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.

THe Perpendicular Line drawn to the extream part of the Diameter is wholly without the Circle, and toucheth the same. All other Lines drawn between it, and the Circumference of the Circle cut it, and go within the same.

Draw by the Point A, which is the extremity of the Diameter AB, the Perpendicular AC: I say first, that all other Points of the same Line, as the Point C, are without the Circle. Draw the Line DC.

Demonstration. Because the Angle DAC of the Triangle DAC, is a Right Angle; DCA shall be acute, and (by the 19^th of the 1^st,) the Side DC shall be greater than the Side DA; thence the Line DC passeth beyond the Circumference of the Circle.

I farther add that the Line CA toucheth the Circle, because in the meeting thereof in the Point A, it cutteth it not, but hath all its other Points without the Circle.

I say also that there cannot be drawn any other Line from the Point A, under∣neath CA, which shall not cut the Circle; as for example, the Line EA. Draw the Perpendicular DI.

Demonstration. Seeing the Angle IAB is Acute, the Perpendicular drawn from the Point D shall be on the same Side with the Angle IAB (by the 17^th of the 1^st.) Let it be DI; the Angle DIA is Right, and the Angle IAD Acute; AD shall be greater than DI; thence the Line DI, doth not reach the Circumference, and the Point I is within the Circle.