The degrees of speed acquired by a moveable in descent with natural motion from the same height, along planes inclined in any way whatever, are equal upon their arrival at the horizontal, all impediments being removed.

1. Here you must first note that it has already been established that along any inclinations, the moveable upon its departure from rest increases its speed, or amount of impetus, in proportion to the time, in accordance with the definition given by the Author for naturally accelerated motion.
   
2. Whence, as he has demonstrated in the last preceding proposition, the spaces passed are in the squared ratio of the times, and consequently of the degrees of speed.
   
3. Whatever the [ratio of] impetuses at the beginning [nella prima mossa], that proportionality will hold for the degrees of the speeds gained during the same time, since both [impetuses and speeds] increases in the same ratio during the same time.
   
4. Now let the height of the inclined plane AB above the horizontal be the vertical AC, the horizontal being CB.
   
5. Since, as we concluded earlier, the impetus of a moveable along the vertical AC is, to its impetus along the incline AB, as AB is to AC, [then] in the incline AB take AD as the third proportional of AB and AC; the impetus [to move] along AC is, to the impetus [to move] along AB (that is, [to move] along AD), as [AB is to AC or as] AC is to AD.
   
6. Hence the moveable, in the same time that it passes the vertical space AC, would also pass the space AD along the incline AB (the momenta being as the spaces); and the degree of speed at C will have to the degree of speed at D the same ratio that AC has to AD.
   
7. But the speed at B is to the speed at D as the time through AB is to the time through AD, by our definition of accelerated motion; and the time through AB is to the time through AD as AC (the mean proportional between BA and AD) is to AD, by the last corollary to Proposition II.
   
8. Therefore the speeds at B and C [both] have to the speed at D the same ratio that AC has to AD, and hence [the speeds at B and C] are equal; which is the theorem intended to be demonstrated.
   
9. From this we may more conclusively prove the Author's ensuing Proposition III, in which he makes use of the [earlier] postulate; this [theorem] states that the time along the incline has to the time along the vertical the same ratio that the incline has to the vertical.
   
10. So let is say: If BA is the time along AB, (29) the time along AD will be the mean proportional between these [AB and AD], that is, AC, by the second corollary to Proposition II.
    
11. But if AC is the time along AD, it will also be the time along AC, since AD and AC are run through in equal times.
    
12. And since if BA is the time along AB, AC will be the time along AC, then it follows that as AB is to AC, so is the time along AB to the time along AC.
    
13. By the same reasoning it will be proved that the time along AC is to the time along some other incline, AE, as AC to AE; therefore, by equidistance of ratios, the time along incline AB is to the time along incline AE homologously as AB is to AE, etc.
    
14. As Sagredo will readily see [later], the Author's Proposition VI could be immediately proved from the same application of this theorem.
    
15. But enough for now of this digression, which has perhaps turned out to be too tedious, though it is certainly profitable in these matters of motion.
    
16. Sagr. And not only greatly to my taste, but most essential to a complete understanding of that principle.
    
17. Salv. Then I shall resume the reading of the text.