Some of the works Newton had already read used infinite series to attack a variety of questions—for one, chasing down ever more precise values for that bane of middle school geometers, π(the number used to calculate the circumference of circles). Newton, with his exceptional ability to speak both algebra and geometry, began to use infinite series to work out how curves behave. One of his favorite tricks was to think about the area under a curve—all that space on the graph between the curve and the x axis—and then build out a series that would add together smaller and smaller patches of that area until the sum of all those terms approached the entire territory and number being sought. Newton applied that idea to a wide range of different curves. He wrote out sequences. He plugged in numbers. He exhausted himself in calculation, cranking his exercises out to fifty decimal places and more.
He totaled his sums—and then discovered what modern mathematics calls the generalized binomial theorem. This result allowed Newton to solve a wide range of specific algebraic equations, including, most significantly, the problem of the area beneath a curve (called quadrature), not just for one shape at a time, but for whole classes of curves. It was a discovery that became one of the pillars of modern mathematics.
As he played with his series, he noticed that in some of them each step in the calculation added a smaller and smaller amount to the total. Extending the operation by hand—row after row of numbers, a strangely beautiful triangle, growing across the page—produced a better and better fit to the ultimate answer. The endpoint, well beyond the stamina of even so heroic a numbers-cruncher as Newton, was obvious: the last terms in such series must dwindle toward nothing. Toward, but never all the way there, an infinitely small approach to zero.
Newton’s calculation of logarithms
Newton was not the first to ponder such infinitesimals. The Greek philosopher Zeno had played with the idea in his famous paradox: the race between the hero Achilles and a tortoise. With a fine sense of fair play, fleet-footed Achilles gave his opponent a head start. According to Zeno, that meant that no matter how much faster Achilles ran, he’d never overtake the tortoise. His reasoning was that in the time it took him to reach where the tortoise had just been, the reptile would move a little farther. When Achilles moved to that point, the tortoise would have moved again, and so on, forever. That increment of distance could get as small as you like, Zeno said, but it would never quite disappear. Hence, the tortoise would beat Achilles every time.
That’s obviously absurd: in real life, an Achilles would charge past a tortoise, no matter how generous the head start. As early as Aristotle, logicians offered formal arguments to refute Zeno. But neither philosophical rigor nor common sense could erase the uneasiness produced by the idea of ever-smaller quantities. Many, like Descartes, simply didn’t want to wrestle with an increment so tiny that it was effectively but not quite zero. Galileo knew that there was something vital about that infinitude but quailed before its mystery, “incomprehensible to our understanding.” That is: there was, as yet, no established mathematical procedure that could use quantities indistinguishable from nothing to demonstrate that yes, in fact, Achilles smokes the tortoise every time. Newton himself, in his first months at Woolsthorpe, was often perplexed, straining to interpret the difference between almost zero and zero itself. But he didn’t linger in the tangled metaphysics of a nothing that wasn’t quite nothing. Instead, he put it to use.
In one case, Newton wanted to be able to identify how much a curve was curving at any point: how steep it might be, and how that steepness—Newton called it “the crookedness in lines”—changed at each point along the figure. Here, he used infinitesimals to produce a straight line whose slope could be calculated and that touched the curve at just that one point and no other—what’s called a tangent.
Such problems took Newton unequivocally beyond classical approaches, in which the curves he was trying to understand had been examined as whole, finished phenomena, parabolas or ellipses or anything else of interest as the object of study. But Newton’s thinking in the last months of 1665 employed his genuinely new way of seeing, in which the mathematical objects he analyzed emerged in the solutions to equations, point after point accumulating along the figure.