposed that there is the same reason of AB to CD, as of B to D: thence, by Exchange (ac∣cording to the 16th.) there shall be the same reason of AB to B, as of CD to D; and by Conversion of reason, there shall be the same reason of AB to A, as of CD to C: and again, by Exchange there shall be the same reason of AB to CD, as of A to C.

USE.

IN the Rule of Fellowship this Proposi∣tion is sometimes made use of; for they do not make use of the Rule of Three for every Associate, but do content themselves for to give to the last the Remainder of the Gain; supposing, that if there be the same Reason of the whole Sum of the Stock to the whole profit, as of the Stock of one As∣sociate, to his part of the Gain; there will be the same reason of the Stock remaining, to the Remainder of the Gain.

The 20th. and 21th. Propositions are not necessary.