Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (14 page)

Figure 3.1. The paradox of Aristotle’s wheel. From Galileo,
Discourses
, Day 1
(
Le Opere di Galileo Galilei
, vol. 8, Edizione Nazionale [Florence: G. Barbera, 1898], p. 68)

Salviati investigated the question of the continuum by way of a medieval paradox known as Aristotle’s wheel, although it had nothing to do with that ancient philosopher, and its revelation was anything but Aristotelian. Imagine, Salviati suggests to his friends, a hexagon
ABCDEF
and a smaller hexagon
HIJKLM
within it and concentric with it, both around the center
G
. And suppose, furthermore, that we extend the side
AB
of the large hexagon to a straight line
AS
, and the parallel side of the smaller hexagon into the parallel line
HT
. Next we rotate the large hexagon around the point
B
, so that the side
BC
comes to rest on the segment
BQ
of the line
AS
. When this happens the smaller hexagon will also rotate until the side
IK
comes to rest on the segment
OP
of the line
HT
. There is a difference, Salviati points out, between the line created by the rotating large hexagon and the line created by the smaller hexagon: the larger hexagon is creating a continuous line, because the segment
BQ
is placed right next to the segment
AB
. The smaller hexagon’s line, however, has gaps, because between the segments
HI
and
OP
there is a space
IO
where the hexagon in its rotation never touches the line. If we complete a full rotation of the large hexagon along the line
AS
, it will create a continuous segment whose length is equal to the hexagon’s perimeter. At the same time, the smaller hexagon will travel a distance approximately equal in length along the line
HT
, but the line it creates will not be continuous: it will be composed of the six sides of the hexagon, with six equal gaps between them.

Now, what is true of hexagons, according to Salviati, is true of any polygon, even one with 100,000 sides. Rolling it along will create a straight line equal in length to its circumference, whereas a smaller but similar polygon inside it will trace a line of equal length, but composed of 100,000 segments interspersed with 100,000 empty spaces. But what will happen if we replace those finite polygons with a polygon with an infinite number of sides—in other words, a circle? As the lower part of Aristotle’s wheel shows, rolling the circle one full revolution will trace a line
BF
equal to the circle’s circumference, and the inner circle, meanwhile, will trace a line of equal length along
CE
while completing its own revolution. In this the circles are no different from the polygons. But here’s the problem: the length of the line
CE
is equal to that of
BF
, a line created by a circle with a greater circumference. How can the smaller circle create a line longer than its own circumference? The answer, according to Salviati, is that the seemingly continuous line
CE
is, just like the line created by the rotating polygons, interspersed with empty spaces that contribute to its length. The line created by the smaller polygon of 100,000 sides is composed of 100,000 segments separated by 100,000 gaps; it follows that the line traced by the smaller circle is composed of an infinite number of segments separated by an infinite number of empty spaces.

By pushing Aristotle’s wheel to its illogical limit, Galileo arrived at a radical and paradoxical conclusion: a continuous line is composed of an infinite number of indivisible points separated by an infinite number of minuscule empty spaces. This supported both his theory of the structure of matter and his view that material objects are held together by the vacuum that pervades them. It provided a new way of thinking about the material world and also pointed to a new vision of mathematics, one in which, according to Salviati, any “continuous quantity is built up of absolutely indivisible atoms.” The inner structure of the mathematical continuum is indistinguishable from the threads of rope, the fibers inside blocks of wood, or the atoms that make up a smooth surface: it is composed of tightly compressed indivisibles with empty spaces between them. For Galileo, the mathematical continuum was modeled on physical reality.

Galileo’s approach was troubling to contemporary mathematicians, as it went directly against the well-established paradoxes that had guided the treatment of the continuum since antiquity. He did, nonetheless, have at least one prominent supporter in fellow Lincean Luca Valerio (1553–1618), who had been inducted into the Accademia dei Lincei at Galileo’s urging. Valerio was professor of rhetoric and philosophy at the Sapienza University in Rome, and widely acknowledged as one of the leading mathematicians in Italy. In his
De centro gravitatis
of 1603 and
Quadratura parabola
of 1606 he had experimented extensively with indivisibles, which enabled him to determine the centers of gravity for plane figures and solids.

But Valerio learned his mathematics among the Jesuits of the Collegio Romano, under the tutelage of Clavius himself, and with this prominent group, Galileo’s mathematical atomism found no favor. For the Jesuits, indivisibles represented the exact inverse of the proper and correct approach to mathematics. The Jesuits, it will be recalled, valued mathematics for the strict rational order it imposed upon a seemingly unruly universe. Mathematics, and in particular Euclidean geometry, represented the triumph of mind over matter and reason over the untamed material world, and reflected the Jesuit ideal not only in mathematics but also in religious and even political matters. By anchoring his mathematical speculations in an intuition of the structure of matter rather than in self-evident Euclidean postulates, Galileo turned this order on its head. The composition of the mathematical continuum, according to Galileo, could be derived from the composition of ropes and the inner structure of a piece of wood, and it could be interrogated by imagining a wheel rolling over a straight surface. In place of the Jesuit approach, Galileo proposed that geometrical objects such as planes and solids were little different from the material objects we see around us. Instead of mathematical reason imposing order on the physical world, we have pure mathematical objects created in the image of physical ones, incorporating all their incoherence. Clavius, needless to say, would not have been pleased.

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.”