FULL CIRCLE

In the year 1604, Galileo Galilei formulated a law of falling bodies in a letter to his friend, Paolo Sarpi. "I have arrived at a proposition," he wrote, "which is most natural and evident, and assuming it, I can demonstrate the rest; namely, that spaces traversed in natural motion are in the squared proportion of the times, and consequently the spaces traversed in equal times are as the odd numbers beginning with unity. And the principle is this, that the naturally moving body increases its velocity in the proportion that it is distant from the origin of motion." This is a curious statement. For the first part is right, but one cannot explain how Galileo knew it, since it does not in fact follow from the principle, which is wrong. Under uniform acceleration, velocity varies directly as time, not distance, and any schoolboy learns the correct law by rote as either or both of two equations,

s = ½ gt2 and s = ½vt.

Even when he finally did get it right, Galileo could not so express it. Algebra had yet to be adapted to description of continuously developing quantities. He disposed only of the resources of ordinary language and of the geometry of Euclid and Archimedes. In 1632, after years of reflection and not a little frustration, he explained the law in Dialogue on the Two Chief Systems of the World, the great Copernican argument over which the Roman Catholic Church humiliated him; and there he repeated, "that the distances passed by the body departing from its rest are to each other in double proportion of the times in which those distances are measured."

To that, Sagredo, the receptive interlocutor, now responds, "This is truly admirable; and do you say there is a mathematical demonstration for it?" And Galileo gratifies the request he has invited by expressing the relation between velocity, distance, and time as a triangle. Falling from rest at A, the body picks up speed through "infinite degrees of velocity." The time of fall is laid off on the vertical AC. E Perpendiculars (DH, EI, etc.) represent the velocity after time AD, DE, etc., and the whole triangle is "the mass and sum of the whole velocity, with which in the time AC it passed such a certain space." Or, to put it otherwise, the area of the triangle (½ vt) measures the distance traversed. And to find the distances travelled by a body moving at uniform velocity (BC), the triangle may be doubled into a rectangle (ACBM).

But though perfectly correct, this must still seem painful and clumsy to the modern reader. Velocity appears as one variable and time as the other, whereas it has become customary to think of velocity rather as a ratio of distance to time. Moreover, it measures the linear distance s by an area. Nor does the geometry yet derive the law in the form which relates distance to acceleration (s = ½ gt2). In 1638, Galileo published his final and scientifically his finest work, Discourses on Two New Sciences. There at last he achieved an explicit statement of both common forms of the law. The discussions of the "Third Day" work towards a renewed demonstration of the velocity-time relationship, in more elegant geometrical form than in the Dialogue, and in less elegant language. Next, Galileo proved what until now he had only asserted: the distances are as the squares of the times. This was far more difficult. He had to formulate graphically what he called "uniformly difform motion," that is to say, a dynamical proposition involving acceleration in the essentially static forms of plane geometry.

He represented the "flow of time" by simple extension, the line AB, on which AD and AE measure any two intervals. To the right, the line HI stands for the path of descent at uniform acceleration, so that HL is the distance traversed in time AD, HM in AE, etc. These things being so, then "I say that the space MH to the space HL is in the duplicate ratio that time AE has to time AD." For, construct AC at any angle to AB. Then DO, EP, etc. will again represent maximum velocity at corresponding time. It followed from the previous (mean-speed) theorem that the spaces are equal which are traversed by one body at uniform acceleration from rest, and a second moving at a constant velocity which is one-half the maximum attained by the first. Thus, the distances of fall HL and HM would be equal to those traversed in times AD and AE at constant velocities one-half of DO and EP respectivity. But it had already been shown that the distances passed by two bodies in uniform motion are to each other as the product of the ratio of the velocities into the ratio of the times. Now, since EP is to OD as AE is to AD, then the ratio of velocities is in this case the same as the ratio of the times. "Therefore, the ratio of the spaces traversed is as the square of the ratio of the times. Q. E. D."

At this point, Salviati, who speaks for Galileo, stops the dialogue as if a light had dawned: "Please suspend your lecture for a moment while I speculate on a certain idea that has just now occurred to me." And he puts the two forms of the law together. AI represents time again, AF is at any angle, and C is the mid-point of AI. Then (to condense the argument a bit), if the body falls freely to C, BC will be the maximum velocity, and the distance will be measured by the rectangle of uniform velocity erected on the base EC equal to ½ CB.

Moreover, if the body continued its descent at constant velocity BC, then in the interval CI it would cover twice the distance that it had described in time AC, starting from rest. But since the body is under uniform acceleration, its velocity during the time CI will increase by an amount FG equal to the parallel of the triangle BFG, which is equal to ABC. Then, adding to velocity GI (equal to BC) half of FG, which is the maximum velocity attained through acceleration, one gets the uniform velocity with which the same space would have been described in the time CI. And perhaps the drift is apparent without further paraphrasing. The rectangular areas which represent the space described increase in successive time intervals, "as the odd numbers beginning with unity, 1, 3, 5; ... and in general, the spaces traversed are in the duplicate ratio of the times, i.e., as the squares of these times."

These figures represent the earliest integrations applied to developing physical quantities and may be taken, therefore, to symbolize the germ from which has grown a mathematical science, not alone of proportions, but of nature. For there was nothing novel about expressing uniform motion in the abstract as a ratio of change in geometrical quantities. Galileo's first triangle of motion was a mathematical commonplace, generally called the Merton Rule after the school of...