Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (10 page)

for the great pains … which he tooke in the
Gregorian Calendar
, by which both the peace of the Church and the Civill businesses have been egregiously troubled: nor hath heaven itself escaped his violence, but hath ever since obeied his appointments: so that S. Stephen, John Baptist, & all the rest, which have been commanded to work miracles at certaine appointed daies … do not now attend till the day come, as they were accustomed, but are awaked ten daies sooner, and constrained by him to come downe from heaven to do that businesse.

The biting satire does not mask the anti-Catholic Donne’s sincere consternation at having to succumb to the Pope’s reordering of both religious and civil time.

Protestant princes were forced into an unpleasant choice: they could either accept the Gregorian calendar, and thereby implicitly acknowledge the universal authority of the Pope, or they could reject it, and knowingly retain an embarrassingly erroneous calendar. Feeling cornered, they reacted with understandable confusion. Queen Elizabeth I of England at first announced that she would go along with the reform, only to backtrack in the face of opposition from the Church of England. The Gregorian reform would not come to the British Isles until 1752. The Dutch Republic split, with some provinces adopting the reform immediately while others retained the Julian calendar until 1700; and Sweden went back and forth between the two calendars until finally settling on the Gregorian in 1753. Farther east, the Russian Orthodox Church, whose quarrel with the Pope predated Luther’s by seven hundred years, also held on to the Julian calendar, until the Bolsheviks, who could not be suspected of being papal agents, imposed the reformed calendar in 1918. The last European nation to adopt the Gregorian calendar was Greece, in 1923, almost three and a half centuries after Clavius and his associates completed their work. By enacting a calendar that effectively took over the world, the Roman Church exhibited commanding authority, whereas its rivals showed only weakness and confusion, and the inherent limitations of their national churches.

The calendar reform was precisely the kind of triumph the Society of Jesus was laboring to achieve. Here was a perfect example of the Catholic Church imposing truth, order, and regularity upon an unruly world. Like St. Ignatius in Rubens’s masterpiece, Pope Gregory was bringing the light of universal truth to the people, who had long suffered in darkness and confusion. The reaction to the reforms confirmed this: wherever the Pope’s command was law, order, peace, and truth prevailed; wherever heretics and schismatics ruled, error, confusion, and strife persisted. Nothing could better illustrate the justice of the approach that was at the core of the Jesuit worldview. Here, the Jesuits believed, was a template for the ultimate triumph of the Roman Church.

The decisive victory of the Roman Church in the matter of the calendar seemed all the more striking when compared to the stalemate that prevailed in other areas of theological dispute. Catholics, for example, believed that the grace of God was bestowed upon sinners only through the holy Church and its sacraments, enacted by an ordained priest. Protestants, conversely, believed in a “priesthood of all believers,” meaning that God would bestow his grace directly upon them. Catholics believed that Christ was physically present in the bread and wine during the sacrament of the Mass. Protestants believed that Christ was either present everywhere (Luther) or that the Mass was a mere commemoration of his sufferings (Zwingli). Catholics believed that God would take into account a man’s good works in this world in determining whether he would be saved or lost. Protestants believed that only faith and divine grace mattered. Catholics believed that the Bible required interpretation by the hierarchy and the traditions of the Church. Protestants believed that the Bible was a clear guide for righteous behavior, accessible to anyone. And so on. What these arguments had (and still have) in common is that they are entirely inconclusive. From Luther’s day to ours, neither side has given an inch, or has seen any reason to.

To be sure, advocates on both sides engaged in passionate, and often violent, debate. They published crude caricatures of each other, depicting Luther as the Devil’s emissary, or the Pope as anti-Christ, and disseminated them as broadly as the new technology of the printing press allowed. They published popular pamphlets denouncing each other’s doctrines as heretical, and catechisms detailing the fundamentals of each faith. They authored learned treatises such as Calvin’s
Institutes of the Christian Religion
or the Jesuit Francisco Suárez’s
Disputationes metaphysicae
, and they engaged, occasionally, in formal debate, as Luther had with Eck in 1519. But despite the effort, time, and resources invested in these battles, neither side was able to impose its position upon the other. What a contrast between this bog of indecision and the glorious, clear-cut victory afforded the Roman Church by the reform of the calendar! If only the secret of the calendrical triumph could be infused into those other fields, the ultimate victory of Pope and Church would be assured.

Clavius believed that he knew what this secret was: mathematics. Theological and philosophical disputes could rage forever, he believed, because there was no universally accepted way to decide who was right and who was wrong. Even when one side possessed the absolute truth (as Clavius believed it did), and the other nothing but error, the adherents of error could still refuse to accept the truth. But mathematics was different: with mathematics, the truth forces itself upon its audience whether they like it or not. One could dispute the Catholic doctrine of the sacraments, but one could not deny the Pythagorean theorem; and no one could challenge the correctness of the new calendar, based as it was on detailed mathematical calculations. Here, Clavius believed, was a key to the ultimate triumph of the Church.

THE CERTAINTY OF MATHEMATICS

Clavius elaborated his views on mathematics in an essay that he attached to his edition of Euclid, which first came out in 1574, just as the commission on the calendar was getting down to work. Entitled simply “In disciplinas mathematicas prolegomena” (“Introductory Essay on the Mathematical Sciences”), it is in fact a passionate appeal for recognition of the power of the mathematical sciences and their superiority over other disciplines. If “the nobility and the excellence of a science is to be judged by the certainty of the demonstrations that it uses,” Clavius wrote, then “without a doubt the mathematical disciplines have the first place among all others”: “They demonstrate everything in which they see a dispute by the strongest reasons, and they confirm it in such a way that they engender true knowledge in the minds of the hearers, and completely remove any doubt.” Mathematics, in other words, imposes itself on the minds of its hearers and compels even the most recalcitrant among them to accept its truths.

“The theorems of Euclid,” he continues, “and the rest of the mathematicians,”

still today as for many years past, retain in the schools their true purity, their real certitude, and their strong and firm demonstrations … And thus so much do the mathematical disciplines desire, esteem, and foster the truth that they reject not only whatever is false, but even anything merely probable, and they admit nothing that does not lend support and corroboration to the most certain demonstrations.

But the case is very different with the other so-called “sciences.” Here, Clavius argues, the intellect deals with a “multitude of opinions” and a “variety of views on the truth of the conclusions that are being assessed.” The result is that whereas mathematics leads to certainty that ends all debate, other fields leave the mind confused and uncertain. Indeed, Clavius continues, commenting on the inherent inconclusiveness of nonmathematical fields, “how far all this is from mathematics, I think no one admits.” “There can be no doubt,” he concludes, “but that the first place among the other sciences should be conceded to mathematics.”

Rigorous, orderly, and irresistible, mathematics was for Clavius the embodiment of the Jesuit program. By imposing truth and vanquishing error, it established a fixed order and certainty in place of chaos and confusion. It should be remembered, however, that when Clavius is speaking of “mathematics,” he has something quite specific in mind. Certainly the arithmetic in use by merchants and traders had its place, as did the emerging new science of algebra, which teaches one how to solve quadratic, cubic, and quartic equations. But the true model of mathematical perfection for Clavius was geometry, as presented in Euclid’s great opus
The Elements
. It was the only mathematical field, he believed, that captured the power and truth of the discipline in its most distilled form. When Clavius wished to emphasize the eternal truth of mathematics, he cited “the demonstrations of Euclid,” and surely it is no coincidence that of all his textbooks on the many mathematical fields, he chose to append his “Prolegomena” to his edition of Euclid.

Composed around 300 BCE,
The Elements
is arguably the most influential mathematical text in history. But not because it presented new and original results:
The Elements
was based on the work of earlier generations of geometers, and most of its results were likely well known to practicing mathematicians. What was revolutionary about Euclid’s work was its systematic and rigorous method. It begins with a series of definitions and postulates that are so simple as to be self-evidently true. According to one definition, “A figure is that which is contained by any boundary or boundaries”; according to one postulate, “all right angles are equal to one another”; and so on. From these seemingly trivial beginnings, Euclid moves step by step to demonstrate increasingly complex results: that the base angles of an isosceles triangle are equal; that in a right triangle, the sum of the squares of the two sides containing the right angle is equal to the square of the third side (the Pythagorean theorem); that in a circle, the angles in the same segment are all equal to one another; and so on. At each step, Euclid does not just argue that his result is plausible or likely, but demonstrates that it is absolutely true and cannot be otherwise. In this manner, layer by layer, Euclid constructs an edifice of mathematical truth, composed of interconnected and unshakably true propositions, each dependent on the ones that precede it. As Clavius points out in the “Prolegomena,” it was the sturdiest edifice in the kingdom of knowledge.

For a taste of the Euclidean method, consider Euclid’s proof of proposition 32 in book 1: that the sum of the angles of any triangle is equal to two right angles—or, as we would say, 180 degrees. Euclid, at this point, has already proven that when a straight line falls on two parallel lines, it creates the same angles with one parallel line as with the other (book 1, proposition 29). He makes good use of this theorem here:

Proposition 32: In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles.

Proof:

Let ABC be a triangle and let one side of it be produced to D. I say that the exterior angle ACD is equal to the two interior and opposite angles CAB, ABC, and the three interior angles of the triangle, ABC, BCA, CAB, are equal to two right angles.

Figure 2.1. The sum of the angles in a triangle.

For let CE be drawn through the point C, parallel to the line AB.

Then, since AB is parallel to CE and AC falls on both of them, the alternate angles BAC and ACE are equal to one another.

Again, since AB is parallel to CE, and the straight line BD falls on them, the exterior angle ECD is equal to the interior and opposite angle ABC.

But the angle ACE was also proved equal to BAC. It follows that the whole angle ACD (composed of ACE and ECD) is equal to the two interior and opposite angles BAC and ABC.

Let the angle ACB be added to each; it follows that the sum of the angles ACB and ACD are equal to the sum of the interior angles of the triangle, ABC, BCA, CAB.

But since the angles ACB and ACE are equal to two right angles, it follows that the angles of the triangle, ABC, BCA, CAB are also equal to two right angles.

Q.E.D.

Euclid’s proof here is fundamentally simple: He extends the triangle’s side
BC
to the point
D
, and then draws a parallel line to
AB
through the opposite corner
C
. Using what he has already proven about the properties of parallels, he transfers the triangle’s angles
A
and
B
to the line
BD
next to the angle
C
, thus showing that the three angles together combine to form a straight line—that is, 180 degrees. But even in this simple proof, all the elements that make Euclid so compelling are clearly present. The proof is based on previous ones, in this case, the unique properties of parallels; from there, it proceeds systematically, step by step, showing clearly that each small step is logically correct and necessary; and ultimately, it arrives at its conclusion, which is absolutely true and universal. Not only the specific triangle
ABC
has angles that combine to 180 degrees, but every triangle that ever was, will be, or can be will show the exact same characteristic. Finally, the proof of proposition 32 and every other Euclidean proof is a microcosm of Euclid’s geometry as a whole. Just as each proof is composed of small logical steps, so the proofs themselves are but small steps in the edifice that is Euclidean geometry. And like each proof alone, geometry as a whole is universally and eternally true, ordering the world and governing its structure everywhere and always.