Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

Demonstration.

Though we have demonstrated this Truth more than once in the foregoing Proposition; yet here we will confirm it again as follows. Having described on each side of the Square BE a Se∣micircle, which will all necessarily touch one another in one point, and be equal to the Semicircle, BAC, if you conceive as many Triangles inscribed also equal to BAC; it will be evident that the Square BE will contain the said 4 Triangles; and besides the little Square FGHI, whose side FI, v. g. is the difference be∣tween the greater side of the Triangle CI, and the less CF, (for because the less side CF=BA, lying in the first Semicircle, if it be continued to I in the se∣cond Semicircle makes CI=CA the greater side of the other Triangle, and so in the others. From thence it is evident, That as the Angles ABC, and ACB together make one right one; so likewise BCF (= CBA) and ECF make also one right one; and consequently ECF is =ACB, and the Arch and the Line EI= to the Arch and the Line AB, &c.) Where∣fore, if the greatest side of the given Triangle BC or BD, &c. be called a, and AC, b and the least AC, or CF, &c. be called c; the 〈◊〉〈◊〉 of the side BC, will be =aa, and the Area of each Triangle ½ bc: and so the 4 Triangles together 2 bc: but the side of the middle little Square will be b−c, and its Square bb+cc=2bc: Wherefore if you add to this the 4 Trian∣gles▪

The Sum of all, i. e. the whole Square BE will be bb+cc=aa. Q. E. D.