In explaining why those curves were as they were, the Fellows of Cambridge would instinctively use Euclid’s geometry: the earth is a sphere. Its orbit around the sun is an ellipse-you get an ellipse by constructing a vast imaginary cone in space and then cutting through it with an imaginary plane; the intersection of the cone and the plane is the ellipse. Beginning with these primitive objects (viz. the tiny sphere revolving around the place where the gigantic cone was cut by the imaginary plane), these geometers would add on more spheres, cones, planes, lines, and other elements-so many that if you could look up and see ’em, the heavens would turn nearly black with them-until at last they had found a way to account for the curves that Newton had drawn on the wall. Along the way, every step would be verified by applying one or the other of the rules that Euclid had proved to be true, two thousand years earlier, in Alexandria, where everyone had been a genius.

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 OPPORTUNITYto 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 down-river, 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 fluxion-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.”