Chap. 2. PHILOSOPHY. 89

serence just in the fame time, as would be imployed by thebody in falling perpendicularly down through the diameterC A. But the time in which the body will descend throughthe arch, is different from the time, which it would take upin falling through the line A B.

60 . It has been thought by some, that because in verysmall arches this correspondent straight line differs but littlefrom the arch itself; therefore the descent through thisstraight line would be performed in such small arches nearlyin the fame time as through die arches themselves: so thatif a pendulum were to swing in small arches, half the timeof a single swing would be nearly equal to the time, in whicha body would fall perpendicularly through twice the lengthof the pendulum. That is, the whole time of the swing, ac-cording to this opinion, will be four fold die time requiredfor the body to fall through half the length of the pendu-lum ; because the time of the body’s falling down twice thelength of the pendulum is half the time required for the fallthrough one quarter of this space, that is through half thependulum’s length. However there is here a mistake ; forthe whole time of the swing, when the pendulum movesthrough small arches, bears to the time required for a bodyto fall down through half the length of the pendulum verynearly the lame proportion, as the circumference of a circlebears to its diameter ; that is very nearly the proportion ofz yy to 11Z, or little more than the proportion of 3 to 1.If the pendulum takes so great a swing, as to pass over an archequal to one sixth part of the whole circumference of the

N circle,