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Body, ler fall from any gVen Height, down tö the Center, soppofing theGravitation to Increasc, as in the Fifth Property ; hüt considering the Imall-ness of Height, to which any Projett can bc niade Afcencl, and over howlittle an Arch of the Globe it can be Cast, by any of our Engincs, we maywell cnough luppole the Gravity Equal throughout, and the Dcscents of Pro-jetts in Parallel Lines , which, in Truth, are towards the Center, the Diste-rence being so finali as by no means to be disoovered in Practice.

Prop. I.] The VeloCitics of Falling Bodies, are Pros ortionatc to the Times fron: Propestiotts con-

, ,. . ,. , -

This tollows, tor that the stRion or Gravity being contmual, in every space Bodies, ant theof Time, the Falling Body reeeives a nevv Impulso, Equal to what it had be- Mo, ‘ on °f P '°~fore, in the fame Space of Time, reeeived from the säme Power For i n-iwu. x.stance, in the First Second of Time, the Falling Body has acquired a Vek-city, which in that Time would carry it to a ccrtain Distance, sopposo z r.

Foot, and were there no new Force, would Descend at that rate with anEquable Motion ; but in the next Second of Time, the tarne Power of Gra-vity continually Acting thereon, superaddsa New Velocity equal to the former ;so that at the end of two Seconds, the Velocity is double to what it was at theend of the First ; and aster the tarne mariner may it be proved to be Triple,at the end of the Tbird Second, and so on. Wherefore the Velocities of Fal-ling Bodies, are Proportionale to the Times of their Falls. E. D.

Prop. II ] The Spaces deferibed by the Fall of a Body , are as the Squarcs of theTimes from the beginning of the Fad.

Demonstration. ] Let A B represont the Time of the Fall of a Body, B C,Perpendicular to A B, the Velocity acquired at the End of the Fall, and 171.draw the Line A C ; then Divide the Line A B, representing the Time, intoas many equal Parts as you pleasc, as b, b, b, b, Öc. and through these Pointsdraw the Lines, b c, b c, b c, b c, &c. Parallel to BC; ’tis manifest th^tthesoveral Lines, b c, represont the several Velocities of the Falling Body, in suchparts of the Time, as A b, is of A B, by the Former Proposition. Itis evidentlikewise, that the Area, ABC, is the Sum of all the Lines b c. being taken,according to the Method of Indivißbles infinitely many j so that the AreaABC, represents the Sum of all the Velocities between none and B C, lüpposedinfinitely many ; which Sum is the Space Delcdnded In the Time representedby A B. And by the tarne reason, the Areas A b c, will represont the SpacesDeseended in the Times A b ; so then the Spaces Deseended in the TimesAB, Ab, are as the Areas of the Triangles, ABC, Abc, which by the'aoth of the 6th of Euclid are as the S quare s of their Homologous Sides A B,

Ab, that is to säy, of the Times wherefore the Defcents of Falling Bodies, ‘

are as the Squares of the Times of their Fad. Q. E. D.

Prop. III.] The Velocity which a Falling Body acquires in any Space of Time, isdouble to that, wherewith it would have mcved the Space Deseended by an EquableMotion, in the fame Time .

Cccc i Demonßration.

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