Men of Mathematics

by E. T. Bell

  As a child Newton was not robust and was forced to shun the rough games of boys his own age. Instead of amusing himself in the usual way, Newton invented his own diversions, in which his genius first showed up. It is sometimes said that Newton was not precocious. This may be true so far as mathematics is concerned, but if it is so in other respects a new definition of precocity is required. The unsurpassed experimental genius which Newton was to exhibit as an explorer in the mysteries of light is certainly evident in the ingenuity of his boyish amusements. Kites with lanterns to scare the credulous villagers at night, perfectly constructed mechanical toys which he made entirely by himself and which worked—waterwheels, a mill that ground wheat into snowy flour, with a greedy mouse (who devoured most of the profits) as both miller and motive power, workboxes and toys for his many little girl friends, drawings, sundials, and a wooden clock (that went) for himself—such were some of the things with which this “un-precocious” boy sought to divert the interests of his playmates into “more philosophical” channels. In addition to these more noticeable evidences of talent far above the ordinary, Newton read extensively and jotted down all manner of mysterious recipes and out-of-the-way observations in his notebook. To rate such a boy as merely the normal, wholesome lad he appeared to his village friends is to miss the obvious.

  The earliest part of Newton’s education was received in the common village schools of his vicinity. A maternal uncle, the Reverend William Ayscough, seems to have been the first to recognize that Newton was something unusual. A Cambridge graduate himself, Ayscough finally persuaded Newton’s mother to send her son to Cambridge instead of keeping him at home, as she had planned, to help her manage the farm on her return to Woolsthorpe after her husband’s death when Newton was fifteen.

  Before this, however, Newton had crossed his Rubicon on his own initiative. On his uncle’s advice he had been sent to the Grantham Grammar School. While there, in the lowest form but one, he was tormented by the school bully who one day kicked Newton in the stomach, causing him much physical pain and mental anguish. Encouraged by one of the schoolmasters, Newton challenged the bully to a fair fight, thrashed him, and, as a final mark of humiliation, rubbed his enemy’s cowardly nose on the wall of the church. Up till this young Newton had shown no great interest in his lessons. He now set out to prove his head as good as his fists and quickly rose to the distinction of top boy in the school. The Headmaster and Uncle Ayscough agreed that Newton was good enough for Cambridge, but the decisive die was thrown when Ayscough caught his nephew reading under a hedge when he was supposed to be helping a farmhand to do the marketing.

  While at the Grantham Grammar School, and subsequently while preparing for Cambridge, Newton lodged with a Mr. Clarke, the village apothecary. In the apothecary’s attic Newton found a parcel of old books, which he devoured, and in the house generally, Clarke’s stepdaughter, Miss Storey, with whom he fell in love and to whom he became engaged before leaving Woolsthorpe for Cambridge in June, 1661, at the age of nineteen. But although Newton cherished a warm affection for his first and only sweetheart all her life, absence and growing absorption in his work thrust romance into the background, and Newton never married. Miss Storey became Mrs. Vincent.

  * * *

  Before going on to Newton’s student career at Trinity College we may take a short look at the England of his times and some of the scientific knowledge to which the young man fell heir. The bullheaded and bigoted Scottish Stuarts had undertaken to rule England according to the divine rights they claimed were vested in them, with the not uncommon result that mere human beings resented the assumption of celestial authority and rebelled against the sublime conceit, the stupidity, and the incompetence of their rulers. Newton grew up in an atmosphere of civil war—political and religious—in which Puritans and Royalists alike impartially looted whatever was needed to keep their ragged armies fighting. Charles I (born in 1600, be headed in 1649) had done everything in his power to suppress Parliament; but in spite of his ruthless extortions and the villainously able backing of his own Star Chamber through its brilliant perversions of the law and common justice, he was no match for the dour Puritans under Oliver Cromwell, who in his turn was to back his butcheries and his roughshod march over Parliament by an appeal to the divine justice of his holy cause.

  All this brutality and holy hypocrisy had a most salutary effect on young Newton’s character: he grew up with a fierce hatred of tyranny, subterfuge, and oppression, and when King James later sought to meddle repressively in University affairs, the mathematician and natural philosopher did not need to learn that a resolute show of backbone and a united front on the part of those whose liberties are endangered is the most effective defense against a coalition of unscrupulous politicians; he knew it by observation and by instinct.

  To Newton is attributed the saying “If I have seen a little farther than others it is because I have stood on the shoulders of giants.” He had. Among the tallest of these giants were Descartes, Kepler, and Galileo. From Descartes, Newton inherited analytic geometry, which he found difficult at first; from Kepler, three fundamental laws of planetary motion, discovered empirically after twenty two years of inhuman calculation; while from Galileo he acquired the first two of the three laws of motion which were to be the cornerstone of his own dynamics. But bricks do not make a building; Newton was the architect of dynamics and celestial mechanics.

  As Kepler’s laws were to play the rôle of hero in Newton’s development of his law of universal gravitation they may be stated here.

  I. The planets move round the Sun in ellipses; the Sun is at one focus of these ellipses.

  [If S, S’ are the foci, P any position of a planet in its orbit, SP + S’P is always equal to AA’, the major axis of the ellipse: fig., page 94.]

  II. The line joining the Sun and a planet sweeps out equal areas in equal times.

  III. The square of the time for one complete revolution of each planet is proportional to the cube of its mean [or average] distance from the Sun.

  These laws can be proved in a page or two by means of the calculus applied to Newton’s law of universal gravitation:

  Any two particles of matter in the universe attract one another with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Thus if m, M are the masses of the two particles and d the distance between them (all measured in appropriate units), the force of attraction between them is where k is some constant number (by suitably choosing the units of mass and distance k may be taken equal to 1, so that the attraction is simply ).

  For completeness we state Newton’s three laws of motion.

  I. Every body will continue in its state of rest or of uniform [unaccelerated] motion in a straight line except in so far as it is compelled to change that state by impressed force.

  II. Rate of change of momentum [“mass times velocity,” mass and velocity being measured in appropriate units] is proportional to the impressed force and takes place in the line in which the force acts.

  III. Action and reaction [as in the collision on a frictionless table of perfectly elastic billiard balls] are equal and opposite [the momentum one ball loses is gained by the other].

  The most important thing for mathematics in all of this is the phrase opening the statement of the second law of motion, rate of change. What is a rate, and how shall it be measured? Momentum, as noted, is “mass times velocity.” The masses which Newton discussed were assumed to remain constant during their motion—not like the electrons and other particles of current physics whose masses increase appreciably as their velocity approaches a measurable fraction of that of light. Thus, to investigate “rate of change of momentum,” it sufficed Newton to clarify velocity, which is rate of change of position. His solution of this problem—giving a workable mathematical method for investigating the velocity of any particle moving in any continuous manner, no matter how erratic—gave him the master key to the whol
e mystery of rates and their measurement, namely, the differential calculus.

  A similar problem growing out of rates put the integral calculus into his hands. How shall the total distance passed over in a given time by a moving particle whose velocity is varying continuously from instant to instant be calculated? Answering this or similar problems, some phrased geometrically, Newton came upon the integral calculus. Finally, pondering the two types of problem together, Newton made a capital discovery: he saw that the differential calculus and the integral calculus are intimately and reciprocally related by what is today called “the fundamental theorem of the calculus”—which will be described in the proper place.

  * * *

  In addition to what Newton inherited from his predecessors in science and mathematics he received from the spirit of his age two further gifts, a passion for theology and an unquenchable thirst for the mysteries of alchemy. To censure him for devoting his unsurpassed intellect to these things, which would now be considered unworthy of his serious effort, is to censure oneself. For in Newton’s day alchemy was chemistry and it had not been shown that there was nothing much in it—except what was to come out of it, namely modern chemistry; and Newton, as a man of inborn scientific spirit, undertook to find out by experiment exactly what the claims of the alchemists amounted to.

  As for theology, Newton was an unquestioning believer in an allwise Creator of the universe and in his own inability—like that of the boy on the seashore—to fathom the entire ocean of truth in all its depths. He therefore believed that there were not only many things in heaven beyond his philosophy but plenty on earth as well, and he made it his business to understand for himself what the majority of intelligent men of his time accepted without dispute (to them it was as natural as common sense)—the traditional account of creation.

  He therefore put what he considered his really serious efforts on attempts to prove that the prophecies of Daniel and the poetry of the Apocalypse make sense, and on chronological researches whose object was to harmonize the dates of the Old Testament with those of history. In Newton’s day theology was still queen of the sciences and she sometimes ruled her obstreperous subjects with a rod of brass and a head of cast iron. Newton however did permit his rational science to influence his beliefs to the extent of making him what would now be called a Unitarian.

  * * *

  In June, 1661 Newton entered Trinity College, Cambridge, as a subsizar—a student who (in those days) earned his expenses by menial service. Civil war, the restoration of the monarchy in 1661, and uninspired toadying to the Crown on the part of the University had all brought Cambridge to one of the low-water marks in its history as an educational institution when Newton took up his residence. Nevertheless young Newton, lonely at first, quickly found himself and became absorbed in his work.

  In mathematics Newton’s teacher was Dr. Isaac Barrow (16301677), a theologian and mathematician of whom it has been said that brilliant and original as he undoubtedly was in mathematics, he had the misfortune to be the morning star heralding Newton’s sun. Barrow gladly recognized that a greater than himself had arrived, and when (1669) the strategic moment came he resigned the Lucasian Professorship of Mathematics (of which he was the first holder) in favor of his incomparable pupil. Barrow’s geometrical lectures dealt among other things with his own methods for finding areas and drawing tangents to curves—essentially the key problems of the integral and the differential calculus respectively, and there can be no doubt that these lectures inspired Newton to his own attack.

  The record of Newton’s undergraduate life is disappointingly meager. He seems to have made no very great impression on his fellow students, nor do his brief, perfunctory letters home tell anything of interest. The first two years were spent mastering elementary mathematics. If there is any reliable account of Newton’s sudden maturity as a discoverer, none of his modern biographers seems to have located it. Beyond the fact that in the three years 1664-66 (age twenty one to twenty three) he laid the foundation of all his subsequent work in science and mathematics, and that incessant work and late hours brought on an illness, we know nothing definite. Newton’s tendency to secretiveness about his discoveries has also played its part in deepening the mystery.

  On the purely human side Newton was normal enough as an undergraduate to relax occasionally, and there is a record in his account book of several sessions at the tavern and two losses at cards. He took his B.A. degree in January, 1664.

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  The Great Plague (bubonic plague) of 1664-65, with its milder recurrence the following year, gave Newton his great if forced opportunity. The University was closed, and for the better part of two years Newton retired to meditate at Woolsthorpe. Up till then he had done nothing remarkable—except make himself ill by too assiduous observation of a comet and lunar halos—or, if he had, it was a secret. In these two years he invented the method of fluxions (the calculus), discovered the law of universal gravitation, and proved experimentally that white light is composed of light of all the colors. All this before he was twenty five.

  A manuscript dated May 20, 1665, shows that Newton at the age of twenty three had sufficiently developed the principles of the calculus to be able to find the tangent and curvature at any point of any continuous curve. He called his method “fluxions”—from the idea of “flowing” or variable quantities and their rates of “flow” or “growth.” His discovery of the binomial theorem, an essential step toward a fully developed calculus, preceded this.

  The binomial theorem generalizes the simple results like

  (a + b)2 = a2 + 2ab + b2, (a + b)3 = a3 + 3a2b + 3ab2 + b3,

  and so on, which are found by direct calculation; namely,

  where the dots indicate that the series is to be continued according to the same law as that indicated for the terms written; the next term is

  If n is one of the positive integers 1, 2, 3 . . . , the series automatically terminates after precisely n + 1 terms. This much is easily proved (as in the school algebras) by mathematical induction.

  But if n is not a positive integer, the series does not terminate, and this method of proof is inapplicable. As a proof of the binomial theorem for fractional and negative values of n (also for more general values), with a statement of the necessary restrictions on a,b, came only in the nineteenth century, we need merely state here that in extending the theorem to these values of n Newton satisfied himself that the theorem was correct for such values of a,b as he had occasion to consider in his work.

  If all modern refinements are similarly ignored in the manner of the seventeenth century it is easy to see how the calculus finally got itself invented. The underlying notions are those of variable, function, and limit. The last took long to clarify.

  A letter, say s, which can take on several different values during the course of a mathematical investigation is called a variable; for example s is a variable if it denotes the height of a falling body above the earth.

  The word function (or its Latin equivalent) seems to have been introduced into mathematics by Leibniz in 1694; the concept now dominates much of mathematics and is indispensable in science. Since Leibniz’ time the concept has been made precise. If y and x are two variables so related that whenever a numerical value is assigned to x there is determined a numerical value of y, then y is called a (one-valued, or uniform) function of x, and this is symbolized by writings y =f(x).

  Instead of attempting to give a modern definition of a limit we shall content ourselves with one of the simplest examples of the sort which led the followers of Newton and Leibniz (the former especially) to the use of limits in discussing rates of change. To the early developers of the calculus the notions of variables and limits were intuitive; to us they are extremely subtle concepts hedged about with thickets of semimetaphysical mysteries concerning the nature of numbers, both rational and irrational.

  Let y be a function of x, say y = f(x). The rate of change of y with respect to x, or, as it is called, the derivat
ive of y with respect to x, is defined as follows. To x is given any increment, say Δx (read, “increment of x”), so that x becomes x + Δx, and f(x), or y, becomes f(x + Δx). The corresponding increment, Δy, of y is its new value minus its initial value; namely, Δy = f(x + Δx) —f(x). As a crude approximation to the rate of change of y with respect to x we may take, by our intuitive notion of a rate as an “average,” the result of dividing the increment of y by the increment of x, that is,

  But this obviously is too crude, as both of x and y are varying and we cannot say that this average represents the rate for any particular value of x. Accordingly, we decrease the increment Δx indefinitely, till, “in the limit” Δx approaches zero, and follow the “average” all through the process: Δy similarly decreases indefinitely and ultimately approaches zero; but does not, thereby, present us with the meaningless symbol but with a definite limiting value, which is the required rate of change of y with respect to x.

  To see how it works out, let f(x) be the particular function x2, so that y = x2. Following the above outline we get first

  Nothing is yet said about limits. Simplifying the algebra we find

  Having simplified the algebra as far as possible, we now let Δx approach zero and see that the limiting value of is 2x. Quite generally, in the same way, if y = xn, the limiting value of is nxn–l, as may be proved with the aid of the binomial theorem.

  Such an argument would not satisfy a student today, but something not much better was good enough for the inventors of the calculus and it will have to do for us here. If y = f(x), the limiting value of (provided such a value exists) is called the derivative of y with respect to x, and is denoted by This symbolism is due (essentially) to Leibniz and is the one in common use today; Newton used another () which is less convenient.