Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (41 page)

9

Mathematics for a New World

AN INFINITY OF LINES

Wallis was the only mathematician among the Society’s founders, and it therefore fell to him to address the problematic status of mathematics. He fully shared his associates’ abhorrence of dogmatism, and in his autobiography he took pride in his moderation and openness to diverse opinions, even when they conflicted with his own views. “It hath been my endeavour all along,” he wrote by way of a summary, “to act by moderate principles, between the extremities on either hand … without the fierce and violent animosities usual in such cases against all that did not act just as I did, knowing that there were many worthy persons engaged on either side.”

Yet, as a mathematician and Savilian Professor, Wallis was committed to a field that had traditionally prided itself on its inflexible methodology and the absolute and incontestable truth of its results. How was one to reconcile this with the moderation and flexibility he cherished as a member of the Royal Society? Wallis’s solution was simple, and radical: he created a new kind of mathematics. Unlike traditional mathematics, this new approach would proceed not through rigorous deductive proof but through trial and error; its results would be extremely probable but not irrefutably certain, and they would be validated not through “pure reason” but by consensus, just like the public experiments conducted at the Royal Society. Ultimately, his mathematics would be judged not by its logical perfection, but by its effectiveness at producing new results.

His mathematics, in other words, was not modeled on Euclidean geometry, the grand logical edifice that had inspired Clavius, Hobbes, and innumerable others over two millennia; it was, rather, designed to emulate the experimental approach practiced at the Royal Society. If Wallis succeeded, he would free mathematics of its association with dogmatism and intolerance and resolve the long-standing objections his fellows at the Society had toward the field. It would be a new “experimental mathematics,” powerful and effective in the service of science, but serving as a model for tolerance and moderation rather than dogmatic rigidity. And at its very core would be the concept of the infinitely small.

The singular nature of Wallis’s approach is apparent from the very first theorem of the very first work he wrote and published as Savilian Professor, “On Conic Sections” (
De sectionibus conicis
).

I suppose, to begin with (according to Bonaventura Cavalieri’s
Geometry of Indivisibles
) that any plane is made up, so to speak, of infinite parallel lines. Or rather (as I prefer) of an infinite number of parallelograms of equal height, the altitude of each one being
of the entire height, or an infinitely small aliquot part (the sign
∞
denoting an infinite number); so that the altitude of all equal the height of the figure.

Immediately we are in the highly unorthodox world of Wallis’s infinitesimal mathematics. Like Cavalieri and Torricelli before him, Wallis considered planes as quasi-material objects made up of an infinite number of lines stacked on top of one another, not as the abstract concepts of Euclidean geometry. That this conflicted with the classical paradoxes of Zeno, and with the problem of incommensurability, was obvious to any mathematician reading the tract, and both Hobbes and the French mathematician Pierre de Fermat were quick to point this out. But Wallis was unimpressed with these obvious criticisms. His notion that plane figures are composed of lines was derived from Cavalieri’s famous analogy of a plane to a piece of cloth made of threads, as well as from the Jesuat’s practice of viewing a plane as an aggregate of lines. He therefore simply referred the reader to Cavalieri, who supposedly had already dealt with all objections and moved on. Wallis even invented a sign to mark the number of infinitesimals that make up the plane and their magnitudes, respectively,
∞
and
.

With these basic tools in hand, Wallis then proceeds to demonstrate the power of his approach by proving an actual theorem:

Since a triangle consists of an infinite number of arithmetically proportionate lines or parallelograms, beginning with a point and continuing to the base (as is clear from the discussion): then the area of the triangle is equal to the base times half the altitude.

Needless to say, Wallis did not need to provide a complex proof in order to determine that the area of a triangle is half its base times its height. The purpose of the proof was not to prove the result, but quite the opposite: to demonstrate the validity of his unconventional approach, by showing that it led to a correct and familiar result. Once he had established the reliability of his method, he could then use it to resolve more challenging and unfamiliar problems.

The statement that the lines composing a triangle are “arithmetically proportionate” calls for some explanation. What Wallis means is that if lines are drawn through a triangle parallel to its base, and if those lines are equally spaced along the triangle’s height, then the lengths of the lines form an arithmetic progression. For example, if a line is drawn halfway between the apex of the triangle and the base, its length will be half that of the base, forming the arithmetical series (0,
, 1), for the apex, the line, and the base, respectively. If the height is divided into three, and lines are drawn at the one-third and two-third marks, their lengths will form the series (0,
,
, 1); if the height is divided into ten, the lengths will be (0,
,
,
 …
, 1), and so on. This holds true regardless of how many parts the height is divided into, as long as those parts are at an equal distance from one another. In his proof, Wallis assumes that this principle holds true even if the height is divided into an infinite number of parts.