Chap. 5. PHILOSOPHICAL DISCOVERIES. 197

of force to overcome those resistances. All motion, there-fore, must be abated and gradually languish in our mechanicalengines, unless they be supplied by new and repeated insty-cnces of the power.

15. It is very well known, that, when allowance is madefor die defect of elasticity in bodies, for attrition, and the re-sidence of the medium, these conclusions are perfectly agree-able to experience; and therefore serve to confirm the ge-neral laws of motion with their corollaries, and our methodsof reasoning from them.

C H A P. V.

Of the motion of projeEliles in vacuo ; of the cycloid , and themotion of a pendulum in it *.

LEMMA I.

Suppose the motion of a body to be uniformly accelerated ;let the time be represented by the right sine a m, ( Plate IV.Fig. I.) and any part of it by a k, draw m n, k l perpen-diculars to a m in m and k, and a n intersecting them in n andl : then the velocities acquired in the times a m, a k, reckonedfrom the beginning of the motion, will be as the perpendicu-lars m n, k l, but the spaces described in these times will be asthe areas amn, akl.

* To render the second book more complete, we have added this supplementsfrom two pieces which the author used to give his scholars. The substance o f themis taken from the learned Mr . Cotes's tracts, primed at the end of his HarmoniaMensurarum.

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