The Quadrature of the Circle, the Square Root of Two, and the Right-Angled Triangle

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1874 edition. Excerpt: ...AF, or of the angle A CF. Every sine is half the chord of double the arc. Thus the sine FG is the half of FH, which is the chord of the arc FAB, double of FA. The chord which subtends the sixth part of the circumference, or the chord of 60, is equal to the radius (Loomis' Geom., Prop. IV., Book VI.); hence the sine of 30 is equal to half of the radius. The versed sine of an arc is that part of the diameter intercepted between the sine and the arc. Thus GA is the versed sine of the arc AF. The tangent of an arc is the line which touches it at one extremity, and is terminated by a line drawn from the center through the other extremity. Thus AI is the tangent of the arc AF, or the angle ACF. The secant of an arc is the line drawn from the center of the circle through one extremity of the arc, and is limited by the tangent drawn through the other extremity. Thus CI is the secant of the arc AF, or of the angle ACF. The cosine of an arc is the sine of the complement of that arc. Thus the arc DF, being the complement of AF, FK is the sine of the arc DF, or the cosine of the arc AF. The cotangent of an arc is the tangent of the complement of that arc. Thus, DL is the tangent of the arc DF, or the cotangent of the arc AF. The cosecant of an arc is the secant of the complement of that arc. Thus CL is the secant of the arc DF, or the cosecant of the arc AF. In general, if we represent any angle by A3 Cos. A = sine (90--L). Cot. A = tang. (90--A). Cosee. A = see. (90--). Since, in a right-angled triangle either of the acute angles is the complement of the other, the sine, tangent, and secant of one of these angles is the cosine, cotangent, and cosecant of the other. The sine, tangent, and secant of an arc are equal to the sine, tangent, and secant of...