SIMP. In truth, I find more pleasure in this simple and clear argument of Sagredo than in the Author’s demonstration, which to me appears rather obscure; thus, I am convinced that matters are as described, once having accepted the definition of uniformly accelerated motion. But as to whether this acceleration is that which nature employs in the case of falling bodies, I am still doubtful. So it seems to me, not only for my own sake but also for all those who think as I do, that this would be the proper moment to introduce one of those experiments— and there are many of them, I understand—which correspond in several ways to the conclusions demonstrated.
SALV. The request which you make, like a true scientist,23 is a very reasonable one. For this is the custom—and properly so—in those sciences where mathematical demonstrations are applied to natural phenomena; this is seen in the case of perspective, astronomy, mechanics, music, and others, which by sense experience confirm the principles that become the foundations of the entire superstructure. I hope therefore it will not appear to be a waste of time if we discuss at considerable length this first and most fundamental question upon which hinge numerous consequences; of these we have in this book only a small number, placed there by the Author, who has done so much to open a pathway hitherto closed to minds of a speculative turn. As far as experiments go, they have not been neglected by the Author; and often, in his company, I have myself performed the tests to ascertain that the acceleration of naturally falling bodies is that above described.
We took a piece of wooden molding or scantling, about twelve cubits long, half a cubit wide, and three inches thick; on its edge we cut a channel a little more than one inch in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball. [213] Having placed this board in a sloping position, by lifting one end some one or two cubits above the other, we rolled the ball, as I was just saying, along the channel, noting, in a manner presently to be described, the time required to make the entire descent. We repeated this experiment many times in order to measure the time with an accuracy such that the deviation between two measurements never exceeded one-tenth of a pulse beat. Having performed this operation and having assured ourselves of its reliability, we now rolled the ball only one-quarter the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former. Next we tried other distances, comparing the time for the whole length with that for half, or with that for two-thirds, or three-fourths, or indeed for any fraction; in such experiments, repeated a full hundred times, we always found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane, i.e., of the channel along which we rolled the ball. We also observed that the times of descent, for various inclinations of the plane, bore to one another precisely that ratio that, as we shall see later, the Author had predicted and demonstrated for them.
For the measurement of time, we employed a large vessel of water placed in an elevated position. To the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length. The water thus collected after each descent was weighed on a very accurate balance. The differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.
SIMP. I would like to have been present at these experiments. But feeling confidence in the care with which you performed them, and in the fidelity with which you relate them, I am satisfied and accept them as true and most certain.
SALV. Then we can resume our reading and proceed.
[214] Corollary 2: Secondly, it follows that, starting from any initial point, if we take any two distances, traversed in any time intervals whatsoever, these time intervals bear to one another the same ratio as one of the distances to the mean proportional of the two distances.
That is, if from the initial point S we take two distances ST and SV and their mean proportional is SX, the time of fall through ST is to the time of fall through SV as ST is to SX; and the time of fall through SV is to the time of fall through ST as SV is to SX. For since it has been shown that the spaces traversed are in the same ratio as the squares of the times; and since, moreover, the ratio of the space SV to the space ST is the square of the ratio SV to SX; it follows that the ratio of the times of fall through SV and ST is the ratio of the distances SV and SX.
Scholium: The above corollary has been proven for the case of vertical fall. But it holds also for planes inclined at any angle; for it is to be assumed that along these planes the velocity increases in the same ratio, that is, in proportion to the time, or, if you prefer, as the series of natural numbers.24
Here, Sagredo, I should like, if it be not too tedious to Simplicio, to interrupt for a moment the present reading in order to make some additions on the basis of what has already been proved and of what mechanical principles we have already learned from our Academician. This addition I make for the greater confirmation of the truth of the principle which we have considered above by means of probable arguments and experiments; and what is more important, for the purpose of deriving it geometrically, after first demonstrating a single lemma that is fundamental in the study of impetus.
SAGR. If the advance which you propose to make is such as will confirm and fully establish these sciences of motion, I will gladly devote to it any length of time. Indeed, I shall not only be glad [215] to have you proceed, but I beg of you at once to satisfy the curiosity which you have awakened in me concerning this particular point. And I think that Simplicio is of the same mind.
SIMP. Quite right.
SALV. Since then I have your permission, let us first of all consider this notable fact—that the momenta or speeds of one and the same moving body vary with the inclination of the plane. The speed reaches a maximum along a vertical direction, and for other directions it diminishes as the plane diverges from the vertical. Therefore the impetus, strength, energy, or, one might say, the momentum of descent of the moving body is diminished by the plane upon which it is supported and along which it rolls.
For the sake of greater clearness, erect the line AB perpendicular to the horizontal AC; next draw AD, AE,AF, etc., at different inclinations to the horizontal. Then I say that all the impetus of the falling body is along the vertical and is a maximum when it falls in that direction; the momentum is less along DA and still less along EA, and even less yet along the more inclined FA. Finally, on the horizontal CA the impetus vanishes altogether; the body finds itself in a condition of indifference as to motion or rest; it has no inherent tendency to move in any direction and offers no resistance to being set in motion. offers no resistance to being set in motion. For just as a heavy body or system of bodies cannot of itself move upwards, or recede from the common center toward which all heavy things tend, so it is impossible for any body of its own accord to assume any motion other than one that carries it nearer to the aforesaid common center. Hence, along the horizontal, by which we understand a surface every point of which is equidistant from this same common center, the body will have no impetus or momentum whatever.
[216] This change of impetus being clear, it is here necessary for me to explain something which our Academician wrote when in Padua, embodying it in a treatise on mechanics prepared solely for the use of his students, and proving it at length and conclusively when considering the origin and nature of that marvelous instrument, the screw. What he proved is the manner in which the impetus varies with the inclination of the plane, as for instance that of the plane FA, one end of which is elevated through a vertical distance FC. This direction FC is that along which the impetus of a heavy body and the momentum of descent become maximum; let us try t
o determine what ratio this momentum bears to that of the same body moving along the incline FA. This ratio, I say, is the inverse of that of the aforesaid lengths. This is the lemma preceding the theorem which I hope to demonstrate later.
It is clear that the impetus of a falling body is equal to the least resistance or force sufficient to hinder it and stop it. In order to measure this force or resistance, I propose to use the weight of another body. Let us place upon the plane FA a body G connected to the weight H by means of a string passing over the point F; then the body H will ascend or descend, along the perpendicular, the same distance which the body G moves along the incline FA; but this distance will not be equal to the rise or fall of G along the vertical, in which direction alone G, like other bodies, exerts its resistance. This is clear. For consider that the motion of the body G from A to F in the triangle AFC is made up of a horizontal component AC and a vertical component CF; and remember that this body experiences no resistance [217] to motion along the horizontal (because by such a motion the body neither gains nor loses distance from the common center of heavy things, which distance is constant along the horizontal); then it follows that resistance is met only in consequence of the body rising through the vertical distance CF. Since then the body G in moving from A to F offers resistance only in so far as it rises through the vertical distance CF, while the other body H must fall vertically an amount equivalent to the entire distance FA; and since this ratio is maintained whether the motion be large or small, the two bodies being tied together; hence, we are able to assert positively that in case of equilibrium (namely, when the two bodies are at rest) the momenta, the velocities, or their propensities to motion, i.e., the spaces that would be traversed by them in equal times, must be in the inverse ratio to their weights.25 This is what has been demonstrated in every case of mechanical motion. Thus, in order to hold the weight G at rest, one must give H a weight smaller in the same ratio as the distance CF is smaller than FA.
If we do this, namely, we let the ratio of the weight G to the weight H be the same as FA to FC, then equilibrium will occur, that is, the weights H and G will have equal moments and the two bodies will come to rest. And since we are agreed that the impetus, energy, momentum, or propensity to motion of a moving body is as great as the least force or resistance sufficient to stop it; and since we have found that the weight H is capable of preventing motion in the weight G; it follows that the lesser weight H, whose entire moment is along the perpendicular FC, will be an exact measure of the partial moment which the larger weight G exerts along the inclined plane FA. But the measure of the total moment of the body G is its own weight, since to prevent its fall it is only necessary to balance it with an equal weight, provided this second weight be free to move vertically. Therefore, the partial impetus or moment of G along the incline FA will bear to the maximum and total impetus of this same body G along the perpendicular FC the same ratio as the weight H to the weight G; this ratio is, by construction, the same which the height FC of the incline bears to the length FA. We have here the lemma which I proposed to demonstrate and which, as you will see, has been assumed by our Author in the second part of the sixth proposition of the present treatise.
SAGR. From what you have shown thus far, it appears to me that one might infer, arguing by perturbed equidistance of ratios, that the moments of one and the same body moving along planes differently inclined but having the same vertical height, such as FA and FI,26 are to each other inversely as the lengths of the planes.
[218] SALV. Perfectly right. This point established, I pass to the demonstration of the following theorem: If a body falls freely along smooth planes inclined at any angle whatsoever but of the same height, the speeds which it has when reaching the bottom are equal, provided that all impediments are removed.
First we must recall the fact that on a plane of any inclination whatever a body starting from rest gains speed, or quantity of impetus, in direct proportion to the time, in agreement with the definition of naturally accelerated motion given by the Author. Hence, as he has shown in the preceding proposition, the distances traversed are proportional to the squares of the times and therefore to the squares of the speeds. Whatever impetus is gained at the first instant, the increments of speed during the same time will be respectively the same, since in each case the gain of speed is proportional to the time.
Let AB be an inclined plane whose height above the horizontal BC is the vertical AC. As we have seen above, the impetus of a body falling along the vertical AC is to the impetus of the same body along the incline AB as AB is to AC. On the incline AB, lay off AD, the third proportional to AB and AC; then the impetus along AC is to that along AB (i.e., along AD) as the length AC is to the length AD. Therefore, the body will traverse the space AD, along the incline AB, in the same time which it would take in falling the vertical distance AC (since the moments are in the same ratio as the distances); and the speed at C is to the speed at D as the distance AC is to the distance AD. But according to the definition of accelerated motion, the speed at B is to the speed of the same body at D as the time required to traverse AB is to the time required for AD; and according to the last corollary of the second proposition, the time for passing through the distance AB bears to the time for passing through AD the same ratio as the distance AC (the mean proportional between AB and AD) to AD. Accordingly the two speeds at B and C each bear to the speed at D the same ratio, namely, that of the distance AC to AD; hence they are equal. This is the theorem which I set out to prove.
From the above we are better able to demonstrate the following third proposition of the Author, in which proposition he employs the preceding principle: The time required to traverse an incline is to that required to fall through the vertical height of the incline in the same ratio as the length of the incline to [219] its height.
For, according to the second corollary of the second proposition, if AB represents the time required to pass over the distance AB, the time required to pass the distance AD will be the mean proportional between these two distances and will be represented by the line AC; but if AC represents the time needed to traverse AD, it will also represent the time required to fall through the distance AC, since the distances AC and AD are traversed in equal times; consequently, if AB represents the time required for AB, then AC will represent the time required for AC. Hence, the times required to traverse AB and AC are to each other as the distances AB and AC.
By the same reasoning it can be shown that the time required to fall through AC is to the time required for any other incline AE as the length AC is to the length AE; therefore, by equidistance of ratios, the time of fall along the incline AB is to that along AE as the distance AB is to the distance AE, etc.27
One might, by applying this same theorem, as Sagredo will readily see, immediately demonstrate the sixth proposition of the Author. But let us end this digression here, which Sagredo has perhaps found rather tedious, though I consider it quite important for the theory of motion.
SAGR. On the contrary it has given me great satisfaction, and indeed I find it necessary for a complete grasp of that principle.
SALV. I will now resume the reading of the text.
[§10.8 Day IV: The Parabolic Path of Projectiles]28
[268] SALV. Once more, Simplicio is here on time. So let us, without rest, take up the question of motion. Here is the text of our Author “On the Motion of Projectiles”:
In the preceding pages we have discussed the properties of uniform motion and of motion naturally accelerated along planes of all inclinations. I now propose to set forth those properties that belong to a body whose motion is compounded of two other motions, namely, one uniform and one naturally accelerated; these properties, well worth knowing, I propose to demonstrate in a rigorous manner. This is the kind of motion seen in a moving projectile; its origin I conceive to be as follows.
Imagine any particle projected along a horizontal plane without friction. Then we know, from what has been more fully explained in
the preceding pages, that this particle will move along this same plane with a motion that is uniform and perpetual, provided the plane has no limits. But if the plane is limited and elevated, then the moving particle, which we imagine to be a heavy body, will on passing over the edge of the plane acquire, in addition to its previous uniform and enduring motion, a downward propensity due to its own weight; and so the resulting motion, which I call projection, is compounded of one that is uniform and horizontal and another that is downward and naturally accelerated. We now proceed to demonstrate some of its properties, the first of which is as follows.
[269] Theorem 1, Proposition 1: A projectile that is carried by a uniform horizontal motion compounded with a naturally accelerated downward motion describes a path that is a semiparabola.
SAGR. Here, Salviati, it will be necessary to stop a little while for my sake and, I believe, also for the benefit of Simplicio; for it so happens that I have not gone very far in my study of Apollonius29 and am merely aware of the fact that he treats of the parabola and other conic sections, without an understanding of which I hardly think one will be able to follow the proof of other propositions depending upon them. Since even in this first beautiful theorem the author finds it necessary to prove that the path of a projectile is a parabola, I imagine we shall have to deal with this kind of curve, and so it will be absolutely necessary to have a thorough understanding, if not of all the properties which Apollonius has demonstrated for these figures, at least of those that are needed for the present treatment.
The Essential Galileo Page 45