[Draka translation]

1. [Enunciation] If the same moveable is carried from rest on an inclined plane, and also along a vertical of the same height, the times of the movements will be to one another as the lengths of the plane and of the vertical.
   
2. Let the inclined plane AC and the vertical AB each have the same altitude above the horizontal CB, that is, the line BA.
   
3. I say that the time of descent along plane AC has, to the time of fall of the same moveable along the vertical AB, the same ratio that the length of plane AC has to the length of vertical AB.
   
4. Assume any lines DG, EI, and FL parallel to the horizontal CB; it follows from our postulate that the degrees of speed acquired by the moveable from the first beginning of motion, A, to the points G and D, are equal, since their approaches to the horizontal are equal; likewise, the speeds at points I and E are the same, as are the speeds at L and F.
   
5. Now, if not only these, but parallels from all points of the line AB are supposed drawn as far as line AC, the momenta or degrees of speed at both ends of each parallel are always matched with each other.
   
6. Thus the two spaces AC and AB are traversed at the same degrees of speed.
   
7. But it has been shown that if two spaces are traversed by a moveable which is carried at the same degrees of speed, then whatever ratio those spaces have, the times of motion have the same [ratio]. (fn 30)
   
8. Therefore the time of motion through AC is to the time through AB as the length of plane AC is to the length of vertical AB; which was to be demonstrated.
   
9. Sagr. It appears to me that the same can be very clearly and briefly concluded, since it has already been shown that the overall [somma del] accelerated motion (fn 31) of passage through AC (and AB) is that of the equable motion whose degree of speed is one-half the maximum degree, [at] CB.
   
10. Therefore, the two spaces AC and AB being [considered as] passed with the same equable motion, it is manifest by Proposition I of Book I that the times of [these] passages will be as the spaces themselves.
    
11. [fn 30] The plural, “degrees of speed,” shows that reference is not directly to Prop. I on uniform motion; rather, this appears to be an extension of that proposition to instantaneous speeds by reasoning similar to the argument used in rejecting proportionality of speeds to distances in free fall (p. 203 and note 15, above).
    
12. Strictly speaking, the extension had not been “shown,” though it follows easily from such an argument and the general definition of equal speeds as those in which proportional distances are traversed in proportional times; cf. Dialogue, p. 24 (Opere, VII, 48).
    
13. [fn 31] The notion of a “total” or "overall" motion employed here in reference to Theorem I, above, had been preceded in Galileo’s thought by a notion of total or overall speed; see Dialogue, p. 229 (Opere, VII, 256).
    
14. Traces of that earlier concept survive in the scholium to Prop. XXIII, below; see also note 20 to the Added Day on percussion, below.