Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  Frontispiece of Tacquet’s Cylindricorum et annularium libri IV. (Photograph courtesy of the Huntington Library)

  The Cylindricorum et annularium is Tacquet’s most celebrated work, the one that established his reputation as one of Europe’s most original and creative mathematicians. As it turned out, it may have been a bit too “original and creative” for his superiors: when Tacquet sent a copy of the book to the newly appointed superior general, Goswin Nickel, the general’s response was surprisingly cool. After thanking the mathematician and congratulating him on the book, Nickel added that it would be better if Tacquet applied his impressive gifts to producing textbooks of elementary geometry for use by students at the Society’s colleges, rather than original works aimed at a select audience of professional mathematicians. Nickel wrote this not out of hostility to Tacquet, but as a way of giving voice to the Jesuits’ suspicion of novelty, and their belief that the role of mathematics was to establish a fixed, unchanging order. Quite likely he was also uncomfortable with the fact that Tacquet made use of indivisibles in his work, if only as a means of discovery, not of proof. In any case, Tacquet, a good soldier in the Army of Christ, obeyed. From then on he published no more original work, but concentrated instead on producing textbooks, some of which were of such quality that they became standards in the field for over a century.

  Born in 1612, Tacquet was far younger than Guldin or Bettini, and so had witnessed the spread of infinitesimals beyond the Alps, with Roberval in France and Wallis in England leading the charge. Bettini, with no international reputation to protect, could afford to bluntly denounce indivisibles as “hallucinations.” But Tacquet could not be so disrespectful of an approach that was gaining favor among his colleagues in northern Europe. So it may have been the pressure of circumstances, it may have been the fact that, unlike Bettini, Tacquet bore no personal grudge against Cavalieri and his followers, or it may simply have been a matter of personal temperament, but when Tacquet turned his attention to the question of indivisibles, his tone was far more restrained than Bettini’s shrill denunciations.

  In his critique, Tacquet is respectful, even deferential, toward his rivals. He refers to Cavalieri as “a noble geometer,” and insists that he “does not wish to detract from the deserved glory” of Cavalieri’s “most beautiful invention.” Tacquet knew of what he spoke, because he was himself deeply familiar with the work of Cavalieri and Torricelli, and was no less capable than they of using their method to arrive at new results. But once one gets beyond his congenial style and mathematical mastery, it becomes clear that Tacquet’s opposition to the infinitely small is just as unyielding as that of the bruising Bettini. “I cannot consider the method of proof by indivisibles as either legitimate or geometrical,” he states flatly at the opening of his discussion of indivisibles. “It proceeds from lines to surfaces, from surfaces to solids, and applies to the surface the equality or proportion obtained from the lines, and transfers what was obtained from the surfaces to the solid.” “By this method,” he concludes, “nothing can be proven by anyone.”

  Unlike Guldin or Bettini, Tacquet did not think that the method of indivisibles was completely useless. It was, for him, a practical tool for finding out new geometrical relations and testing them. But one should never mistake a result arrived at by indivisibles for a properly proven geometrical truth. “If a theorem is proposed, that is proven by no other method than indivisibles, I will always doubt its truth until it is shown that the proof can be redone through homogenes” (homogenes being Tacquet’s term for classical-type demonstrations). Reasoning by indivisibles, he points out, though sometimes useful, is just as likely to lead to erroneous and absurd results as it is to true ones, and therefore must never be trusted.

  In the substance of his critique, Tacquet follows closely in the footsteps of his predecessors Guldin and Bettini. He gladly concedes that a line can be formed by moving indivisibles—a line by a moving point, a surface by a moving line, a solid by a moving surface. But this does not mean that a quantity is composed of indivisibles, since accepting such an idea, he insists, would be the death of geometry. So, in the end, it was the refined and urbane Fleming André Tacquet, and not the coarse and quarrelsome Italian Mario Bettini, who provided a synopsis of the Jesuit position on indivisibles: if indivisibles were not destroyed, then geometry itself would be. There was no middle ground.

  THE HIDDEN CAMPAIGN

  The Jesuit campaign against the infinitely small in the seventeenth century proceeded on several parallel tracks. The legal track was mostly carried out in the decrees of the Revisors General, supported by direct orders from the superior generals, and punishments to recalcitrant subordinates. The mathematical track was carried out by the order’s professional mathematicians Guldin, Bettini, and Tacquet. Their role was to discredit infinitesimals on purely mathematical grounds, while advocating for the methods of the ancients. The towering prestige of the Jesuits and the solidarity of the Society’s mathematicians, who supported one another’s positions, ensured that both the official decrees and the mathematical opinions would resonate far beyond the confines of the order.

  There is much, nonetheless, that we do not know about the decades-long campaign against the infinitely small: How many mathematicians who privately supported the method of indivisibles chose to keep quiet for fear of Jesuit reprisals? How many were denied university appointments because of their suspected allegiance to the forbidden doctrine? How many aspiring mathematicians simply turned away from the infinitely small, fearing that their professional prospects would suffer if they supported it? This hidden facet of the Jesuit campaign, conducted through personal interaction, private correspondence, and institutional pressure, is very hard to pin down with any certainty. But we do know enough to get a taste of the hostility and pressure faced by mathematicians who supported the infinitely small in Italy, the land where the power of the Jesuits was greatest.

  The Society’s influence in Italy was deep and pervasive. Even in the 1620s, when Jesuit power was at a low ebb, Cavalieri tried for years to secure for himself a university chair, before finally being appointed professor at the University of Bologna in 1629. At least one of these rebuffs, at Parma in 1626, was due to stiff Jesuit opposition. Torricelli developed his mathematics privately in the 1630s but was never a serious candidate for any university post, and he published his work only after being installed in the Medici court in Florence. It seems more than likely that the hidden hand of the resurgent Jesuits was at work here, reaching out to extinguish any opportunity the brilliant young mathematician had to establish himself in the community of scholars.

  Things only got worse for Cavalieri and Torricelli’s friends and students, those who, under normal conditions, would have carried on their trailblazing work. This generation included many talented mathematicians, but none of them (with a single exception) found it possible to continue down the road laid out by his teachers. Take, for example, Cavalieri’s student at Bologna Urbano d’Aviso (b. 1618), who wrote an admiring biography of his teacher, but who, when it came to mathematics, contented himself with authoring an elementary textbook on astronomy. Another Cavalieri student, Pietro Mengoli (1626–84), succeeded Cavalieri to the mathematics chair at Bologna and was a subtle and talented mathematician. But he was also a conservative one, who avoided indivisibles and later retreated entirely from mathematics to engage in solitary religious meditations. Giannantonio Rocca (1607–56), Cavalieri’s friend and the man who warned him against publishing his polemic against Guldin as a dialogue, was also noted for his skill as a mathematician, and Cavalieri even included some of his results in his Exercitationes. But Rocca himself never published a line on indivisibles.

  The case is much the same with Torricelli’s associates. Vincenzo Viviani (1622–1703) was, along with Torricelli, Galileo’s friend and companion in his final years. The two worked together in Florence after Galileo’s death, and Viviani ultimately succeeded Torricelli as mathematician to the Medici court. Viviani a
lways considered himself Galileo’s student and intellectual heir, and wrote a biography of the Florentine that serves as the basis for all modern ones. But when it came to mathematics, Viviani’s work was almost entirely in the classical mold: he translated the ancient classics and issued new editions of Apollonius’s Concis and Euclid’s Elements, but only very rarely did he refer to indivisibles, and then only to repeat well-known results, such as the quadrature of the parabola. By his later years, he seemed to have given up even on that: when Leibniz in 1692 published a solution to certain mathematical problems left standing by Galileo, Viviani harshly criticized him for making use of infinitesimals. By this time, it seems, even Galileo’s students in Italy had accepted, and perhaps internalized, the ban on the infinitely small.

  Antonio Nardi, another of Torricelli’s friends, wrote extensively on mathematics and in support of the method of indivisibles—but never published, despite his repeatedly stated intent to do so. All that remains of his work are thousands of pages in the archives of the Biblioteca Nazionale Centrale in Florence, of which not a word ever saw light. And then there was the case of Michelangelo Ricci, a student of Torricelli and Castelli in Rome in the 1630s who, remarkably, rose to become a cardinal of the Church. Ricci was a talented and well-regarded mathematician who was also, as his letters show, an admirer of Galileo, Cavalieri, and Torricelli and an enthusiastic practitioner of the method of indivisibles. But he, too, kept his mathematical preferences under wraps and published nothing on the subject.

  The silence of Mengoli and Nardi, Viviani and Ricci, tells the story of the slow suffocation and ultimate death of a brilliant Italian mathematical tradition. Cavalieri, the oldest of Galileo’s mathematical disciples, was fortunate enough to secure a university position in the 1620s, when the Galileans were ascendant in Rome. The younger Torricelli encountered a far harsher environment in the 1630s, but was saved from oblivion by the remarkable stroke of luck that summoned him to Arcetri in time to be appointed Galileo’s successor. But for those who wished to follow in their footsteps, the enmity of the Society of Jesus ensured that no such miracles were at hand. No city or prince wished to risk the wrath of the Jesuits, and as a result, no university chairs or positions of honor at princely courts were in the offing for supporters of the infinitely small. So they kept their silence, corresponding among themselves and with mathematicians abroad, but never publishing their work or drawing attention to themselves. And once they, too, passed from the scene, there was no one left in Italy to carry the torch of the infinitely small.

  THE LAST STAND OF THE INFINITELY SMALL

  Before the champions of the infinitely small in Italy gave up the fight and conceded defeat to their enemies, they made one final stand in defense of their mathematical approach. It was conducted with wit and a fearless spirit by the last Italian mathematician to advocate infinitesimals openly: Brother Stefano degli Angeli, of the order of the Jesuats of St. Jerome. Angeli was born in Venice and entered the Jesuat order at a young age. His intellectual gifts were apparently recognized early on, because at the age of twenty-one he was sent to the Jesuat house in Ferrara to teach literature, philosophy, and theology. After a year or so, possibly due to ill health, he was moved again, this time to Bologna. There he met the man who would help shape the rest of his life and career: the prior of the Bologna house, his Jesuat brother Bonaventura Cavalieri.

  When they met in the mid-1640s, Cavalieri was already a famous man in mathematical circles, known as the father of the method of indivisibles. He was laboring on his response to Guldin’s critique, but he was also in bad health, suffering severely from the gout that would take his life in 1647. In Angeli he found a friend and a disciple, one who embraced his mathematical approach with enthusiasm, and soon showed himself a talented mathematician in his own right. It is easy to imagine the two of them together, cloaked in the white habit and leather belt of their order, the middle-aged man and his young disciple walking daily through the busy streets of Bologna from the Jesuat house to the ancient university. Were they speculating, as they walked, about the composition of the continuum or a new approach for calculating the area inside a spiral? Were they debating the best response to Guldin and how to present it? Or were they lamenting the latest salvo from the Jesuits? We will never know, of course, but it is more than likely that all these topics came up between them. What we do know is that the two formed a strong bond, and that Angeli came to see himself as the guardian of Cavalieri’s legacy. When in the last months of his life Cavalieri became too ill to attend to the publication of the Exercitationes, it was Angeli who made the final edits and saw the book through the press.

  After Cavalieri’s death Angeli was transferred again, likely at his own request, and spent the next five years as rector of the Jesuat house in Rome. It was an impressive promotion for a young man who was no more than twenty-four, and it was no doubt aided by the strong support of his late mentor. Angeli was, at the time, already an accomplished mathematician, so it is significant to note that during his entire stay in Rome he did not publish a thing. We are already familiar with this pattern from the experience of Torricelli, who spent the decade of the 1630s in Rome deeply engaged in mathematics but who began publishing only when he was safely ensconced in the Medici court in Florence. The Eternal City, world headquarters of the Society of Jesus and home of the Collegio Romano, was not a place in which one could freely advocate the doctrine of the infinitely small.

  But in 1652, Angeli was transferred back to his native city of Venice, where he was appointed provincial councilor (definitore) for his order. It would have been a welcome move, for Venice was a very good place for one seeking shelter from the long arm of the Society of Jesus. This is because back in 1606 the city got into a dispute with Pope Paul V over its right to try and punish ecclesiastics, and the Pope, enraged that the city leaders were infringing on his authority, excommunicated the city. The Venetian Senate, however, was undeterred: it demanded that the city’s clergy continue to administer the sacraments despite the interdict, and the vast majority of the priests complied. The Jesuits, ever loyal to the Pope, did not, and were consequently expelled from the city. City and Pope reconciled the following year, but the Jesuits remained banned from Venice for the next fifty years. The Black Robes were finally allowed back in 1656, but even after their return, their influence in Venice remained limited. Angeli took full advantage: protected by the leaders of his own order and by a vigilant Venetian Senate that was still suspicious of the Jesuits, he was free to show his true colors, and began publishing on the method of indivisibles.

  When Angeli entered the battle over the infinitely small, he did so with élan and a flair that had not been seen in decades. Cavalieri had tried to appease his critics by straying as little as possible from the classical canon, and later gave up on his provocative anti-Guldin dialogue. Torricelli simply refused to engage the critics of his method, and the others, from Nardi to Ricci, never published their views at all. But Angeli charged into the fray like an avenging angel, determined to strike back at the Jesuits for the stranglehold that was slowly suffocating the method he held dear. His first broadside was included in an “Appendix pro indivisibilibus,” attached to his 1658 book Problemata geometrica sexaginta (“Sixty Geometrical Problems”), and it was aimed directly at Mario Bettini.

  In defending indivisibles, Angeli ridiculed Bettini’s discussion of a paradox presented in Galileo’s Discorsi, in which the circumference of a bowl is shown to be equal to a single point. “Father Mario Bettini of the Society of Jesus,” Angeli wrote, is “a man who because he was the author of the Apiary can be called The Bee.” This is appropriate, he continues, because “just as a bee both makes honey and stings, so does Bettini: he makes honey, in teaching the sweetest doctrine, but he stings what is, according to him, wrong in mathematics.” Unfortunately, Bettini is “an unlucky bee.” Although he “uses his sting to fend off indivisibles, he is nevertheless in danger,” because, as Angeli shows in detail, Galileo’s paradox
proves Bettini’s position untenable.

  Angeli’s comparison of Bettini to a confused bee is mocking enough, but he is not yet done with the Jesuit. He quotes the passage in which Bettini calls the method of indivisibles “counterfeit philosophising” (“similitudinem philosophantium”), and exclaims, “far, far be it from me to wish to make my geometrical theorems useless.” Seeing an opening, Angeli pounces: “Note here, reader, how this author, on falling in with indivisibles, cries out as if he were met with demons: Far, far be it from me, etc.” Bettini is here a hysterical exorcist trying to fend off demonic indivisibles with furious incantations. But as to substance, Angeli concludes, “he adds nothing new but spite.”

  The hyperbolic Bettini was perhaps easy quarry, but neither was the more formidable Tacquet spared Angeli’s sharp pen. In the preface to his De infinitis parabolis of 1659, Angeli describes how a few days after the publication of his previous book, in which he faced down the Jesuit Bettini, he wandered into the Venetian bookstore Minerva. There he came across Cylindrica et annularia, the work of another, “most deserving mathematician of the same Society.” Flipping through the pages, he by chance came across a passage in which the author “carps on indivisibles,” claiming that they are neither legitimate nor geometrical. Angeli claims that he had never previously heard of the book or known of its critique of indivisibles, but that is highly unlikely. He was extremely well versed in the mathematical output of his contemporaries, and later in the preface, he cites the Frenchmen Jean Beaugrand and Ismael Boulliau, the Englishman Richard White, and the Dutchman Frans van Schooten, as well as his fellow Italians. It stretches credulity that he was not familiar with the work of Tacquet, the leading Jesuit mathematician of the day, or Tacquet’s views on indivisibles, until he stumbled upon them in a Venetian bookstore. Angeli’s plea of ignorance is rather a rhetorical pose, aimed at presenting him as an impartial scholar reacting to outrageous claims made by Bettini and Tacquet. The long and bitter history that had pitted him against the Jesuits for decades is left unmentioned.