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 V. THEOREM.

IN every Isosceles Triangle, the Angles which are above the Base are equal, as also those which are underneath.

Let ABC be an Isosceles Triangle, viz, that the side AB, AC, be equal, I say that the Angle ABC, ACB, are equal, as also the Angles GBC, HCB, which ly under the Base BC: Let there be imagined another Triangle DEF, having an Angle D equal to the Angle A; and the Sides DE, DF, equal to AB, AC: Now since the Sides AB, AC, are equal, the Four Lines AB, AC, DE, DF, shall be equal.

Demonstration, because the Sides AB, DE, AC, DF, are equal, as

also the Angles A and D, if we put the Triangle DEF, on the Triangle ABC, the Line DE shall fall upon AB and DF, on AC and FE, on BC; (by the Fourth) therefore the Angle DEF shall be equal to ABC, and since one part of the Line DE falls on AB, the whole Line DI shall fall on AG; otherwise Two streight Lines would contain a space, therefore the Angle IEF shall be equal to the Angle GBC: Now if you should turn the Triangle DEF, and make the Line DF to fall on AB, and DE, on AC; because the Four Lines AB, DF, AC, DE, are equal, as also the Angles A and D; the Two Triangles ABC, DEF, shall ly exactly on each other, and the Angles ACB, DEF, HCB, IEF, shall be equal: Now according to our first Comparison, the Angle ABC was equal to DEF, and GBC to IEF; therefore the Angles ABC, ACB, which are equal to the same DEF, and GBC, HCB, which are also equal to the same IEF, are also equal among themselves.

I thought fit not to make use of Euclid's Demonstration, because it being too difficult for young Learners to Apprehend, they are often discouraged to proceed.