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 IV. THEOREM.

TWo Lines drawn in a Circle, cut not each other equally, if the Point of their Intersection be not in the Center.

If the Lines AC, BD, cut each other in the Point I, which is not the Cen∣ter of the Circle, they will not cut each other equally. In the first place, if one of them should pass through the Center, it is evident, that the same cannot be divided into two equal Parts, but by the Center. If neither of them pass through the Center, as BD, EI, draw the Line AIC, which passeth through the Center.

Demonstration. If the Line AC di∣vide the Line BD into two equal Parts in I, the Angles AID, AIB, would be equal (by the 2^d.) In like manner, if the Line EG was divided into two equal Parts, in the Point I, the Angle AIG would be right: thence the Angles AIB, AIG, would be right, and con∣sequently equal, which cannot be, the one being part of the other. Lastly, the Line AIC, which passeth through the Centers, would then be perpendi∣cular to the Lines BD, GI, if both of them were divided equally in the Point I.

USE.

BOth of those Propositions are of use in Trigonometry: hereby is demonstra∣ted, that the half Chord of an Arch is per∣pendicular to the Semidiameter; and by consequence, it is the sine of the half Arch: by which also it is demonstrable, that the Side of a Triangle hath the same Propor∣tion as the sines of their opposite Angles. We furthermore make use of this Proposi∣tion to find the Excentricity of the Circle which the Sun describeth in a Year.