This follows, for that the Action of _Gravity_ being _continual_, in every Space of Time, the falling Body receives a new Impulse, equal to what it had before, in the same Space of Time, received from the same Power: For Instance, in the first Second of Time, the falling Body has acquired a _Velocity_, which in that time would carry it to a certain Distance, suppose 32 Foot, and were there no new Force, would descend at that rate with an _equable Motion_: But in the next Second of Time, the same Power of _Gravity_ continually acting thereon, superadds a new _Velocity_ equal to the former; so that at the end of two Seconds, the _Velocity_ is double to what it was at the end of the first, and after the same manner may it be proved to be triple, at the end of the third Second, and so on. Wherefore the _Velocities_ of _falling Bodies_, are proportionate to the Time of their _Falls_, _Q. E. D._ [Illustration: _Plate 4. pag. 310_] _Prop. II._ The _Spaces_ described by the Fall of a Body, are as the _Squares_ of the Times, from the beginning of the _Fall_. _Demonstration._ Let AB (_Fig. 9. Tab. 4._) represent the _Time_ of the _Fall_ of a _Body_, BC perpendicular to AB, the _Velocity_ acquired at the end of the _Fall_, and draw the Line AC; then divide the Line AB representing the Time, into as many equal Parts as you please, as b, b, b, b, _&c._ and through these Points draw the Lines bc, bc, bc, bc, _&c._ parallel to BC, 'tis manifest that the several Lines, bc, represent the several _Velocities_ of the falling Body, in such Parts of the _Time_ as Ab is of AB, by the former Proposition. It is evident likewise, that the _Area_ ABC is the Sum of all the Lines bc being taken, according to the Method of _Indivisibles_, infinitely many; so that the _Area_ ABC represents the Sum of all the _Velocities_, between none and BC supposed infinitely many; which Sum is as the Space descended in the Time represented by AB. And by the same Reason the _Areas_ Abc, will represent the Spaces descended in the Times Ab; so then the Spaces descended in the Times AB, Ab, are as the _Areas_ of the _Triangles_ ABC, Abc, which by the 20th of the 6 of _Euclid_, are as the _Squares_ of their _Homologous Sides_ AB, Ab, that is to say, of the _Times_: Wherefore the Descents of _falling Bodies_, are as the _Squares_ of the Times of their _Fall_, _Q. E. D._ _Prop. III._ The _Velocity_ which a _falling Body_ acquires in any Space of time, is double to that, wherewith it would have moved the Space, descended by an equable Motion, in the same _time_.