Bernouilli’s Brachistichrone:
An Exemplary Case of the “Science of the Moments of Becoming”
In response to Kepler’s call for the development of a mathematics appropriate to non-uniform motion, Leibniz invented a new form of geometry of position, that he called, the “infinitesimal calculus”. While a horror may well up in the minds of some at these words, such terrors can be calmed, were one to realize, that the source of this consternation, is due entirely to the Aristotelean assault on Leibniz, by Newton, Euler, Lagrange, Cauchy and their mindless adherents, who imposed on Leibniz’ beautiful invention, the scowling, constipated formalism of his enemies.
What Kepler’s discovery required, was a geometry that measured the position of the planet, with respect to the principle of change governing the planet’s motion. The achievements of the Greeks proved insufficient, as those investigations sought to determine positions with respect to other positions. What Leibniz supplied was a geometry of position that determined position with respect to a principle of change.
To attune our minds to Leibniz’ invention, turn to another investigation of the geometry of position, developed during the same time, under the leadership of J.S. Bach. As Bach’s compositions demonstrate, musical notes are not positions, that in turn determine intervals, which in turn determine scales and then keys, and then the whole well-tempered system. As any listener to a Bach composition can easily recognize, the position of any note, is an ambiguity, that becomes less ambiguous, as the composition unfolds, and the intervals so generated, and their inversions, are heard with respect to the well-tempered system of bel canto polyphony as a whole. It is the change, with respect to the whole well-tempered system, that determines the notes, not the notes that determine the change.
So too, with a planet in a Keplerian orbit. The position of the planet at any given moment, is a function of the harmonic ordering of the solar system. Two positions of a planet mark off an interval of an orbit, but that interval is not defined by the positions, rather, the positions are defined by the change that occurs in that interval of action. Since in a non-uniform orbit, the speed and trajectory of the planet is always changing, Kepler demanded a means to measure that change at each moment. Leibniz delivered, developing his new geometry of position, i.e., the infinitesimal calculus.
As mentioned in the last installment of this series, a good pedagogical example of Leibniz’ discovery, is its application in John Bernoulli’s discovery of the brachistichrone curve. (What follows is a summary of the concepts of Bernoulli’s construction. It will require some work on the part of the reader, and is intended to be read in conjunction with Bernoulli’s original essay, an English translation of which can be found in D.E. Smith’s, “Source Book of Mathematics”.
In 1697, Bernoulli put out a challenge in Leibniz’ Acta Eruditorum, to all mathematicians in the world. The problem was stated:
“Mechanical Geometrical Problem on the Curve of Quickest Descent.”
“To determine the curve joining two given points, at different distances from the horizontal and not on the same vertical line, along which a mobile particle acted upon by its own weight and starting its motion from the upper point, descends most rapidly to the lower point.”
The prize promised was not gold or silver, “for these appeal only to base and venal souls, for which we may hope for nothing laudable, nothing useful for science. Rather, since virtue itself is its own most desirable reward and fame is a powerful incentive, we offer the prize, fitting for the man of noble blood, compounded of honor, praise, and approbation; thus we shall crown, honor and extol, publicly and privately, in letter and by word of mouth the perspicacity of our great Apollo.”
As Bernoulli pointed out, the problem posed could not be solved, even by the method of maxima and minima of Fermat. For in those cases, Fermat sought the maximum and minimum from among a given set of quantities or loci, such as the point of maximum curvature of a conic section. Instead, Bernoulli’s problem was to find a minimum curve, among an infinity of possible paths. Every position on this sought after curve, was determined by a principle of change. So, what had to be discovered was, from a given a principle of change, i.e., least-time, how are the positions of the body determined. This is the equivalent of finding the correct orbit of a planet, not merely a possible one. Or to put it in metaphysical terms, “How can we know, how a falling body knows, to find the path of least descent?”
As you will see, Bernouilli was not posing an abstract mathematical puzzle, for the mere sake of befuddling others, the solution to this problem led to important discoveries in mechanics, as well as metaphysics.
Bernoulli’s attack on this problem began with what he called “Fermat’s metaphysical principle”, that light always seeks out the path of least time. It was a discovery of the ancient Greeks, that when light was reflected from a mirror, the path it took was the shortest length. However, when light was refracted by traveling through different media, such as water and air, the path of the light was not the shortest length. Fermat, discovering that the velocity of light was slower in denser media, demonstrated that the light changed its direction at the boundary between the two media, so as to follow the path of least time. This, of course, was consistent with the Greek discovery. In reflection, since the light travels through only one medium and therefore doesn’t change velocity, the shortest path, is also the path of least time. But, when there’s a change in medium, the light travels the shortest path in space-time, or the path of least-time.
Bernoulli’s approach was to follow the light, so to speak, to the path of least time. If the path of a ray of light traveling through a medium, whose density is continuously changing, according the same principle as that of a body falling under gravity, the the least time path of the light, will be the same as the least time path of the body.
But, how to discover the path, when we only know the principle of change, and have no positions to which to orient? At each moment along the light’s path, the light would be changing its speed and direction, such that its overall travel took the least time. Thus, similar to the motion of a planet, at each such moment, the light was ceasing to be what it was, and becoming what it will be. At each moment, the position of the light was a function of the principle of maintaining the least-time path.
Fermat had shown, that as light moved from a rarer to a denser medium, it slowed down, and its path became more vertical. For example, if light were traveling through air to water, the angle its path made with a vertical line, changed at the boundary between the air and water. If the angle its path made with the vertical in the air changed, the angle it made with the vertical under the water changed accordingly. However, the two angles did not change proportionally. Rather, they changed such that the sines of the angles were always in the same proportion.
So, at each “moment of becoming” along the light’s path, the light’s velocity and trajectory were changing, such the sine of the angle the light’s path made at that moment, was always proportional to the sines of the angles at all such “moment’s of becoming.”
To find the brachistichrone, Bernoulli thought of the medium in the following way:
“If we now consider a medium which is not uniformly dense but is as if separated by an infinite number of sheets lying horizontally one beneath another, whose interstices are filled with transparent material of rarity increasing or decreasing according to a certain law; then it is clear that a ray which may be considered as a tiny sphere travels not in a straight but instead in a certain curved path. This path is such that a particle traversing it with velocity continuously increasing or diminishing in proportion to the rarity, passes from point to point in the shortest time.”
Under this idea, at each horizontal sheet, the speed and direction of the light changes. The principle under which its speed and direction changes at each horizontal sheet, Leibniz called, the differential. The totality of all such differentials, (what Leibniz called the integral), is the sought after brachistichrone curve.
From one “moment of becoming” to the next, the position of the light changes, as it passes vertically from one density to the next. Each such vertical change in position is accompanied by a horizontal change in position, that corresponds to the sine of the angle of inclination at each “moment of becoming”. (Bernoulli’s geometrical construction of the above can be found on p. 652 of Smith.) Bernoulli adopted Leibniz’ notation for these ideas, calling the vertical change, dy, the horizontal change, dx, and the resulting change in the path of the light, dz. The proportion between the vertical and the horizontal, dy:dx, and the resulting change in the path, dz, is a function of the rate at which the density of the medium is changing.
Bernoulli shows, that if the density of the medium is changing according to the rate at which a body falls under its own weight, (specifically, that the velocity changes according to the square root of the vertical distance) then the resulting curve is a cycloid. “…you will be petrified with astonishment when I say that this cycloid, the tautochrone of Huygens is our required brachistochrone…” he declared.
But, Bernoulli noted that this was not a discovery of a particular physical phenomenon, but a metaphysical discovery of a universal principle:
“Before I conclude, I cannot refrain from again expressing the amazement which I experienced over the unexpected identity of Huygen’s tautochrone and our brachistochrone. Furthermore, I think it is noteworthy that this identity is found only under the hypothesis of Galileo so that even from this we may conjecture that nature wanted it to be thus. For, as nature is accustomed to proceed always in the simplest fashion, so here she accomplishes two different services through one and the same curve, while under every other hypothesis two curves would be necessary the one for oscillations of equal duration the other for quickest descent. If, for example, the velocity of a falling body varied not as the square root but as the cube root of the height falalen through, then the brachistochrone would be algebraic, then tautochrone on the other hand transcendental; but if the velocity varied as the height fallen through then the curves would be algebraic, the one a circle, the other a straight line.”