Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  What remains is to determine the area of the figure OGRQ (equal to the part of the circle outside the spiral) and compare it to the area of the entire circle. Cavalieri does this in two stages: First, using classical geometrical methods, he shows that the curve OGR is a parabola. Then, using indivisibles, he shows that the area of the triangle ORQ is equal to the area of the entire circle. This is clear if we consider the area of the circle to be made up of the circumferences of successive concentric circles, starting at the center (radius “0”), and culminating at the rim (radius AE). Placing the lengths of all these circumferences side by side, Cavalieri argued, produces the triangle ORQ. Now, he had previously shown that the area defined by a half parabola (OGRQ) is two-thirds the area of the enclosing triangle ORQ. Since ORQ is equal to the area of the entire circle, and OGRQ is equal to the area of the circle that lies outside the spiral, it follows that the area of the circle inside the spiral takes up the remaining area of the circle, or one-third of it. QED.

  Cavalieri’s proof of the area enclosed inside a spiral showed that his method could deal with the areas and volumes of geometrical figures, issues that were at the forefront of mathematical research at the time. Indeed, it demonstrated that indivisibles went to the very heart of geometrical questions, in a way that Euclidean proofs could not: indivisibles not only proved that certain relations held true, but also showed why this was the case. The two triangles that make up a parallelogram are equal because they are made up of the same indivisible lines; an Archimedean spiral encompasses one-third of its enclosing circle because its indivisible curves can be rearranged into a parabola. Whereas Euclidean proofs deduced necessary truths about geometrical figures, indivisibles allowed mathematicians to peer into the inner sanctum of geometrical figures and observe their hidden structure.

  THE CAUTIOUS INDIVISIBLIST

  Yet radical though his method was, Cavalieri was, by temperament and conviction, a conservative and quite orthodox mathematician. Deeply conscious of the logical conundrums presented by infinitesimals, he tried to burnish his orthodox credentials by staying as close as possible to the traditional Euclidean style of presentation. He also incorporated certain unwieldy restrictions into his method in an attempt to circumvent the paradoxes.

  The internal tension within Cavalieri’s work comes through in a letter he wrote to the aging Galileo in June of 1639. He had recently received a copy of Galileo’s Discourses, and he was writing to thank the old master for his bold endorsement of indivisibles. Quoting the Roman poet Horace, Cavalieri compared Galileo to “the first to dare to steer the immensity of the sea, and plunge into the ocean,” and then continued:

  It can be said that with the escort of good geometry and thanks to the spirit of your supreme genius, you have managed to easily navigate the immense ocean of indivisibles, of vacuum, of light, and of a thousand other hard and distant things that could shipwreck anyone, even the greatest spirit. Oh how much the world is in your debt for having paved the road to things so new and so delicate!… and as for me, I will not be a little obliged to you, since the indivisibles of my Geometry will gain indivisible lustre from the nobility and clarity of your indivisibles.

  So far so good—Cavalieri is showering his old master with praise, and basking in the glow of his approval. Then, without warning, he takes a step back and renounces the very doctrine for which he has just praised Galileo. “I did not dare to affirm that the continuum is composed of indivisibles,” he writes. All he did, he insists, was to show “that between the continua there is the same proportion as between the collection of indivisibles.”

  Cavalieri here comes remarkably close to disavowing his own indivisibles. Whereas in his books he boldly compared a geometrical plane to a piece of cloth woven with threads, and a solid to a book composed of pages, he now implies that he didn’t really mean it. He took no position, he suggests, on the true composition of the mathematical continuum. All he did was introduce a new entity called “all the lines” of a plane figure or “all the planes” of a solid. Now, if a proportion exists between “all the lines” of one figure and “all the lines” of another, then, he claims, the same proportion exists between the areas of the two figures. And the same is true of “all the planes” of solids.

  Cavalieri, assailed by critics, was insisting that he was agnostic on the thorny question of the composition of the continuum. His method, he insisted, was legitimate regardless of whether the continuous magnitudes were composed of indivisibles. He even avoided using the offending term itself. Remarkably, despite the fact that his most famous work is called Geometry by Way of Indivisibles (Geometria indivisibilibus), and although he discusses indivisibles in the methodological and philosophical passages of his works, he never actually mentions the term in his mathematical demonstrations, where the concept is always rendered as “all the lines” or planes. He placed strict limitations on the kinds of indivisibles that were allowed, and went out of his way to make his work appear traditional and orthodox, by presenting it in a traditional Euclidean mode of postulates, demonstrations, and corollaries. As for new and previously unknown results, Cavalieri avoided them altogether.

  All, however, was to no avail. Cavalieri’s contemporaries, whether hostile or sympathetic, simply did not believe his claim that he was undecided on the question of the composition of the continuum. His method, they thought, spoke for itself, and it clearly depended on the notion that continuous magnitudes are made up of infinitesimal components. Why would we be interested in a magnitude called “all the lines” if we didn’t implicitly assume that these lines comprised a surface? Why would we compare “all the planes” of one solid with “all the planes” of another if we didn’t think that they constituted their respective volumes? Cavalieri’s bold metaphors of the cloth and the book, which openly endorse indivisibles, they found creative and inspiring, leading to ever-new discoveries. The cautious disclaimers that followed led only to an unwieldy terminology and a cumbersome method that largely negated the power and promise of indivisibles.

  In the coming years, mathematicians who disliked Cavalieri’s method, as did the Jesuits Paul Guldin and André Tacquet, denounced him for his violation of the traditional canons; those who welcomed his approach, as the Italian Evangelista Torricelli and the Englishman John Wallis did, claimed to be his followers while freely making use of infinitesimals with complete disregard for the Jesuat’s carefully thought-out constraints. No one, but truly no one, actually followed Cavalieri’s restrictive system.

  Cavalieri’s name and his books were often cited by mathematicians when they came under attack by critics of infinitesimals. The heavy and unwieldy volumes, with their contorted Latin, Euclidean structure, and air of solemn authority, provided some cover to later adherents of infinitesimal methods. They thought it was safe to point to the Jesuat master as the source of their system, and the one who had resolved all its difficulties in his learned volumes. After all, as they knew well, hardly anyone actually read Cavalieri’s books.

  GALILEO’S LAST DISCIPLE

  In the end it was Cavalieri’s younger contemporary, the brilliant Evangelista Torricelli, who took infinitesimals where the Jesuat would not go. Born in 1608 to a family of modest means, most likely in the city of Faenza, in northern Italy, young Evangelista moved to Rome at the age of sixteen or seventeen and there fell in love with mathematics. As he wrote in 1632 to Galileo, he did not receive a formal mathematical education but “studied alone, under the direction of the Jesuit fathers.” Yet it was the Benedictine monk Benedetto Castelli—the same who had encouraged Cavalieri in his mathematical studies in Pisa—who was most influential in the young man’s choice of vocation. Unlike his teacher Galileo, Castelli seemed to enjoy mentoring, and kept an eye out for promising young mathematicians. Now a professor at the Sapienza University in Rome, he took Torricelli under his wing and introduced him to the work of Galileo and Cavalieri.

  In September 1632, no doubt with Castelli’s encouragement, Torricelli wrote to Galileo
, introducing himself as “a mathematician by profession, though still young, a student of Father Castelli for the past six years.” The Dialogue on the Two Chief World Systems had appeared only a few months before, and the series of events that would lead to Galileo’s condemnation and house arrest the following year was already under way. Torricelli begins by assuring the old master that Castelli takes every opportunity to defend the Dialogue, in order to avoid an “inconsiderate decision.” He then moves to establish his own credentials as a geometer and astronomer, and a dedicated follower of Galileo’s. “I was the first in Rome,” he writes,

  to have studied your book assiduously and in detail … I did so with the pleasure that you can imagine for one who, already having a good enough experience of the geometry of Apollonius, of Archimedes, of Theodosius, and having studied Ptolemy and seen almost all of Tycho, Kepler, and Longomontanus, I adhered finally to Copernicus … and professed my attachment to the Galilean school.

  Unfortunately for Torricelli, “ardent Galilean” proved to be a precarious identity in Rome once the Dialogue and its author were condemned less than a year later. This likely explains why we hear nothing of Torricelli for nearly a decade thereafter. He remained in Rome, pursued his mathematical work in private, studied Galileo’s Discourses on Two New Sciences, which appeared in 1638, and generally kept a low profile. He reappears only in March 1641, when Castelli obtained permission to visit Arcetri, and wrote to Galileo to announce the good news. He will bring with him, he promised, a manuscript by the young Torricelli, who had been his student ten years before. “You will see,” he flatters the old man, “how the road you have opened to the human spirit is followed by a very virtuous man. He shows us how fruitful and rich is the grain you have sown in this subject of motion; you will also see that he brings honor to the school of your Excellency.”

  The visions of open roads and fields of grain likely appealed to the lonely old man, who had been confined to his house for the past eight years. But it was the brilliance of Torricelli’s work that had the greatest effect on Galileo. He was deeply impressed with what Castelli had shown him, and asked to meet the young mathematician. Castelli, for his part, was moved by Galileo’s frailty and near blindness, and was concerned that he may not have long to live. Together they hatched a plan to bring Torricelli to Arcetri to serve as Galileo’s secretary and help him edit and publish his latest works. Having received the invitation in early April, Torricelli wrote back to say that he was overcome and “confused” by the great honor done to him. Nevertheless, he seemed in no hurry to leave bustling Rome and join the old master in his lonely retreat. He made repeated excuses, but finally, in the fall of 1641, he packed up his belongings and traveled to Galileo’s Arcetri villa. There he spent his days editing the “fifth day” of the Discourses, to be added to the four days of dialogue that were published in 1638.

  Only three months after Torricelli’s arrival, his mission came to an abrupt end. In the early days of 1642, Galileo came down with heart palpitations and a fever, and on January 8, at the age of seventy-seven, the old master breathed his last. As a man condemned for “vehement heresy,” he was interred in a small side room of the Basilica of Santa Croce in Florence, only to be moved to a place of honor in the central basilica a century later. Torricelli, meanwhile, was packing his things once more for the return journey to Rome when he received a startling offer: he could stay in Florence as Galileo’s successor, and become mathematician to the Grand Duke of Tuscany and professor of mathematics at the University of Pisa. The offer did not include Galileo’s position as court “philosopher,” most likely because it was Galileo’s insistence on his right, as philosopher, to pronounce on the structure of the world that had gotten him into trouble with the Church. But even without this additional accolade, the offer presented Torricelli with the opportunity of a lifetime: a secure position with a generous salary, the chance to pursue his studies without interruption, and public recognition as heir to the greatest scientist in Europe. He accepted without hesitation.

  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 th
ese 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.