3
 
To Resolve Problems by Motion

CAMBRIDGE IN 1664 HAD for the first time in its history a professor of mathematics, Isaac Barrow, another former Trinity College sizar, a decade older than Newton. Barrow had first studied Greek and theology; then left Cambridge, learned medicine, more theology, church history, and astronomy, and finally turned to geometry. Newton attended Barrow’s first lectures. He was standing for examinations that year, on his way to being elected a scholar, and it was Barrow who examined him, mainly on the Elements of Euclid. He had not studied it before. At Stourbridge Fair he found a book of astrology and was brought up short by a diagram that required an understanding of trigonometry¹—more than any Cambridge student was meant to know. He bought and borrowed more books. Before long, in a few texts, he had at hand a précis of the advanced mathematics available on the continent of Europe. He bought Franz van Schooten’s Miscellanies and his Latin translation of Descartes’s difficult masterpiece, La Géométrie; then William Oughtred’s Clavis Mathematicæ and John Wallis’s Arithmetica Infinitorum.² This reading remained far from comprehensive. He was inventing more than absorbing.

At the end of that year, just before the winter solstice, a comet appeared low in the sky, its mysterious tail blazing toward the west. Newton stayed outdoors night after night, noting a path against the background of the fixed stars, watching till it vanished in the light of each dawn, and only then returned to his room, sleepless and disordered. A comet was a frightening portent, a mutable and irregular traveler through the firmament. Nor was that all: rumors were reaching England of a new pestilence in Holland—perhaps from Italy or the Levant, perhaps from Crete or Cyprus.

Hard behind the rumors came the epidemic. Three men in London succumbed in a single house; by January the plague, this disease of population density, was spreading from parish to parish, hundreds dying each week, then thousands. Before the outbreak ran its course, in little more than a year, it killed one of every six Londoners.³ Newton’s mother wrote from Woolsthorpe:

Isack

received your leter and I perceive you
           letter from me with your cloth but
none to you your sisters present thai
love to you with my motherly lov
you and prayers to god for you I
your loving mother

     hanah
     wollstrup may the 6. 1665⁴

The colleges of Cambridge began shutting down. Fellows and students dispersed into the countryside.

Newton returned home. He built bookshelves and made a small study for himself. He opened the nearly blank thousand-page commonplace book he had inherited from his stepfather and named it his Waste Book.⁵ He began filling it with reading notes. These mutated seamlessly into original research. He set himself problems; considered them obsessively; calculated answers, and asked new questions. He pushed past the frontier of knowledge (though he did not know this). The plague year was his transfiguration.⁶ Solitary and almost incommunicado, he became the world’s paramount mathematician.

Most of the numerical truths and methods that people had discovered, they had forgotten and rediscovered, again and again, in cultures far removed from one another. Mathematics was evergreen. One scion of Homo sapiens could still comprehend virtually all that the species knew collectively. Only recently had this form of knowledge begun to build upon itself.⁷ Greek mathematics had almost vanished; for centuries, only Islamic mathematicians had kept it alive, meanwhile inventing abstract methods of problem solving called algebra. Now Europe became a special case: a region where people were using books and mail and a single language, Latin, to span tribal divisions across hundreds of miles; and where they were, self-consciously, receiving communications from a culture that had flourished and then disintegrated more than a thousand years before. The idea of knowledge as cumulative—a ladder, or a tower of stones, rising higher and higher—existed only as one possibility among many. For several hundred years, scholars of scholarship had considered that they might be like dwarves seeing farther by standing on the shoulders of giants, but they tended to believe more in rediscovery than in progress. Even now, when for the first time Western mathematics surpassed what had been known in Greece, many philosophers presumed they were merely uncovering ancient secrets, found in sunnier times and then lost or hidden.

With printed books had come a new metaphor for the world’s organization. The book was a container for information, designed in orderly patterns, encoding the real in symbols; so, perhaps, was nature itself. The book of nature became a favorite conceit of philosophers and poets: God had written; now we must read.⁸ “Philosophy is written in this grand book—I mean the universe—which stands continually open to our gaze,” said Galileo. “But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics.…”⁹

But by mathematics he did not mean numbers: “Its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.”

The study of different languages created an awareness of language: its arbitrariness, its changeability. As Newton learned Latin and Greek, he experimented with shorthand alphabets and phonetic writing, and when he entered Trinity College he wrote down a scheme for a “universal” language, based on philosophical principles, to unite the nations of humanity. “The Dialects of each Language being soe divers & arbitrary,” he declared, “a generall Language cannot bee so fitly deduced from them as from the natures of things themselves.”¹⁰ He understood language as a process, an act of transposition or translation—the conversion of reality into symbolic form. So was mathematics, symbolic translation at its purest.

For a lonely scholar seeking his own path through tangled thickets, mathematics had a particular virtue. When Newton got answers, he could usually judge whether they were right or wrong, no public disputation necessary. He read Euclid carefully now. The Elements—transmitted from ancient Alexandria via imperfect Greek copies, translated into medieval Arabic, and translated again into Latin—taught him the fundamental program of deducing the properties of triangles, circles, lines, and spheres from a few given axioms.¹¹ He absorbed Euclid’s theorems for later use, but he was inspired by the leap of Descartes’s Géométrie, a small and rambling text, the third and last appendix to his Discours de la Méthode.¹² This forever joined two great realms of thought, geometry and algebra. Algebra (a “barbarous” art, Descartes said,¹³ but it was his subject nonetheless) manipulated unknown quantities as if they were known, by assigning them symbols. Symbols recorded information, spared the memory, just as the printed book did.¹⁴ Indeed, before texts could spread by printing, the development of symbolism had little point.

With symbols came equations: relations between quantities, and changeable relations at that. This was new territory, and Descartes exploited it. He treated one unknown as a spatial dimension, a line; two unknowns thus define a plane. Line segments could now be added and even multiplied. Equations generated curves; curves embodied equations. Descartes opened the cage doors, freeing strange new bestiaries of curves, far more varied than the elegant conic sections studied by the Greeks. Newton immediately began expanding the possibilities, adding dimensions, generalizing, mapping one plane to another with new coordinates. He taught himself to find real and complex roots of equations and to factor expressions of many terms—polynomials. When the infinite number of points in a curve correspond to the infinite solutions of its equation, then all the solutions can be seen at once, as a unity. Then equations have not just solutions but other properties: maxima and minima, tangents and areas. These were visualized, and they were named.

Early Newton drawings of apparatus. (illustration credit 3.1)

No one understands the mental faculty we call mathematical intuition; much less, genius. People’s brains do not differ much, from one to the next, but numerical facility seems rarer, more special, than other talents. It has a threshold quality. In no other intellectual realm does the genius find so much common ground with the idiot savant. A mind turning inward from the world can see numbers as lustrous creatures; can find order in them, and magic; can know numbers as if personally. A mathematician, too, is a polyglot. A powerful source of creativity is a facility in translating, seeing how the same thing can be said in seemingly different ways. If one formulation doesn’t work, try another.

Newton’s patience was limitless. Truth, he said much later, was “the offspring of silence and meditation.”¹⁵

And he said: “I keep the subject constantly before me and wait ’till the first dawnings open slowly, by little and little, into a full and clear light.”¹⁶

Newton’s Waste Book filled day by day with new research in this most abstract of realms. He computed obsessively. He worked out a way to transform equations from one set of axes to any alternative frame of reference. On one page he drew a hyperbola and set about calculating the area under it—“squaring” it. He stepped past the algebra Descartes knew. He would not confine himself to expressions of a few (or many) terms; instead he constructed infinite series: expressions that continue forever.¹⁷ An infinite series need not sum to infinity; rather, because the terms could grow smaller and smaller, they could close in on a goal or limit. He conceived such a series to square the hyperbola—

—and carried out the calculation to fifty-five decimal places: in all, more than two thousand tiny digits marching down a single page in orderly formation.¹⁸ To conceive of infinite series and then learn to manipulate them was to transform the state of mathematics. Newton seemed now to possess a limitless ability to generalize, to move from one or a few particular known cases to the universe of all cases. Mathematicians had a glimmering notion of how to raise the sum of two quantities, a + b, to some power. Through infinite series, Newton discovered in the winter of 1664 how to expand such sums to any power, integer or not: the general binomial expansion.

(illustration credit 4.1)

He relished the infinite, as Descartes had not. “We should never enter into arguments about the infinite,” Descartes had written.

For since we are finite, it would be absurd for us to determine anything concerning the infinite; for this would be to attempt to limit it and grasp it. So we shall not bother to reply to those who ask if half an infinite line would itself be infinite, or whether an infinite number is odd or even, and so on. It seems that nobody has any business to think about such matters unless he regards his own mind as infinite.¹⁹

Yet it turns out that the human mind, though bounded in a nutshell, can discern the infinite and take its measure.

A special aspect of infinity troubled Newton; he returned to it again and again, turning it over, restating it with new definitions and symbols. It was the problem of the infinitesimal—the quantity, impossible and fantastic, smaller than any finite quantity, yet not so small as zero. The infinitesimal was anathema to Euclid and Aristotle. Nor was Newton altogether at ease with it.²⁰ First he thought in terms of “indivisibles”—points which, when added to one another infinitely, could perhaps make up a finite length.²¹ This caused paradoxes of dividing by zero:

—nonsensical results if 0 is truly zero, but necessary if 0 represents some indefinitely small, “indivisible” quantity. Later he added an afterthought—

        (that is undetermined)

Tis indefinite how greate a sphære may be made how greate a number may be reckoned, how far matter is divisible, how much time or extension wee can fansy but all the Extension that is, Eternity, are infinite.²²

—blurring the words indefinite and undetermined by applying them alternately to mathematical quantities and degrees of knowledge. Descartes’s reservations notwithstanding, the infinitude of the universe was in play—the boundlessness of God’s space and time. The infinitesimal—the almost nothing—was another matter. It might have been simply the inverse problem: the infinitely large and the infinitely small. A star of finite size, if it could be seen at an infinite distance, would appear infinitesimal. The terms in Newton’s infinite series approached the infinitesimal. “We are among infinities and indivisibles,” Galileo said, “the former incomprehensible to our understanding by reason of their largeness, and the latter by their smallness.”²³

Newton was seeking better methods—more general—for finding the slope of a curve at any particular point, as well another quantity, related but once removed, the degree of curvature, rate of bending, “the crookedness in lines.”²⁴ He applied himself to the tangent, the straight line that grazes the curve at any point, the straight line that the curve would become at that point, if it could be seen through an infinitely powerful microscope. He drew intricate constructions, more complex and more free than anything in Euclid or Descartes. Again and again he confronted the specter of the infinitesimal: “Then (if hs & cd have an infinitely little distance otherwise not) …”; “… (which operacon cannot in this case bee understood to bee good unlesse infinite littleness may bee considered geometrically).…”²⁵ He could not escape it, so he pressed it into service, employing a private symbol—a little o—for this quantity that was and was not zero. In some of his diagrams, two lengths differed “but infinitely little,” while two other lengths had “no difference at all.” It was essential to preserve this uncanny distinction. It enabled him to find areas by infinitely partitioning curves and infinitely adding the partitions. He created “a Method whereby to square those crooked lines which may bee squared”²⁶—to integrate (in the later language of the calculus).

As algebra melded with geometry, so did a physical counterpart, the problem of motion. Whatever else a curve was, it naturally represented the path of a moving point. The tangent represented the instantaneous direction of motion. An area could be generated by a line sweeping across the plane. To think that way was to think kinetically. It was here that the infinitesimal took hold. Motion was smooth, continuous, unbroken—how could it be otherwise? Matter might reduce to indivisible atoms, but to describe motion, mathematical points seemed more appropriate. A body on its way from a to b must surely pass through every point between. There must be points between, no matter how close a is to b; just as between any pair of numbers, more numbers must be found. But this continuum evoked another form of paradox, as Greek philosophers had seen two thousand years before: the paradox of Achilles and the tortoise. The tortoise has a head start. Achilles can run faster but can never catch up, because each time he reaches the tortoise’s last position, the tortoise has managed to crawl a bit farther ahead. By this logic Zeno proved that no moving body could ever reach any given place—that motion itself did not exist. Only by embracing the infinite and the infinitesimal, together, could these paradoxes be banished. A philosopher had to find the sum of infinitely many, increasingly small intervals. Newton wrestled with this as a problem of words: swifter, slower; least distance, least progression; instant, interval.

That it may be knowne how motion is swifter or slower consider: that there is a least distance, a least progression in motion & a least degree of time.… In each degree of time wherein a thing moves there will be motion or else in all those degrees put together there will be none:… no motion is done in an instant or intervall of time.²⁷

A culture lacking technologies of time and speed also lacked basic concepts that a mathematician needed to quantify motion. The English language was just beginning to adapt its first unit of velocity: the term knot, based on the sailor’s only speed-measuring device, the log line heaved into the sea. The science most eager to understand the motion of earthly objects, ballistics, measured the angles of gun barrels and the distances their balls traveled, but scarcely conceived of velocity; even when they could define this quantity, as a ratio of distance and time, they could not measure it. Galileo, when he dropped weights from towers, could make only the crudest estimates of their velocity, though he used an esoteric unit of time: seconds of an hour. Newton was struck by the ambition in his exactitude: “According to Galileo an iron ball of 100 lb. Florentine (that is 78 lb. at London avoirdupois weight) descends 100 Florentine braces or cubits (or 49.01 Ells, perhaps 66 yds.) in 5 seconds of an hour.”²⁸

In the autumn of 1665 he made notes on “mechanical” lines, as distinguished from the merely geometric. Mechanical curves were those generated by the motion of a point, or by two such motions compounded: spirals, ellipses, and cycloids. Descartes had considered the cycloid, the curve generated by a point on a circle as the circle rolls along a line. He regarded this oddity as suspect and unmathematical, because it could not (before the calculus) be described analytically. But such artifacts from the new realm of mechanics kept intruding on mathematics. Hanging cables or sails in the wind traced mechanical curves.²⁹ If a cycloid was mechanical, it was nevertheless an abstraction: a creature of several motions, or rates, summed in a certain way. Indeed, Newton now saw ellipses in different lights—geometrical and analytical. The ellipse was the effect of a quadratic formula. Or it was the closed line drawn in the dirt by the “gardener’s” construction, in which a loose cord is tied to two pegs in the ground: “keeping it so stretched out draw the point b about & it shall describe the Ellipsis.”³⁰ Or it was a circle with extra freedom; a circle with one constraint removed; a squashed circle, its center bifurcating into a pair of foci. He devised procedures for drawing tangents to mechanical curves, thus measuring their slopes; and, in November, proposed a method for deducing, from two or more such lines, the corresponding relation between the velocities of two or more moving bodies.³¹

He found tangents by computing the relationship between points on a curve separated by an infinitesimal distance. In the computation, the points almost merge into one, “conjoyne, which will happen when bc = O, vanisheth into nothing.”³² That O was an artifice, a gadget for the infinitesimal, as an arbitrarily small increment or a moment of time. He showed how the terms with O “may be ever blotted out.”³³ Extending his methods, he also quantified rates of bending, by finding centers of curvature and radii of curvature.

A geometrical task matched a kinetic task: to measure curvature was to find a rate of change. Rate of change was itself an abstraction of an abstraction; what velocity was to position, acceleration was to velocity. It was differentiation (in the later language of the calculus). Newton saw this system whole: that problems of tangents were the inverse of problems of quadrature; that differentiation and integration are the same act, inverted. The procedures seem alien, one from the other, but what one does, the other undoes. That is the fundamental theorem of the calculus, the piece of mathematics that became essential knowledge for building engines and measuring dynamics. Time and space—joined. Speed and area—two abstractions, seemingly disjoint, revealed as cognate.

Repeatedly he started a new page—in November 1665, in May 1666, and in October 1666—in order to essay a system of propositions needed “to resolve Problems by motion.”³⁴ On his last attempt he produced a tract of twenty-four pages, on eight sheets of paper folded and stitched together. He considered points moving toward the centers of circles; points moving parallel to one another; points moving “angularly” or “circularly”—this language was unsettled—and points moving along lines that intersected planes. A variable representing time underlay his equations—time as an absolute background for motion. When velocity changed, he imagined it changing smoothly and continuously—across infinitesimal moments, represented by that o. He issued himself instructions:

Set all the termes on one side of the Equation that they become equall to nothing. And first multiply each terme by so many times as x hath dimensions in the terme. Secondly multiply each term by so many times  … & if there bee still more unknowne quantitys doe like to every unknowne quantity.³⁵

Time was a flowing thing. In terms of velocity, position was a function of time. But in terms of acceleration, velocity was itself a function of time. Newton made up his own notation, with combinations of superscript dots, and vocabulary, calling these functions “fluents” and “fluxions”: flowing quantities and rates of change. He wrote it all several times but never quite finished.

In creating this mathematics Newton embraced a paradox. He believed in a discrete universe. He believed in atoms, small but ultimately indivisible—not infinitesimal. Yet he built a mathematical framework that was not discrete but continuous, based on a geometry of lines and smoothly changing curves. “All is flux, nothing stays still,” Heraclitus had said two millennia before. “Nothing endures but change.” But this state of being—in flow, in change—defied mathematics then and afterward. Philosophers could barely observe continuous change, much less classify it and gauge it, until now. It was nature’s destiny now to be mathematized. Henceforth space would have dimension and measure; motion would be subject to geometry.³⁶

Far away across the country multitudes were dying in fire and plague. Numerologists had warned that 1666 would be the Year of the Beast. Most of London lay in black ruins: fire had begun in a bakery, spread in the dry wind across thatch-roofed houses, and blazed out of control for four days and four nights. The new king, Charles II—having survived his father’s beheading and his own fugitive years, and having outlasted the Lord Protector, Cromwell—fled London with his court. Here at Woolsthorpe the night was strewn with stars, the moon cast its light through the apple trees, and the day’s sun and shadows carved their familiar pathways across the wall. Newton understood now: the projection of curves onto flat planes; the angles in three dimensions, changing slightly each day. He saw an orderly landscape. Its inhabitants were not static objects; they were patterns, process and change.

What he wrote, he wrote for himself alone. He had no reason to tell anyone. He was twenty-four and he had made tools.