SALV. You are quite too modest, pretending ignorance of facts which not long ago you acknowledged as well known—I mean at the time when we were discussing the strength of materials and needed to use a certain theorem of Apollonius that gave you no trouble.
SAGR. I may have chanced to know it; or I may possibly have assumed it since it was needed only once in that discussion. But now when we have to follow all these demonstrations about such curves, we ought not, as they say, to swallow it whole, and thus waste time and energy.
SIMP. And then, even if Sagredo were, as I believe, well equipped for all his needs, I do not understand even the elementary terms; for although our philosophers have treated the motion of projectiles, I do not recall their having described the path of a projectile except to state in a general way that it is always a curved line, unless the projection be vertically upwards. Thus, if [270] the little geometry I have learned from Euclid since our previous discussion does not enable me to understand the demonstrations that are to follow, then I shall be obliged to accept the theorems on faith, without fully comprehending them.
SALV. On the contrary, I desire that you should understand them from the Author himself, who, when he allowed me to see this work of his, was good enough to prove for me two of the principal properties of the parabola because I did not happen to have at hand the books of Apollonius. These properties, which are the only ones we shall need in the present discussion, he proved in such a way that no prerequisite knowledge was required. These theorems are, indeed, proved by Apollonius, but after many preceding ones, which would take a long time to follow. I wish to shorten our task by deriving the first property purely and simply from the mode of generation of the parabola and proving the second immediately from the first.
Beginning now with the first, imagine a right cone, erected upon the circular base ibkc with apex at l. The section bac of this cone made by a plane drawn parallel to the side lk is the curve that is called a parabola. The base of this parabola bc cuts at right angles the diameter ik of the circle ibkc, and the axis ad is parallel to the side lk. Now having taken any point f in the curve bfa, draw the straight line fe parallel to bd. Then, I say, the square of bd is to the square of fe in the same ratio as the axis ad is to the portion ae.
Now, through the point e pass a plane parallel to the circle ibkc, producing in the cone a circular section whose diameter is the line geh. Since bd is at right angles to ik in the circle ibk, the square of bd is equal to the rectangle formed by id and dk; so also in the upper circle that passes through the points gfh, the square of fe is equal to the rectangle formed by ge and eh; hence the square of bd is to the square of fe as the rectangle id-dk is to the rectangle ge-eh. And since the line ed is parallel to hk, the line eh, being parallel to dk, is equal to it; therefore the rectangle id-dk is to the rectangle ge-eh as [271] id is to ge, that is, as da is to ae; hence also the rectangle id-dk is to the rectangle ge-eh, that is, the square of bd is to the square of fe, as the axis da is to the portion ae. QED.
The other proposition necessary for this discussion we demonstrate as follows. Let us draw a parabola whose axis ca is prolonged upwards to a point d; from any point b draw the line bc parallel to the base of the parabola; if now the point d is chosen so that da equals ca, then, I say, the straight line drawn through the points b and d will be tangent to the parabola at b.
For imagine it were possible that this line cuts the parabola above or that its prolongation cuts it below; then through any point g on this line draw the straight line fge. Since the square of fe is greater than the square of ge, the square of fe will bear a greater ratio to the square of bc than the square of ge to that of bc; and since, by the preceding proposition, the square of fe is to that of bc as the line ea is to ca; it follows that the line ea will bear to the line ca a greater ratio than the square of ge to that of bc, or, than the square ofedto that ofcd (the sides of the triangles deg and dcb being proportional). But the line ea is to ca, or da, in the same ratio as four times the rectangle ea-ad is to four times the square of ad, or, what is the same, to the square of cd (since this is four times the square of ad). Hence four times the rectangle ea-ad bears to the square of cd a greater ratio than the square of ed to the square of cd. But that would make four times the rectangle ea-ad greater than the square of ed. This is false, the fact being just the opposite, because the two portions ea and ad of the line ed are not equal. Therefore the line db touches the parabola without cutting it. QED.
SIMP. Your demonstration proceeds too rapidly and, it seems to me, you keep on assuming that all [272] of Euclid’s theorems are as familiar and available to me as his first axioms, which is far from true. For example, you just sprang upon us that four times the rectangle ea-ad is less than the square of ed because the two portions ea and ad of the line ed are not equal; and this brings me little composure of mind, but rather leaves me in suspense.
SALV. Indeed, all real mathematicians assume on the part of the reader perfect familiarity with at least the Elements of Euclid. Here it is necessary in your case only to recall the proposition of Book II30 in which he proves that when a line is cut at two points into equal and unequal parts respectively, the rectangle formed on the unequal parts is less than that formed on the equal (i.e., less than the square on half the line), by an amount that is the square of the segment between the two cut points; from this it is clear that the square of the whole line, which is equal to four times the square of the half, is greater than four times the rectangle of the unequal parts. In order to understand the following portions of this treatise it will be necessary to keep in mind the two elementary theorems from conic sections which we have just demonstrated; these two theorems are indeed the only ones which the Author uses. We can now resume the reading of the text and see how he demonstrates his first proposition, in which he shows that a projectile undergoing motion compounded of uniform horizontal motion and naturally accelerated fall describes a semiparabola.
Let us imagine an elevated horizontal line or plane ab along which a body moves with uniform speed from a to b. Suppose this plane to end abruptly at b; then at this point the body will, on account of its weight, acquire also a natural motion downwards along the perpendicular bn. Draw the line be along the plane ab to represent the flow, or measure, of time; divide this line into a number of segments, bc, cd, de, representing equal intervals of time; and from the points c, d, e, let fall lines that are parallel to the perpendicular bn. On the first of these lay off any distance ci; [273] on the second a distance four times as long, df; on the third, one nine times as long, eh; and so on, in proportion to the squares of cb, db, eb, or, we may say, in the squared ratio of these same lines. Accordingly we see that while the body moves from b to c with uniform speed, it also falls perpendicularly through the distance ci, and at the end of the time interval bc it finds itself at the point i. In like manner at the end of the time interval bd, which is the double of bc, the vertical fall will be four times the first distance ci; for it has been shown in a previous discussion that the distance traversed by a freely falling body varies as the square of the time. In like manner the space eh traversed during the time be will be nine times ci. Thus it is evident that the distances eh, df, ci will be to one another as the squares of the lines be, bd, bc. Now, from the points i, f, h draw the straight lines io, fg, hl parallel to be; these lines hl, fg, io are equal to eb, db, and cb, respectively; so also are the lines bo, bg, bl respectively equal to ci, df, and eh; furthermore, the square of hl is to that of fg as the line lb is to gb, and the square of fg is to that of io as gb is to ob; therefore the points i, f, h lie on one and the same parabola. In like manner it may be shown that if we take equal time intervals of any size whatever, and if we imagine the body to be carried by a similar compound motion, its positions at the end of these time intervals will lie on one and the same parabola. QED.
This conclusion follows from the converse of the first of the two propositions given above. For, having drawn a parabola through the points b and
h, any other two points, f and i, not falling on the parabola must lie either within or without; consequently the line fg is either longer or shorter than the line that terminates on the parabola. Therefore the square of hl will not bear to the square of fg the same ratio as the line lb to gb, but a greater or smaller. The fact is, however, that the square of hl does bear this same ratio to the square of fg. Hence the point f does lie on the parabola, and so do all the others.
SAGR. One cannot deny that the argument is new, subtle, and conclusive. It also rests upon various assumptions, namely, that the horizontal motion remains uniform, that the vertical motion continues to be accelerated downwards in proportion to the square of the time, and that such motions and velocities as these combine without altering, disturbing, or hindering each other, so that as the motion proceeds the path of the projectile does not change into a different curve. But this, in my opinion, [274] is impossible. For the axis of the parabola along which we suppose the natural motion of a falling body to take place stands perpendicular to a horizontal surface and ends at the center of the earth; and since the parabola deviates more and more from its axis, no projectile can ever reach the center of the earth or, if it does, as seems necessary, then the path of the projectile must transform itself into some other curve very different from the parabola.
SIMP. To these difficulties, I may add others. One of these is that we suppose the horizontal plane, which slopes neither up nor down, to be represented by a straight line as if each point on this line were equally distant from the center. This is not the case, for as one starts from the middle of the line and goes toward either end, one departs farther and farther from the center of the earth and so is constantly going uphill; whence it follows that the motion cannot remain uniform through any distance whatever, but must continually diminish. Besides, I do not see how it is possible to avoid the resistance of the medium, which must destroy the uniformity of the horizontal motion and change the law of acceleration of falling bodies. These various difficulties render it highly improbable that a result derived from such unreliable assumptions should hold true in practical experience.
SALV. All these difficulties and objections which you urge are so well founded that it is impossible to remove them; and as for me, I am ready to admit them all, which indeed I think our Author would also do. I grant that these conclusions proved in the abstract will be different when applied in the concrete and will be false to this extent, that neither will the horizontal motion be uniform, nor will the natural acceleration be in the ratio assumed, nor will the path of the projectile be a parabola, etc. But, on the other hand, I ask you not to begrudge our Author that which other eminent men have assumed, even if not strictly true.
The authority of Archimedes alone will satisfy everybody. In his works on mechanics and on the quadrature of the parabola, he takes for granted that the beam of a balance or steelyard is a straight line, every point of which is equidistant from the common center of all heavy bodies, and that the strings by which heavy bodies are suspended are parallel to each other. Some consider this assumption permissible because, in practice, our instruments and the distances [275] involved are so small in comparison with the enormous distance from the center of the earth that we may consider a minute of arc on a great circle as a straight line, and may regard the perpendiculars let fall from its two extremities as parallel. For if in actual practice one had to consider such small quantities, it would be necessary first of all to criticize the architects who presume, by the use of a plumb line, to erect high towers with parallel sides. I may add that, in all their discussions, Archimedes and the others considered themselves as located at an infinite distance from the center of the earth, in which case their assumptions were not false, and therefore their conclusions were absolutely correct. When we wish to apply our proven conclusions to distances which, though finite, are very large, it is necessary for us to infer, on the basis of demonstrated truth, what correction is to be made for the fact that our distance from the center of the earth is not really infinite, but merely very great in comparison with the small dimensions of our apparatus. The largest of these will be the range of our projectiles—and here we need consider only the artillery— which, however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth; and since these paths terminate upon the surface of the earth, only very slight changes can take place in their parabolic shape, which, it is conceded, would be greatly altered if they terminated at the center of the earth.
As to the perturbation arising from the resistance of the medium, this is more considerable and does not, on account of its manifold forms, submit to fixed laws and exact description. Thus, if we consider only the resistance which the air offers to the motions studied by us, we shall see that it disturbs them all and disturbs them in an infinite variety of ways corresponding to the infinite variety in the form, weight, and velocity of the projectiles. As to velocity, the greater this is, the greater will be the resistance offered by the air; also resistance will be greater as the moving bodies become less dense. Thus, although the falling body ought to be accelerated in accordance with the rule of distance being proportional to the square of the duration of its motion, yet no matter how heavy the body is, if it falls from a very considerable height, the resistance of the air will be such as to eventually prevent any increase in [276] speed and render the motion uniform; and in proportion as the moving body is less dense, this uniformity will be attained more quickly and from smaller heights. Even horizontal motion, which would be uniform and constant if no impediment were offered, is altered by the resistance of the air and finally ceases; and here again, the less dense the body, the quicker the process.
Of such effects of weight, velocity, and also shape, which are infinite in number, it is not possible to give any exact description. Hence, in order to handle this matter in a scientific way, it is necessary to cut loose from these difficulties, to discover and demonstrate the theorems in the case of no impediments, and to use them and apply them with such limitations as experience will teach. The advantage of this method will not be small, for the material and shape of the projectile may be chosen as dense and round as possible, so that it will encounter the least resistance in the medium; and the spaces and velocities will be small enough for the most part that we shall be easily able to correct them with precision. Indeed, in the case of those projectiles we use, thrown from a sling or crossbow, and made of dense material and round in shape or of lighter material and cylindrical in shape (such as arrows), the deviation from an exact parabolic path is quite imperceptible. Furthermore, if you will allow me a little greater liberty, I can show you, by two experiments, that the dimensions of our apparatus are so small that these external and incidental resistances, among which that of the medium is the most considerable, are scarcely observable.
I proceed to the consideration of motions through the air, since it is with these that we are now especially concerned. The resistance of the air exhibits itself in two ways: first by offering greater impedance to less dense than to very dense bodies, and second by offering greater resistance to a body in rapid motion than to the same body in slow motion.
Regarding the first of these, consider the case of two balls having the same dimensions, but one weighing ten or twelve times as much as the other; one, say, of lead, the other of oak, both allowed to fall from an elevation of 150 or 200 cubits. Experiment shows that they will reach the ground with a slight difference in speed, showing us that in both cases the retardation caused by the air is small. For if both balls start at the same moment and at the same elevation, and if the leaden one be slightly retarded and the wooden one greatly retarded, then [277] the former ought to reach the earth a considerable distance in advance of the latter, since it is ten times as heavy; but this does not happen; instead, the gain in distance of one over the other does not amount to the hundredth part of the entire fall. And in the case of a ball of stone weighing only a third or half as much as one
of lead, the difference in their times of reaching the ground will be scarcely noticeable. Now, the impetus acquired by a leaden ball in falling from a height of 200 cubits (which is such that if its motion became uniform the ball would traverse 400 cubits in a time interval equal to that of the fall) is very considerable in comparison with the speeds which we are able to give to our projectiles by the use of bows or other machines (except firearms); so it follows that we may, without noticeable error, regard as absolutely true those propositions which we are about to prove without considering the resistance of the medium.
Passing now to the second case, where we have to show that the resistance of the air for a rapidly moving body is not very much greater than for one moving slowly, ample proof is given by the following experiment. Attach to two threads of equal length—say four or five cubits—two equal leaden balls and suspend them from the ceiling; now pull them aside from the perpendicular, one through 80 or more degrees, the other through not more than 4 or 5 degrees; so that when set free, the first falls, passes through the perpendicular, and describes large but slowly decreasing arcs of 160, 150, 140 degrees, etc., and the other swings through small but also diminishing arcs of ten, eight, six degrees, etc. Here it must be remarked first of all that the first passes through its arcs of 180, 160 degrees, etc., in the same time that the other swings through its ten, eight degrees, etc.; from this it follows that the speed of the first ball is sixteen and eighteen times greater than that of the second; accordingly, if the air offers more resistance to the high speed than to the low, the frequency of oscillation in the large arcs of 180 or 160 degrees, etc., ought to be less than in the very small arcs of ten, eight, four, two degrees, or even one. But this prediction conflicts with experiment. For if two persons start to count the oscillations, one the large and the other the small, they will discover that after counting tens and even hundreds they will not differ by a single oscillation, not even by a fraction of one. This observation justifies the two following propositions, [278] namely, that oscillations of very large and very small amplitude all take the same time, and that the resistance of the air does not affect motions of high speed more than those of low speed, contrary to the opinion which we ourselves entertained earlier.
The Essential Galileo Page 46