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.
Demonstration. Draw the Line EC parallel to AB, and AE parallel to BC in the same Fig. 9. and compleat the Parallelogram ABCE, it is evident that the Area thereof may represent the Space, a Body moved equably with the Velocity BC would describe in the Time AB, and the Triangle ABC represents the Space describ'd by the Fall of a Body , in the same Time AB, by the second Proposition. Now the Triangle ABC is half of the Parallelogram ABCE, and consequently the Space described by the Fall , is half what would have been described by an equable Motion with