The Clockwork Universe

by Edward Dolnick

9 Digby assured his audience that “there is great quantity of it in Ireland.”

  10 Bacon’s zeal for experimentation may have done him in. On a winter’s day when he happened to be in the company of the royal physician, Bacon suddenly had the bright idea that perhaps snow could preserve meat. “They alighted out of the coach and went into a poor woman’s house at the bottom of Highgate hill, and bought a fowl,” wrote the memoirist John Aubrey, and Bacon stuffed the bird with snow. Bacon came down with what proved to be a fatal case of pneumonia. He blamed the snow but noted on his deathbed that the story had a bright side. “As for the experiment itself, it succeeded excellently well.”

  11 The ancient world had clung just as fiercely to the code of secrecy. Legend has it that Pythagoras banished one of his followers (or in some accounts threw him off a boat, drowning him) for “telling men who were not worthy” a dreadful mathematical secret. Hippasus’s sin was revealing to outsiders the discovery that certain numbers (in this case, the square root of 2) cannot be written down precisely (14/10 is close, for instance, but no fraction is exact). The Greeks found this numerical truth horrifying, a rip in the cosmic fabric.

  12 We still see relics of that prejudice against “applied” knowledge today. The historian Paolo Rossi notes that the term “liberal arts” originally came into use to mark off those areas of study deemed proper for a gentleman’s education. These were the fields suited to free men (liberi) rather than to servants or slaves.

  13As a thirteen-year-old, Hooke briefly apprenticed with the famous portrait painter Peter Lely. (It was Lely whom Oliver Cromwell instructed to “paint my picture truly like me,” warts and all.) Hooke’s artistic career came to an early end when he found he was allergic to the paints and oils in Lely’s studio.

  14The esteemed eighteenth-century mathematician Laplace, for example, inspired despair even in his admirers. “I never came across one of Laplace’s ‘Thus it plainly appears,’ ” wrote one, “without feeling sure that I have hours of hard work before me to fill up the chasm and find out and show how it plainly appears.”

  15 “As one of [Thomas More’s] daughters was passing under the bridge,” according to John Aubrey, “looking on her father’s head, said she, ‘That head has lain many a time in my lap, would to God it would fall into my lap as I pass under.’ She had her wish, and it did fall into her lap, and is now preserved in a vault in the cathedral church at Canterbury.”

  16The word disease is a relic of this theory. When the humors fell out of balance, the patient’s ease gave way to dis-ease.

  17 Like James Thurber, who never managed to see anything through a microscope but a reflection of his own eye, Pepys had trouble getting the hang of his microscope. “My wife and I with great pleasure,” he wrote in his diary in August 1664, “but with great difficulty before we could come to find the manner of seeing anything.”

  18 Glanvill provides yet another example of how seventeenth-century scientists simultaneously endorsed new beliefs and clung to old ones. He argued strenuously in favor of science’s new findings and at the same time insisted that spirits, demons, and witches were real. To deny the existence of evil spirits, Glanvill insisted, was to veer dangerously near to saying that only the tangible was real, and that was tantamount to atheism. No witches, no God!

  19 The mystery would only be unraveled around 1800.

  20 The moon gave the Greeks problems. It was a heavenly body, which meant it had to be perfect and unblemished, but no one could miss its patches of light and dark. One attempted explanation: the moon was a perfect mirror and its dark spots were the reflections of oceans on Earth.

  21 The stars will not look exactly the same, mostly because the earth wobbles a bit on its axis, like a spinning top. But the changes are so small that art historians and astronomers, working together, have answered such questions as what the sky over St.-Rémy-de-Provence looked like on June 19, 1889, the night Van Gogh painted “Starry Night.” (Van Gogh stuck remarkably close to reality.)

  22 Galileo’s intellectual offspring espouse the same view today, in virtually identical words. “To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature,” wrote the physicist Richard Feynman. “If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.”

  23Modern-day astronomers have shown that Tycho’s star was a supernova, an exploding star, rather than a new one.

  24Fontenelle’s exuberance was characteristic, though consistency was not his strength. In the next breath he professed, with equal verve, to be worried that the immensity of the universe made his own efforts seem tiny and irrelevant. Like Carl Sagan in our day, he was known as much for enthusiasm as scholarship. Fontenelle lived to be one hundred and scarcely slowed down along the way. Near the end, he met one famous beauty and remarked, “Ah madame, if I were only eighty again!”

  25 Leeuwenhoek was a contemporary of Vermeer. Both men lived in Delft, the two shared a fascination with light and lenses, and Leeuwenhoek served as executor of Vermeer’s will. Some art historians believe that Vermeer’s Astronomer and his Geographer both depict Leeuwenhoek, but no one has been able to prove that Leeuwenhoek and Vermeer ever met.

  26 The microscope that Leeuwenhoek used on that fateful night was put up for auction in April 2009. The winning bidder paid $480,000.

  27 As one of Pythagoras’s followers told the tale, the story began when Pythagoras listened to the sound of hammering as he walked by a blacksmith’s shop. As the blacksmith struck the same piece of iron with different hammers, some sounds were harmonious, others not. The key, Pythagoras found, was whether the weights of the hammers happened to be in simple proportion. A twelve-pound hammer and a six-pound hammer, for instance, produced notes an octave apart.

  28 Augustine did not explain why God did not make the world in 28 days (1 + 2 + 4 + 7 + 14) or 496 days or various other possibilities.

  29 A prime number is one that can’t be broken down into smaller pieces. For example, 2 is prime, and so are 3, 5, and 7; 10 is not prime (because 10 = 2 × 5). Prime numbers get rarer as you count higher and higher, but no matter how big a prime you name, there is always a bigger one.

  30 There are infinitely many choices of a, b, and c that satisfy a2 + b2 = c2. But if you try any power higher than 2—if, for instance, you try to find whole numbers a, b, and c that satisfy a3 + b3 = c3 or a4 + b4 = c4—you will never find a single example that works (discounting the trivial case where a, b, and c are all set equal to 0). The statement that no such example exists is one of the most famous in mathematics. It is known as Fermat’s last theorem, after the mathematician Pierre de Fermat, who jotted it down in the margin of a book in 1637. He had found “a truly marvelous proof,” he scribbled, but “the margin is not large enough” to fit it. No one ever found his proof—presumably he’d made a mistake in his reasoning—and for more than three hundred years countless mathematicians tried and failed to find proofs of their own. Success finally came in 1995, as detailed in Amir Aczel’s Fermat’s Last Theorem.

  31 At the half-moon, for instance, sun, moon, and Earth form a right triangle.

  32 The nineteenth-century German mathematician Carl Gauss, a towering figure in the history of mathematics, believed in the possibility of life on other worlds. Gauss supposedly proposed—the story may well be apocryphal—that since all intelligent beings would eventually discover the same mathematical truths, we could communicate with moon creatures by choosing a vast, empty space in Siberia and planting trees in an enormous diagram of the Pythagorean theorem.

  33Pluto is considerably smaller than the moon, and in 2006 astronomers decided to downgrade it to “minor planet” status.

  34 Not by the human ear, at any rate. God could hear these cosmic harmonies, as dogs can detect whistles pitched too high for human hearing.

  35 The first person to refer to Kepler’s “laws” was Voltaire, in 1738. Scientists eventually fol
lowed his lead.

  36 Tycho, like Galileo, is generally referred to by his first name.

  37 A circle can be thought of as a special ellipse, one in which the two focuses are in the same place.

  38 Ballet dancers and basketball players seem to hang in midair, but that is an illusion. The trick for both dancer and athlete is to throw in a few moves midflight. The eye reads the extra motions as taking extra time.

  39 “Music,” Leibniz wrote, “is the pleasure the human soul experiences from counting without being aware that it is counting.”

  40 To be more accurate, in t seconds a ball falls a distance proportional to t2 inches rather than precisely equal to t2 inches. (It falls, for instance, 3 × t2 inches or 10 × t2 inches or some other multiple, depending on the steepness of the ramp.) Everything I’ve said here carries over to the more general case, but the numbers would be off-putting. For purposes of illustration, I chose the ramp that showed the pattern most clearly.

  41 One prominent historian calls it “incomprehensible” that Greek mathematicians never conceived of graphs. But neither did their intellectual descendants for well over a thousand years. Even an enormous hint went unnoticed. Monks in the Middle Ages invented musical notation, which meant they no longer had to commit countless chants to memory. “The musical staff was Europe’s first graph,” noted the historian Alfred Crosby, but several more centuries would pass before scientists saw that they, too, could use graphs to depict changes in time.

  42 The question of whether vacuums could exist spurred long, angry debates. The invention of the air pump did not settle the debate, in the view of Leibniz and some others, because even if a jar no longer contained air it might still contain some more ethereal fluid. Leibniz and Descartes both maintained that the very notion of a vacuum was nonsensical—how could there be a place containing nothing at all, when the meaning of the word place is “the location where something happens to be”? Newton and Pascal insisted just as vehemently that vacuums were real. Descartes contended, cattily, that the only vacuum was in Pascal’s head.

  43 If the first block were 1 inch thick, the next ½ inch, then ⅓, ¼, 1/5, and so on, the tower would climb infinitely high (although it would rise excruciatingly slowly).

  44 A sequence may attain its goal. The sequence 1, 1, 1, . . . has the number 1 as its limit. But a “typical” sequence draws ever nearer to its goal without actually touching it. The sequence .9, .99, .999, . . . never reaches its limit, which is the number 1.

  45 Gilbert and Sullivan’s Major-General knew it well, along with much else. “About binomial theorem I’m teeming with a lot o’ news / With many cheerful facts about the square of the hypotenuse.”

  46 Unbeknownst to Leibniz, the English mathematician and astronomer Thomas Harriot had been the first to discuss binary numbers, decades before. But Harriot never published any of his work, and his papers went unseen until the late 1700s. It turns out that Harriot had recorded a number of other firsts as well; Harriot turned a telescope to the sky a few weeks before Galileo did.

  47 Sometimes the right notation can even hint at a deep, surprising insight. Simply using decimal notation, and then adding column by column, suggests that 1 + .1 + .01 + .001 + . . . = 1.11111 . . . , and not infinity.

  48 Most people “know” not just that an apple fell but that it bonked Newton on the head.

  49 Another exotic import, tea, had arrived at about the same time, although coffee caught on first. On September 25, 1660, Pepys wrote in his diary that “I did send for a cup of tee (a China drink) of which I never had drank before.”

  50 The statement if a planet travels in an ellipse, then it follows an inverse-square law is different from the statement if a planet follows an inverse-square law, then it travels in an ellipse. It might have been that one was true but the other was not. If someone owns a dog, then he owns a pet is true; if someone owns a pet, then he owns a dog is not. In this case it was clear to Newton (though bewilderingly obscure to others) that if one statement was true, the other had to be true as well.

  51 Sixteen hundred years before Newton, Plutarch wrote that Archimedes grew so absorbed in his thoughts that he “would often forget his food and neglect his person” and have to be “carried by absolute violence to bathe.”

  52 In time, this bewilderment died away. Darwin noted impatiently that, although his critics demanded that he explain where intelligence and awareness come from, nobody demanded a similar account of gravity. “Why is thought being a secretion of brain more wonderful than gravity a property of matter?” he asked.

  53 The debate over whether we should look at scientists’ characters and motives, or if it is only their findings that matter, continues today. “Science doesn’t work because we’re all nice,” a NASA climatologist declared in November 2009, in the midst of a dispute over global warming. “Newton may have been an ass, but the theory of gravity still works.”