CHAPTER ONE

The First Anticipated Return: Halley’s Comet 1758

OUR STORY will begin with a comet, a new method of mathematics, and a seemingly intractable problem. The comet is the one that appeared over Europe in August 1682, the comet that has since been named for the English astronomer Edmund Halley (1656–1742). This comet emerged in the late summer sky and, according to observers at Cambridge University, hung like a beacon with a long, shimmering tail above the chapel of King’s College. To that age, comets were mysterious visitors, phenomena that appeared at irregular intervals with no obvious explanation. Their origins, substance, and purpose were matters of pure speculation. Some thought that they were wayward stars. Others suggested that they might originate in the atmosphere, each a burning piece of Helios’s chariot, perhaps, that had been caught between the earth and the moon.

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.¹ 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

3a. Computers for first return of Halley’s Comet: Alexis-Claude Clairaut

3b. Computers for first return of Halley’s Comet: Joseph-Jérôme Le Français de Lalande

3c. Computers for first return of Halley’s Comet: Nicole-Reine Étable de la Brière Lepaute

Joseph Lalande and Nicole-Reine Lepaute had an unconventional relationship that reflected the unconventional work in which they were engaged. Lalande never married; “he was a supremely ugly man,” wrote Alder, yet he “loved women, especially brilliant women, and promoted them in both word and deed.”²¹ The two met when Lalande, acting as a representative of the Académie des Sciences, called upon Lepaute’s husband and asked to examine the royal clocks. Lalande had been asked to prepare a report on clockmaking, and he returned regularly to the Lepaute apartment. He soon came to appreciate that Mme Lepaute had substantial mathematical talent and that her “astronomical tables were next to her household account books.” He asked Nicole-Reine Lepaute to prepare a table for his report on the royal clocks, a request that started a thirty-year collaboration. “She endured my faults,” wrote Lalande of his partner, “and helped to reduce them.”²²

Clairaut’s computing plan was a step-by-step process that treated Jupiter and Saturn as if they were the hour and minute hands of a giant clock, ticking their way around the Sun. Each step of the calculation would advance the two planets a degree or two in their orbits. The procedure then required an adjustment of the orbits based upon the gravitational pull between the two planets and the sun. Saturn might be pulled a bit closer to the Sun, or Jupiter might be pushed forward in its orbit. On this celestial clock, the comet was a second hand that circled on a long, skinny ellipse in the opposite direction of the planets. The calculation of this orbit could be done after the calculations for Jupiter and Saturn, for though the two planets pulled on every step of the comet, the comet had no substantial effect on the planets. This part of the computation would advance the comet a small distance along the ellipse, compute the forces from the two giant planets, and adjust the orbit accordingly.²³ Lalande and Lepaute handled the first part of the computation, the three-body problem involving Saturn and Jupiter, while Clairaut took the orbit of the comet itself.

The three friends began their computations of Halley’s comet in June 1757. They worked at a common table in the Palais Luxembourg using goose-quill pens and heavy linen paper.²⁴ Lalande and Lepaute worked at one side of the table and handed their results to Clairaut. Presumably, they wore the formal court dress of palace residents: coats and knee breeches for the men, a dress with underskirts for Lepaute, and powdered wigs for all. They would have begun their work late in the morning, as palace residents were rarely early risers. The computations would take them through the early afternoon and into the evening, when dimming lights and fatiguing hands discouraged further efforts. At times, they would compute while they ate their meals. When servants appeared with food, the three would simply push their papers and ink pots aside to make room for the dishes and continue with their work.²⁵

In addition to stepping Halley’s comet through its orbit, Clairaut took the responsibility of checking all of the calculations for errors. With three people contributing to the task, the project offered many opportunities for arithmetical mistakes. Identifying the source of such mistakes was difficult, and even a small error might be compounded in the calculations and ultimately render the final result meaningless. Critics of scientific research had pointed to such errors as a weakness of mathematical methods and suggested that numerical techniques were too fallible to capture the nature of the universe. Such critics included the English writer and social commentator Jonathan Swift (1667–1745), who attacked, in his satire Gulliver’s Travels, both the general validity of mathematics and the specific accuracy of Edmund Halley’s calculations.

Gulliver’s Travels appeared about thirty-five years before Clairaut began his computations. At the time of its publication, Halley had just become the Astronomer Royal of England. One section of the book describes an astronomer-king, much like Halley, who governs a mythical land of scientists called Laputa. In Laputa, the residents decorate their clothes with astronomical symbols and keep one eye turned always to the heavens. This astronomer-king eats food cut into geometrical shapes and gets so absorbed in calculation that he requires a servant to arouse his attention. His subjects are absent-minded, selfish, and contemptuous of practical issues. They “are dexterous enough upon a piece of paper,” the fictional Gulliver notes, “yet in the common actions and behaviour of life, I have not seen a more clumsy, awkward, and unhandy people.” They are deeply afraid of a comet, easily recognizable as Halley’s, which they believe will destroy the earth in its next passage.²⁶

As Swift mocks the community of scientists, he directly attacks scientific calculation by equating the work of computation with the craft of a foolish and obstinate tailor. This tailor is asked to prepare a new suit of clothes for Gulliver and uses the full range of astronomical tools to create the pattern. Gulliver reports that the tailor “first took my altitude by a quadrant,” a device slightly larger than Halley’s sextant, “and then, with a rule and compasses, described the dimensions and outlines of my whole body.” When Gulliver returns to accept the new clothes, he records that the tailor “brought my clothes very ill made, and quite out of shape, by happening to mistake a figure in the calculation.” The tailor makes no effort to correct the mistake, and Gulliver notes that “such accidents [occur] very frequently and [are] little regarded.”²⁷

In the summer of 1757, French translations of Gulliver’s Travels could be easily found at Paris booksellers, though it does not seem to be the sort of book that Alexis Clairaut would have enjoyed. He had a methodical nature, one that would have carefully sought to avoid producing a suit of “clothes very ill made” or a cometary orbit ill calculated. He devised procedures that would check every step of the computation. Some values were double-computed; others were checked with graphs.²⁸ The equations themselves were checked with two preliminary calculations. Starting with observations from the 1531 return, the three workers computed a full orbit and compared the final positions of the comet with the data from 1607. After adjusting the equations, Clairaut, Lalande, and Lepaute repeated the process. Beginning with the 1607 observations, they calculated a second orbit and compared the final values with Halley’s observations of the 1682 return.²⁹ These computations tripled the amount of work, but the process gave Clairaut, Lalande, and Lepaute a measure of confidence in their results.

The daily routine of computation, the effort to advance the comet eight or ten degrees in its orbit, tested the stamina of the three. “It is difficult to comprehend the courage which was demanded by this enterprise,” Joseph Lalande later wrote. He complained that “as a result of this hard work, I acquired an illness which, for the rest of my life, shall be with me.” By contrast, Nicole-Reine Lepaute carried her share of the burden without complaint. “Her ardor is surprising,” remarked Clairaut. Lalande confirmed this opinion, stating that “we would not, without her, have dared undertake this enormous work.”³⁰

The “enormous work” kept Clairaut, Lalande, and Lepaute busy at their computing table through late September. By then, astronomers were searching the night sky, expecting to catch the first gleam of the comet. Already several bodies had been falsely identified as the returning comet of 1682. Each passing day brought more pressure upon Clairaut, until he decided that he would have to increase the speed of the calculations. As the group began computing the last segment of the orbit, the part that came closest to the sun and planets, Clairaut simplified the equations by removing the terms that accounted for the gravity of Saturn. He believed that the influence of Saturn was small in that part of the orbit and that the revised calculation would still give a reasonable approximation of the comet’s motion.³¹

On the fourteenth of November, Clairaut presented the results to the Académie des Sciences. He predicted that the comet would reach its perihelion, the point closest to the sun, in mid-April. His calculations showed that the date would be April 15, 1758, but he knew that it was unlikely that the calculation captured perfectly the movement of the comet.³² “You can see with what caution I make such an announcement,” he said, “since so many small quantities, which must be neglected in methods of approximation, can change the time by a month.”³³ His results suggested that the comet might round the sun as early as March 15 or as late as May 15.

For six weeks, the computation stood as a major accomplishment while astronomers watched and waited for the first observation of the comet. No sign of the comet was found in November and none in the first weeks of December, but on Christmas Day 1757, a German astronomer sighted the fuzzy glow of the comet nucleus. By the middle of January, enough observations had been made of the approaching body to show that the calculation had overestimated the date of the comet’s arrival. The comet reached its perihelion on March 13, a few days outside of the interval that Clairaut had presented to the Académie des Sciences.³⁴ As the news of the discrepancy began to spread, astronomers started to assess the calculation and debate its meaning. The most public critic of the computation was French mathematician Jean Le Rond d’Alembert (1717–1783). D’Alembert was no social commentator in the manner of Jonathan Swift, but an established astronomer and an editor of the Encyclopédie, the French Enlightenment’s grand catalog of knowledge. He argued that Clairaut’s work was not based upon well-defined principles of scientific inquiry but was only a rough and unsubstantiated approximation.³⁵ He denounced what he called the “spirit of calculation” and claimed that the computation was more “laborious than deep.”³⁶

D’Alembert’s criticisms sparked an extended argument among French scientists, an exchange that was fueled in no little part by the egos and ambitions of the two protagonists.³⁷ From the tenor of the debate, it seems clear that even Clairaut had more faith in Newton’s theory of gravitation than in his own calculation of the comet’s perihelion. Clairaut never suggested that Newton’s theory needed to be adjusted to fit the prediction; instead, he returned to his figures and attempted to find some mistake that had caused the discrepancy. Under the best of circumstances, this is a dangerous endeavor, for there is a tremendous temptation to bend the equations in order to fit the data. With the perspective of modern astronomy, we know that Clairaut did not account for the influences of Uranus and Neptune, two large planets that were unknown in 1757. Without these planets, the equations could produce the correct date for the perihelion only if they misstated the influence of Jupiter and Saturn. Clairaut never took his adjustments that far, though he did produce a better prediction. A modern analysis of the result concluded that there were still many problems in the computation and that Clairaut benefited from the “fortuitous cancellation of opposing errors.”³⁸ After two years of public rancor, Clairaut decided that he was tired of the debate, and he ended the controversy by begging d’Alembert, who clearly had no interest in large calculations, to “leave in peace those who did.”³⁹

In the most charitable light, Clairaut had improved Halley’s work by a factor of ten. By the time Halley had completed his research, he had recognized that the comet had a variation in its return of approximately two years, about 600 days. Clairaut’s calculation missed the actual date of return by 33 days. As he might have undershot the date of perihelion by an equal amount, the rough accuracy of his calculations was twice 33, or 66 days. Beyond the simple accuracy of his result, Clairaut’s more important innovation was the division of mathematical labor, the recognition that a long computation could be split into pieces that could be done in parallel by different individuals. In spite of d’Alembert’s criticism, the astronomers of the mid-eighteenth century recognized that Clairaut’s division of labor was an important contribution to astronomical practice.⁴⁰ Clairaut never undertook another calculation of equal complexity, but his two assistants, Joseph Lalande and Nicole-Reine Lepaute, were involved with computation for the rest of their careers. In 1759, Lalande became the director of the Connaissance des Temps, an astronomical almanac published by the Académie des Sciences, and he appointed Nicole-Reine Lepaute as his assistant. The two of them prepared tables for the Connaissance des Temps that predicted the positions of the stars, the sun, the moon, and the planets. These tables were easier to calculate than the orbit of Halley’s comet, as they were based on a long history of data and as they could be corrected from observations taken throughout the year. Likewise, the division of labor for these calculations was simpler than it had been for Halley’s comet. Lalande prepared the computing plans and checked the results, while Lepaute calculated the values for the tables.

Lalande called Lepaute his “assistant without equal,”⁴¹ but of course, she was anything but equal to him. Lalande was able to advance from his position and would eventually be appointed a professor of astronomy and director of the Paris Observatory. Lepaute had no such opportunities and would spend fifteen years as a computer for the Connaissance des Temps. However, even with its limitations, the appointment to the Connaissance des Temps had certain advantages for Lepaute. It gave her an official standing among French scientists, a rare accomplishment for a woman. During her fifteen-year career, she found that she could occasionally do some astronomical work outside of the Connaissance des Temps. In 1764, she published a map under her own name that predicted the extent and duration of an upcoming solar eclipse. She was apparently quite proud of that work, for she asked that it be included in a portrait that she gave to Lalande.⁴²

“No scientific discovery is named after its original discoverer,” wrote the historian Stephen Stigler.⁴³ Even if Clairaut, Lalande, and Lepaute were the first astronomers to divide the labor of scientific calculation, their names did not travel with their contribution. Others would rediscover the idea without knowing of Halley’s comet or the computations that were done at a table in the Palais Luxembourg. Still, the work of those three scientists during the summer and fall of 1757 identified a pattern that touched three or four generations of human computers, a pattern that divided calculations into independent pieces, assembled the results from each piece into a final product, and checked that result for errors.