Pythagorus

by Kitty Ferguson

  [3]By ‘perfect number’ they did not mean what the Pythagoreans had meant when they identified 10 as the perfect number. A perfect number by more modern standards (found already in Nicomachus) is a number the sum of whose divisors equals the number. The number 6 is the smallest perfect number: 1+2+3=6.

  [4]A planet’s period is the time it takes to complete one orbit.

  [5]Aurelian was reading from a mistranslation of the Book of Job.

  [6]Regino’s description sounds very much like Aristotle’s, which means he must indeed have got it through Boethius. Regino lived before the reintroduction of Aristotle to Latin Europe.

  [7]In the mid-twentieth century, there was still one expert, Vincenzo Capparelli, who was convinced that Pythagoras invented Arabic numerals (Vincenzo Capparelli, La sapienza di Pitagora [Padua: CEDAM, 1941]).

  [8]Most who used an abacus were still using Roman numerals, the English exchequer as late as the sixteenth century! (H. G. Koenigsberger, Medieval Europe, 400–1500 [Harlow, England: Longman Group, 1987], p. 202.)

  [9]T. S. Eliot echoed those sentiments when he suggested that to those who say we shouldn’t read the old authors since we know so much more than they did, we should answer, ‘And they are what we know.’

  CHAPTER 15

  ‘Wherein Nature shows herself most excellent and complete’

  Fourteenth–Sixteenth Centuries

  In the fourteenth century, most educated people in Europe regarded foreign languages as completely impenetrable and unlearnable, so the author Francesco Petrarca (Petrarch) was being venturesome when he decided to learn Greek. He engaged a teacher, a monk named Barlaam of Seminara, but the project was not a success and Petrarch was fated to go on lamenting that he would never arrive at the best understanding of philosophy because his Greek was not good enough.

  He was disarmingly modest. Perhaps he did, as he claimed, merely chuckle when he was an old man and heard the news – it was being repeated all over Venice and beyond – that four young aristocrats, who had dined and drunk exceedingly well, had off-handedly dismissed him as ‘certainly a good man but a scholar of poor merit’. In a letter written just a few years before that Venetian slight, Petrarch described himself:

  You must realise, my friend, how far I fall short, in my own opinion and in reality, from this judgement of yours: I am not any of the things you attribute to me. What am I? A scholar? Hardly that. A backwoodsman, a solitary man in the habit of muttering foolishness in the shade of lofty beech trees, and – the height of arrogance and rashness! – wielding a shaky pen under a laurel sapling. Fervent in labour, but not happy with the results; lover of letters, but not fully versed in them; adherent of no sect, but greedy for truth. That is hard to find, and I, feeble in my search, often lose confidence and, fearing error, accept doubt in place of truth. Thus I have gradually become a squatter in the academy, and, like many, I have rallied to that humble band that claims nothing, holds nothing certain, doubts everything except what it is sacrilege to doubt.8

  Some of the ‘lofty beech trees’ among whom Petrarch muttered his foolishness were Augustine and Cicero, Aristotle and Plato (he read them in Latin translations), and Pythagoras, whom he knew through those other authors.

  Collecting works from the classical period, tracking down manuscripts and early copies, had become the fashion among those sufficiently educated and wealthy, and the acquisition of something interesting was a matter of great excitement to share with like-minded friends. Petrarch’s own large library reflected that fashion and his love of learning, but, for all his modesty, the library he stored in his head was vaster than most men’s collections. He read more than anyone else, remembered most of it verbatim, and had a habit of imagining himself personally involved in history and literature. As one commentator wrote,

  Since he was such a keen observer of actual life and so lovingly devoted to the investigation of the human heart, all the records of the past became a living reality to him, and he felt himself sharing in the drama as if he had an active part in the cast. It was not just a whim that he, the untiring letter writer, started to ‘correspond’ with characters of ancient times, as if they could answer him. When he read their works, he almost forgot that they were long since dead.1

  No wonder Shakespeare so often found inspiration and material for his plays in Petrarch. Through Shakespeare and others who read Petrarch, he played an influential role in shaping future culture.

  Petrarch was no fan of the Pythagorean doctrine of reincarnation, which he thought was an example of the way a wise and brilliant man can be perfectly capable of coming up with nonsense. ‘Who does not know’, he wrote, ‘that Pythagoras was a man of exalted genius? However, we also know his Metempsychosis. I am amazed beyond belief that this idea could spring up in the brain, not of a philosopher, but even of any human being.’ Pythagoras’ claim to have been Euphorbus in an earlier life was ‘an empty lie’ and ‘deceitful pretense’. But then Petrarch also scorned Democritus’ suggestion that ‘heaven and earth, and all things in general, consist of atoms’.2

  Petrarch, as imagined by engraver Rob Hart, 1835

  A few pages after his disparaging words, Petrarch turned around and referred to Pythagoras in reverential tones as ‘the most ancient of all natural philosophers’. No one knows where he got the quotation that he attributed to Pythagoras and used to defend not only the Christian faith but also Plato and Moses from those who ‘blind and deaf as they are, do not even listen to Pythagoras, who asserts that “it is the virtue and power of God alone to achieve easily what Nature cannot, since He is more potent and efficient than any virtue or power, and since it is from Him that Nature borrows her powers.”’3 Petrarch did not believe that Pythagoras had actually written this, or, indeed, anything, but he thought that others had written down ‘what he expounded in his conversations’.

  Petrarch is often called the first humanist. He trusted God so devoutly and completely that he felt free to leave the deepest religious issues alone and concentrate instead on philosophy, which he preferred to define as the study of the art of happiness and living well.4 Pythagoras, Plato, and Christianity seemed a natural, logical continuum to him.

  In the middle of the next century, the fifteenth, no less a personage than Lorenzo de Medici lent his patronage to an attempt to re-create Plato’s Academy at the villa of his acquaintance Marsilio Ficino, near Florence. The Accademia Platonica was Ficino’s brainchild and dream. He translated all of Plato’s works into Latin directly from the Greek, wrote commentaries on them, and gathered a group of writers, thinkers, and artists to study them in a congenial setting. When Ficino had also finished translating Porphyry, Iamblichus, Proclus, and Plotinus, those who knew no Greek could read nearly the entire surviving output of the Platonic and neo-Platonic writers in Latin. It is a pity that Petrarch had lived a century too early to enjoy all these works in translation!

  One of Ficino’s Academy members was the artist Botticelli, whose painting Primavera was supposed to be a visual metaphor for the music of the spheres, relating mythological creatures to planetary orbits and the notes of an octave in music. Ficino himself developed an elaborate system of heavenly music. He was also interested in the early church fathers and, like Petrarch, thought that Platonic doctrine and reasoning (which he thought were divinely inspired) were in harmony with Christianity, having particular value in that they could provide independent confirmation of Christian beliefs in a manner that would satisfy those among Ficino’s contemporaries who were of a sceptical and even atheistic frame of mind. He gave a Pythagorean/Platonic spin to his treatment of the fall and salvation of man, referring to the belief that the earthly existence of the soul is an exile from its divine home. The Pythagoreans and Platonists agreed, he wrote, that ‘because of a certain old disease of the human mind, everything that is very unhealthy and difficult befalls us; but
, if anyone should restore the soul to its previous condition, then immediately all will be set in order.’ To Ficino, that sounded like humanity in its fallen state looking towards the salvation of Jesus, in Christian doctrine. A yearning to turn back to God was built into human nature:

  Just as when an element is situated outside its proper location, its power and natural inclination toward that natural place are preserved together with its nature, in so far as it is able at some time to return to its own region; so, they [the Pythagoreans and Platonists] think, even after man has wandered from the right way, the natural power remains to him of returning first to the path, then to the end.5

  Ficino agreed with those neo-Pythagoreans who had concluded that the same primordial wisdom had emerged in different ages and cultures. The truth of philosophy, religion, and natural science, in all times and places, was, at some deep, so far unplumbed level, one consistent truth. This, Ficino thought, was a manifestation of the ‘unity’ that the Pythagoreans had held so in awe.

  In the city of Parma during this same period, the musician and physician Giorgio Anselmi (some thought he was also a magician) developed the first system since Eriugena’s to take into account the fact that the planets change their distances from Earth. In Anselmi’s cosmic musical plan, a planet produced not one tone but many different notes as its distance changed, so that each planet sang its own song. All the planet songs together produced magnificent counterpoint and harmony. Though no music of his time went beyond a three-octave range, Anselmi’s planetary scale, calculated from the planets’ periods, was eight octaves long from the stars to the Moon.

  Ficino’s younger Florentine friend Giovanni Pico, Count of Mirandola (known as Pico della Mirandola), was fond of using the phrase, the ‘ancient theology of Pythagoras’. He regarded Pythagoras as no less than a Christian sage and connected the peace promised by Jesus – ‘Come unto me, ye that labour, and I will give you peace, which the world and nature cannot give’ – with a Pythagorean ‘longed-for’ peace in which

  all minds do not merely accord in one intellect that is above every intellect but in some inexpressible fashion become absolutely one. This is that friendship which the Pythagoreans say is the end of all philosophy. This is the peace of God, which the [Christmas] angels descending to earth announced to men of good will, that by this peace men themselves, ascending into heaven, might become angels.6

  Until that time, ‘Let us desire this peace for our friends, for our age, for every house into which we enter, for our souls.’

  Pico did not always write so clearly and simply. One of his more impenetrable documents was ‘Fourteen Conclusions after Pythagorean Mathematics’,7 which arose out of his fascination with ‘the method of philosophising through numbers’ as it was taught by ‘Pythagoras, Philolaus, Plato, and the first Platonists’. Aristotle would have summoned his Delian diver!

  Unity, duality, and what is, are the causes of numbers. Unity is the cause of unitary numbers, duality, of generative numbers, what is, of substantial numbers.

  In participating numbers some are species of numbers, others unions of species.

  Where the unity of the point moves to the duality of the binary, there the first triangle is.

  He who knows the series I, II, III, IV, V, XII, will possess exactly providence’s distribution.

  By one, three, and seven we understand, in Pallade, the unification of the separate, the causative and beatifying power of intellect.

  The triple proportion, Arithmetic, Geometric, and Harmonic, indicates to us three daughters of Themidos, symbolising judgement, justice, peace.

  The secret of straight, reflected, and refracted lines in the science of perspective reminds us of triple nature: intellectual, animal, and corporeal.

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

  Irascibleness is in the proportion of a fifth to the concupiscent nature.

  Reason is in the proportion of a fourth to anger.

  The judgement of the senses in music is not to be heeded, only of the intellect.

  In the numbering of forms we should not go beyond forty.

  Any equilateral plane number can symbolise the soul.

  Any linear number can symbolise God.

  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.[1] 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 rationalised 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’.9 [2]

  In 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.[3] 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.10

  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 Ren
aissance 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 systematise 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.[4]

  In 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’.11 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-centred 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 ‘centre of the universe’, his proposal was a huge demotion.