Isaac Newton (7 page)

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.

4
 
Two Great Orbs

H
ISTORIANS CAME TO SEE
Newton as an end-point: the “culmination” and “climax” of an episode in human affairs conventionally called the Scientific Revolution. Then that term began to require apologies or ironic quotation marks.
¹
Ambivalence is appropriate, when one speaks of the turning point in the development of human culture, the time when reason triumphed over unreason. The Scientific Revolution is a story, a narrative frame laid down with hindsight. Yet it exists and existed, not just in the backward vision of historians but in a self-consciousness among a small number of people in England and Europe in the seventeenth century. They were, as they thought, virtuosi. They saw something new in the domain of knowledge; they tried to express the newness; they invented academies and societies and opened channels of communication to promote their break with the past, their
new
science.

We call the Scientific Revolution an epidemic, spreading across the continent of Europe during two centuries: “It would come to rest in England, in the person of Isaac Newton,” said the physicist David Goodstein. “On the way north, however, it stopped briefly in France.…”
²
Or a
relay race, run by a team of heroes who passed the baton from one to the next:
COPERNICUS
to
KEPLER
to
GALILEO
to
NEWTON
. Or the overthrow and destruction of the Aristotelian cosmology: a worldview that staggered under the assaults of Galileo and Descartes and finally expired in 1687, when Newton published a book.
³

For so long the earth had seemed the center of all things. The constellations turned round in their regular procession. Just a few bright objects caused a puzzle—the planets, wanderers, like gods or messengers, moving irregularly against the fixed backdrop of stars. In 1543, just before his death, Nicolaus Copernicus, Polish astronomer, astrologer, and mathematician, published the great book
De Revolutionibus Orbium Coelestium
(“On the Revolutions of the Heavenly Spheres”). In it he gave order to the planets’ paths, resolving them into perfect circles; he set the earth in motion and placed an immobile sun at the center of the universe.
⁴

Johannes Kepler, looking for more order in a growing thicket of data, thousands of painstakingly recorded observations, declared that the planets could not be moving in circles. He suspected the special curves known to the ancients as ellipses. Having thus overthrown one kind of celestial perfection, he sought new kinds, believing fervently in a universe built on geometrical harmony. He found an elegant link between geometry and motion by asserting that an imaginary line from a planet to the sun sweeps across equal areas in equal times.
⁵

Galileo Galilei took spy-glasses—made by inserting spectacle makers’ lenses into a hollow tube—and pointed them upward toward the night sky. What he saw both inspired and
disturbed him: moons orbiting Jupiter; spots marring the sun’s flawless face; stars that had never been seen—“in numbers ten times exceeding the old and familiar stars.”
⁶
He learned, “with all the certainty of sense evidence,” that the moon “is not robed in a smooth polished surface but is in fact rough and uneven.” It has mountains, valleys, and chasms. (He also thought he had detected an atmosphere of dense and luminous vapors.)