Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (11 page)

It was clear to Clavius that Euclid’s method had succeeded in doing precisely what the Jesuits were struggling so hard to accomplish: imposing a true, eternal, and unchallengeable order upon a seemingly chaotic reality. The diverse world we see around us, made of seemingly limitless shapes, colors, and textures, might appear to us as chaotic and unruly. But thanks to Euclid, we know better: all this diversity and apparent chaos is in fact strictly ordered by the eternal and universal truths of geometry. Antonio Possevino, a Jesuit, papal nuncio (ambassador), and friend and collaborator of Clavius, made this point in his
Bibliotheca selecta
of 1591, where he argues that

if anyone mentally conceives of God as wisest and as Geometrical architect for all … he will understand that the world had been joined by God from all substances and from the whole of matter; but since he wished to leave nothing discordant and unordered, but to adorn it with ratio, measurement, and number … therefore the Craftsman of the world imitated the fairest and eternal exemplar.

God had imposed geometry upon unruly matter, and hence the eternal rules of geometry prevail everywhere and always.

Mathematics, and geometry in particular, was for Clavius an expression of the highest Jesuit ideals and provided a clear road map for the Society as it struggled to build a new Catholic order. In some instances mathematics could be used directly to enhance the power of the Church, as was the case with the reform of the calendar. In other instances mathematics could serve as an ideal model for true knowledge, which the other disciplines could aspire to emulate. Either way, to Clavius one thing was clear: mathematics could no longer languish as an afterthought in the Jesuit empire of learning, but must become a core discipline of the curriculum and a key component in the formation of Jesuits.

CLAVIUS AGAINST THE THEOLOGIANS

The road to establishing mathematics as a core discipline in the Jesuit curriculum was a difficult one. In the first place, Clavius had to deal with those of his colleagues who simply did not believe that mathematics deserved the high position in which he wished to place it. Ignatius, they pointed out, had not placed much stock in mathematics, and the authorities he had prescribed were not particularly favorable to mathematics. Aquinas, Ignatius’s chosen theological authority, had only limited use for simple mathematics; Aristotle, the Jesuits’ guide in philosophy, assigned mathematics a far smaller role than did his teacher and philosophical rival Plato; and in Aristotelian physics and biology, mathematics played no part at all.

The most outspoken of Clavius’s opponents at the Collegio Romano seems to have been the theologian Benito Pereira, the same Jesuit who had proclaimed that one must always “adhere to the old and generally accepted opinions.” “My opinion,” Pereira declared in 1576, just as Clavius was launching into the project of calendar reform, “is that mathematical disciplines are not proper sciences.” The problem with mathematics, according to Pereira, is that its demonstrations are weak, and consequently, it does not produce true knowledge, referred to in the philosophical language of the time as
scientia
. This is because proper demonstrations, according to Aristotle, proceed from true causes—those rooted in the essential nature of the objects discussed. For example, the classic syllogism

All men are mortal

Socrates is a man

Therefore Socrates is mortal

proceeds from the fact that mortality is an essential part of being human. But nothing like this, Pereira argues, exists in mathematics, because mathematical demonstrations do not take into account the essence of things. Instead, they point to complex relations between numbers, lines, figures, etc.—all interesting in themselves, no doubt, but lacking the logical force of a demonstration from true causes. The use of parallel lines, for example, might reveal to us that the sum of the angles of a triangle is equal to two right angles, but the parallel lines did not
cause
this to be true. For all intents and purposes, Pereira suggests, mathematics doesn’t even have a true subject matter; it merely draws connections between different properties. If one seeks strong demonstrations, one must turn elsewhere: to the syllogistic demonstrations of Aristotelian physics, which are almost entirely devoid of mathematics.

Not so, retorted Clavius in the “Prolegomena.” The subject of mathematics is matter itself, since all mathematics is “immersed” in matter. This, he argues, puts mathematics in a distinguished place in the order of knowledge: both immersed in matter and abstracted from it, mathematics is halfway between physics, which deals only with matter, and metaphysics, which deals with things separated from matter. Mathematics, according to Clavius, should not aspire to equality with metaphysical theology, which deals with things such as the soul and salvation. But it is, nonetheless, clearly in a superior position to the Aristotelian physics favored by Pereira. Whether Clavius won the argument is a matter of opinion. Contemporaries thought that he at least held his own, and that is really all he needed. His rising prestige as the Society’s representative on the calendar commission did more than his logical and rhetorical powers to bolster his arguments, and in any case, he was more interested in actual pedagogical reform than in abstract philosophical debate. That is where he directed his fight, and that is where he would ultimately win it.

Clavius laid out his plans for raising the profile of mathematics in the Society in a document called “Modus quo disciplinas mathematicas in scholis Societatis possent promoveri” (“The ways in which the mathematical disciplines could be promoted in the Society’s schools”), which he circulated around 1582, shortly after the calendar commission had completed its work. In order for the program to succeed, he argued, it was first necessary to raise the prestige of the field in the eyes of the students. This would require some cooperation from his colleagues, and he did not hesitate to take direct aim at those he suspected of sabotaging his efforts. He clearly had Pereira and his allies in mind when he complained that reliable sources had informed him that certain teachers openly mocked the mathematical sciences. “It will contribute much,” he wrote, to the promotion of mathematics

if the teachers of philosophy abstain from those questions which do not help in the understanding of natural things and very much detract from the authority of the mathematical disciplines in the eyes of students, such as those in which they teach that the mathematical sciences are not sciences [and] do not have demonstrations …

“Experience teaches,” he added acidly, “that these questions are a great hindrance to students and of no service to them.”

Apart from countering the pernicious influence of hostile colleagues, Clavius also made positive suggestions for the advancement of mathematics in the Society’s schools. First and foremost, he argued, master teachers must be found “with uncommon erudition and authority,” since without those, students “seem unable to be attracted to the mathematical disciplines.” In order to produce a cadre of such capable professors, Clavius suggested establishing a special school, where the most promising mathematics students in the Jesuit colleges would be sent to pursue higher studies. Later on, once they took up their regular teaching positions, the graduates of the school “should not be taken up with many other occupations,” but be left to focus on mathematical instruction. To counter antimathematical prejudice, it was extremely important that these highly trained mathematicians be treated by their colleagues with the utmost respect, and invited to take part in public disputations alongside the professors of theology and philosophy. The prestige of mathematics, he explained, required this: “pupils up to now seem almost to have despised these sciences for the simple reason that they think that they are not considered of value and are even useless, since the person who teaches them is never summoned to public acts with the other professors.”

Then as now, students were very quick to pick up on which subjects and teachers were valued and which were not, and it was close to impossible for instructors in an undervalued field to get the students to take them seriously. Today it is more likely to be teachers of philosophy and the humanities who complain that their fields are disrespected by instructors in the prestigious mathematical sciences. But even if the roles of the different disciplines are roughly reversed today, the dynamic is still much the same.

THE EUCLIDEAN KEY

Staffing the colleges with qualified teachers was one thing. Giving them something to teach, however, was another, and here again Clavius stepped in with a proposal. Already in 1581 he wrote up a detailed mathematical curriculum, which he called “Ordo servandus in addiscendis disciplinis mathematicis”—literally, “The order to be kept in learning the mathematical disciplines.” His complete curriculum consisted of twenty-two lesson sets spread over three years of study, a plan that ultimately proved too ambitious to be generally implemented. In the Jesuit colleges, theology and philosophy still came first. Nevertheless, this did not prevent Clavius from pushing hard to introduce as much of his curriculum as possible into the schools.

The first, most important, and key component of Clavius’s curriculum was inevitably Euclidean geometry. Any incoming student would start out by studying the first four books of Euclid, which deal with plane geometry. He would then study the fundamentals of arithmetic, before moving on to astronomy, geography, perspective, and music theory, among others, each according to the accepted authority on the subject: Jordanus de Nemore on arithmetic, Sacrobosco on astronomy, Ptolemy on geography, and so on. But he would return time and again to the greatest master of the mathematical sciences, Euclid, until he had thoroughly mastered the entire thirteen books of
The Elements
. It was a logical sequence of studies, but for Clavius it also represented a deeper ideological commitment. Geometry, being rigorous and hierarchical, was, to the Jesuit, the ideal science. The mathematical sciences that followed—astronomy, geography, perspective, music—were all derived from the truths of geometry, and demonstrated how those truths governed the world. Consequently, Clavius’s mathematical curriculum did not just teach the students specific competencies. More important, it demonstrated how absolute eternal truths shape the world and govern it.

Clavius spent much of the final thirty years of his life trying to implement this program. Initially he hoped to incorporate his plan into the Society’s
Ratio studiorum
, the master document of Jesuit college education that had been in the works for decades. A draft produced at the Collegio Romano in 1586 so closely follows Clavius’s suggestions that he likely authored its mathematics chapter himself. It proposed, for example, that a mathematics professor “who could be Father Clavius” should teach a three-year advanced course in mathematics to instruct future Jesuit teachers in the field. A later draft from 1591 repeated much of the same language, and even warned, as Clavius had, against teachers who would subvert the authority and importance of mathematics. The final version of the
Ratio
, issued in 1599 and officially approved, was drier and shorter than its more florid predecessors, but it, too, accepted the general trend of Clavius’s proposals. Each student would study the basics of Euclid’s
Elements
, and thereafter learn more advanced topics sporadically. In addition, “those apt and inclined to mathematics should be trained privately, after the course.” Clavius, in the end, had not gotten his own school of mathematics, but he still got much of what he wanted.

Clavius’s dogged advocacy of mathematics was never limited to the question of the curriculum; he also threw himself into the daunting project of writing new textbooks to replace the medieval texts in use in the Society’s schools. While these were considered authoritative, they also dated back hundreds of years, and presented material in a style that was unlikely to appeal to sixteenth-century students. In 1570, Clavius published the first edition of his commentary on Sacrobosco’s
Tractatus de Sphaera
, the standard medieval astronomy textbook, and in 1574 the first of many editions of his commentary on Euclid. These were followed by books on the theory and practice of the gnomon—the vertical part of a sundial—in 1581; the astrolabe—used for measuring the height of a star above the horizon—in 1581; practical geometry (1604); and algebra (1608). The textbooks were often clothed in the guise of commentaries on the traditional texts, such as Euclid’s
Elements
and Sacrobosco’s
de Sphaera
, and indeed they did retain the core teachings of their sources (such as Sacrobosco’s assumption that the sun revolves around the Earth). Nevertheless Clavius’s editions were in effect new books, bringing in new and up-to-date topics, emphasizing applications, and presenting the materials in a clear and appealing manner. They saw many editions throughout the sixteenth and seventeenth centuries, and remained the standard textbooks in the Jesuit schools well into the seventeen hundreds.

The project closest to Clavius’s heart, however, was the establishment of a mathematics academy at the Collegio Romano. Initially, in the 1570s and ’80s this was an informal group of select mathematics students who gathered around Clavius to study advanced topics. But in the early 1590s, Clavius managed to convince his friend the theologian Robert Bellarmine, who was the Collegio’s rector at the time, to formalize the arrangement. Thereafter members of the academy were exempted from other duties for a year or two during their studies, and allowed to concentrate exclusively on mathematics. In 1593, General Acquaviva lent his own authority to the arrangement, decreeing that the best mathematics students in the Jesuit network of colleges would be sent to Rome to study with Father Clavius. The result was that Clavius was soon the leader of a group of young mathematicians who were not only competent teachers, but brilliant mathematicians in their own right. Among them was the statesmanlike father Christoph Grienberger, who was Clavius’s successor at the Collegio; the fiery father Orazio Grassi (1583–1654), who famously tangled with Galileo over the nature of comets; Father Gregory St. Vincent (1584–1667); and Father Paul Guldin (1577–1643)—all of them among the foremost European mathematicians of their generation. In 1581, Clavius had complained that Jesuits were ignorant of mathematics and fell silent when it was discussed. But owing almost entirely to his dogged and tireless leadership, only a few decades later, Jesuits were setting the standard for the study of mathematics in Europe.