It is deduced, second, that if at the beginning of motion there are taken any two spaces whatever, run through in any [two] times, the times will be to each other as either of these two spaces is to the mean proportional space between the two given spaces.

From the beginning of motion, S, take two spaces, ST and SV of which the mean proportional shall be SX, the time of fall through ST will be to the time of fall through SV as ST is to SX or let us say that the time through SV is to the time through ST as VS is to SX. Since it has been demonstrated that the spaces run through are in the duplicate ratio of the times (or what is the same thing, are as the squares of the times), the ratio of space VS to space ST is the doubled ratio of VS to SX, or is the same as that of the squares of VS and SX. It follows that the ratio of times of motion through SV and ST are as the spaces, or the lines, VS and SX.