Although Galileo’s support of infinitesimals gave them a degree of visibility and respectability that no other endorsement could have achieved, he himself made little use of indivisibles in his actual mathematical work. One of the few exceptions is in his famous argument on the distance traversed by a free-falling body. Suppose, Salviati proposes in Day 3 of Discourses, a body is placed at rest at point C, and then accelerates at a constant pace, as in free fall, until it reaches point D. Now let the line AB represent the total time it takes that body to get from C to D, and the line BE, perpendicular to AB, represent the body’s greatest speed, which it reaches at D. Draw a line from A to E, and lines parallel to BE at regular intervals between AB and AE. Each of these lines, Salviati argues, represents the speed of the object at a particular moment during its steady acceleration. Since there is an infinite number of points on AB, each representing an instant in time, there is also an infinite number of such parallel lines, which together fill in the triangle ABE. The sum of all the speeds at every point, furthermore, is equivalent to the total distance traversed by the object during the time AB.
Figure 3.2. Galileo on uniformly accelerated bodies. From Discourses, Day 3. (Ed. Naz., vol. 8, p. 208)
Now, Salviati says, if we take a point F halfway between B and E, and draw a line through it parallel to AB, and a line AG parallel to BF intersecting with it, then the rectangle ABFG is equal in area to the triangle ABE. But just as the area of the triangle represents the distance traversed by a body moving at a uniformly accelerated speed, so the area of the rectangle represents the distance covered by a body moving at a fixed speed. It follows, Salviati concludes, that the distance covered in a given time by a body that begins at rest and uniformly accelerates is equal to the distance covered by a body moving at a fixed speed for the same amount of time, if the speed is half the maximum speed reached by the accelerated body.
Known as the law of falling bodies, it is one of the first things any student learns today in a high school physics class, but in its time, it was nothing short of revolutionary. It was the first quantitative mathematical description of motion in modern science, and it laid the foundations for the modern field of mechanics—and, in effect, modern physics. Galileo was well aware of the importance of the law, and he included it in two of his most popular works, the Dialogue of 1632 and the Discourses of 1638. Although it relies mostly on Euclidean geometrical relations, it does show Galileo’s willingness to assume that a line is composed of an infinite number of points. That was precisely the question posed to him by Cavalieri in 1621, and whatever answer Galileo gave him, the young monk was not discouraged. During the 1620s he took the idea of the infinitely small and turned it into a powerful mathematical tool that he called the method of indivisibles. The name stuck.
THE DUTIFUL MONK
Cavalieri was born in Milan in 1598 to a family that was likely respectable and possibly even noble, but of modest means. His parents named him Francesco, but he took the name Bonaventura at the age of fifteen, when he became a novice in the order of the Apostolic Clerics of St. Jerome, more commonly known as the Jesuats. Only a single letter distinguishes the Jesuats from Ignatius’s famed Jesuits, but the two societies could hardly have been more different. Whereas the Jesuits were a modern order, forged in the crucible of the Reformation crisis, the Jesuats dated back to the fourteenth century and were the product of the fierce piety of the decades that followed the Black Death. Whereas the Jesuits were a dynamic force whose schools and missions encompassed the globe, the Jesuats were a local Italian order, respected for their work with the sick and dying but wholly lacking the ambition of Ignatius’s followers. The forming of a Jesuit, as we have seen, could take decades, but the training of a Jesuat was a much briefer affair: in 1615, at the age of seventeen and two years into his novitiate, Cavalieri pronounced his vows and donned the white habit and a dark leather belt that identified him as a full-fledged member of the order. A few months later he left his home city of Milan for the Jesuat house in Pisa.
We do not know if the move to Pisa was Cavalieri’s idea or that of his superiors, but it would turn out to be an auspicious one for the young Jesuat and for mathematics. “I am proud, and will always be,” he wrote many years later to fellow mathematician Evangelista Torricelli (1608–47), “of having received under the serenity of that sky the first aliments and elements of mathematics.” The instigator of his budding fascination with mathematics was Benedetto Castelli (1578–1643), Galileo’s former student and lifelong friend and supporter, who was at the time the professor of geometry at the University of Pisa. Castelli introduced Cavalieri both to Galileo’s work in physics and mathematics and, in due course, to the great Florentine himself. In 1617, Cavalieri moved to Florence, where, aided by the influence of his Milanese patron, Cardinal Federico Borromeo, he joined the circle of disciples and admirers around Galileo at the Medici court. “With your help,” the cardinal wrote to Galileo, Cavalieri “will reach that level in his profession which we can already perceive from his singular inclinations and ability.”
The following year, Cavalieri returned to Pisa, where he began giving private lectures in mathematics, substituting for Castelli, who was drafted into Grand Duke Cosimo’s household to serve as tutor to his sons. Cavalieri was now a professional mathematician in all but title, but for the next decade his life was torn between his field of choice and his duties to the Jesuat order. In 1619 he applied for the mathematics chair at the University of Bologna, which had been vacant since Giovanni Antonio Magini died two years before. Only the active support of Galileo could have secured such a prestigious position for so young an applicant, but Galileo seemed reluctant to intervene, so the opportunity slipped away. Instead, in 1620, Cavalieri was recalled to the Jesuat house in Milan, where he also became deacon to Cardinal Borromeo. Far from the brilliant Medici court, Cavalieri found that his talents were not always appreciated. “I am now in my own country,” he wrote to Galileo, “where there are these old men who expected of me great progress in theology, as well as in preaching. You can imagine how unwillingly they see me so fond of mathematics.”
Despite his growing immersion in mathematics, Cavalieri was serious about his religious vocation. He set out to study theology, and soon made up for lost time, “to the great wonder of everyone.” As a result, and also thanks to the cardinal’s support, he rose quickly in the ranks of the order, and in 1623 he was named prior of the Jesuat monastery of St. Peter in the town of Lodi, not far from Milan. Three years later he was promoted to prior of St. Benedict’s monastery in the larger city of Parma. Yet all the while, Cavalieri was casting about for a position as a professional mathematician. In 1623 he renewed his efforts to secure the professorship in Bologna, but the Bolognese Senate, while not rejecting his appeal outright, repeatedly asked him for more and more samples of his work. When his old mentor Castelli was appointed to the mathematics chair at Sapienza University in 1626, Cavalieri sensed an opportunity. But despite his taking leave of his duties to promote his case, and spending six months in Rome with Galileo’s friend and fellow Lincean, the influential Giovanni Ciampoli (1589–1643), nothing came of it. Back in Parma, he approached the Jesuit fathers who ran the University of Parma, but as he wrote to Galileo after the fact, they would not allow a mere Jesuat, not to mention a student of Galileo, to teach in the university.
It was not until 1629 that the tide finally turned for Cavalieri. Galileo, at long last warming to his student’s cause, declared that “few scholars since Archimedes, and perhaps nobody, have gone so deeply and profoundly into the understanding of geometry” as Cavalieri. The Bolognese Senate was duly impressed, and on August 25 it offered the vacant chair of mathematics at the University of Bologna to the Jesuat. Having spent a full decade trying to secure the position, Cavalieri did not hesitate: he quickly moved into the Jesuat house in Bologna, and began lecturing at the university that same October. He would stay in that city for the remaining nineteen years of his life, living in the monastery
and teaching at the university. Although a young man by modern standards, he was in failing health and suffering from repeated bouts of gout, which made travel extremely difficult. Only once during those years did he venture from his adopted city, and it was for the only cause that could have lured him from the comfort of his daily routine: it was in 1636, when he visited Galileo during the old master’s long and lonely years of house arrest.
The decade between Cavalieri’s stay in Pisa and his appointment as professor in Bologna was an uncomfortable one for the young monk, but it was also his most mathematically productive period. In fact, nearly all the original proofs for which he became known, and even much of the actual text of his books, date to these itinerant years. Once settled in Bologna, he was weighed down by his teaching duties, as well as by the demands of the senate, which required its professor of mathematics to produce a steady stream of astronomical and astrological tables. Even so, the industrious monk managed to publish Lo specchio ustorio (“The Burning Mirror”) in 1632, Geometria indivisibilibus (“Geometry by Indivisibles”) in 1635, and Exercitationes geometricae sex (“Six Geometric Exercises”) in 1647. These works, conceived and largely written during the 1620s, established Cavalieri’s reputation as a mathematician, and the leading proponent of infinitesimals.
ON THREADS AND BOOKS
Just as Galileo began his mathematical theorizing on the continuum with a discussion on the inner composition of ropes and blocks of wood, Cavalieri, too, founded his mathematical method on our material intuitions: “It is manifest,” he writes, “that plane figures should be conceived by us like cloths woven of parallel threads; and solids like books, composed of parallel pages.” Any surface, no matter how smooth, is in fact made up of minuscule parallel lines, arrayed side by side; and any three-dimensional figure, no matter how solid it appears, is nothing but a stack of razor-thin planes, one on top of the other. These thinnest of slices, equivalent to the smallest components, or atoms, of material figures, Cavalieri called indivisibles.
As he was quick to point out, there are important differences between physical objects and their mathematical cousins: a cloth and a book, he noted, are composed of a finite number of threads and pages, but planes and solids are made up of an indefinite number of indivisibles. It is a simple distinction that lies at the heart of the paradoxes of the continuum, and whereas Galileo glossed over the matter in the Discourses, the more cautious Cavalieri brought it to the fore. Even so, it is clear that Cavalieri, like Galileo, began his mathematical speculations not with abstract universal axioms, but with lowly matter. From there he moved upward, generalizing our intuitions of the material world and turning them into a general mathematical method.
For a taste of Cavalieri’s method, consider proposition 19 in the first exercise of the Exercitationes:
If in a parallelogram a diagonal is drawn, the parallelogram is double each of the triangles constituted by the diagonal.
Figure 3.3. Cavalieri, Exercitationes, p. 35, prop. 19. (Bologna: Iacob Monti, 1647)
This means that if a diagonal FC is drawn for the parallelogram AFDC, the area of the parallelogram is double the area of each of the triangles FAC and CDF. If one approaches the proof in a traditional Euclidean manner, then it is almost trivial: the triangles FAC and CDF are congruent, because, first, they share the side CF; second, the angle ACF is equal to the angle CFD (because AC is parallel to FC); and third, the angle AFC is equal to DCF (because AF is parallel to CD). Since the two triangles together compose the parallelogram, and since, being congruent, they are equal in area, it follows that the area of the parallelogram is double that of each of them. QED.
Cavalieri, of course, knew all this very well, and he likely would not have wasted a theorem in his book on proving something so elementary. But he was after something else, so he proceeded differently:
Let equal segments FE and CB be marked off from points F and C along the sides FD and CA respectively. And from the points E and B mark segments EH and BM, parallel CD, which cross the diagonal FC at points H and M respectively.
Cavalieri then shows that the small triangles FEH and CBM are congruent, because the sides BC and FE are equal, angle BCM is equal to EFH, and angle MBC is equal to FEH. It follows that the lines EH and BM are equal.
In the same way we show of the other parallels to CD, namely those that are marked in equal distances from the point F and C along the sides FD and AC, that they are also equal between themselves, just as the extremes, AF and CD, are equal. Therefore all the lines of the triangle CAF are equal to all the lines of the triangle FDC.
Since “all the lines” of one triangle are equal to “all the lines” of the other, Cavalieri argues, their areas are equal, and the parallelogram is double the area of each of them. QED.
The contrast between Cavalieri’s proof and the traditional Euclidean demonstration is stark. The Euclidean proof began with the universal characteristics of a parallelogram, which are themselves derived from Euclid’s self-evident postulates. From this universal beginning it moved step by logical step to establish the relations in this particular case—that of a parallelogram divided into two triangles. It shows, in essence, that the universal laws of reasoning require that the two triangles be equal. But Cavalieri refuses to proceed from such abstract universal principles, and begins instead with a material intuition: what, he asks, is the area of each triangle made of? His answer, based on a rough analogy to a piece of cloth, is that it is composed of parallel lines laid out neatly side by side. To find out the total area of each triangle, he then proceeds to “count” the lines that make it up. Since there is an infinity of lines in each surface, literally counting is impossible, but Cavalieri shows that their number and size are nevertheless the same from one triangle to the other, and hence the areas of the two triangles are equal.
The point of Cavalieri’s proof is to show not that the theorem is true—which is obvious—but rather why it is true: the two triangles are equal because they are composed of the same number of identical indivisible lines placed side by side. And it is precisely this material take on geometrical figures that distinguishes Cavalieri’s approach from the classical Euclidean one. The Euclidean approach orders geometrical objects, and ultimately the world, through its universal first principles and its logical method. Cavalieri’s approach, in contrast, begins with an intuition of the world as we find it, and then proceeds to broader and more abstract mathematical generalizations. It can rightly be called “bottom-up” mathematics.
Cavalieri’s parallelogram proof showed that his method of indivisibles worked, but not that there was any advantage to adopting it. Quite the contrary: he offered a long and convoluted demonstration of a theorem that could be proved in one or two lines using the traditional Euclidean approach. If all Cavalieri’s proofs went to such great lengths to accomplish so little, it is unlikely that he would have found many followers to adopt his approach. But this of course was not the case: the parallelogram proof demonstrated the reliability of indivisibles. To demonstrate their power, Cavalieri turned to more difficult challenges.
The “Archimedian spiral,” known since antiquity, is produced by a point traveling at a fixed speed along a straight line, while the line itself rotates at a fixed angular speed around the point of origin. In the diagram, the curve is traced by a point traveling steadily from A to E, while the line AE itself is rotating at a fixed rate around the central point A. After a single revolution the spiral arrives at point E, and encloses a “snail-shaped” area AIE inside the larger circle MSE, whose radius is AE. Cavalieri set out to prove that the area enclosed within the spiral AIE is one-third the area of the circle MSE. Archimedes had used his own ingenious approach to demonstrate that this was so. Cavalieri, however, approached the problem in a novel, intuitive manner, using indivisibles to transform the complex spiral into the familiar and well-understood parabola.
Figure 3.4. Cavalieri’s calculation of the area enclosed inside a spiral. From Cavalieri, Geometria indivisibil
ibus libri VI, prop. 19. (Bologna: Clementis Ferroni, 1635)
Cavalieri posits a rectangle, OQRZ, in which the side OQ is equal to the radius AE of the circle MSE, and the side QR is equal to the circle’s circumference. Returning to the spiral, he then picks a random point V along AE and generates a circle IVT around the central point A. The circle IVT has two parts: one, VTI, is outside the area enclosed by the spiral; the other, IV, is inside the spiral. He takes the length VTI (external to the spiral) and places it as a straight line KG inside the rectangle and parallel to QR, with K a point on the line OQ, and OK (that is, the distance of K from O) equal to the radius AV. He then does the same for every point along AE, taking the portion of its circle that is outside the spiral and placing its length in its proper place along the side OQ, inside the rectangle. Each point along AE has an equivalent point along OQ, with a straight line emanating from it representing that portion of the circle that lies outside the spiral. In the end, all the circular lines forming the area AES outside the spiral are equal to all the straight lines composing the area OGRQ inside the rectangle. Consequently, according to Cavalieri, the area enclosed by OGRQ is equal to the area inside the circle MSE that is not contained in the spiral.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 11