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 kinematic philosophy which flourished in that ancient Oxford College during the fourteenth century. Moreover, the mean speed theorem reduces to the law of acceleration, and needed rather to be stated helpfully than to be discovered. Everything, indeed, or nearly everything, that Galileo put together may be found in the writings of one or another of the late scholastics, in the aphorisms of Leonardo da Vinci, or in the works of some predecessor among the Renaissance mathematicians. But only Galileo, and he only after many a false start, developed the judgment and intuition and feel for the physical to select the elements of a physics from this olla podrida of mathematical techniques and philosophical assertions. His was the transforming touch of the mathematical physicist, the first of his kind, who would really change a situation instead of simply entering a discussion. That touch reveals itself thrice over in these passages. First, he derived the rule of uniform acceleration in a form applicable to freely-falling bodies. Then he included it in a general statement containing both the velocity-time and the acceleration-time-squared measures of path. Finally he applied it to the real case in nature and therein lay his genius.

For only Galileo would have given the discussion the turn it takes immediately after the last of these, his mathematical demonstrations. Simplicio, who upholds the Aristotelian case, bows before the force of geometry, "so that 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 one meets in nature in the case of falling bodies, I am still doubtful; and 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 illustrate in several ways the conclusions reached." And Galileo reports on the famous experiments on inclined planes which he had imagined, and some of which he may quite probably have actually performed.

* * *

These were portentous triangles. To the historian thinking broadly about the recent destiny and future prospects of western civilization, it may well appear that our own culture, in which whatever our temperament we are bound to live, is set off from those of Asia, Africa, and the world of antiquity by two fundamental factors. From one of these it emerged: its religious chrysalis was Christianity, investing history with the promise of fulfillment of a sort. The other it produced: the most dynamic, distinctive, and influential creation of the western mind is a progressive science of nature. Only there in the technical realm, indeed, does the favorite western idea of progress hold any demonstrable meaning. No one understands political power better than Machiavelli did. Picasso cannot conclusively be held a better or worse artist than Leonardo was. But every college freshman knows more physics than Galileo knew, whose claim is higher than any other's to the honor of having founded modern science, and more too than Newton did, whose mind was the most powerful ever to have addressed itself to nature.

In its early days, science was distinct from technology, springing rather from thought and philosophy than from craftsmanship. Nowadays, however, and indeed for the last century and more, science has merged ever more intimately with technology, so arming it with power, so enhancing its capacities, that no words, nor any fears or dreams, may exaggerate what depends upon the employment. Nor is the future of our own world of the West alone in play through this, its great invention. Perhaps the historian may be pardoned a single prophecy, if it comes at the beginning of a book before his tale has made him pompous. Anxious though our moments are, today is not the final test of wisdom among statesmen or virtue among peoples. The hard trial will begin when the instruments of power created by the West come fully into the hands of men not of the West, formed in cultures and religions which leave them quite devoid of the western sense of some ultimate responsibility to man in history. That secular legacy of Christianity still restrains our world in some slight measure, however self-righteous it may have become on the one side, and however vestigial on the other. Men of other traditions can and do appropriate our science and technology, but not our history or values. And what will the day hold when China wields the bomb? And Egypt? Will Aurora light a rosy-fingered dawn out of the East? Or will Nemesis?

Albert Einstein once remarked that there is no difficulty in understanding why China or India did not create science. The problem is rather why Europe did, for science is a most arduous and unlikely undertaking. The answer lies in Greece. Ultimately science derives from the legacy of Greek philosophy. The Egyptians, it is true, developed surveying techniques and conducted certain surgical operations with notable finesse. The Babylonians disposed of numerical devices of great ingenuity for predicting the patterns of the planets. But no Oriental civilization graduated beyond technique or thaumaturgy to curiosity about things in general. Of all the triumphs of the speculative genius of Greece, the most unexpected, the most truly novel, was precisely its rational conception of the cosmos as an orderly whole working by laws discoverable in thought. The Greek transition from myth to knowledge was the origin of science as of philosophy. Indeed, knowledge of nature formed part of philosophy until they parted company in the scientific revolution of the seventeenth century.

In our own world, science continues to be what it was in Greece, conceptual thought mediating between consciousness and nature. But it is also something more. It has become determinate instead of simply speculative. For the scientific revolution reversed the direction in which information flows, and added body to the structure of communication. Greek science was subjective, rational, and purely intellectual. It started inside the mind whence concepts like purpose, soul, life, and organism were projected outward to explain phenomena in the familiar terms of self-knowledge. In those terms the success of an explanation depended only on its universality and capacity to satisfy the reason. Greek science scarcely knew experiment and never thought to move beyond curiosity to power. Modern science, on the other hand, is impersonal and objective. It takes its starting points outside the mind in nature and winnows observations of events which it gathers under concepts, to be expressed mathematically if possible and tested experimentally by their success in predicting new events and suggesting new concepts. Modern science has not abandoned rationality, but it is first of all metrical and experiential. Related to this is its association with technology as a continuation of that generalized thrust toward mastery of the world which began in the West with the Renaissance. Modern science, finally, seeks both to comprehend and control nature — though according to the positivist school of philosophers, whose persuasion dominates at the moment, comprehension is an illusory goal. For them prediction and control are everything.

A true revolution brings fundamental change through rebellion against constituted authority, but it is clear from the history of revolutions that to repudiate a debt is not to escape it. So it was that the creation of modern science in the Renaissance was at once a rebirth of Greek science and a bursting of its confines. To separate the new from the old in the Renaissance is always difficult, for humanists steeped in classical learning found antique words for new ideas. It is, however, no falsification of a complex situation — it is rather a first approximation toward resolving it — to say that science stirred into new life under the inspiration of Plato working against the cramping of learning within a fossilized Aristotelianism.

By Galileo's time, the science and authority of Aristotle had led the western mind a long way to a dead end. Aristotle's was the most capacious of philosophies. In principle it explained everything, dealing rather in reasons than structures, and preferring categories over abstractions. For example, Galileo could describe mathematically how a stone would fall under ideal conditions. He could not say why it fell. Aristotle's physics, on the contrary, could not measure its motion. But this was not to be expected in a real world of friction and complexity where ideal conditions never occur. Aristotle could do more important things. He could explain why a stone fell, why sparks flew upward, and why the stars ran round in their courses.

Beneath its physical manifestation as translation, Aristotelian motion is metaphysical, an instance of change, an evidence of imperfection. Change is the act of things realizing their potentialities in a world striving ever to fulfill its creator's will toward order, which is toward the good, so far as its corruption permits. In an orderly cosmos there is by definition a place for everything. Heavy things belong at the bottom. To say that the stone falls because it is of the class of things which are heavy constitutes, therefore, a full explanation. So, too, fire rises because it is light. The locus of that element is in the ethereal region, with air below it, water below that, and crude earth massed round the center. But what of an arrow? Here a distinction of motions must be introduced. Its motion is not natural but forced, not orderly but disorderly and violent. It must have a cause. Logic requires effects not to outlast their causes. Therefore, every motion against nature presupposes a moving agent, and demands explanation.

What, then, moves the arrow after it has parted from the bow-string? In a philosophy which is nothing if not universal, to have no answer would be to have no science, and after some hesitation, Aristotle meets the dynamical difficulty with the air. It is the surrounding medium closing in behind the projectile which urges it along its way. There can, therefore, be no motion in a void. There cannot even be natural motion, for in this case the medium serves to retard the body. In a vacuum a stone would fall instantaneously. Since that is absurd, nature knows no void, and the world must be a plenum, finite and by later standards rather small. But the inadmissibility of the void goes deeper than abhorrence of the vacuum. It goes all the way to the foundations. There is no such thing as place in a void, and the goal of this philosophy was to define the right and necessary place for every species of being according to the purpose that it served. Nor can there be existence in the nothing. In a void no stone could tell where to go, no flame find the way to leap. The very notion of direction or order would become meaningless. To admit the void is to accept the reign of chaos, wherein whirl is king, in lieu of our own world full of meaning.