Men of Mathematics (19 page)

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 whole 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.

*  *  *

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
= a
²
+ 2ab + b
²
, (a
+
b
)
³
= a
³
+ 3
a
²
b
+ 3
ab
²
+ b
³
,

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.