Isaac hadn’t studied Euclid that much, and hadn’t cared enough to study him well. If he wanted to work with a curve he would instinctively write it down, not as an intersection of planes and cones, but as a series of numbers and letters: an algebraic expression. That only worked if there was a language, or at least an alphabet, that had the power of expressing shapes without literally depicting them, a problem that Monsieur Descartes had lately solved by (first) conceiving of curves, lines, et cetera, as being collections of individual points and (then) devising a way to express a point by giving its coordinates—two numbers, or letters representing numbers, or (best of all) algebraic expressions that could in principle be evaluated to generate numbers. This translated all geometry to a new language with its own set of rules: algebra. The construction of equations was an exercise in translation. By following those rules, one could create new statements that were true, without even having to think about what the symbols referred to in any physical universe. It was this seemingly occult power that scared the hell out of some Puritans at the time, and even seemed to scare Isaac a bit.
By 1664, which was the year that Isaac and Daniel were supposed to get their degrees or else leave Cambridge, Isaac, by taking the very latest in imported Cartesian analysis and then extending it into realms unknown, was (unbeknownst to anyone except Daniel) accomplishing things in the field of natural philosophy that his teachers at Trinity could not even comprehend, much less accomplish—they, meanwhile, were preparing to subject Isaac and Daniel to the ancient and traditional ordeal of examinations designed to test their knowledge of Euclid. If they failed these exams, they’d be branded a pair of dimwitted failures and sent packing.
As the date drew nearer, Daniel began to mention them more and more frequently to Isaac. Eventually they went to see Isaac Barrow, the first Lucasian Professor of Mathematics, because he was conspicuously a better mathematician than the rest. Also because recently, when Barrow had been traveling in the Mediterranean, the ship on which he’d been passenger had been assaulted by pirates, and Barrow had gone abovedecks with a cutlass and helped fight them off. As such, he did not seem like the type who would really care in what order students learned the material. They were right about that—when Isaac showed up one day, alarmingly late in his academic career, with a few shillings, and bought a copy of Barrow’s Latin translation of Euclid, Barrow didn’t seem to mind. It was a tiny book with almost no margins, but Isaac wrote in the margins anyway, in nearly microscopic print. Just as Barrow had translated Euclid’s Greek into the universal tongue of Latin, Isaac translated Euclid’s ideas (expressed as curves and surfaces) into Algebra.
Half a century later on the deck of Minerva, that’s all Daniel can remember about their Classical education; they took the exams, did indifferently (Daniel did better than Isaac), and were given new titles: they were now scholars, meaning that they had scholarships, meaning that Newton would not have to go back home to Woolsthorpe and become a gentleman-farmer. They would continue to share a chamber at Trinity, and Daniel would continue to learn more from Isaac’s idle musings than he would from the entire apparatus of the University.
ONCE HE’S HAD THE OPPORTUNITY to settle in aboard Minerva, Daniel realizes it’s certain that when, God willing, he reaches London, he’ll be asked to provide a sort of affidavit telling what he knows about the invention of the calculus. As long as the ship’s not moving too violently, he sits down at the large dining table in the common room, one deck below his cabin, and tries to organize his thoughts.
Some weeks after we had received our Scholarships, probably in the Spring of 1665, Isaac Newton and I decided to walk out to Stourbridge Fair.
Reading it back to himself, he scratches out probably in and writes in certainly no later than.
Here Daniel leaves much out—it was Isaac who’d announced he was going. Daniel had decided to come along to look after him. Isaac had grown up in a small town and never been to London. To him, Cambridge was a big city—he was completely unequipped for Stourbridge Fair, which was one of the biggest in Europe. Daniel had been there many times with father Drake or half brother Raleigh, and knew what not to do, anyway.
The two of us went out back of Trinity and began to walk downstream along the Cam. After passing by the bridge in the center of town that gives the City and University their name, we entered into a reach along the north side of Jesus Green where the Cam describes a graceful curve in the shape of an elongated S.
Daniel almost writes like the integration symbol used in the calculus. But he suppresses that, since that symbol, and indeed the term calculus, were invented by Leibniz.
I made some waggish student-like remark about this curve, as curves had been much on our minds the previous year, and Newton began to speak with confidence and enthusiasm—demonstrating that the ideas he spoke of were not extemporaneous speculation but a fully developed theory on which he had been working for some time.
“Yes, and suppose we were on one of those punts,” Newton said, pointing to one of the narrow, flat-bottomed boats that idle students used to mess about on the Cam. “And suppose that the Bridge was the Origin of a system of Cartesian coordinates covering Jesus Green and the other land surrounding the river’s course.”
No, no, no, no. Daniel dips his quill and scratches that bit out. It is an anachronism. Worse, it’s a Leibnizism. Natural Philosophers may talk that way in 1713, but they didn’t fifty years ago. He has to translate it back into the sort of language that Descartes would have used.
“And suppose,” Newton continued, “that we had a rope with regularly spaced knots, such as mariners use to log their speed, and we anchored one end of it on the Bridge—for the Bridge is a fixed point in absolute space. If that rope were stretched tight it would be akin to one of the numbered lines employed by Monsieur Descartes in his Geometry. By stretching it between the Bridge and the punt, we could measure how far the punt had drifted downriver, and in which direction.”
Actually, this is not the way Isaac ever would have said it. But Daniel’s writing this for princes and parliamentarians, not Natural Philosophers, and so he has to put long explanations in Isaac’s mouth.
“And lastly suppose that the Cam flowed always at the same speed, and that our punt matched it. That is what I call a flux-ion—a flowing movement along the curve over time. I think you can see that as we rounded the first limb of the S-curve around Jesus College, where the river bends southward, our fluxion in the north-south direction would be steadily changing. At the moment we passed under the Bridge, we’d be pointed northeast, and so we would have a large northwards fluxion. A minute later, when we reached the point just above Jesus College, we’d be going due east, and so our north-south fluxion would be zero. A minute after that, after we’d curved round and drawn alongside Midsummer Commons, we’d be headed southeast, meaning that we would have developed a large southward fluxion—but even that would reduce and tend back towards zero as the stream curved round northwards again towards Stourbridge Fair.”
He can stop here. For those who know how to read between the lines, this is sufficient to prove Newton had the calculus—or Fluxions, as he called it—in ‘65, most likely ‘64. No point in beating them over the head with it…
Yes, beating someone over the head is the entire point.
Banks of the River Cam
1665
Almost five thousand years agone, there were pilgrims walking to the Celestial City, as these two honest persons are; and Beelzebub, Apollyon, and Legion, with their companions, perceiving by the path that the pilgrims made that their way to the City lay through this town of Vanity, they contrived here to set up a fair; a fair wherein should be sold all sorts of vanity, and that it should last all the year long. Therefore at this Fair are all such merchandise sold, as houses, lands, trades, places, honours, preferments, titles, countries, kingdoms, lusts, pleasures, and delights of all sorts, as whores, bawds, wives, husbands, children, masters, servants, lives, blood, bodies, souls, silver, go
ld, pearls, precious stones, and what not.
And moreover, at this Fair there is at all times to be seen jugglings, cheats, games, plays, fools, apes, knaves, and rogues, and that of all sorts.
—JOHN BUNYAN, The Pilgrim’s Progress
IT WAS LESS THAN AN hour’s walk to the Fair, strolling along gently sloped green banks with weeping willows, beneath whose canopies were hidden various prostrate students. Black cattle mowed the grass unevenly and strewed cow-pies along their way. At first the river was shallow enough to wade across, and its bottom was carpeted with slender fronds that, near the top, were bent slightly downstream by the mild current. “Now, there is a curve whose fluxion in the downstream direction is nil at the point where it is rooted in the bottom—that is to say, it rises vertically from the mud—but increases as it rises.”
Here Daniel was a bit lost. “Fluxion seems to mean a flowing over time—so it makes perfect sense when you apply the word to the position of a punt on a river, which is, as a matter of fact, flowing over time. But now you seem to be applying it to the shape of a weed, which is not flowing—it’s just standing there sort of bent.”
“But Daniel, the virtue of this approach is that it doesn’t matter what the actual physical situation is, a curve is ever a curve, and whatever you can do to the curve of a river you can do just as rightly to the curve of a weed—we are free from all that old nonsense now.” Meaning the Aristotelian approach, in which such easy mixing of things with obviously different natures would be abhorrent. All that mattered henceforth, apparently, was what form they adopted when translated into the language of analysis. “Translating a thing into the analytical language is akin to what the alchemist does when he extracts, from some crude ore, a pure spirit, or virtue, or pneuma. The fœces—the gross external forms of things—which only mislead and confuse us—are cast off to reveal the underlying spirit. And when this is done we may learn that some things that are superficially different are, in their real nature, the same.”
Very soon, as they left the colleges behind, the Cam became broader and deeper and instantly was crowded with much larger boats. Still, they were not boats for the ocean—they were long, narrow, and flat-bottomed, made for rivers and canals, but with far greater displacement than the little punts. Stourbridge Fair was already audible: the murmur of thousands of haggling buyers and sellers, barking of dogs, wild strains from bagpipes and shawms whipping over their heads like twists of bright ribbon unwinding in the breeze. They looked at the boat people: Independent traders in black hats and white neck-cloths, waterborne Gypsies, ruddy Irish and Scottish men, and simply Englishmen with complicated personal stories, negotiating with sure-footed boat-dogs, throwing buckets of mysterious fluids overboard, pursuing domestic arguments with unseen persons in the tents or shacks pitched on their decks.
Then they rounded a bend, and there was the Fair, spread out in a vast wedge of land, bigger than Cambridge, even more noisy, much more crowded. It was mostly tents and tent people, who were not their kind of people—Daniel watched Isaac gain a couple of inches in height as he remembered the erect posture that Puritans used to set a better example. In some secluded parts of the Fair (Daniel knew) serious merchants were trading cattle, timber, iron, barrelled oysters—anything that could be brought upriver this far on a boat, or transported overland in a wagon. But this wholesale trade wanted to be invisible, and was. What Isaac saw was a retail fair whose size and gaudiness was all out of proportion to its importance, at least if you went by the amount of money that changed hands. The larger avenues (which meant sluices of mud with planks and logs strewn around for people to step on, or at least push off against) were lined with tents of rope-dancers, jugglers, play-actors, puppet shows, wrestling-champions, dancing-girls, and of course the speciality prostitutes who made the Fair such an important resource for University students. But going up into the smaller byways, they found the tables and stalls and the cleverly fashioned unfolding wagons of traders who’d brought goods from all over Europe, up the Ouse and the Cam to this place to sell them to England.
Daniel and Isaac roamed for the better part of an hour, ignoring the shouts and pleadings of the retailers on all sides, until finally Isaac stopped, alert, and sidestepped over to a small folding display-case-on-legs that a tall slender Jew in a black coat had set up. Daniel eyed this Son of Moses curiously—Cromwell had readmitted these people to England only ten years previously, after they’d been excluded for centuries, and they were as exotic as giraffes. But Isaac was staring at a constellation of gemlike objects laid out on a square of black velvet. Noting his interest, the Kohan folded back the edges of the cloth to reveal many more: concave and convex lenses, flat disks of good glass for grinding your own, bottles of abrasive powder in several degrees of coarseness, and prisms.
Isaac signalled that he would be willing to open negotiations over two of the prisms. The lens-grinder inhaled, drew himself up, and blinked. Daniel moved round to a supporting position behind and to the side of Isaac. “You have pieces of eight,” the circumcised one said—midway between an assertion and a question.
“I know that your folk once lived in a kingdom where that was the coin of the realm, sir,” Isaac said, “but…”
“You know nothing—my people did not come from Spain. They came from Poland. You have French coins—the louis d’or?”
“The louis d’or is a beautiful coin, befitting the glory of the Sun King,” Daniel put in, “and probably much used wherever you came from—Amsterdam?”
“London. You intend to compensate me, then, with what—Joachimsthalers?”
“As you, sir, are English, and so am I, let us use English means.”
“You wish to trade cheese? Tin? Broadcloth?”
“How many shillings will buy these two prisms?”
The Hebraic one adopted a haggard, suffering look and gazed at a point above their heads. “Let me see the color of your money,” he said, in a voice that conveyed gentle regret, as if Isaac might have bought some prisms today, and instead would only get a dreary lesson in the unbelievable shabbiness of English coinage. Isaac reached into a pocket and wiggled his fingers to produce a metallic tromping noise that proved many coins were in there. Then he pulled out a handful and let the lens-grinder have a glimpse of a few coins, tarnished black. Daniel, so far, was startled by how good Isaac was at this kind of thing. On the other hand, he had made a business out of lending money to other students—maybe he had talent.
“You must have made a mistake,” said the Jew. “Which is perfectly all right—we all make mistakes. You reached into the wrong pocket and you pulled out your black money*—the stuff you throw to beggars.”
“Ahem, er, so I did,” Isaac said. “Pardon me—where’s the money for paying merchants?” Patting a few other pockets. “By the way, assuming I’m not going to offer you black money, how many shillings?”
“When you say shillings, I assume you mean the new ones?”
“The James I?”
“No, no, James I died half a century ago and so one would not normally use the adjective new to describe pounds minted during his reign.”
“Did you say pounds?” Daniel asked. “A pound is rather a lot of money, and so it strikes me as not relevant to this transaction, which has all the appearances of a shilling type of affair at most.”
“Let us use the word coins until I know whether you speak of the new or the old.”
“New meaning the coins minted, say, during our lifetimes?”
“I mean the Restoration coinage,” the Israelite said, “or perhaps your professors have neglected to inform you that Cromwell is dead, and Interregnum coins demonetized these last three years.”
“Why, I believe I have heard that the King is beginning to mint new coins,” Isaac said, looking to Daniel for confirmation.
“My half brother in London knows someone who saw a gold CAROLUS II DEI GRATIA coin once, displayed in a crystal case on a silken pillow,” Daniel said. “People have begun to call the
m Guineas, because they are made of gold that the Duke of York’s company is taking out of Africa.”
“I say, Daniel, is it true what they say, that those coins are perfectly circular?”
“They are, Isaac—not like the good old English hammered coins that you and I carry in such abundance in our pockets and purses.”
“Furthermore,” said the Ashkenazi, “the King brought with him a French savant, Monsieur Blondeau, on loan from King Louis, and that fellow built a machine that mills delicate ridges and inscriptions into the edges of the coins.”
“Typical French extravagance,” Isaac said.
“The King really did spend more time than was good for him in Paris,” Daniel said.
“On the contrary,” the forelocked one said, “if someone clips or files a bit of metal off the edge of a round coin with a milled edge, it is immediately obvious.”
“That must be why everyone is melting those new coins down as fast as they are minted, and shipping the metal to the Orient…?” Daniel began,
“…making it impossible for the likes of me and my friend to obtain them,” Isaac finished.
“Now there is a good idea—if you can show me coins of a bright silver color—not that black stuff—I’ll weigh them and accept them as bullion.”
“Bullion! Sir!”