The Music of Pythagoras (37 page)

Until that time, “Let us wish for this peace for our friends, for our century . . . for every home into which we go,” he wrote.

Pico did not always write so clearly and simply. One of his more impenetrable documents was “Fourteen Conclusions after Pythagorean Mathematics,”
⁸
which arose out of his fascination with “the method of philosophizing through numbers” as it was taught by “Pythagoras, Philolaus, Plato, and the first Platonists.”
⁹
Aristotle would have summoned his Delian diver!

1. Unity, duality, and that which is, are the causes of numbers: One, of unitary numbers; two, of generative ones; that which is, of substantial ones.

2. In participated numbers some are species of numbers, others unions of species.

3. Where the unity of the point proceeds to the alterity of the binary, there the triangle first exists.

4. Whoever knows the series of 1, 2, 3, 4, 5, 12, will possess precisely the distribution of providence.

5. By 1, 3, and 7 we understand the unification of the separate in Pallas: the causative and beatifying power of the intellect.

6. The threefold proportion—Arithmetical, Geometrical, and Harmonic—represents to us the three daughters of Themis, being the symbols of judgment, justice, and peace.

7. By the secret of straight, reflected, and refracted lines in the science of perspective we are reminded of the triple nature: intellectual, animal, and corporeal.

8. Reason is in the proportion of an octave to the concupiscent nature.

9. The irascible nature is in the proportion of a fifth to the concupiscent.

10. Reason is in the proportion of a fourth to anger.

11. In music the judgment of the sense is not to be heeded: only that of the intellect.

12. In numbering forms we should not exceed 40.

13. Any equilateral plane number may symbolize the soul.

14. Any linear number may symbolize the gods.

Not surprisingly, when the twenty-three-year-old Pico went to Rome and offered to debate another of his lists,
Nine Hundred Conclusions
, there were no takers. Like the “Fourteen Conclusions,” the
Nine Hundred
were short sentences, covering the subjects of scholastic and earlier theology, Arabic and Platonic philosophy, the Chaldean Oracles, the Zoroastrian Magi, and Orphic doctrines.
*
All, Pico insisted, were reconcilable with one another, and he was prepared to debate anyone who disagreed. Truth was universal. What might seem to be opposing schools of thought and doctrine really were all the same primordial wisdom of humankind, sharing a common truth.

Pico’s interest was piqued by the Jewish Cabalistic literature, in which words and numbers serve as a form of mystical code. Cabala is a form of Jewish mysticism that, though it had roots as early as the first century
A.D
., fully emerged in the twelfth century. Though a text of Merkava mysticism (a precursor of Cabala) had included a creation story with ten divine numbers, and one of the most important Cabalistic texts, the twelfth-century
Sefer ha-bahir
(“Book of Brightness”), introduced into Judaism the idea of the transmigration of souls, in neither case was there a known link with Pythagoras. But another man who immersed himself in the Cabala at about the same time as Pico, insisted there was a connection. Johann Reuchlin, a German humanist, set out to combine the study of Hebrew, Greek, theology, philosophy, and the Cabala, and to link it all with the name of Pythagoras. He wrote to Pope Leo X that, just as Ficino had so admirably done for Plato in Italy, he would “complete the work with the rebirth of Pythagoras in Germany.” He rationalized the connection with the Cabala by drawing attention to the (questionable) fact that “the philosophy of Pythagoras was drawn from the teachings of Chaldean science.”
^10
†

I
N THE SAME
century when Ficino set up his Florentine academy and Pico issued his intellectual challenges, their older contemporary Leon Battista Alberti, inspired by the work of the ancient Roman Vitruvius,
was insisting on beautiful proportions in buildings and applying Pythagorean principles to architecture. Books on architecture seemed to come in sets of four or ten volumes—two good Pythagorean choices. Vitruvius had written his “Ten” in the first century
B.C
., and, Alberti produced his “Ten” in 1485.
*
They were translated from Latin into Italian in the mid-sixteenth century. Alberti liked to use what he thought were Pythagorean ideas and extend them in ways of his own:

I am every day more and more convinced of the truth of the Pythagorean saying, that Nature is sure to act consistently, and with a constant analogy in all her operations. From whence I conclude that the numbers by means of which the agreement of sounds affects our ears with delight, are the very same which please our eyes and mind. We shall therefore borrow all our rules for the finishing of our proportions from the musicians, who are the greatest masters of this sort of numbers, and from those things wherein nature shows herself most excellent and complete.
¹¹

Alberti divided the kinds of areas to be measured in an architectural design into three categories: short, medium, and long. The Pythagorean ratios were the only ones that he applied to the “short” or “simple” areas: The shortest was a square; the next an area that started with a square and then added on a third again as much space, making a ratio of 3 to 4 between the square and the total area.

The last also started with a square and added on half again as much space, making a ratio of 2 to 3 between the square and the total area.

For larger areas, Alberti used proportions that went beyond these ratios, but all could, in one way or another, be linked to them.

Though Alberti was one of the most important theorists of architecture in the Renaissance and also one of that era’s greatest practitioners, his achievements were by no means confined to architecture. He was truly a “Renaissance man”—a moral philosopher, a major contributor to the techniques of surveying and mapping, a pioneer in cryptography, and the first to systematize and set down the rules for drawing a three-dimensional picture on a two-dimensional surface, establishing principles that would underlie perspective drawing from that time forward. Nevertheless, it was arguably in architecture that he had his most lasting impact, not only because of the splendid buildings he designed, but also because his Ten Books, with their Pythagorean principles, were read and studied by all Renaissance architects after him, including Andrea Palladio, perhaps the most influential architect of all time.
*

I
N THE EARLIER
part of Alberti’s century, Nicholas of Cusa, born in 1401, had been considering a startling, fresh approach to structure on a much larger scale: the entire cosmos. Though his name sounds Italian, Nicholas was the son of a boatman on the Mosel River. He received his religious training with a devotional group of laymen in the Netherlands and his university education at Heidelberg and Cologne. Later, as a university scholar and a cardinal of the Catholic church, Nicholas not only found Christian faith and classical philosophy compatible, but that compatibility became for him a fertile ground from which to begin innovative thinking in other areas of knowledge. He decided that God was infinite, and the universe had no limit other than God . . . so the universe was infinite too. Contrary to what most people believed (they had learned it from Aristotle), he insisted that the universe was not made of different types of substance at different levels, such as the impure region near Earth and the pure region of the celestial spheres. The universe was homogeneous. The stars were “each like the world we live in, each a particular area in one universe, which contains as many such areas as there are
uncountable stars.”
¹²
Nicholas was sure that Earth was a star like the Sun and the other stars, and it moved. This was not the orthodox, Ptolemaic/Aristotelian stationary-Earth-centered astronomy that was being taught in the universities! Nicholas worked his ideas up in a highly original, mathematics-based system. He did not suggest another body to usurp the importance of the Earth, but even without nominating a competitor for “center of the universe,” his proposal was a huge demotion.

Nicholas believed the human mind had innate power to know things and to acquire knowledge, and, like Aristotle, he thought that knowledge had to be acquired directly from nature and experience. He also believed that learning about nature and the universe required the use of numbers and the study of numerical proportion and ratios. He was fond of the Pythagorean practice of applying numbers to many aspects of life. In his treatise “On Catholic Concordance” he used the order of the heavens as a model for harmony in the church; and in his book
Of Learned Ignorance
he drew a parallel between the search for truth and converting a square to a circle.

Nicholas, like Alberti, was a Renaissance man. He drew up a map of Europe and was the first to prove that air has weight. He apparently never worried whether his ideas about the arrangement of the cosmos might conflict with church doctrine. It seems he had no reason for concern. The church never condemned or criticized him.

Astronomy was about to take an even more decidedly Pythagorean turn. In 1495, twenty-two-year-old Nicolaus Copernicus and his older brother Andreas journeyed south from their native Poland and “walked across the Alps”—their destination Bologna, seat of Italy’s oldest university. Nicolaus had completed four years at the Jagiellonian University in Kraków, which was renowned for its astronomy. If a student intended to continue his education after he had finished the quadrivium and the trivium, he chose an area of study and went to a university that specialized in that. Nicolaus’ uncle and guardian, an influential man who became bishop of Warmia, was apparently worried that his nephew was developing a keen interest in astronomy. Hoping that the Italian sunshine and the stimulating intellectual community of the University of Bologna would turn the young man’s interest in a better direction, he insisted Nicolaus go to Bologna, famous for its law faculty. (Copernicus did eventually receive a doctorate in canon law, the law of the church, although not from Bologna.)

While studying in Bologna, Copernicus met the university’s leading scholars and teachers of astronomy and astrology, and also a mathematician named Maria de Novara, whose influence was probably the most valuable of all that Copernicus carried away with him from these years. Novara was a neo-Platonist and a close younger associate of the men of Ficino’s academy in Florence. His neo-Platonism was decidedly Pythagorean. He fervently believed in the need to uncover the simple mathematical and geometric reality that underlies the apparent complexity of nature, and he insisted that nothing so complicated and cumbersome as Ptolemaic astronomy could possibly be a correct representation of the cosmos. His young friend Copernicus came to agree.

No new astronomical discovery, nor any better or more accurate observations of the heavens, caused Copernicus to discard Ptolemaic Earth-centered astronomy and replace it with a system in which the Sun was at the center. Though over the long passage of years the errors produced by the Ptolemaic system had made it less and less accurate in predicting planetary positions, no observational instrument during Copernicus’ lifetime was accurate enough to show whether the Copernican system solved this problem. The telescope would not appear until early in the seventeenth century, and the astronomical observations that Copernicus made himself were often less accurate than those of Hellenistic and Islamic astronomers centuries before him.

The early Pythagoreans, in the wake of their discovery of the ratios of musical harmony, had gone off in wild and misguided directions to decide there had to be ten bodies in the cosmos, disregarding the fact that there was no evidence of that number’s correctness, running ahead of nature, and arriving at the wrong conclusions. And here was Copernicus, doing something of the same kind, for when he decided that Ptolemaic astronomy could not be correct, he did so largely for reasons other than physical evidence. The beginning of the scientific revolution was perhaps not so scientific—not in the way we most commonly think of “scientific.”