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 II. THEOREM.

A Streight Line drawn from one point of the Circumference of a Circle to another, shall fall within the same.

Let there be drawn from the Point B, in the Circumference, a Line to the Point C. I say that it shall fall whol∣ly within the Circle. To prove that it cannot fall without the Circle, as BVC; having found the Center there∣of, which is A, draw the Lines AB, AC, AV.

Demonstration. The Sides AB, AC, of the Triangle ABC, are equal: whence (by the 5^th. of the 1^st.) the An∣gles ABC, ACB, are equal. And seeing the Angle AVC, is exteriour in respect of the Triangle AVB, it is greater than ABC, (by the 16^th. of the 1^st.) it shall be also greater than the Angle ACB. Thence (by the 19^th of the 1^st.) in the Triangle ACV, the side AC, opposite to the greatest An∣gle AVC, is greater than AV; and by consequence, AV cannot reach the circumference of the Circle, seeing it

is shorter than AC, which doth but reach the same; wherefore the Point V is within the Circle: the same may be proved of any Point in the Line AB; and therefore the whole Line AB falls within the Circle.

USE.

IT is on this Proposition that are ground∣ed those which demonstrate that a Circle toucheth a streight Line but only in one Point: for if the Line should touch two Points of its Circumference, it would be then drawn from one Point of its Cir∣cumference to the other; and consequent∣ly, would fall within the Circle, accord∣ing to this Proposition; although its De∣finition saith, that it cutteth not its Cir∣cumference. Theodosius demonstrateth after the same manner, That a Globe can∣not touch a Plain but in one single Point; otherwise the Plain would enter into the Globe.