When Computers Were Human (3 page)

The only aspects of the 1682 comet that could be studied with certainty were its position against the fixed stars of night and the length of its tail. The young Edmund Halley recorded both measurements on at least seven distinct nights that summer. He was a gentleman of private life, possessed of an independent income and a new house in a prosperous village just north of London. His collection of scientific instruments included a sextant, a small telescope mounted on an arc of a circle, which allowed him to measure the distance of the comet's head from nearby stars. His measurements were not in miles or meters or light-years but in degrees of an angle. His home marked the joint of that celestial angle. One leg stretched from the earth to the head of the comet. The second leg reached to a star, the end of the tail, or some other reference point. The work required patience and a steady hand. By the time the comet vanished, Halley had traced its path across the sky and recorded the advance and retreat of the tail. At the time, it was not entirely clear what Halley might do with these measurements. If they had been the measurements of a planet, he might have computed an orbit, but few believed that comets moved in ellipses around the sun as the
planets did. Halley had other interests to pursue, so he put his comet data away for future use.

2. Halley's comet over Cambridge, 1682

The new method of mathematics was calculus, a subject then known in England as fluxions. Calculus is the mathematics of physical activity, the mathematics of change. It probes the nature of movement by dividing it into smaller and smaller steps and then reassembling these tiny units into surprisingly elegant and simple expressions. The techniques of calculus had their origin in an attempt to explain the motion of the planets by physical laws rather than by the arbitrary actions of superhuman beings. The
English proponent of calculus was Isaac Newton (1642–1727), who developed the method while he was writing his masterwork,
Philosophiae Naturalis Principia Mathematica
(
The Mathematical Principles of Natural Philosophy
), a book commonly called
Principia
. In
Principia
, Newton explained that he was attempting to analyze “the motions of the planets, the comets, the moon, and the sea,” the last term referring to the movements of the tides.
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In the central part of the book, Newton considered the motion of two objects under the influence of a single universal force, which he called gravity. The two objects might be the moon and the earth, a planet and the sun, or even a comet and some other celestial object. In these circumstances, Newton argued that gravity impels the bodies to follow certain kinds of paths: the gentle bend of the hyperbola, the tight hairpin of a parabola, and the cyclical orbit of an ellipse.

The intractable problem appeared when the calculus of Newton met the comet data of Halley. Halley called upon Newton in 1684, when
Principia
was nothing more than a collection of notes. He helped Newton prepare the final manuscript for publication in 1687 and promoted Newton's ideas at the Royal Society, the central organization of seventeenth-century English science. Though he frequently thought about the problems of comets and astronomy, he let thirteen years pass after his initial observations in 1682 before he undertook a serious analysis of his data. During those intervening years, he had other problems to keep him busy. He served as clerk to the Royal Society and as the editor of its journal,
Philosophical Transactions
. He also studied a number of other scientific problems, such as the design of diving bells and the mathematics of finance.

In September 1695, Halley returned to his comet data and attempted to validate the statements that Newton had made about comets in
Principia
. Newton had speculated that comets moved in parabolas around the sun, narrow curves that started at a distant point in the universe, sped past the earth, turned sharply at the sun, and then rushed back to the void whence they came. It seemed a plausible theory, but he had never done the analysis to verify it.
²
Halley spent about a month working with the measurements from four different comets, trying to identify the path that each object made through the solar system. From an individual comet, he would select three observations, each recorded on a different day. From these numbers he computed the parameters of a parabolic curve. Newton had done this sort of work with graphs, but after a little practice Halley could report, “I am now become so ready at the finding a Cometts orb by calculation.”
³
Once he had calculated the parabola, he adjusted the curve by comparing it to the other observations of the comet. If he found that all of the observations were close to the parabola, he would conclude that he had found the proper path. If he discovered that some of them fell at a distance from the curve, he would attempt
to adjust the parameters in order to bring the parabola closer to the observations.

The procedure worked well for the first three comets: one observed by Newton in 1664, a second that Halley had observed just before the 1682 comet, and a third that had appeared shortly after.
⁴
Each of these objects seemed to followed a parabolic curve. When Halley began to work on the 1682 comet, the comet that he had observed from his home, he altered his methods. He chose to fit the data to a closed ellipse rather than an open-ended parabola. Halley's biographer has noted that this idea did not come from calculation but was “based upon somewhat inspired insight.”
⁵
Halley had noted that the 1682 comet followed a path that had been traversed by two earlier comets, one observed in 1531 by the German astronomer Peter Apian (1495–1552) and a second recorded in 1607 by Johannes Kepler. With his 1682 data, Halley computed the values for an elliptical orbit and then compared the curve to the earlier observations. Pleased with the results, he wrote to Newton, “I am more and more confirmed that we have seen that Comett now three times since ye Year 1531.”
⁶

Though he was certain that the 1682 comet orbited the Sun, Halley recognized that his calculations did not prove his claim. His work did not address a substantial inconsistency in his data. Seventy-six years separated Apian's observations from those of Kepler. Only seventy-five years passed between Kepler's sighting and Halley's data from 1682. The analysis suggested that the comet should have a fixed period, that it should return without fail every seventy-five years. Halley speculated that the discrepancy might be caused by the gravitational pull of the outer planets, forces which could easily disturb the orbit of the comet and change the date of its return. Writing to Newton, he asked, “When your more important business is over, I must entreat you to consider how far a Comet's motion may be disturbed by the Centers of Saturn and Jupiter, particularly in its ascent from the Sun.”
⁷

Newton responded quickly, but his reply was vague and unhelpful. “How far a comet's motion may be disturbed,” he wrote to Halley, “cannot be affirmed without knowing the Orb of ye Comet & times of its transit through ye Orbs of [the two planets].”
⁸
Once Saturn and Jupiter became part of the equations, the calculations were no longer straightforward and could not be handled by a single astronomer in his spare minutes and hours. The Sun, Saturn, and Jupiter form a three-body system, three objects moving through space, each exerting an influence upon the other two. Newton had been unable to find a simple expression that described the motion of such a system, even though he had been able to find solutions for two bodies in motion. In his best effort, he had devised an approximation that crudely described the movement of three bodies,
but this expression was not precise enough to explain the variation in the comet's period.
⁹

The lack of a simple solution to the three-body problem stymied Halley's calculations, but it did not shake his faith. He freely discussed his ideas in public and published his theory of comets in
Astronomiae Cometicae Synopsis
(
A Synopsis of the Astronomy of Comets
).
¹⁰
In this book, he claimed that he could “undertake confidently to predict the return” of the comet in 1758. Some scholars noted a lack of mathematical rigor in Halley's analysis and questioned this claim. Responding to the criticism, Halley weakened his statements, claimed that the comet might return at any time within a 600-day period that began in 1757, and replaced his confident prediction with a sentence that began, “I think, I may venture to foretell” the return of the comet.
¹¹

From time to time, Halley tried to improve his predictions for the 1758 return. He made little progress, as he was unwilling, or perhaps unable, to refine his estimates into a detailed computation. His final effort occurred in about 1720, just before he became Astronomer Royal and director of the Royal Observatory in Greenwich. For this calculation, he had a new approximate solution for the three-body problem of Saturn, Jupiter, and the Sun. From this solution, he deduced that the comet was pulled farther from the Sun after its 1682 return and hence would require more time to traverse its path. It was one more crude estimate, but it would stand as his final word on the subject. In his last revision of his
Astronomiae Cometicae
, which was published after his death in 1742, he announced that his comet would return “about the end of the year 1758, or the beginning of the next.”
¹²
With this opinion on the subject, he bequeathed the comet to future generations. “Having touched upon these things,” he wrote, “I shall leave them to be discussed by the care of posterity, after the truth is found out by the event.”
¹³

Posterity made the return of Halley's comet a test for Newton's theory of gravitation. Newton's “followers have, from his principles, ventured even to predict the returns of several [comets],” wrote the Scottish philosopher Adam Smith (1723–1790), “particularly of one which is to make its appearance in 1758.” If scientists could predict the date of return, they would take the agreement between prediction and observation as evidence that Newton's ideas on gravity were correct. If the predicted date did not coincide with the actual date, then they would conclude that other forces were at work in the universe. Smith believed that Newton's analysis was probably correct. “His system,” he stated, “now prevails over all opposition, and has advanced to the acquisition of the most universal empire that was ever established in philosophy.” However, Smith was not willing to accept the prediction for Halley's comet without a proper test. “We must wait for that time before we can determine,
whether his philosophy corresponds as happily to [comets] as to all the [planets].”
¹⁴

A thorough test of the gravitational theory required computational techniques beyond the mathematics that Halley had used for his initial analysis of the comet. Newton's calculus would never provide a simple way to describe the motion of three or more bodies and hence would never give an accurate date for the comet's return. The only way to determine the comet's orbit was to substitute brawn for brain, to divide the comet's progress into tiny steps, analyze the forces pulling on the comet, and then combine these steps into a serviceable whole through the tedious process of summation. “What immense labor,” wrote one astronomer, “what geometrical knowledge did not this task require?”
¹⁵
Among the astronomers that followed Edmund Halley, few even considered undertaking the labor. Only one, a French mathematician named Alexis-Claude Clairaut (1713–1765), made a serious attempt to predict the date of the 1758 return, an attempt that required both a computational technique beyond those developed by Newton and the means of dividing the work among computing assistants.

Clairaut was described by his contemporaries as “ambitious,” “vivacious by nature,” and “successful in society.”
¹⁶
He had already made a reputation as a mathematician by extending Newton's calculus and developing a computational method of handling the three-body problem. He had used his method to find solutions to other problems in astronomy, but the challenge of Halley's comet had a special appeal. The comet was well known to astronomers in France and England. Observers in both countries were scanning the sky for a fuzzy speck of light that would be the first sign of the comet. If he could predict the point of the comet's first appearance, Clairaut would become famous indeed. Such a calculation had many practical problems, not the least being the ability of local weather to obscure the night sky. Instead of predicting the first observation of the comet, Clairaut computed the date of the perihelion, the date the comet made its closest approach to the Sun.
¹⁷

Clairaut decided to undertake the calculation sometime in the spring of 1757. He may have been encouraged by two friends, Joseph-Jérôme Le Français de Lalande (1732–1807) and Nicole-Reine Étable de la Brière Lepaute (1723–1788), with whom he divided the computations. Joseph Lalande was a young astronomer, a scientist near the start of his career. He had studied in a seminary to become a Jesuit priest and had nearly taken orders, but, as the historian Ken Alder has noted, he “had an insatiable thirst for fame.”
¹⁸
He left the clerical life to make his reputation as an astronomer. By the time he was twenty-one, he had undertaken a major astronomical project that combined the efforts of the Paris and Berlin observatories. He had also been elected to the French scientific society, the Académie des Sciences. Nicole-Reine Lepaute came from the thin stratum of wealthy French bourgeoisie.
¹⁹
She had been born in the Palais Luxembourg and had been educated by her parents. “In her early childhood, she devoured books, [and] passed nights at readings,” wrote Lalande, who also claimed that she was “the only woman in France who [had] a true knowledge of astronomy.” When she was twenty-five, she had married the
royal clockmaker, Jean André Lepaute (1709–1789). It seems to have been a generous marriage, one that allowed Mme Lepaute some freedom to exercise her scientific skill. Lalande recorded that “she observed, she calculated, and she described the inventions of her husband.”
²⁰