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 XXXI. THEOREM.

IF several Magnitudes are in a greater Reason than a like number of other Magnitudes Ranked in the same order, the first of the first Rank shall have a greater Reason to the last, than the first of the se∣cond Rank to the last.

16.	10.	3.	9.	6.	2.
A,	B,	C,	D,	E,	F.

If there be a greater Reason of A to B, than of D to E; and if B have a greater Rea∣son to C, than E to F; there shall be a greater Reason of A to C, than of D to F.

Demonstration. Seeing there is a grea∣ter Reason of A to B than of D to E: there shall be also a greater Reason of A to D, than of B to E. And because there is a greater Reason of B to C, than of E to F; there shall be also a greater Reason of B to E, than of C to F. Thence there shall be a greater Reason of A to D, than of C to F; and by ex∣change (by the 27th.) there will be a grea∣ter Reason of A to C, than of D to F.