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 XIX. THEOREM.

THe Perpendicular drawn to a touch Line at the Point of touching Passeth through the Center.

Let the Line AB touch the Circle in the Point D; and let the Line DC be Perpendicular to AB: I say that DC passeth through the Center. For if it passeth not through it, drawing from the Center to the Point D a Line; it would be Perpendicular to AB: and so Two Lines Perpendicular to the same, would be drawn to the same Point D; which cannot be.

USE.

THe use of touch Lines is very com∣mon in Trigonometry, which hath obliged us to make a Table thereof, it serveth us to measure all sorts of Tri∣angles, even Spherical. We have in Opticks some Propositions founded on those

••ouch Lines, as when we are to determine the part of a Globe which is enlightned. The Theorie of the Phases of the Moon is esta∣blished on the same Doctrine, as also that Celebrated Problem by which Hipparchus findeth the distance of the Sun, by the diffe∣rence of the true and apparent Quadra∣tures. In the Gnomonicks, the Italian and Babylonian hours, are often Tangent Lines. We measure the Earth by a Line which toucheth its Surface. And we take in the Art of Navigation, a touch Line to be our Horizon.