Proposition 2, Theorem 2

If a moveable descends from rest in uniformly accelerated motion, the spaces run through in any times whatever are to each other as the duplicate ratio of their times; that is, are as the squares of those times.

1. Let the flow of time from some first instant A be represented by the line AB, in which let there be taken any two times, AD and AE.
   
2. Let HI be the line in which the uniformly accelerated moveable descends from point H as the first beginning of motion, let space HL be run through in the first time AD and HM be the space throught which it descends in time AE.
   
3. I say that space MH is to space HL is to space HL in the duplicate ratio of time EA to time AD.
   
4. Or let us say that spaces MH and HL have the same ratio as do the squares of EA/ and AD.
   
5. Draw line AC at any angle with AB.
   
6. From points D and E draw the parellels DO and LP of which DO will represent the maximum degree of speed acquired at instant D of time -AD/, and PE the maximum degree of speed acquired at instant E of time AE.
   
7. Since it was demonstrated above that as to spaces run through those are equal to one another of which one is traversed by a moveable in uniformly accelerated motion from rest, and the other is traversed in the same time by a moveable carried in equable notion whose speed is one-half the maximum acquired in the accelerated motion, it follows that spaces MH and LH are the same that would be traversed in times EA and DA in equable motions whose speeds are as the halves of PE and OD. Therefore if it is shown that these spaces MH and LH are in the duplicate ratio of the times EA and DA, what is intended will be proved.
   

Now in Proposition IV of Book I ["On Uniform Motion," above] it was demonstrated that the spaces run through by moveables carried in equable motion have to one another the rotio compounded from the ratio of speeds and from the ratio of times. Here, indeed, the ratio of speeds is the same as the ratio of times, since the ratio of one-half PE, to one-half OD, or of PE to OD, is that of AE, to AD. Hence the ratio of spaces run through is the duplicate ratio of the times; which was to be demonstrated.

It also follows from this that this same ratio of spaces is the duplicate ratio of the maximum degrees of speed, that is, of lines PE and OD, since PE is to OD as EA is to DA.