Sherlock Holmes and the Pythagoreans
[Written on loose sheets of paper in the hand of John H. Watson, dated March 29th, 1883 --Ed.\
It was the 27th of the unusually cold March of 1883 and, as the rain fell outside 221B Baker St., I sat with Sherlock Holmes in our small parlour. It was past eight in the evening, and it seemed unlikely that my services would be called upon, or that Holmes would receive one of the mysterious visitors whose intriguing problems had resulted in so many of our adventures together. As the evening wore on, and Holmes languidly sat by the fire, I noticed him more than once cast a glance at his cocaine bottle.
Though the deleterious effects of cocaine on memory and health are becoming ever better known, I had had but little success in convincing my friend of the danger of its effects. And so I felt some relief when Holmes reached for the cocaine bottle, allowed his hand to play briefly over its lid, and then withdrew his hand and stood up, with a curious smile on his thin lips.
“Watson,” he said, “you are no doubt aware of my interest in history.”
I confessed that I was aware of it, for I had often had occasion to consult his extensive collection.
“I read history,” he continued, “in no small part for the pleasure of a game I call ‘historical detection’. I thought I might engage a case that has for some time interested me, and I should welcome your assistance, if you would like to join me.”
I expressed my great interest in doing so, and Holmes went to his bookshelf and drew from it a number of leather bound volumes which he laid on the table. He then began to pace to and fro and, while pacing, began to talk.
“Very well Watson. I will now give you the police report, as it were, upon the death that we are to investigate. We shall then proceed to an examination of the witnesses. But what can be roughly agreed upon is this. The great Greek philosopher, Pythagoras -”
“Just a moment Holmes,” I intervened. “I had always thought that Plato coined the term ‘philosophy’.”
“A common enough mistake,” Holmes said with a smile. “But it was Pythagoras who coined the term, almost a hundred and fifty years before Plato’s birth. [See Iamblichus 70 --Ed.\ Pythagoras, you recall, was born around the beginning of the 6th century, probably around 570 BC. We know little about his life, and there is much evidence that is contradictory. What seems certain, however, is that he was born on Samos, but left the island, no doubt on account of the ascendancy of the tyrant, Polycrates. His Samian compatriot, Herodotus, likely did the same thing.”
Holmes paused in his pacing to find his pipe. As he talked he filled it with tobacco, tamping the dried leaves with the curious silver statuette of the Hindu elephant god Ganesh, which was given to him by the Sultan of Bengieh in connection with the now well-known affair of the weeping dervish, a matter in which Holmes had been of some small service to that potentate.
“When Pythagoras left Samos,” Holmes continued, “he traveled to Croton in Northern Italy. The town still exists, though the Italians call it Crotone. By this time he had developed the philosophy which was to do him so much credit. The inhabitants of Croton were said to have welcomed him with open arms.”
Holmes paused and looked up from his pipe. “I grant that there were many wonders in the ancient world. But I should be most surprised at that of a town evincing much interest at the arrival of a philosopher. It is perhaps not so surprising that Iamblichus, who tells us this, goes immediately on to explain that Pythagoras could speak the language of the beasts.” [See Iamblichus 64ff --Ed.\
“But surely in the time of man’s innocency-” I began to protest.
“Perhaps, perhaps,” said Holmes, smiling, and waving away my objection. “At any rate, by the turn of the century these innocents had had enough, and drove Pythagoras and his followers out of the city. He resettled in another coastal Italian town, Metapontum.”
“Ah, another case like that of Socrates. I suppose it was envy and suspicion that caused the great man to be expelled,” I said.
“In part it may have been envy,” replied Holmes, “but Pythagoras - whether by converting the young aristocrats or in some other way - had gained political control over a large area before his expulsion from Croton. The revolt against him seems to have been plotted by democrats, who were unwilling to be governed by a secret society. And it is said that one Pythagorean who favoured the cause of the revolutionaries was a young man, perhaps at the time only a boy, called Hippasus.”
I confessed that I was unfamiliar with the name.
“Indeed,” said Holmes, “the name has been all but forgotten. But it seems that this man remained a Pythagorean even after the revolt, when Pythagoras had relocated to Metapontum, which was in fact the city of Hippasus’ birth.
“We know,” Holmes continued, “that Hippasus performed certain experiments. For while Pythagoreanism had undeniably mystical teachings, prohibiting certain foods, certain customs, and defending the transmigration of souls, what Pythagoras is most remembered for are his teachings on number. He believed, as you recall, that the universe is, in its deepest elements, nothing but numbers. Thus the apparent order of things is structured by a deeper order, an order which is perfectly rational, and in which everything can be expressed in ratios. In consequence the Pythagoreans revered numbers, especially those from one to ten. This reverence was nourished by experimentation aiming to discover numbers existing, as it were, in the wild. Geometry, of course, offered the chance to discover numbers in the shapes we find all around us. Another such area was music theory. Hippasus seems to have had interests in both subjects. He is known to have theorized about pentagons and pentagrams, specifically about the dodecahedron, a figure made up of twelve pentagons placed within one another, and which occurred naturally in the crystal and pyrite found in Italy. [See von Fritz 256 for a confirmation of Holmes’ insightful observation. --Ed.\ And he was interested in music and sound. Hippasus experimented with the sounds of bronze disks and vessels partially filled with water. [See e.g. Aristoxenus fr. 90 -- Ed.\ So much seems uncontroversial.”
“I hope,” said I, my head swimming at the thought of these mathematical figures, “that we shall not need to inquire too closely into figures of twelve pentagons, or other mathematical obscurantia.”
“No,” said Holmes, “or at least these are not the first objects of our detection. We shall rather be inquiring into the details of Hippasus’ rather untimely death, which was followed in short order by the end of the Pythagorean movement. For it seems that sometime after the death of Pythagoras, Hippasus was out on a ship and died under rather suspicious circumstances.” He paused, and let his hands run over the volumes which he had laid upon the table. “Perhaps I have said enough. Let us begin to search out the witnesses.”
Holmes picked up a slender volume from the table. “Our first witness is Iamblichus. He has the most to tell us, but take care Watson, for he has the greatest stake in the matter as well. For Iamblichus was himself a Pythagorean, or as his movement was better known, a ‘neo-Pythagorean’. Iamblichus of Chalcedon lived roughly from 245-325 AD, and so his news about the original Pythagoreans was hardly firsthand. But Iamblichus considered Platonism and Pythagoreanism to be inseparably connected, and reckoned himself an heir to both movements. He was therefore keenly aware that he must explain how the Pythagorean movement, which he so admired, vanished from the ancient world.
“While Iamblichus is not overtly hostile to Hippasus, his name comes up again and again in connection with the troubles of the Pythagoreans. Hippasus is implicated in the democratic revolution in Croton. [See Iamblichus 119 --Ed.\ Iamblichus tells us that Hippasus wrote a mystical text, which he pretended had been written by Pythagoras. The object of this plot was to discredit Pythagoras. [See Iamblichus 143 --Ed.\
And then Hippasus is implicated in the discovery of irrational numbers.
‘Concerning Hippasus, they say that he was a Pythagorean, and because he was the first to construct and publish the sphere of twelve pentagons [the dodecahedron\, he died at sea for this act of impiety. They add that although he gained the reputation for this discovery, it really belongs, as does everything else, to ‘the master’. This is how they refer to Pythagoras, for they never refer to him by name.’ [Iamblichus DK 184a4. Holmes does not offer a translation here, so I have taken the liberty of inserting one from Waterfield 108, T36 --Ed.\
Iamblichus goes into detail about the demise of Hippasus elsewhere, though when he does he never refers to him by name. But it is clearly the same man.”
Holmes read me this passage as well.
‘Nor did [the Pythagoreans\ think fit either to speak or to write in such a way that their conceptions might be obvious to the first comer; for the very first thing Pythagoras is said to have taught is that, being purified from all intemperance, his disciples should preserve the doctrines they had heard in silence. It is accordingly reported that he who first divulged the theory of commensurable and incommensurable quantities to those unworthy to receive it, was by the Pythagoreans so hated that they not only expelled him from their common association, and from living with him, but also for him they constructed a tomb, as for one who had migrated from the human into another life. It is also reported that the Divine Power was so indignant with him who divulged the teachings of Pythagoras that he perished at sea, as an impious person who divulged the method of inscribing in a sphere the dodecahedron, one of the so-called solid figures, the composition of the icostagonus. But according to others this is what happened to him who revealed the doctrine of the irrational and incommensurable quantities.’ [Iamblichus Life of Pythagoras 34. Again, Watson supplies no translation, so I have inserted Guthrie 116 --Ed.\
“In many years of detection I have observed, Watson, that the gods very rarely throw errant philosophers overboard. And given that there was, according to Iamblichus, ample motive, and that the Pythagoreans were so thoughful as to build Hippasus a tomb ahead of time, there are some grounds for suspicion in connection with his passing.”
Holmes then put the book down on the table.
“So Watson,” he said, “what are your impressions of the witness?”
“I think, Holmes,” I said, “that Iamblichus seems to be in some confusion about matters. He does not seem to be quite sure what to ascribe to Hippasus. Then again, he may see a connection between irrational numbers and dodecahedra, but I confess that I am not very sure of either concept, not to mention their conjunction.”
As I was saying this, Holmes nodded assent to each point. “Indeed Watson, Iamblichus does seem to be uncertain about Hippasus. In part I believe this stems from an attempt on his part to account for his own standing in the history of philosophy. For Iamblichus was unsure how to fit the neo-Pythagoreans into the Pythagorean movement.
“The problem was this. After the death of its founder, the movement fractured into traditionalists, known as the ‘akousmatikoi’ and innovators, called the ‘mathematikoi’. These latter forged ahead with Pythagorean mathematical investigations. But the akousmatikoi attempted to preserve the Pythagorean teachings just as they had heard them - and hence their name. Iamblichus thought that the neo-Pythagoreans ought to follow all the teachings of Pythagoras, and hence be akousmatikoi. But in so thinking he was vexed by, as scholars believe, discovering that Hippasus was thought to be an akousmatikos, not a mathematikos. And so, despite being unable to explain why, Iamblichus refers to Hippasus as a mathematikos. And if I am not mistaken, it is this stream of contradictory thought about Hippasus which you diagnose here, my dear Watson.
“To your second question, concerning the link between the dodecahedron and irrational numbers. You will recall, of course, that an irrational number is one that cannot be expressed as a fraction composed of two integers. Irrationals are numbers like ‘π’, numbers ending in decimals that trail on forever without repetition. Now the connection between irrationals and dodecahedra is a fairly simple one. When a pentagram is inscribed in a pentagon, it is possible to find within it a triangle. And a triangle can be used to prove the existence of an irrational number, using Pythagoras’ own theorem, that the sum of the squares of the two shorter sides of a triangle is equal to the sum of the square of the hypotenuse. For instance, suppose that the length of both the shorter sides is 1. Thus, the squares are also 1. Added together, that makes 2. 2 is thus the square of the hypotenuse. How then to get at the length of the hypotenuse? We must find the square root of 2. But there’s the rub, my dear Watson, for the square root of 2 is irrational. It begins 1.41421356237 and goes on, well, it goes on forever. [For a more contemporary account of how Hippasus might have derived his view from dodecahedra, see von Fritz 257 --Ed.\ The trouble is that the existence of any such numbers makes Pythagorean views on number virtually untenable. The recent discovery of that German, Professor Cantor, that the vast majority of numbers are irrational, seems to utterly refute it.” [Holmes must be referring, not to Cantor’s well-known diagonal argument (unpublished for another eight years), but likely to Cantor’s 1874 article in Crelle’s Journal für die Reine und Angewandte Mathematik --Ed.\
“But why is that,” I asked, though I was beginning to discern the difficulty.
“Why, it is elementary, Watson,” said Holmes. “What makes Pythagoreanism attractive is that the relation between one thing and another can be expressed in a ratio. But irrational numbers are incommensurable with other numbers: no ratio will express the relation they bear one another. They have no common standard of measurement. And so if there are irrational numbers, the beautiful and elegant Pythagorean universe vanishes, and gives way to the same drab universe we seem to live in, one where a chair and a dog and a comet have nothing at all in common, no common unit of measurement at all.”
“I see, Holmes,” I said. “So Hippasus let the cat out of the bag, did he?”
“Well, perhaps he did. This at least is what Iamblichus would have us believe. And Iamblichus is our most prolix witness. Nevertheless, we have yet to hear two more ancients.”
Holmes picked a new volume from the table. “Professor Yonge has once again produced an excellent volume. He has translated the lives of Diogenes Laertius, that philosophical gossip monger of the early second century AD. Holmes turned to page 371. Ah, here it is. Diogenes shall enlighten us on four further points.
‘I. Hippasus was a citizen of Metapontum, and a pupil of Pythagoras.
II. He used to say that the time of the changes of the world was definite, and that the universe also was finite, and in a state of perpetual motion.
III. Demetrius, in his treatise on People of the same Name, says that he left no writings behind him.
IV. There were two people of the name of Hippasus; this man, and another who wrote an account of the Constitution of the Lacedaemonians, in five books. And he was himself a Lacedaemonian.’
Alas Watson, this does not give us much. But every small piece of information may help in the detection of crime. And it is worthy of note that Diogenes elsewhere confirms the existence of a rumour to the effect that Hippasus wrote a mystical book, and attributed it to Pythagoras.” [See Diog. Laert. 8.7 --Ed.\
I was forced to agree that I could not see much of value in Diogenes’ testimony.
Holmes went on. “Our final witness is the most silent and yet the nearest to the events of Hippasus’ death. But this last witness, who is Aristotle, is also the most august. Yet all that Aristotle tells us, here Holmes glanced at the edition of the Metaphysics he was holding, is this, that Heraclitus and Hippasus both maintained that the first principle of all things is fire. [Met. 984a7 --Ed.\
“Well Watson, there it is. We’ve heard from all three ancient witnesses. And now, unless you would like to call any others, it is time for your verdict. Was Hippasus murdered? And, if so, why?”
I sat for a moment gathering my thoughts. And then I said “Well Holmes, it seems to me that the evidence shows first, that Hippasus was likely murdered and second, that it was an act of philosophical self-defence. It seems evident that Hippasus plotted against the Pythagoreans while Pythagoras was still in power and then, after the old man’s death, attempted to destroy the movement. He spread slanders about Pythagoras, as both Iamblichus and Diogenes Laertius agree. And, finally, what Aristotle tells us is that his own views were completely at odds with Pythagoreanism. Fire as a basic principle indeed! So, Holmes, it seems to me that in the death of Hippasus, the spirit of Pythagoras was avenged.”
After I had finished speaking Holmes sat back in his chair and puffed on his pipe, until in the smoke the contours of his gaunt frame were all but lost in his leather armchair. “Indeed Watson, he said, it may be that you are right. Everything you describe is true of the Pythagoreans and yet-” and Holmes lapsed into a silence which stretched out for minutes.
Then, just as I had begun to think that Holmes had given up on the enterprise entirely, he suddenly sat up straight in his chair and went on talking as though there had been no pause at all, “...and yet, I would be willing to declare that Hippasus was more faithful than any other to Pythagoras and his teachings, and that he died because of it.”
“What’s this Holmes!” I cried. “Surely you cannot be serious. After all that you have told me, after all that we have heard together, we have seen that no man was less a Pythagorean than Hippasus.”
“Yes,” said Holmes, “so the evidence might seem to demonstrate. And yet I believe I can show you that the truth of the matter is very different. But first, I believe I have one more witness to call out. And this one, for a change, may allow for a little cross examination.”
I was about to protest that this was hardly sporting, if Holmes had another witness in reserve ready to turn the matter on its head. But then he rapped his pipe on the wooden table, like a judge tapping a gavel, and announced “I call to the stand, Dr. John Watson.”
“Holmes,” I said, “really this is too much. You know as well as I do that my knowledge of the classics is far less than yours. What could I possibly have to offer?”
“We shall see Watson, we shall see,” was my friend’s reply.
And then, adopting a judicial tone, he asked, “Now Dr. Watson, do you confess to knowing of Hippasus?”
“Well yes,” I replied, “but only because you told me, Holmes.”
“Indeed,” said Holmes. “but I was only the intermediate cause of your knowledge. Surely you’ll grant that had it not been for our three sources, you would not know anything at all about Hippasus.”
“Of course I grant it,” I said, “but what can this have to do with Hippasus’ death?”
“It has everything to do with it, my dear Watson, for our sources have something very important in common. They are all philosophers, or historians of philosophy.”
“Of course they are,” said I, rather indignantly, “but what difference does it make?”
“Only this: nothing in your account of Hippasus’ crime and punishment indicates anything about his skill as a philosopher. And yet Diogenes Laertius and Aristotle, the two philosophers who have the least stake in the welfare of Pythagoreanism in any of its forms, are only interested in the thought of Hippasus. And Iamblichus never denies that Hippasus was a good philosopher, or that his work was valuable.”
“Yes yes Holmes,” I responded, “and surely one might give an account of Hippasus the son or Hippasus the husband too. But we cannot take such things into consideration in the question of his crimes against Pythagoreanism.”
“Ah but this is where you go wrong Watson. For we know virtually nothing about Hippasus’ crimes. The vast preponderance of our evidence concerns his philosophical speculations.”
Here I told Holmes that I had entirely lost the thread of the argument.
“Very well,” he said, “let me approach the matter a little differently. We know that Hippasus was a philosopher of some note. But the evidence also shows that he was a Pythagorean. And even Iamblichus, who surely would have liked to do so, does not contest this. Instead, Iamblichus accuses Hippasus of breaking a rule of Pythagoreanism. But the question lingers, how could a Pythagorean have believed that fire, and not number, was the first principle, or that all things changed?
“Had Hippasus been a fool, one might dismiss this inconsistency as the thought of someone who misunderstood the teachings of his master. And yet we discover that, although no book was ever reliably attributed to him, the thought of Hippasus is remembered by Aristotle, and by Diogenes Laertius. This indicates that his thought must have been extraordinary indeed.
“Hippasus was then a remarkable philosopher, whose connection to the Pythagorean movement - it would seem - remained unbroken until his death. And this is sufficient, I think, for me to narrate some details of Hippasus’ life and death.
Holmes reclined once again in his chair, and moving his hands to illustrate his tale, began to talk. “Such was the promise of the young Hippasus that, though he had plotted against Pythagoras in Croton, Pythagoras did not cast him away. And over the years, the fiery young democrat became one of the venerable sage’s foremost students. Hippasus experimented widely on naturally occurring numbers. And whether it was he or Pythagoras who made the discovery, irrational numbers were soon discovered. But Hippasus was a faithful Pythagorean, and searched Pythagoras’ mystical thought for a way of incorporating this discovery.
“What would you have done, Watson? Suppose you had discovered a contradiction in the science of medicine - wouldn’t you try to see how much of it could be saved? Hippasus believed he had found such a way to save Pythagoreanism. The structures of all things were not the bare ratios Pythagoras had at first imagined. The structure of things was an ever-changing process. Diogenes puts it better than I can, Watson: ‘He used to say that the time of the changes of the world was definite, and that the universe also was finite, and in a state of perpetual motion.’ Fire burns, Watson, it serves a purpose, and follows an order, but can you discern it? This was Hippasus’ solution, that the structure of things was as elusive as the dancing flame.
“Hippasus’ view may not have been very different from that of Pythagoras at the end of his life. But I grant that it is not as reassuring as the idea that all is rational number. And it may be that there is more similarity between Hippasus and Heraclitus than Aristotle points out: both believed the world was a flux, and neither had many followers.
“The end of the story is then this. Pythagoreanism was a political as well as a philosophical movement. Though Hippasus was too clever to be defeated in argument, his turn toward mysticism was emptying the Pythagorean ranks. Iamblichus tells us that Hippasus’ crime was revealing Pythagorean secrets. But if Hippasus devoted his life to finding a way around irrational numbers, why bother keeping their existence secret? They were no part of Pythagoreanism, but a problem that the school overcame.
“As you have witnessed, Watson, Hippasus’ memory was even more persistent than the man. Some Pythagorean probably did throw Hippasus overboard. But the movement ended soon afterward, and ancient philosophers remembered him as the Pythagorean of flux. Despite the fact that he left no writings, his brilliance was undeniable. The only foe cunning enough, and patient enough to remove his mark upon the world, was time.”
[The following brief note was written at the end of the story in a different hand --Ed.\
I was unable to find any continuation of this story in my husband’s papers. While he may have intended to extend or even publish it, the manuscript is dated March 29th, and on the 1st of April my husband and his friend Sherlock Holmes became involved in the ghastly case of Dr. Roylott, which my husband introduced to the public as the ‘Adventure of the Speckled Band’. I believe that my husband never returned to the story above.
[Mary Watson, widow of John Watson --Ed.\
Works Cited by the Editor
Diogenes Laertius. The lives and Opinions of Eminent Philosophers. Trans. C. D. Yonge. London: H. G. Bohn, 1853.
Von Fritz, Kurt. The Discovery of Irrational Numbers by Hippasus of Metapontum. The Annals of Mathematics. 46:2 (1945) 242-264.
Guthrie, Kenneth. The Pythagorean sourcebook and library: An Anthology of Ancient Writings which relate to Pythagoras and Pythagorean philosophy. Grand Rapids: Phanes Press, 1987.
Waterfield, Robin. The first Philosophers: the Presocratics and Sophists. Oxford, New York: Oxford University Press, 2000.