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:
In the parallelogram ABCD in which the side AB is greater than the side BC, a diameter BD is traced with a point E along it, and EF and EG parallel to AB and BC respectively, then EF is greater than EG, and the same is true for all other similar parallels. Therefore all the lines similar to EF in the triangle ABD are greater than all the lines similar to EG in the triangle CDB, and therefore the triangle ABD is greater than the triangle CDB. Which is false, because the diameter BD divides the parallelogram down the middle.
Figure 3.8. Torricelli, the paradox of the parallelogram. Based on E. Torricelli, Opera omnia, vol. 1, part 2, p. 417.
The conclusion that the two halves of the rectangle differ in size is absurd, but it seems to follow easily from the concept of indivisibles. What is to be done? Ancient mathematicians, well aware that infinitesimals could lead to such contradictions, simply banned them from mathematics. Cavalieri reintroduced indivisibles but tried to deal with such contradictions by inscribing rules into his procedures to ensure those contradictions would not arise. For example, he insisted that, in order to compare “all the lines” in one figure with “all the lines” in the other, the lines in both figures must all be parallel to a single line he called the “regula.” Since the lines EF and EG in Torricelli’s paradox are not parallel, then Cavalieri could claim that they should not be compared at all, and the paradox can be averted. In practice, however, Cavalieri’s artificial limitations were ignored both by his followers, who saw them as inconvenient hindrances, and by his critics, who did not believe that they resolved the fundamental problem.
Torricelli took a different approach. Instead of trying to evade the paradox, he made a sustained effort to understand it and what it meant for the structure of the continuum. His conclusion was startling: The reason all the short lines parallel to EG produce an area equal to the same number of longer lines parallel to EF is that the short lines are “wider” than the long lines. More broadly, according to Torricelli, “that indivisibles are all equal to each other, that is that points are equal to points, lines are equal in width to lines, and surfaces are equal in thickness to surfaces, is an opinion that seems to me not only difficult to prove, but in fact false.” This is a stunning idea. If some indivisible lines are “wider” than others, doesn’t that mean that they can in fact be divided, to reach the width of the “thin” lines? And if indivisible lines have a positive width, doesn’t it follow that an infinite number of them would add up to an infinite magnitude—not to the finite area of the triangles ADB and CDB? And the very same applies to points with a positive size and surfaces with a “thickness.” The assumption seems absurd, but Torricelli insisted that his paradoxes indicated that there was no other explanation. And not only that: he founded his entire mathematics approach on precisely this idea.
In order to transform this basic insight into a mathematical system, it was not enough to say in principle that indivisibles differed in size from one another; it was necessary to determine by precisely how
much they differed from one another. For this, Torricelli turned once again to the paradox of the parallelogram. In the diagram, the same number of long lines EF and short lines EG produce exactly the same total area. For this to be true, the short lines EG need to be “wider” by exactly the same proportion as the lines EF are “longer.” That in turn is the ratio of BC to BA, which is, in other words, the slope of the diagonal BD. At a stroke, Torricelli transformed a rather dubious speculation about the composition of the continuum into a quantifiable and usable mathematical magnitude.
Torricelli then showed exactly how to make mathematical use of indivisibles with “width” by calculating the slope of the tangent of a class of curves that we would characterize as ym = kxn, and that he called an “infinite parabola.” In this he went well beyond Cavalieri, who calculated areas and volumes enclosed in geometrical curves, but never their tangents. Indeed, Cavalieri’s insistence on comparing only conglomerations of “all the lines” or “all the planes” left no room for the delicate calculation of tangents, which are slopes calculated at single indivisible points. But Torricelli’s more flexible method, which distinguished between the magnitudes of different indivisibles, made this possible. He first pointed to the figures ABEF and CBEG in the paradox of the parallelograms. The two figures, known as “semi-gnomons,” are equal in area because they complete the equal triangles DFE and EGD to the equal triangles ADB and CDB. This will always be true, furthermore, no matter where the point E is positioned on the diagonal DB, even as it is moved to the point B itself. Accordingly, the line BC is equal in area, or “quantity,” to the line AB, even though the line AB is longer. This is the case because, just like the semi-gnomon CBEG, the indivisible line BC is “wider” than AB by precisely the same ratio that AB is longer.
Now, as long as we are dealing with straight lines, such as the diagonal BD, the semi-gnomons are always equal, and the “width” of the indivisibles is given by the simple ratio of the slope. But what happens if, instead of a straight line, we are given a generalized parabola, which in modern terms would be given as ym = kxn? In this “infinite parabola,” the semi-gnomons are no longer equal, but they do hold a fixed relationship. As Torricelli proved using the classic method of exhaustion, if the segment on the curve is very small, the ratio of the two semi-gnomons is as. And if the width of the semi-gnomons is only a single indivisible, then the “size” of the indivisible lines that meet at the curve is as.
Figure 3.9. Semi-gnomons meeting at a segment of an “infinite parabola.” If the segment is very small, or indivisible, then the ratio of the areas of the semi-gnomons is as .
This result enabled Torricelli to calculate the slope of the tangent at every point on the “infinite parabola,” shown as the curve AB in figure 3.10. Torricelli’s key insight is that at the point B, where the two indivisible lines BD and BG meet the curve, they also meet the straight line that is the curve’s tangent at that point. And whereas the “area” of the two indivisibles is as relative to the curve, it is equal relative to the straight tangent—if the tangent is extended to become the diagonal of a rectangle. Accordingly, in Figure 3.10, the ratio of the “areas” of BD and BG is , but the ratio of the “areas” of BD and BF is 1. This means that the ratio of the “areas” of BF and BG is . Now, BF and BG have the same “width” in Torricelli’s scheme, because they both meet the curve BF (or its tangent) at point B at precisely the same angle. The difference between the two segments is only in their lengths, and it follows that the length BF is to the length BG as . Now, BF is equal to ED, and BG is equal to AD, and therefore the ratio of the abscissa ED of the tangent to the abscissa AD of the curve is , or, more simply, ED = . Therefore the slope of the tangent at point B is . In this way, the slope of an “infinite parabola” can be known at any given point on the “infinite parabola,” based on its abscissa and ordinate.
Figure 3.10. Torricelli’s calculation of the slope of an “infinite parabola.”
The significance of Torricelli’s procedure here extends beyond the ingenuity of the proof itself (which is considerable), and to the challenge it posed to the mathematical tradition. Since ancient times, mathematicians had shied away from paradoxes, treating them as insurmountable obstacles, and a sign that their calculations had reached a dead end. But Torricelli parted ways with this venerable tradition: instead of avoiding paradoxes, he sought them out and harnessed them to his cause. Galileo had speculated about the infinitesimal structure of the continuum, but qualified his remarks by admitting that the continuum was a great “mystery.” Cavalieri did his best to avoid paradoxes and to conform to traditional canons, even at the cost of making his method unwieldy. But Torricelli unapologetically used paradoxes to devise a precise and powerful mathematical tool. Instead of banishing the paradox of the continuum from the realm of mathematics, Torricelli placed it at the discipline’s heart.
Despite its clear logical perils, Torricelli’s method made a profound impression on contemporary mathematicians. Although constantly skirting the edges of error, it was also flexible and remarkably effective. In the hands of a skilled and imaginative mathematician, it was a powerful tool that could lead to new and even startling results. In the 1640s it spread quickly to France, where it was developed by the likes of Gilles Personne de Roberval and Pierre de Fermat (1601–65), who corresponded directly with Torricelli. The Minim father Marin Mersenne, who was the central node of the European “Republic of Letters,” also corresponded with Torricelli, and then spread the Italian’s method on to England, where Wallis and Barrow mistakenly attributed it to Cavalieri. Quickly disseminating across the Continent, Torricelli’s radical practice encapsulated the power and the promise, as well as the dangers, of the new infinitesimal mathematics.
Torricelli did not enjoy his newfound prominence for long. On October 5, 1647, he fell ill, and less than three weeks later, on October 25, he was dead at the age of thirty-nine. In an hour of lucidity shortly before his death, Torricelli instructed his executors to deliver his manuscripts to Cavalieri in Bologna, so that he could publish what he saw fit. But it was too late: on November 30, just over a month after Torricelli breathed his last, Cavalieri, too, was dead from the gout that had afflicted him for many years. Within a few short years Italian mathematics was deprived of its guiding light Galileo and of his two chief mathematical disciples. In the span of a few decades these three had transformed the face of mathematics, opening up new avenues of progress, and possibilities that were eagerly seized upon by mathematicians across Europe. A generation later their “method of indivisibles” would be transformed into Newton’s “method of fluxions” and Leibniz’s differential and integral calculus.
In their own land, however, Galileo, Cavalieri, and Torricelli would have no successors. For just as Italian mathematics was being deprived of the leadership of Galileo and his disciples, the tide in Italy was turning decisively against their brand of mathematics. The Society of Jesus, which had long viewed the method of indivisibles with suspicion, had swung into action. In a fierce decades-long campaign, the Jesuits worked relentlessly to discredit the doctrine of the infinitely small and deprive its adherents of standing and voice in the mathematical community. Their efforts were not in vain: as 1647 was drawing to a close, the brilliant tradition of Italian mathematics was coming to an end as well. It would be centuries before the land of Galileo, Cavalieri, and Torricelli was once again home to creative mathematicians of the highest rank.
4
“Destroy or Be Destroyed”: The War on the Infinitely Small
THE DANGERS OF THE INFINITELY SMALL
The Jesuit mathematician André Tacquet (1612–60) was, by the standards of his time, a man of the world. Although he may never have left his native Flanders, his network of correspondents spanned Europe’s religious divide, reaching to Italy and France, but also to Protestant Holland and England. Only months before his death he entertained the Dutch polymath Christiaan Huygens, who had traveled to Antwerp with the express purpose of meeting Tac
quet, by then regarded as one of the brightest mathematical stars ever to come out of the Society of Jesus. The two spent only a few days together, but got along so well that the Jesuit was convinced that he had managed to lure Huygens to the Catholic faith. (He hadn’t.) But ultimately it was not Tacquet’s personal charm, but rather his mathematical excellence that transcended seventeenth-century prejudices. In England, Henry Oldenburg, secretary of the Royal Society of London and no friend of the Jesuits, spent so much time describing Tacquet’s Opera mathematica at the Society’s meeting in January 1669 that he felt compelled to apologize to the fellows for abusing their patience. But it was, he insisted, “one of the best books ever written on mathematics.”
Tacquet’s claim to mathematical fame rested chiefly on his 1651 book Cylindricorum et annularium libri IV (“Four Books on Cylinders and Rings”), in which he showed a complete mastery of the full mathematical arsenal available in his day. He calculated the areas and volumes of geometrical figures using both classical approaches and the new methods developed by his contemporaries and immediate predecessors. But when it came to indivisibles, the usually mild-mannered Jesuit turned blunt:
I cannot consider the method of proof by indivisibles as either legitimate or geometrical … many geometers agree that a line is generated by the movement of a point, a surface by a moving line, a solid by a surface. But it is one thing to say that a quantity is generated from the movement of an indivisible, a very different thing to say that it is composed of indivisibles. The truth of the first is altogether established; the other makes war upon geometry to such an extent, that if it is not to destroy it, it must itself be destroyed.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 13