Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (16 page)

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.

What remains is to determine the area of the figure
OGRQ
(equal to the part of the circle outside the spiral) and compare it to the area of the entire circle. Cavalieri does this in two stages: First, using classical geometrical methods, he shows that the curve
OGR
is a parabola. Then, using indivisibles, he shows that the area of the triangle
ORQ
is equal to the area of the entire circle. This is clear if we consider the area of the circle to be made up of the circumferences of successive concentric circles, starting at the center (radius “0”), and culminating at the rim (radius
AE
). Placing the lengths of all these circumferences side by side, Cavalieri argued, produces the triangle
ORQ
. Now, he had previously shown that the area defined by a half parabola (
OGRQ
) is two-thirds the area of the enclosing triangle
ORQ
. Since
ORQ
is equal to the area of the entire circle, and
OGRQ
is equal to the area of the circle that lies
outside
the spiral, it follows that the area of the circle
inside
the spiral takes up the remaining area of the circle, or one-third of it.
QED.

Cavalieri’s proof of the area enclosed inside a spiral showed that his method could deal with the areas and volumes of geometrical figures, issues that were at the forefront of mathematical research at the time. Indeed, it demonstrated that indivisibles went to the very heart of geometrical questions, in a way that Euclidean proofs could not: indivisibles not only proved that certain relations held true, but also showed
why
this was the case. The two triangles that make up a parallelogram are equal because they are made up of the same indivisible lines; an Archimedean spiral encompasses one-third of its enclosing circle because its indivisible curves can be rearranged into a parabola. Whereas Euclidean proofs deduced necessary truths about geometrical figures, indivisibles allowed mathematicians to peer into the inner sanctum of geometrical figures and observe their hidden structure.

THE CAUTIOUS INDIVISIBLIST

Yet radical though his method was, Cavalieri was, by temperament and conviction, a conservative and quite orthodox mathematician. Deeply conscious of the logical conundrums presented by infinitesimals, he tried to burnish his orthodox credentials by staying as close as possible to the traditional Euclidean style of presentation. He also incorporated certain unwieldy restrictions into his method in an attempt to circumvent the paradoxes.

The internal tension within Cavalieri’s work comes through in a letter he wrote to the aging Galileo in June of 1639. He had recently received a copy of Galileo’s
Discourses
, and he was writing to thank the old master for his bold endorsement of indivisibles. Quoting the Roman poet Horace, Cavalieri compared Galileo to “the first to dare to steer the immensity of the sea, and plunge into the ocean,” and then continued:

It can be said that with the escort of good geometry and thanks to the spirit of your supreme genius, you have managed to easily navigate the immense ocean of indivisibles, of vacuum, of light, and of a thousand other hard and distant things that could shipwreck anyone, even the greatest spirit. Oh how much the world is in your debt for having paved the road to things so new and so delicate!… and as for me, I will not be a little obliged to you, since the indivisibles of my
Geometry
will gain indivisible lustre from the nobility and clarity of your indivisibles.

So far so good—Cavalieri is showering his old master with praise, and basking in the glow of his approval. Then, without warning, he takes a step back and renounces the very doctrine for which he has just praised Galileo. “I did not dare to affirm that the continuum is composed of indivisibles,” he writes. All he did, he insists, was to show “that between the continua there is the same proportion as between the collection of indivisibles.”

Cavalieri here comes remarkably close to disavowing his own indivisibles. Whereas in his books he boldly compared a geometrical plane to a piece of cloth woven with threads, and a solid to a book composed of pages, he now implies that he didn’t really mean it. He took no position, he suggests, on the true composition of the mathematical continuum. All he did was introduce a new entity called “all the lines” of a plane figure or “all the planes” of a solid. Now, if a proportion exists between “all the lines” of one figure and “all the lines” of another, then, he claims, the same proportion exists between the areas of the two figures. And the same is true of “all the planes” of solids.

Cavalieri, assailed by critics, was insisting that he was agnostic on the thorny question of the composition of the continuum. His method, he insisted, was legitimate regardless of whether the continuous magnitudes were composed of indivisibles. He even avoided using the offending term itself. Remarkably, despite the fact that his most famous work is called
Geometry by Way of Indivisibles
(
Geometria indivisibilibus
), and although he discusses indivisibles in the methodological and philosophical passages of his works, he never actually mentions the term in his mathematical demonstrations, where the concept is always rendered as “all the lines” or planes. He placed strict limitations on the kinds of indivisibles that were allowed, and went out of his way to make his work appear traditional and orthodox, by presenting it in a traditional Euclidean mode of postulates, demonstrations, and corollaries. As for new and previously unknown results, Cavalieri avoided them altogether.

All, however, was to no avail. Cavalieri’s contemporaries, whether hostile or sympathetic, simply did not believe his claim that he was undecided on the question of the composition of the continuum. His method, they thought, spoke for itself, and it clearly depended on the notion that continuous magnitudes are made up of infinitesimal components. Why would we be interested in a magnitude called “all the lines” if we didn’t implicitly assume that these lines comprised a surface? Why would we compare “all the planes” of one solid with “all the planes” of another if we didn’t think that they constituted their respective volumes? Cavalieri’s bold metaphors of the cloth and the book, which openly endorse indivisibles, they found creative and inspiring, leading to ever-new discoveries. The cautious disclaimers that followed led only to an unwieldy terminology and a cumbersome method that largely negated the power and promise of indivisibles.

In the coming years, mathematicians who disliked Cavalieri’s method, as did the Jesuits Paul Guldin and André Tacquet, denounced him for his violation of the traditional canons; those who welcomed his approach, as the Italian Evangelista Torricelli and the Englishman John Wallis did, claimed to be his followers while freely making use of infinitesimals with complete disregard for the Jesuat’s carefully thought-out constraints. No one, but truly no one, actually followed Cavalieri’s restrictive system.

Cavalieri’s name and his books were often cited by mathematicians when they came under attack by critics of infinitesimals. The heavy and unwieldy volumes, with their contorted Latin, Euclidean structure, and air of solemn authority, provided some cover to later adherents of infinitesimal methods. They thought it was safe to point to the Jesuat master as the source of their system, and the one who had resolved all its difficulties in his learned volumes. After all, as they knew well, hardly anyone actually read Cavalieri’s books.

GALILEO’S LAST DISCIPLE

In the end it was Cavalieri’s younger contemporary, the brilliant Evangelista Torricelli, who took infinitesimals where the Jesuat would not go. Born in 1608 to a family of modest means, most likely in the city of Faenza, in northern Italy, young Evangelista moved to Rome at the age of sixteen or seventeen and there fell in love with mathematics. As he wrote in 1632 to Galileo, he did not receive a formal mathematical education but “studied alone, under the direction of the Jesuit fathers.” Yet it was the Benedictine monk Benedetto Castelli—the same who had encouraged Cavalieri in his mathematical studies in Pisa—who was most influential in the young man’s choice of vocation. Unlike his teacher Galileo, Castelli seemed to enjoy mentoring, and kept an eye out for promising young mathematicians. Now a professor at the Sapienza University in Rome, he took Torricelli under his wing and introduced him to the work of Galileo and Cavalieri.

In September 1632, no doubt with Castelli’s encouragement, Torricelli wrote to Galileo, introducing himself as “a mathematician by profession, though still young, a student of Father Castelli for the past six years.” The
Dialogue on the Two Chief World Systems
had appeared only a few months before, and the series of events that would lead to Galileo’s condemnation and house arrest the following year was already under way. Torricelli begins by assuring the old master that Castelli takes every opportunity to defend the
Dialogue
, in order to avoid an “inconsiderate decision.” He then moves to establish his own credentials as a geometer and astronomer, and a dedicated follower of Galileo’s. “I was the first in Rome,” he writes,

to have studied your book assiduously and in detail … I did so with the pleasure that you can imagine for one who, already having a good enough experience of the geometry of Apollonius, of Archimedes, of Theodosius, and having studied Ptolemy and seen almost all of Tycho, Kepler, and Longomontanus, I adhered finally to Copernicus … and professed my attachment to the Galilean school.

Unfortunately for Torricelli, “ardent Galilean” proved to be a precarious identity in Rome once the
Dialogue
and its author were condemned less than a year later. This likely explains why we hear nothing of Torricelli for nearly a decade thereafter. He remained in Rome, pursued his mathematical work in private, studied Galileo’s
Discourses on Two New Sciences
, which appeared in 1638, and generally kept a low profile. He reappears only in March 1641, when Castelli obtained permission to visit Arcetri, and wrote to Galileo to announce the good news. He will bring with him, he promised, a manuscript by the young Torricelli, who had been his student ten years before. “You will see,” he flatters the old man, “how the road you have opened to the human spirit is followed by a very virtuous man. He shows us how fruitful and rich is the grain you have sown in this subject of motion; you will also see that he brings honor to the school of your Excellency.”

The visions of open roads and fields of grain likely appealed to the lonely old man, who had been confined to his house for the past eight years. But it was the brilliance of Torricelli’s work that had the greatest effect on Galileo. He was deeply impressed with what Castelli had shown him, and asked to meet the young mathematician. Castelli, for his part, was moved by Galileo’s frailty and near blindness, and was concerned that he may not have long to live. Together they hatched a plan to bring Torricelli to Arcetri to serve as Galileo’s secretary and help him edit and publish his latest works. Having received the invitation in early April, Torricelli wrote back to say that he was overcome and “confused” by the great honor done to him. Nevertheless, he seemed in no hurry to leave bustling Rome and join the old master in his lonely retreat. He made repeated excuses, but finally, in the fall of 1641, he packed up his belongings and traveled to Galileo’s Arcetri villa. There he spent his days editing the “fifth day” of the
Discourses
, to be added to the four days of dialogue that were published in 1638.

Only three months after Torricelli’s arrival, his mission came to an abrupt end. In the early days of 1642, Galileo came down with heart palpitations and a fever, and on January 8, at the age of seventy-seven, the old master breathed his last. As a man condemned for “vehement heresy,” he was interred in a small side room of the Basilica of Santa Croce in Florence, only to be moved to a place of honor in the central basilica a century later. Torricelli, meanwhile, was packing his things once more for the return journey to Rome when he received a startling offer: he could stay in Florence as Galileo’s successor, and become mathematician to the Grand Duke of Tuscany and professor of mathematics at the University of Pisa. The offer did not include Galileo’s position as court “philosopher,” most likely because it was Galileo’s insistence on his right, as philosopher, to pronounce on the structure of the world that had gotten him into trouble with the Church. But even without this additional accolade, the offer presented Torricelli with the opportunity of a lifetime: a secure position with a generous salary, the chance to pursue his studies without interruption, and public recognition as heir to the greatest scientist in Europe. He accepted without hesitation.