Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (17 page)

The next six years were remarkably productive for Torricelli. Previously he was so little known that Galileo had hardly heard of him, and Castelli had had to present him to Galileo as a former student. But with Galileo’s passing and his appointment as mathematician to the Medici court, Torricelli suddenly became one of the leading scientists in Europe. He began a long and fruitful correspondence with French scientists and mathematicians, including Marin Mersenne (1588–1648) and Gilles Personne de Roberval (1602–75), and forged connections with fellow Italian Galileans Raffaello Magiotti (1597–1656), Antonio Nardi (died ca. 1656), and Cavalieri. Inspired by the
Discourses
, he pondered Galileo’s thesis that nature’s
horror vacui
(abhorrence of a vacuum) is what holds objects together. This led him in 1643 to experiments that established that a vacuum could, in fact, exist in nature, and to the construction of the world’s first barometer.

Unlike the work of Galileo and Cavalieri, who published frequently, Torricelli’s work can be found mostly in his correspondence and in unpublished manuscripts he circulated among his friends and colleagues. The sole exception is a book entitled
Opera geometrica
, published in 1644 and containing a collection of essays on subjects ranging from the physics of motion to the area enclosed by a parabola. Some of these, such as Torricelli’s discussion of spheroids, rely on traditional mathematical methods derived from the ancients. The third treatise, however, entitled “De dimensione parabolae” (“On the Dimension of the Parabola”), is anything but traditional: it is Torricelli’s dramatic introduction of his own method of indivisibles.

TWENTY-ONE PROOFS

Surprisingly, given its name, the purpose of “De dimensione parabolae” is not the calculation of the area inside a parabola. This was calculated and demonstrated by Archimedes more than 1,800 years before, and was well known to Torricelli and his contemporaries. It requires no further proof. What the treatise does offer is no fewer than twenty-one different proofs of this familiar result. Twenty-one times in succession, Torricelli poses the theorem that “the area of a parabola is four thirds the area of a triangle with the same base and height,” and twenty-one times he proves it, each time differently. It is likely the only text in the history of mathematics to offer this many different proofs of a single result, and by a wide margin. It is a testament to Torricelli’s virtuosity as a mathematician, but its purpose was different: to contrast the traditional classical methods of proof with the new proofs by indivisibles, thereby showing the manifest superiority of the new method.

Figure 3.5. Torricelli, “De dimensione parabolae”: The area enclosed by the parabola
ABC
is four-thirds the area of the triangle
ABC
.

The first eleven proofs of “De dimensione” conform to the highest standards of Euclidean rigor. To calculate the area enclosed in a parabola, they make use of the classical “method of exhaustion,” attributed to the Greek mathematician Eudoxus of Cnidus, who lived in the fourth century BCE. In this method the curve of the parabola (or a different curve) is surrounded by a circumscribed and a circumscribing polygon. The areas of the two polygons are easy to calculate, and the area enclosed by the parabola lies somewhere in between. As one increases the number of sides of the two polygons, the difference between them becomes smaller and smaller, limiting the possible range of the area of the parabola.

Figure 3.6. The Method of Exhaustion. As the number of sides in the inscribed polygon is increased, its area more closely approximates the area of the parabola. The same is true of a circumscribing polygon.

The proof then proceeds through contradiction: If the area of the parabola is larger than four-thirds of the triangle with the same base and height, then it is possible to increase the number of sides of the circumscribing polygon to the point where the polygon’s area will be smaller than that of the parabola. If the parabola’s area is smaller than that, then it is possible to increase the number of sides of the circumscribed polygon to the point where its area will be larger than that of the parabola. Both these possibilities contradict the assumption that one polygon circumscribes the parabola and that the other is circumscribed by it, and therefore the area of the parabola must be exactly four-thirds of a triangle with the same base and height.
QED
.

Figure 3.7. The parabola segment
ABC
circumscribes the triangle
ABC
and is circumscribed by the triangle
AEC
. As the number of sides of the polygons is increased, as in the trapezoid
AFGC
, the enclosed area more closely approximates the area enclosed by the parabola segment.

While these traditional proofs were perfectly correct, they did, Torricelli pointed out, have some drawbacks. The most obvious one is that proofs by exhaustion require one to know in advance the desired outcome—in this case, the relationship between the areas of a parabola and a triangle. Once the result was known, the method of exhaustion could show that any other relationship would lead to a contradiction, but it offered no clue as to why this relationship holds, or how to discover it. This absence led Torricelli and many of his contemporaries to believe that the ancients possessed a secret method for discovering these relationships, which they then carefully edited out of their published works. (The twentieth-century discovery of Archimedes’s treatise on his nonrigorous method of discovery in the erased text of a tenth-century palimpsest suggests that they may not have been altogether wrong.) The other chief drawback of the classical method is that it is cumbersome, requiring numerous auxiliary geometrical constructions and leading to its conclusion by a roundabout and counterintuitive route. Classical proofs, in other words, might be perfectly correct, but they were far from being useful tools for obtaining new insights.

The last ten proofs of the “De dimensione parabolae” abandoned the traditional mold of the method of exhaustion, making use of indivisibles instead. These, as Torricelli pointed out, were direct and intuitive, showing not only that the results were true, but also
why
they were true, since they were derived directly from the shape and composition of the geometrical figures in question. We have already seen how Cavalieri proved the equivalence of the two triangles making up a parallelogram by showing that they were composed of the same lines, and the equivalence of the areas enclosed by a spiral and a parabola by translating the curved indivisibles of one into the straight indivisibles of the other. Torricelli proposed the same approach for calculating the area of a parabola. The method of indivisibles, according to Torricelli, was a “new and admirable way” for demonstrating innumerable theorems by “short, direct, and positive proofs.” It was “the Royal Road through the mathematical thicket,” compared to which the geometry of the ancients “arouses only pity.”

As Torricelli told it, the “marvelous invention” of indivisibles belonged entirely to Cavalieri, and his own contribution in the
Opera geometrica
was merely to make it more accessible. More accessible it certainly was, since Cavalieri’s
Geometria indivisibilibus
was notoriously obscure, proceeding through innumerable theorems and lemmas to arrive at even the simplest results. Torricelli, in contrast, jumps directly into his mathematical problems with no rhetorical flourishes, and wastes no ink on either the verbosity or the rigor of Euclidean deduction. “We turn away from the immense ocean of Cavalieri’s
Geometria
,” Torricelli wrote, acknowledging the notorious difficulty of Cavalieri’s text. As for him and his readers, he continued, “being less adventurous we will remain near the shore,” will not bother with elaborate presentations, and will focus instead on reaching results.

Torricelli’s text was so much more user-friendly than Cavalieri’s that it caused considerable confusion to later generations of mathematicians. John Wallis and Isaac Barrow (1630–77) in England, and Gottfried Wilhelm Leibniz (1646–1716) in Germany, all claimed to have studied Cavalieri and learned his method. In fact, their work clearly shows that they studied Torricelli’s version of Cavalieri, believing that it was merely a clear exposition of the original. This arrangement certainly had its advantages: Torricelli, instead of defending his approach, simply refers interested readers to Cavalieri’s
Geometria
, where, he assures them, they will find all the answers they seek. Later mathematicians followed his lead, and when challenged on the problematic premises of indivisibles, they, too, were happy to send their critics to seek their answers in Cavalieri’s ponderous tomes.

A PASSION FOR PARADOX

In fact, there were important differences between Cavalieri’s and Torricelli’s approaches to infinitesimals. Most critically, in Torricelli’s method all the indivisible lines taken together really did make up the surface of a figure, and all the indivisible planes actually composed the volume of a solid. Cavalieri, as will be recalled, worked hard to avoid this identification, speaking of “all the lines” as if they were different from a plane and of “all the planes” as if they were different from a solid. But Torricelli had no such qualms. In his proofs, he moves directly from “all the lines” to “the area itself” and from “all the planes” to “the volume itself,” without bothering with the logical niceties that so concerned his elder. This opened Torricelli up to criticism that he was violating the ancient paradoxes on the composition of the continuum, but the truth is that Cavalieri, for all his caution, was subjected to pretty much the same critiques. At the same time, Torricelli’s directness made his method far more intuitive and straightforward than Cavalieri’s.

The contrast between the two is also manifest in their very different attitudes toward paradox. Cavalieri, the traditionalist, tried to avoid it at all cost, and when confronted with potential paradoxes in his method, he responded with tortured explanations of why they were not actually so. But Torricelli reveled in paradoxes. His collected works include three separate lists of paradoxes, detailing ingenious contradictions that arose if one assumed that the continuum was composed of indivisibles. This might seem surprising for a mathematician who is trying to establish the credibility of a method based precisely on this premise, but for Torricelli the paradoxes served a clear purpose. They were not merely puzzling amusements to be set aside when one engaged in serious mathematics; they were, rather, tools of investigation that revealed the true nature and structure of the continuum. The paradoxes were, in a way, Torricelli’s mathematical experiments. In an experiment, one creates an unnatural situation that pushes natural phenomena to an extreme, thereby revealing truths that are hidden under normal circumstances. For Torricelli, paradoxes served much the same purpose: they pushed logic to the extreme, thereby revealing the true nature of the continuum, which cannot be accessed by normal mathematical means.

Torricelli presented dozens of paradoxes, many of them subtle and complex, but even the simplest one captures the essential problem: