What we have demonstrated for movements run through along verticals is to be understood also to apply to planes, however inclined; for these, it is indeed assumed that the degree of increased speed [accelerations] grows in the same ratio; that is, according to the increase of time, or let us say according to the series of natural numbers from unity. (26)

Salv. Here, Sagredo, I want permission to defer the present reading for a time, though perhaps I shall bore Simplicio, in order that I may explain further what has been said and proved up to this point.

At the same time it occurs to me that, by telling you of some mechanical conclusions reached long ago by our Academician, I can add new confirmation of the truth of that principle which has already been examined by us with probable reasonings and by experiments.

More important, this will be geometrically proved after the prior demonstration of a single lemma that is elementary in the study of impetuses.

Sagr. When you promise such gains, there is no amount of time I should not willingly spend in trying to confirm and completely establish these sciences of motion.

For my part, I not only grant permission to you to satisfy us on this matter, but I even beg you to allay as swiftly as possible the curiosity you have aroused in me.

I think Simplicio feels the same way about this.

Drake footnote 26: The ensuing section was added later (note 19, above) without disturbing the original text order. Dictated by the blind Galileo about October 1638 and revised in November 1639, it was put into dialogue form by Viviani end inserted in the 1655 edition. It was placed at this point rather than with the earlier statement of the postulate because it requires prior demonstration of Prop. II. in which the postulate was not used.

Simp. How can I say otherwise?

Salv. Then, since you give me leave, consider it in the first place as a well-known effect that the momenta or speeds of the same moveable are different on diverse inclined planes, and that the greatest [speed] is along the vertical.

The speed dimishes along other inclines according as they depart more from the vertical and are more obliquely tilted.

Whence the impetus, power [/talento/], energy, or let us say momentum of descent, comes to be reduced in the underlying plane on which the moveable is supported and descends.

The better to explain this, let the line AB be assumed to be erected vertically on the horizontal AC, and then let it be tilted at different inclinations with respect to the horizontal, as at AD, AE, AF, etc.

I say that the impetus of the heavy body for descending is maximal and total along the vertical BA, is less than that along DA, still less along EA, successively diminishes along the more inclined FA, and is finally completely extinguished on the horizontal CA, where the moveable is found to be indifferent to motion and to rest, and has in itself no inclination to move in any direction, nor yet any resistance to being moved.

This it is impossible that a heavy body (or combination thereof) should naturally move upward, departing from the common center toward which all heavy bodies mutually converge [/conspirano/]; and hence it is impossible that these be moved spontaneously except with that motion by which their own center of gravity approaches the said common center. (27)

Whence, on the horizontal, which here means a surface [everywhere] equidistant from the said [common] center, and therefore quite devoid of tilt, the impetus or momentum of the moveable will be null.

This change of impetus assumed, I must next explain something that our Academician, in an old treatise on mechanics written at Padua for the use of his pupils, (28) demonstrated at length and conclusively in connection with his treatment of the origin and character of that marvelous instrument, the screw; namely, the ratio in which this change of impetus along planes of different inclinations takes place.

Given the inclined plane AF, for example, and taking as its elevation above the horizontal the line FC, along which the impetus of a heavy body and its momentum in descent is maximum, we seek the ratio that this momentum has to the momentum of the of the same moveable along the incline FA, which ratio, I say, is inverse to that of the said lengths.

This is the lemma to be put before the theorem that I hope then to be able to demonstrate.

Drake footnote 27. This conception became a fundamental principle in Toricelli's continuation of Galileo's work; cf. E. Toricelli, Opere (Faenza, 1919), II. 105 [??] Comparison with Galileo's dictated text (note 26, above) suggests that this sentence was interpolated by Viviani when he put the argument in dialogue form.

Drake footnote 28. Galileo's treatise On Mechanics was first published in a French translation by Marin Mersenne (1588-1648) in 1634. The original Italian, of which three manuscript forms exist (1593, 1594, and ca. 1600), was posthumously published in 1649.

It is manifest that the impetus of descent of a heavy body is as great as the minimum resistance or force that suffices to fix it and hold it [at rest].

I shall use the heaviness of another moveable for that force and resistance, and [as] a measure thereof.

Let the moveable G, then, be placed on plane FA, tied with a thread which rides over F and is attached to the weight H; and let us consider that the space of the vertical descent or rise of these [H] is always equal to the whole rise or descent of the other moveable, G, along the incline AF – not just to the vertical rise or fall, through which the moveable G (like any other moveable) exclusively exercises its resistance.

That much is evident.

For consider the motion of the moveable G in the triangle AFC (for example, upward from A to F) as composed of the horizontal transversal AC and the vertical CF.

As before, there is no resistance to its being moved along the horizontal, since by beans of such a motion no loss or gain whatever is made with regard to its distance from the common center of heavy things, that being conserved always the same on the horizontal [as defined above].

It follows that the resistance is only with respect to compulsion to go up the vertical CF.

Hence the heavy body G, moving from A to F, resists in rising only the vertical space CF; but that other heavy body H must descend vertically as much as the whole space FA.

And this ratio of ascent and descent remains always the same, being as little or as great as the motion of the said moveables by reason of their connection together.

Thus we may assert and affirm that when equilibrium (that is, rest) is to prevail between two moveables, their [overall] speeds or their proponsions to motion – that is, the spaces they would pass in the same time – must be inverse to their weights [/gravita/], exactly as is demonstrated in all cases of mechanical movements.

Thus, in order to hinder the descent of G, it will suffice that H be as much lighter that G as the space CF is proportionately less than the space FA.

Hence if the heavy body G is, to the heavy body H, as FA is to FC, equilibrium will follow; that is, the heavy bodies H and G will be of equal moments, and the motion of these moveables will cease.

Now, we have agreed that the impetus, energy, momentum, or propensity to motion of a moveable is as much as the minimum force or resistance that suffices to stop it; and it has been concluded that the heavy body H suffices to prohibit motion to the heavy body G; hence the lesser weight H, which exercises its total [static] moment in the vertical FC, will be the precise measure of the partial moment that the greater weight G exercises along the inclined plane FA.

But the measure of the total moment of heavy body G is G itself, since to hinder the vertical descent of a heavy body, there is required the opposition of one equally heavy when both are free to move vertically.

Therefore the partial impetus or momentum of G along the incline FA will be, to the maximum and total impetus of G along the vertical FC, as the weight H is to the weight G, which is (by construction) as the vertical FC (the height of the incline) is to the incline FA itself.

This is what was proposed to be demonstrated as the lemma; and as we shall see, it is assumed by our Author as known in the second part of Proposition VI of the present treatise.

Sagr. İt seems to me that from what you have concluded thus far, it can be easily deduced, arguing by perturbed equidistance of ratios, that the momenta of the same moveable along differently inclined planes having the same height, such as FA and FI, are in the inverse ratio of those same planes.

Salv. A true conclusion. This established, I go on next to demonstrate the theorem itself; that is: