Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (15 page)

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 indivisibilibus
libri VI, prop. 19. (Bologna: Clementis Ferroni, 1635)