The Trouble with Hooke - Newton, Leibniz, and the Greatest Mathematical Clash of All Time - The Calculus Wars

The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time (2006)

Chapter 3. The Trouble with Hooke

1664–1672

See the great Newton, He who first surveyed

The plan by which the universe was made

Saw nature’s simple yet stupendous laws

And prov’d the effects tho’ not explain’d the cause

—Text from a 1787 engraving entitled

The Most Highly Esteemed Sir Isaac Newton

One of the most succinct, if perhaps overly idolizing, descriptions of Newton’s hardworking days at Cambridge appeared three hundred years after he was born. In February 1943, a conference of scholars met in Jerusalem to commemorate the tricentennial of Newton’s birth, and an address given by the masters, fellows, and scholars of Trinity College reads in part, “Here [Newton] labored at his calculations and carried out his experiments. . . . In these precincts he walked in meditation whilst the genius of his mind formed those bursts of experimental activity, when for six weeks at a time the fire in his laboratory scarcely went out night or day.”

The image that we have of the young Newton as a superdiligent mad scientist fits because it was true. He had to work so hard as a young man because one doesn’t easily uncover the secrets of the universe, which is exactly what he did. As the 1943 masters, fellows, and scholars put it, “Law and order in the physical universe were revealed as never before.”

Ironically, the work that Newton did at Cambridge for which he is most remembered is that which he did away from Cambridge, during the time he spent holed up in his family home in Grantham. There, he toiled away for many months figuring out how the universe worked and making one spectacular discovery after another while waiting for the school to reopen. It was his so-called annus mirabilis, or “miraculous year.”

“In those days, I was in the prime of my age for invention and minded mathematics and philosophy more than at any time since,” Newton would later write. One of his biographers makes a valid point that the miraculous year should more properly be called his anni mirabiles, or “miraculous years,” since the time actually encompassed 1665–67.

Often working night and day, Newton rarely placed anything—including food, rest, family, or even his own safety—above his science. He would forget to eat, forget to wash, and grow oblivious to everything around him except his books, notes, and experiments. One story that I like is that Newton’s cat grew quite fat munching on all the food that he left untouched. Another is that because he was interested in light and vision, he stared at the sun for periods of time so that he could observe the “fantasies” of color that would be burned into his field of vision. He did this so many times, the story has it, he had to shut himself in a dark room for days to restore his vision.

Worse were Newton’s attempts to jab behind his eyeball with a bodkin—a sort of long needle—to alter his retina (the layer of cells with light receptors) and see how this affected his vision. “I took a bodkin & put it betwixt my eye & the bone as neare to the backside of my eye as I could: & pressing my eye . . . there appeared severall white, darke & coloured circles,” Newton recorded in his notebook, complete with a hand-drawn diagram showing his hand shoving the bodkin behind his anatomically correct eyeball.

I saw a copy of this notebook on display at the Huntington Gardens Museum in Pasadena in 2005. While I was standing there, a woman and her teenage son were looking at the book and trying to understand . . . what they were looking at exactly.

“What’s a bodkin?” the boy asked his mother.

“It’s some kind of needle,” she said.

I could see the uncertainty on her face even if her son couldn’t, so I jumped in. “It’s a long needle that tailors used to use,” I said. “It’s long, but it has a dull point. They used to use them to poke holes in leather.” Nobody said anything after that.

In inventing calculus, which Newton also did during the anni mirabiles, he never did anything as severe as almost blinding himself, but in later years he would become blind to the possibility of Leibniz’s accomplishments. Newton was a potent mix of brilliance and vanity, and he would later reject the notion that someone else like Leibniz could have accomplished the same thing that he did in these early days.

There is a sense today that the calculus wars were ridiculous because so much of the work that led to the development of calculus and so much of the subsequent work that helped develop calculus into the extensive advanced subject it is today was done by mathematicians other than Newton and Leibniz. Much territory had already been explored in the seventeenth century, and the world was on the brink of finding the calculus. Even though the notion of inevitability of discovery was not as common in the seventeenth century as it is today, when it is quite common for scientists working separately on the same problems to arrive at similar or identical solutions, there is no doubt that calculus was inevitable. All the basic work was done—somebody just needed to take the next step and put it all together. If Newton and Leibniz had not discovered it, someone else would have.

This is not to take anything away from Newton and Leibniz—particularly since they both invented calculus largely by teaching themselves what they needed to know. Cambridge was not a center of mathematics in those days, and Newton was basically on his own. He bought and read a copy of Descartes’ Geometry. In his later days, Newton recounted to John Conduitt, his nephew-in-law, how he would read Descartes for a few pages, get stuck, go back and reread, get stuck again, read more, and on and on until he had mastered the work.

Newton became familiar with the infinite series. These were ways of finding numerical solutions to problems like the area of a geometrical shape by summing up a series of numbers. In England, mathematician John Wallis had already made progress with this type of analysis by the time Newton arrived on the scene. Wallis is a somewhat obscure figure in the history of mathematics, but he was a mathematical titan of his day, and his work greatly influenced Newton. His book, Arithmetica infinitorum, shows some of the first steps in the direction of calculus. In it, he anticipates calculus by seeing the questions that calculus would answer, and he discusses the geometrical ideas of earlier mathematicians who had done some of this work. Reading Wallis’s work on infinite series, Newton was inspired to extend this work and invent a general way to analyze geometric curves with algebra—calculus, essentially.

Newton’s big breakthrough was to view geometry in motion. He saw quantities as flowing and generated by motion. Rather than thinking of a curve as a simple geometrical shape or construction on paper, Newton began to think of curves in real life—not as static structures like buildings or windmills but as dynamic motions with variable quantities.

By the time Newton was elected a scholar of Trinity College on April 28, 1664, he was aware of the difficult problems that the invention of calculus would solve: those interesting but difficult to deal with problems in geometry, such as finding the area under a curve or finding the tangent (the ability to draw a line perpendicular to any point on the curve). Being a scholar meant that he now had a stipend and living-expensed account, and he was no longer the one to fetch bread. He was, at this point, very close to inventing calculus. Within two years, he would have it. But first an apocalypse would intervene.

A comet appeared in the sky the week before Christmas in 1664, and England’s king wondered what it could mean. Charles II, who had been installed a few years before after the failure of the government following Oliver Cromwell’s death, was a superstitious man. He followed astrology, was most watchful of such signs, and was more or less representative of his people in this respect. Many in the city wondered what evil fate the comet might portend. William Lilly, a famous astrologer who published a yearly almanac, prophesied in his 1665 edition that another heavenly omen, a lunar eclipse over England in January, would bring “the sword, famine, pestilence, and mortality or plague.”

As if that weren’t enough, another comet appeared in March 1665. (Actually it was the same comet as the previous December, now on its return trip around the sun.) It’s not hard to imagine the fortune-tellers coming out of the woodwork to walk the streets of London in their velvet jackets and black cloaks, bemoaning doom to the people who followed them. And for once, they were right. A horrible plague ravaged England the following summer, and sixty thousand people died in London alone.

The fortune-tellers were still quack conjurers and flimflam artists. In fact, it was not a terrible stretch to predict that plague would hit England in 1665, because the plague had already been circulating through Europe for a few years. Holland was particularly afflicted in 1663, when a thousand people a week were dying in Amsterdam; and England was not just geographically close to Holland, it was butting heads with its neighbor across the channel. Britain had fought a recent war with the Dutch and was on the verge of going to war again. Already, in 1662, England had seized the Dutch colony of New Amsterdam and changed its name to New York. Conflict in the colonies could be expected to export conflict back to Europe, and plague to England.

Besides, in those times disease was inevitable. It was as much a part of life of Newton’s England as was bad weather. People lived stuffed into slums with poor sanitation. Streets were crowded and had open sewers running down the middle that buzzed with flies in the summer. Half of England’s population survived on a subsistence living, and many people suffered from diseases like rickets, caused by a deficiency in vitamin D. People caught measles, malaria, and dysentery in the summer, and in the “r” months, there was typhus, influenza, and tuberculosis, the “captain of all these man of death,” as John Bunyan called it. And infections cut across all walks of life. Oliver Cromwell probably died of malaria. Smallpox killed Queen Mary II in 1694. James II may have been stricken with syphilis.

Plague was not necessarily the worst of these diseases because it was not ever-present, as many of the others were. But perhaps because it was episodic it was more terrifying. And to catch the plague is a horrible thing. The infection manifests in painfully swollen lymph glands—called bubos, a term from whence the disease name “bubonic” plague comes. Fevers, chills, exhaustion, headaches, and sometimes severe respiratory illness accompany the disease. Outbreaks during the 1630s killed more than half of the population of some cities. Previous to the outbreak in Holland in the 1660s, there had been an epidemic of plague in France in 1647–1649.

Typically outbreaks of plague occur through the rat population. Large numbers of rats will succumb to an epidemic of infection, and if such a population is living in an urban center, their fleas transmit the bacteria to humans. This is what happened in England in the summer of 1665, when a terrible outbreak of bubonic plague ravaged London. “The contagion now growing all about us,” the diarist John Evelyn wrote on August 28, 1665.

By that September, prohibitions against public meetings were in effect everywhere, and by October, one in ten Londoners was dead. “Lord! How empty the streets are and [how] melancholy,” Samuel Pepys wrote on October 16, 1665, “so many poor sick people in the streets . . . and so many sad stories overheard as I walk, everybody talking of this dead, and that man sick, and so many in this place, and so many in that.”

Nor was the plague confined to London. Cambridge, where Newton was in residence, shut its doors in the fall of 1665 because of the epidemic. “[It] pleased almighty God in his just severity to visit this town of Cambridge with the plague of pestilence,” as one contemporary account put it. Newton was forced to retreat to the safety of his country home in Grantham, and he stayed there for more than a year until studies at Cambridge resumed in April 1667.

What emerged from these years is arguably the greatest single body of knowledge any scientist has ever produced in such a short time period. Newton arrived at an understanding of the mechanics of motion, and began working on a mathematical description of the laws of motion. He also made major discoveries concerning optics, fluid mechanics, the physics of tides, and the theory of universal gravitation.

His optical experiments during this time were both beautiful and insightful. He shut himself in a room with no outside light except from a single point source coming from the sun shining through a small hole in the wall. The sun cast a ray of light through the hole, and Newton experimented on the light with a prism. His big breakthrough was understanding that ordinary white light is composed of the spectrum of colors red, orange, yellow, green, blue, indigo, and violet. He also discovered, in careful experiments, that just as a prism can split white light into this spectrum of colors, so can a second prism return the separated colors into white light.

These experiments and others gave Newton the material for his famous book, Opticks. But that was not all. Also during this time he conceptualized the material for his more famous book, the Principia, which he wrote in the 1680s and which would outline the mathematical underpinnings of physical motion and revolutionize physical science. His law of universal gravitation, described in mathematical detail in the Principia, has been called the greatest scientific discovery of all time, and the book continues to be translated from its original Latin today.

It was also in this time that the legend of Isaac Newton and the apple was born. This story is still one of the most enduring tales in the history of science—even though it is probably completely fabricated. Perhaps the only thing that is true about it is that Newton loved apples. The story is no more true than the one about the alligators in the sewers of New York, but it has stuck through the centuries.

Voltaire popularized the tale when he wrote of Newton and the apple almost seventy-five years later. Voltaire’s famous story has Newton walking in a garden when he sees an apple fall to the ground from an apple tree branch. This, wrote Voltaire, caused Newton to fall into a profound meditation upon the cause of the apple’s falling. According to legend, Newton observed that the apple fell as if it would pass toward the center of the Earth (the center of gravity). Why then, the student-scientist wondered, doesn’t the moon fall to earth as well? Perhaps it does. Perhaps it is constantly falling! That, Voltaire claimed, was the inspiration for Newton’s theory of universal gravitation, “the Cause of which had so long been sought, but in vain, by all the Philosophers,” added Voltaire.

The problem with the apple story is that it oversimplifies the process of discovery that Newton was engaged in. There probably was no one eureka moment (or falling apple moment) that gave Newton the insight to develop his theory of universal gravitation, but rather a less glamorous sequence of long moments spent in study reading, writing, thinking, and working it out. Still, in some ways, it would be easier to understand a genius like Newton if he did simply act as the receiver of great and sudden bursts of insight. It saves having to think about just how he went about the actual work, which strains comprehension. Plus the apple is a great symbol for discovery. Sex, food, sin, and the fall of man—all these things are represented by this humble fruit.

An apple tree planted just to the right of the great gate at the front of Trinity College is said to be a descendant of the actual apple tree that Newton supposedly sat under when he worked out universal gravitation. When I was in Cambridge, I observed more than one person gawking at the tree. Perhaps they were looking, like Newton, for some inspiration to come falling like a red delicious description of nature from those mythic branches. Perhaps, like me, they were puzzled with the disappointment of an apple tree in January—no leaves, no fruit, and scraggly branches that bore nothing but tradition to stir interest. Next to the massive gates of Trinity, it seemed small and insignificant—like it couldn’t support the weight of the famous fruit its progenitor produced. Still, apple or no apple, universal gravitation changed science and mathematics forever.

Significantly, Newton also invented calculus in this time—what he called his method of fluxions and fluents. Voltaire’s story of calculus is, incidentally, much less interesting than his story of the apple. “’Tis the Art of numbering and measuring exactly a Thing whose Existence cannot be conceived,” Voltaire explained simply. Calculus is really a set of rules for analyzing and solving, with algebra, problems related to geometrical curves. It was the answer to some of the big questions of mathematicians of the time—questions like how to find the tangent to (or slope of) a curve at any given point, and how to find quadratures, the areas under given curves.

On Halloween day, 1665, Newton sat down and began to write a short treatise he would call “How to Draw Tangents to Mechanical Lines.” A few weeks later he followed this with another paper, “To Find the Velocities of Bodies by the Lines They Describe,” which was another early stab at calculus.

I saw a yellowed copy of the manuscript “How to Draw Tangents” under a glass case at the Huntington Library in Pasadena. Most of the crowd walked by with little more than a passing glance and seemed more impressed with a calculation of logarithms to fifty-five places—something that Newton worked out in his early days. Newton wrote to an acquaintance about this once, “I am ashamed to tell to how many places I carried these computations, having no other business at that time: for then I took really too much delight in these inventions.” The paper has several large triangular columns of numbers—scary to look at if you are trying to make sense of what they mean, but in the context of a museum, quite striking—even artistic in a visionary sort of way.

Newton wrote a manuscript on November 13, 1665, describing his method of calculus with examples. Over the winter, he continued to work on a number of other topics, and he returned to calculus on May 16, 1666, devising a general method with several propositions for solving problems by motion. Finally, in October 1666, he wrote a tract of forty-eight pages with eight propositions with the heading “To Resolve Problems by Motion, these following Propositions are sufficient.” The piece had twelve problems that his methods of analysis could solve directly via his arithmetic methods, including drawing tangents to curves or the instantaneous rate of change (the derivative) at any point along the curve; finding the points of greatest curvature; finding the length of curved lines, finding curved lines whose areas are equal; and finding the area under a curve (the integral) or the area between two curves. This was a real breakthrough.

When he returned to Cambridge in 1667, Newton was a changed man. What he had done, and what Leibniz would repeat a decade later, was to invent one powerful system of mathematics general enough to analyze any curve. At the time Newton was making these discoveries, however, Leibniz still knew nearly nothing of mathematics. On October 2, 1667, Newton received his M.A. from Cambridge and became a Fellow of Trinity College. Strangely, he then set mathematics aside and did nothing more with it for the next two years.

In 1669, he turned once again to mathematics and optics, familiarizing himself with the work of a mathematician in Cambridge named Isaac Barrow. Barrow was the Lucasian professor at Cambridge, a chair founded a few years earlier by Henry Lucas, and Barrow held this chair from 1664 until he stepped down in 1669, passing the distinction to Newton. It had a huge endowment, so Newton got the equivalent of a huge raise and large promotion. Barrow was probably the best colleague Newton could have had, not only for helping him ascend the academic ladder but also because Barrow helped Newton publish, an act toward which Newton had not taken so much as a baby step by the end of the 1660s.

This would all soon change thanks to Barrow and prompted in part by a book published in 1668 by Nicholas Mercator, a German mathematician who lived in London. Mercator’s book introduces the term “natural logarithm” and impressively describes how to solve a particular quadrature problem—integration of the function . This is a trivial problem in calculus today, but it was an elegant and important work when it was published. As impressive as it was, Mercator’s work was a specific and rather elementary example of what Newton could solve using calculus. As Voltaire put it decades later, “Mercator published a Demonstration of this Quadrature, much about which Time, Sir Isaac Newton . . . had invented a general Method to perform, on all geometrical Curves.”

If Voltaire couldn’t help but be impressed three-quarters of a century after the fact, one can only imagine how impressed Newton’s contemporaries would have been if they had read Newton’s work. But almost none of them could because Newton’s work didn’t exist anywhere in print. He had written a few manuscripts in the late 1660s and early 1670s that described calculus. The first of these was a Latin work he wrote in 1669, based on his earlier work from 1666, entitled De Analysi per Aequationes Numero Terminorum Infinitas (On Analysis by Means of Equations Having an Infinite Number of Terms). This book would later play a crucial role in the calculus wars. Newton and his allies would point to the existence of De Analysi as proof that he had developed his calculus years before Leibniz.

De Analysi was supported in the second unfinished book that he wrote in the winter of 1670–1671, Tractatus de Methodis Serierum et Fluxionum (A Treatise of the Methods of Fluxions and Series). Together, these two books were the first writings that contained Newton’s calculus—indeed, the first writings ever to describe calculus. The problem was, he didn’t publish them.

Had he published De Analysi when he wrote it, Newton would have saved himself a lot of trouble, there never would have been a calculus wars, and he would have advanced knowledge much faster than he did by not publishing. But this sounds easier in retrospect than it was at the time. Publishing such a complicated mathematical treatise would have been extremely difficult in the wake of the great fire of London, which destroyed publishing houses along with much of the rest of the city in 1666—a disaster so dramatic that it’s worth describing briefly.

The fire started just after midnight on September 2, 1666, and was apparently the fault of a baker named Thomas Farryner, of old Pudding Lane. But Farryner’s fault might have been anyone’s. London was a tinderbox of a city in those days. Wooden houses were built upon wooden houses, and their floors were covered in dry straw. The building of new houses within the city walls had continued until the point where every street and open space was filled with a sort of kindling of residential urban decay waiting for a match to march hellfire.

No one could have guessed how devastating the fire would be, though. Surveying the fire on Sunday, the morning after it started, Samuel Pepys called it an “infinite great fire” that threatened to burn the entire city. And a few days later, he lamented, “Lord! What a sad sight it was by moonlight, to see the whole city almost on fire.”

John Evelyn bemoaned the dismal sight of the fire in his diary the night after the blaze had started. The next day, he recorded how the fire had worsened: “O the miserable and calamitous spectacle, such as perhaps the whole world has not seen its like since the foundation of it: nor is it to be outdone until the world’s universal conflagration . . . God grant mine eyes may never behold the like, who now saw above ten thousand houses all in one flame. The noise, the crackling and thunder of the impetuous flames, the shrieking of women and children, the hurry of people, and the fall of towers, houses and churches, was like a hideous storm. . . .”

“London was, but is no more,” Evelyn wrote.

Unfortunately, in the early hours of the fire, the residents of the city were concerned more with saving as many goods as they could than with fighting the flames. The fire could have been contained by tearing down the houses in its path, but this was a tough policy to implement. The lord mayor of London, Thomas Bludworth, refused to tear down buildings without the consent of the owners, and, for obvious reasons, few who owned a house that had not already burned would consent to having their property preemptively destroyed. There were direct ways of fighting the flames—bucket brigades and hand-pumped hoses—but these efforts could do little to quell a conflagration that by Sunday was more than a mile long, blazing a path through the city. Sunday night, and all day and night Monday the fire spread.

By then, it was too late. Panic set in and people began to flee the flames. The streets were sick with carts and conveyances. Londoners of every description—men, women, children, animals—and their property moved toward the city gates and the safety of the outside. The Thames River was congested with barges and boats doing the same. For years, London had been a center pulling in new residents from the largely rural population of England, but now the city was a massive human spout, pouring people back into the countryside.

Pepys, to his credit, succeeded in saving the naval offices and the Tower of London by organizing dockworkers to destroy the buildings around the structures. Other parts of London were saved by using gunpowder to destroy large parts of the city that were lying in the fire’s path. But by the time these dramatic measures were taken, it was too late for much of the city. Fueled by strong winds, the flames spread rapidly, and the fate of the city was sealed. By Tuesday, the devastating power of the fire reached the spires of St. Paul’s Cathedral, which dominated the London skyline, and burned it to the ground. Rivers of lead melted off St. Paul’s Cathedral and ran through the streets.

By the time the fire died down, the devastation was massive. Some 373 out of 448 of the city’s acres were scorched. An enormous wealth of property was destroyed, along with 13,200 houses and dozens of churches and municipal buildings. About a sixth of the city’s population was homeless. And yet, as Voltaire later wrote, “To the astonishment of all Europe, London was rebuilt in three years and arose more beautiful, regular, and commodius than it had ever been before.”

The reason why I mention the fire here is not because it’s a good cautionary tale in urban planning or because it’s an inspirational story about the resilience of a population in snapping back after being beaten down, but because it’s a seminal event in the calculus wars. One of the biggest victims of the fire was the publishing industry, seriously damaging the ability of a mathematician such as Newton to publish book-length works. If he were writing a popular pamphlet or clever little handbill, it could have been a different story.

Modern printing was introduced into Europe by Laurens Coster in Holland and Johannes Gutenberg in Germany, and by the seventeenth century, publishing had taken off. The wide availability of books enabled the wealthiest to build libraries, but it also allowed average people to find pamphlets, journals, newspapers, and books on all subjects. Publishing had grown into an industry in Europe, and book sales were exploding there.

Book publishing in London, however, was an industry in crisis when Newton was writing about calculus. Producing a book could be a big risk, since the cost of paper was so high. In the seventeenth century, paper was made from the pulp rendered out of old rags, and the book industry would take big financial hits after plague outbreaks like the one in 1665, because many of the old rags were contaminated by disease and would be burnt instead of pulped, increasing the cost of paper.

Meanwhile, the fire ravaged the city booksellers’ stores and destroyed countless stocks of books—so many, in fact, that publishers couldn’t afford to take the risk of publishing books that they couldn’t quickly sell. As a result, printings rarely exceeded one thousand copies. Typical best sellers of those times were books on religion, for which there was a high demand. This did not bode too well for Newton and other authors of obscure and cryptic mathematics—especially given all their equations and the difficulty in typesetting them. One book that was published in this time, the optical and geometrical lectures of Newton’s mentor Isaac Barrow, is said to have nearly bankrupted the printers.

Thus, for younger and unknown mathematicians like Newton, there was hardly any possibility of publishing a book on mathematics. In fact, De Analysi wasn’t published until Newton was an old man. Instead, he simply gave a copy of it to Isaac Barrow, and De Analysi might have died as a document of no historical importance had it not been for the fact that Barrow was so impressed with it that he wrote to his friend John Collins in London on July 20, 1669: “A friend of mine here that hath a very excellent genius to those things [referring to the book by Mercator], brought me the other day some papers, wherein he hath set down methods of calculating the dimensions of magnitudes like that of Mr. Mercator concerning the hyperbola, but very general.”

A few years later, Newton described these methods himself in a letter to Collins he wrote on December 10, 1672, elaborating his approach to finding tangents to curves: “This Sir, is one particular, or rather a Corollary of a General Method which extends itself without any troublesome calculation, not only to the drawing tangents to all curve lines whether geometrick or mechanick or however related to straight lines or to other curve lines but also to the resolving other abstruser kinds of problems about the crookedness, areas, lengths, centers of gravity of curves &c.”

Collins was so excited when he read De Analysi that he had a copy of it made without Newton’s knowledge. This copy would be one of the central documents offered as proof of Leibniz’s plagiarism during the climax of the dispute years later.

However hard it may have been for Newton to publish a book in the early 1670s, he still had other options. A new kind of publishing was on the rise—the journal—and in London, the journal Philosophical Transactions of the Royal Society had been operating for a few years. It started as a way of keeping track of the papers that were sent to and presented at the Royal Society, and it became a convenient way to publish the latest findings and to keep in touch with discoveries in other parts of the world. This journal was not alone. Several others started in Europe in Newton and Leibniz’s lifetimes. In the late 1660s, when Newton was ready to present the world with his work in mathematics, the Philosophical Transactions would have been the perfect place to do so. Why didn’t Newton have De Analysi or some shorter version of it published in the Transactions? He may very well have done so had everything gone smoothly for him.

Newton wanted to have his optical works presented first. He would start by revealing to the members of the Royal Society one of his great inventions: a telescope that looked like a toy—an early reflecting telescope. Reflecting telescopes are strange-looking instruments, shorter and fatter than traditional telescopes, with the eyepiece on the side rather than at the back.

The model Newton designed and constructed was less than a foot long, the size of a toy, but size didn’t really matter. Barrow demonstrated the reflecting telescope in front of the Royal Society, and it magnified a distant object more than a traditional telescope several times larger. Whereas most small telescopes of the day could magnify objects 12 or 13 times, the much smaller reflecting telescope Newton built could magnify an object “about 38 times,” as he wrote in one description. It was a vast scaling down of the technology of the telescope, and it excited members of the society.

“You have been so generous, as to impart to the Philosophers here, your Invention of contracting Telescopes,” wrote the secretary of the Royal Society to Newton on January 2, 1672. “It having been considered, and examined here by some of the most eminent in Optical Science and practice, and applauded by them, they think it necessary to use some means to secure this invention from the usurpation of foreigners;And therefore have taken care to represent by a scheme that first specimen, sent hither by you, and to describe all the parts of the Instrument, together with its effect, compared with an ordinary, but much larger, [telescope].”

Newton’s reflecting telescope was impressive enough to gain him election to the Royal Society. Thomas Birch, one of the early historians of the Royal Society, wrote in his 1756 History of the Royal Society of London for Improving Natural Knowledge from its First Rise that “on December 21, Mr. I. Newton, Professor of Mathematics at the University of Cambridge, was proposed candidate by the Lord Bishop of Salisbury.” Newton was ecstatic. On January 11, 1672, an issue of Transactions of the Royal Society had a paper that described the design for Newton’s reflecting telescope. By that summer, Newton’s reflecting telescopes were being built on both sides of the English Channel. Had he done nothing else in his life, Newton would probably still be remembered for this early contribution to optics. But he had so much more to contribute, including his extensive mathematical work, which he could have easily published in the society’s journal.

However, he decided he would first follow up his reflecting telescope with a report describing a new theory he had developed on light and colors—something he called “the oddest if not the most considerable detection which hath hitherto been made in the operation of nature.”

His theory may have been new, but the field was anything but. Optics had been vibrant throughout the seventeenth century. Descartes had studied optics and so had several figures who followed him, including older and more accomplished scientists than Newton, like Robert Hooke and Robert Boyle in England, and Leibniz’s mentor Christian Huygens in France.

Newton’s theory was much to the contrary of some of the leading theories of his day, and was a direct challenge to some of these leading scientific minds. To Descartes and others in the seventeenth century, light was like sound—a pulse propagated through a transparent medium, much as sound is really just pressure waves that emanate from a source through the movement of air molecules. Sound ceases to exist in a vacuum, and if you take a bell, stick it in a jar, and pump out the air, it will no longer make a sound when struck. Robert Boyle had demonstrated this to the awe of those who watched, just a few years before. If there is no air, there is no medium to transmit the sound, and many thought that it was the same with light. To Newton’s contemporaries, color was not a characteristic of the light but of the vibration in the medium.

Newton was certainly not ignorant of this view and of the body of previous work that supported it. He had read, understood, and had been inspired by the existing theories of light and color. The problem was that once Newton started experimenting, his respect for his own observations outstripped his respect for previous theories. When he saw that the wave theory of light was in conflict with what he observed in his experiments in 1666 and 1667, he boldly proposed that light is not a wave but a particle—an emission made up of innumerable small particles of light traveling through space. He described them as “multitudes of unimaginably small and swift corpuscles springing from shining bodies.” Newton also developed a new theory of colors, which held that color was not a characteristic of the wave but a characteristic of the light.

Significantly, he discovered that normal light as we know it is heterogeneous in the sense that it is a mixture of different colors—as we would say today, different wavelengths. White light, Newton found, was far from the pure colorless light that people had always assumed but was rather a combination of all the colors of the rainbow. “The most surprising and wonderful composition was that of Whiteness,” Newton wrote in 1672. “There is no one sort of Rays which alone can exhibit this. ’Tis ever compounded, and to its composition are requisite all the aforesaid primary Colours, mixed in a due proportion.”

This was exactly the opposite of what many of his contemporaries would have thought. White light to them was the absence of color, just as white paint was the absence of pigment. If you take paints and mix red and green and blue and yellow and violet together, you will get something dark and ugly. So how in the world could white light be a mixture of all these colors as colored light?

It was according to Newton. Replicating his student experiments, he demonstrated this by darkening his room except for a single source of light, running that point source through a prism and splitting it into the rainbow colors, and then running these through a second prism whereby they were recombined into white light. This was an exciting conclusion—much more so than his mathematical work.

On February 6, 1672, Newton sent a paper describing white light and his other theories to Henry Oldenburg, the secretary of the Royal Society in London, to be published in the Philosophical Transactions of the Royal Society. Newton’s “New Theory about Light and Colours” was published on February 19, 1672. A copy of the letter sent by Newton can still be viewed by visitors to the Royal Society today, as I discovered when I was in London. It contains a cover letter with a florid penmanship announcing, “A discourse of Mr. Isaac Newton, containing his New Theory about Light and Colors, sent by him from Cambridge Febr. 6. 1671/72 for ye Secretary of ye R. Society in order to be communicated to [the body].”

Newton’s paper was read to the society on February 8, 1672. The range of the topics considered by the society on the same day is interesting: after Newton’s paper was read, Wallis read a paper speculating about the moon’s influence on atmospheric pressure and on the barometer. After Wallis, a letter from Naples about tarantula bites was read, written by an Italian named Cornelio. Next, Flamsteed read a letter about the moons of Jupiter, and finally a letter from a German physician, Hanneman, was read, asking about the opinion of the Royal Society Fellows on sanguification and how it is performed. Lunar pressure on the atmosphere, toxic spider bites, gas giant moons, and the ins and outs of bleeding were nothing compared to Newton’s letter in terms of the interest generated.

Newton’s work was the product of several years of novel and meticulously performed experiments, analysis, and refinement. He was not merely describing some part of nature as he saw it, he was seeing that nature be described as it was. His work was an astonishingly bold new way of thinking about light and colors, and it would eventually be recognized as one of his great accomplishments. Presenting it was a baby step toward becoming the greatest British intellectual of his day. In fact, when I was in London, I noticed homage on Newton’s tomb in the form of a cherublike creature playing with a prism.

Now a twenty-eight-year-old Cambridge professor, he was ready to take what should have been a victory lap. But as great an accomplishment as this work would eventually be for Newton, his original 1672 paper instead created trouble. He was forced to endure stinging public criticisms of his optics work by his contemporaries—especially Robert Hooke—and Newton did not have the reputation or prestige that he would later wield against Leibniz to deflect it.

The members of the Royal society showed how seriously they regarded Newton’s work by appointing a committee to look into the paper and write a report thoroughly evaluating it. Hooke was the one to write the report, and he included in it his criticisms of Newton’s conclusions. Not coincidentally, the report protected Hooke’s own intellectual territory.

Hooke was the foremost authority in Britain on optics at the time, and he had been the curator of experiments at the Royal Society for ten years—a position that he rose to not through politicking but brilliance, especially in his work in optics and the application of optics to microscopy. Hooke’s opinion was so highly regarded in London society that after the great fire, he was one of a handful of commissioners chosen by the city for the rebuilding effort.

Hooke was also infamous as one of the most outspoken and intellectually cutthroat of the Royal Society’s members and often wielded the esteem of his position like an ax. In 1672, he set his sights on Newton’s theory of colors, sending the Royal Society a condescending letter claiming to have performed all the experiments himself, prior to Newton. In addition, he concluded the experiments proved that light was a propagating pulse through a transparent medium and color was a refraction of light—exactly what Newton’s work was supposed to be refuting. In other words, Hooke claimed that the difference was not of data but of the interpretation of data.

“I have perused the Excellent Discourse of Mr. Newton about colors and Refractions, and I was not a little pleased with the niceness and curiosity of his observations,” Hooke wrote, “[But the experiments] do seem to me to prove that light is nothing but a pulse or motion propagated through an homogeneous, uniform and transparent medium, and that color is nothing but the disturbance of that light by the communication of that pulse to other transparent mediums.”

This letter, which Hooke had taken all of three or four hours to write, must have been a smashing blow to Newton. Hooke was one of his heroes, and Newton had been greatly influenced by Hooke’s famous book, Micrographia, his seminal studies of the microscopic world—a book that Pepys called “the most ingenious book that I have ever read in my life.” When Newton had read Micrographia, he had been fascinated by the detailed drawings of lenses and lengthy discussions of optics inside the book and had recorded pages and pages of notes on it.

After reading Hooke’s 1672 letter condemning him, Newton spent three months composing a reply, carefully going over his notebooks and other materials and pulling together many different lines of thought to address Hooke’s criticisms in a single lengthy discourse. The brash young twenty-something scientist confronted his elder head on. He wrote pages and pages addressing Hooke’s criticisms point by point. After a few months’ delay, he sent a highly edited version. As in so many other times in his life, Newton showed that his best defense was a strong offense. He opined that Hooke’s theory was “not only insufficient, but in some respects unintelligible.”

Newton essentially believed that objections without experimental results should be rendered invalid. And he had done the experiments. Once separated into component colors, the various colors of light could not be further separated or changed by passing them through a prism.

“I have intercepted [a single colored ray of light] with the colored film of air interceding two compressed plates of glass; transmitted it through colored mediums, and through mediums irradiated with other sort of rays, and diversely terminated it, and yet could never produce any new color out of it,” Newton wrote in his paper. “It would by contracting or dilating become more brisk, or faint, and by the loss of many rays, in some cases very obscure and dark; but I could never see it changed in specie.”

Newton was not the only one who had a hard time getting his novel theories accepted—based on experiments though they were. In fact, this was a common theme in the seventeenth century. Johannes Kepler’s theory that the planets follow elliptical orbits was a hard pill for many of his contemporaries to swallow. Circles were more perfect shapes, the criticism went, so what need would the heavens have of ellipses? This same kind of thinking caused many to question the existence of sunspots after Galileo discovered them. Why would the sun have spots? Galileo faced a similar protest of his discovery of moons that circled Jupiter. Because these moons were invisible to the naked eye, Galileo was ridiculed by at least one Italian scholar, who said in effect that if we couldn’t see them, they would be of no use to us, and therefore couldn’t exist. The critic also made a complicated argument that involved the number seven. New moons would increase the number of planets and moons in the solar system above seven. But there could only be seven planets for the sake of natural harmony—just as there were seven orifices on the human head.

Not all resistance to new ideas was so banal. These were dangerous times for ideas as well as their authors. The inquisition in Rome placed Galileo under house arrest for life, and, after publication of his Dialogue in 1623, banned him from ever publishing again. Descartes left his native France in 1628 for fear that he would be persecuted for writing unpopular ideas, and he stayed in self-imposed exile in the Netherlands until 1644. John Bunyan, who wrote Pilgrim’s Progress, the so-called layman’s bible that was one of the most famous books of the seventeenth century, was locked up from 1660 to 1672 for the seemingly innocuous charge of preaching without a license. Giordano Bruno was burned at the stake in 1600 for daring to put forth unpopular positions.

Newton never faced anything as harsh as burning at the stake, but there is no question that Hooke’s attacks clouded his psyche for decades. And Hooke was not alone in opposing him.

In the months after Newton sent in his paper on colors, other criticisms drifted in from the continent, and Newton responded with a number of letters. He got comments from a Jesuit priest, Father Ignatius Pardies, who was a respected member of the Paris community of scientists. Pardies protested that he simply could not believe that colored rays combined should make white light. His comments were intelligent, valid criticisms that Newton was able to address in kind. Likewise, intelligent comments came in from Huygens, Leibniz’s Paris mentor. However, criticisms of a different nature came from a Belgian named Franciscus Linus, whose greatest legacy seems to be being remembered as a stupid, ignorant, and narrow-minded man.

The effect of the criticism, comments, and correspondence on Newton was to send him back into his turtle shell of Cambridge. He even intimated to Oldenburg that he would prefer to drop out of the Royal Society, and was considering abandoning all experimental research.

The unfortunate victim of all this fighting was Newton’s work on calculus, since Newton always intended to publish his optical and calculus work together. The pain of publishing the former caused him to abandon plans to publish the latter. Because of the trouble with Hooke, Newton lost his taste for publishing altogether. If there was any possibility of his publishing his mathematical works before, there was no longer any question that this could not be done. Though he had invented his fluxional calculus in the mid-1660s, the world would have to wait another two decades before it got a taste of it. And when it did, Newton would not be the author. Until then, Newton became a sort of Greta Garbo of the science world.

Events were transpiring in Europe—a war for France and much of the rest of the Continent was looming on the horizon—that would steer Leibniz first to Paris and then to London—and into a collision course with Newton. Leibniz would display none of Newton’s reservations about publishing or sharing his ideas with others.