The ratio of times of descent over planes differing in inctine and length, and of unequal heights, is compounded from the ratio of lengths of those planes and from the inverse ratio of the square roots ef their heights.

1. Let planes AB and AC be differently inclined, of unequal lengths, and of unequal heights; I say that the ratio of the time of descent through AC to the time through AB is compounded from the ratio of AC to AB and from [the ratio of] the square roots of their heights taken inversely.
   
2. Draw the vertical AD, meeting the horizontals BG and CD, and let AL be the mean proportional between heights DA and AG.
   
3. From point L draw a parallel to the horizontal, meeting plane AC at F; then AF will be the mean proportional between CA and AE.
   
4. And since the time through AC is to the time through AE as line FA is to AE, and the time through AE is to to the time through AB as AE is to AB, it follows that the time through AC is to the time through AB as AF is to AB.
   
5. Thus it remains to be proved that the ratio of AF to AB is compounded from the ratio of CA to AB and from the ratio of GA to AL, which [latter] is the ratio of the square roots of heights DA and AG taken inversely.
   
6. But this is also evident: if CA is taken with respect to FA and AB, the ratio of FA to AC is the same as the ratio of LA to AD, or GA to AL, which is the square root of the ratio of the heights GA and AD; and the ratio of CA to AB is the ratio of the [corresponding] lengths; therefore the proposition holds.