Sir Isaac Newton’s Book I.

88

when it swings through the greater arch C D, shall take up inits swing from C to D a longer time than in swinging from Eto F, when it moves only in that lesser arch ; or the time inwhich the body let fall from C will descend through the archC A is greater than the time, in which it will descend throughthe arch E A, when let fall from E. But the first os thesetimes will not hold the same proportion to the latter, as thefirst arch C A bears to the other arch E A; which will appearthus. Let C G and E H be two horizontal lines. It has beenremarked above a , that the body in falling through the archC A will acquire as great a velocity at the point A, as it wouldhave gained by falling directly down through G A ; and infalling through the arch E A it will acquire in the point A on-ly that velocity, which it would have got in falling throughH A. Therefore, when the body descends through the great-er arch C A, it shall gain a greater velocity, than when it pas-ses only through the lesser; so that this greater velocity will insome degree compensate the greater length of the arch.

yq. The increase of velocity, which the body acquiresin falling from a greater height, has such an effect, that, ifstraight lines be drawn from A to C and E, the body wouldfall through the longer straight line C A just in the same time,as through the shorter straight line E A. This is demonstrat-ed by the geometers, who prove, that if any circle, as ABCD(fig. 49.) be placed in a perpendicular situation; a bodyshall sail obliquely through every line, as A B drawn from thelowest point A in the circle to any other point in the circum-

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