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 XVI. THEOREM.

IF four Lines be proportional, the Rect∣angle comprehended under the first and fourth, is equal to the Rectangle compre∣hended under the second and third, and if the Rectangle comprehended under the ex∣trems, be equal to the Rectangle comprehen∣ded under the means, then are the four Lines proportional.

if the Lines AB, CD, be proportional, that is to say, that there is the same Ratio of A to B, as of C to D; the Rectangle comprehended under the first A, and the fourth D, shall be equal to the Rectangle comprended under B and C.

Demonstration. The Rectangles have their Angles equal, seeing it is Right; they have also their Sides reciprocal; they are thence equal (by the 16th.)

In like manner, if they be equal, their sides are reciprocal; that is to say, that there is the same Ratio of A to B, as of C to D.