From this it is manifest that if there are any number of equal times taken successively from the first instant or beginning of motion, say AD, DE, EF, and FG, in which spaces HL, LM, MN, and NI are traversed, then these spaces will be to one another as are the odd numbers from unity, that is, as 1, 3, 5, 7; but this is the rule [ratio] for excesses of squares of lines equally exceeding one another [and] whose [common] excess is equal to the least of the same lines, or, let us say, of the squares successively from unity. Thus when the degrees of speed are increased in equal times according to the simple series of natural number, the spaces run through in the same times undergo increases according with the series of odd numbers from unity.

Sagr. Please suspend the reading for a bit, while I develop a fancy that has come to my mind about a certain conception. To explain this, and for my own as well as for your clearer understanding, I'll draw a little diagram. I imagine by this line AI the progress of time after the first instant at A; and going from A at any angle you wish, I draw the straight line AF. And joining points I and F, I divide the time AI at the middle in C, and I draw CB parallel to IF, taking CB to be the maximum degree of the speed which, commencing from rest at A, grows according to the increase of the parallels to BC extended in triangle ABC; which is the same as to increase [according] as the time increases.

I assume without argument, from the discussion up to this point, that the space by the moveable falling with its speed increased in the said way is equal to the space that would be passed by the same moevable if it were moved during the same time AC in uniform motion whose degree of speed was equal to EC, one-half of BC. I now go on to imagine the moveable [to have] descended with accelerated motion and to be found at instant C to have the degree of speed BC. It is manifest that if it continued to be moved with the same degree of speed BC, without acceleration further, then in the ensuing time CI it would pass a space double that which it passed in the equal time AC with degree of uniform speed EC, one-half the degree BC. (24) But since the moveable descends with speed always uniformly increased in all equal times, it will add to the degree CB, in the ensuing time CI, those same momenta of speed growing according to the parallels of triangle BFG, equal to triangle ABC; so that to the degree of speed GI there being added one-half the degree FG, the maximum of those [speeds] acquired in the accelerated motion governed by the parallels of triangle BFG, we shall have the degree of speed IN, with which it would be moved with uniform motion during time CI. That degree IN is triple the degree EC convinces [us] that the space passed in the second time CI must be triple that [which was] passed in the first time CA.

And if we assume added to AI a further equal part of time IO, and enlarge the triangle out to APO, then it is manifest that if the motion continued through the whole time IO with the degree of speed IF acquired in the accelerated motion during time AI, this degree IF being quadruple EC, the space passed in time IO would be quadruple that passed in the first equal time AC. Continuing the growth of uniform acceleration in triangle FPQ, similar to that of triangle ABC which, reduced to equable motion, adds the degree equal to EC, and adding QR equal to EC, we shall have the entire equable speed exercised over time IO quintuple the equable [speed] of the first time AC; and hence the space passed [will be] quintuple that [which was] passed in the first time AC.

Thus you see also, in this simple calculation that the spaces passed in equal times by a moveable which, parting from rest, acquires speed in agreemnet with the growth of time, are to one another as the odd numbers from unity, 1, 3, 5; and taking jointly the spaces passed, that which is passed in double the time is four times that passed in the half [i.e., in the given time], and that passed in triple the time is nine times [as great.] And in short, the spaces passed are in the duplicate ratio of the times; that is, are as the squares of those times.

Drake's footnote 24: This "double-distance" rule was in fact not found by Galileo until after his odd-number rule, so that here the order of presentation follows his order of discovery. Cf. scholium to Prop. XXIII, below, and Prop. XXV.

Simp. Really I have taken more pleasure from this simple and clear reasoning of Sagredo's than from the (for me) more obscure demonstration of the Author, so that I am better able to see why the matter must proceed in this way, once the definiton of uniformly accelerated motion has been postulated and accepted. But I am still doubtful whether this is the acceleration employed by nature in the motion of her falling heavy bodies. Hence, for my understanding and for that of other people like me, I think that it would be suitable at this place [for you] to adduce some experiment from those (of which you have said that there are many) that agree in various cases with the demonstrated consclusions.

Salv. Like a true scientist, you make a very reasonable demand, for this is usual and necessary in those sciences which apply mathematical demonstrations to physical conclusions, as may be seen among writers on optics, astronomers, mechanics, and others who confirm their principle with sensory experiences, those being foundations of all the resulting structure. I do not want to have it appear a waste of time [superfluo] on our part, [as] if we had reasoned at excessive length about this first and chief foundation upon which rests an immense framework of infinitely many conclusions – of which we have only a tiny part put down in this book by the Author, who will have gone far to open the entrance and portal that has until now been closed to speculateve minds. Therefore as to the experiments: the Author has not failed to make them, and in order to be assured that the acceleration of heavy bodies falling naturally does follow the ratio expounded above, I have often made the test [prova] in the following manner, and in his company.

In a wooden beam or rafter about twelve braccia long, half a braccio wide, and three inches thick, a channel was rabbeted in along the narrowest dimension, a little over an inch wide and made very straight; so that this would be clean and smooth, there was glued within it a piece of vellum, as much smoothed and cleaned as possible. In this there was made to descend a very hard bronze ball, well rounded and polished, the beam having been tilted by elevating one end of it above the horizontal plane from one to two braccia, at will. As I said, the ball was allowed to descend along [per] the said groove, and we noted (in the manner I shall presently tell you) the time that it consumed in running all the way, repeating the same process many times, in order to be quite sure as to the amount of time, in which we never found a difference of even the tenth part of a pulse-beat. (25)

Drake's footnote 25: Actual results obtained by procedures similar to Galileo's vindicate his claim as to their reliability. His manuscript records of another type of inclined plane experiment show him to have obtained results within one percent of modern theoretical values.

This operation being precisely established, we made the same ball descend only one-quarter the length of this channel, and the time of its descent being measured, this was found always to be precisely one-half the other. Next making the experiment for other lengths, examining now the time for the whole length [in comparison] with the time of one-half, or with that of two-thirds, or of three-quarters, and finally with any other division, by experiments repeated a full hundred times, the spaces were always found to be to one another as the squares of the times. And this [held] for all inclinations of the plane; that is, of the channel in which the ball was made to descend, where we observed also that the times of descend for diverse inclinations maintained among themselves accurately that ratio that we shall find later assigned and demonstrated by our Author.

As to the measure of time, we had a large pail filled with water and fastened from above, which had a slender tube affixed to its bottom, through which a narrow thread of water ran; this was received in a little beaker during the entire time that the ball descended along the channel or parts of it. The little amounts of water collected in this way were weighed from time to time on a delicate balance, the differences and ratios of the weights giving us the differences and ratios of the times, and with such precision that, as I have said, these operations repeated time and again never differed by any notable amount.

Simp. It would have given me great satisfaction to have been present at these experiments. But being certain of your diligence in making them and your fidelity in relating them, I am content to assume them as most certain and true.

Salv. Then we may resume our reading, and proceed.