Monday 30 November 2015

Eratosthenes and the well

Eratosthenes is one of the most famous individuals of the Greco-Roman world -- and justly so: he was a leading expert in cartography, philology, mythography, ethnography, geometry, and astronomy. In an age of nepotism, his raw talent and hard work got him headhunted by Ptolemy III while he was still living in Athens.

Nowadays, he is most famous for making a reasonably accurate calculation of the circumference of the earth, using 3rd century BCE data and methods. This was a celebrated feat in his own time too. The island of Elephantine, at Syene (modern Aswan), had a new hieroglyphic symbol for its name created, apparently in honour of his calculation, in the form of a plumb bob and try square.
for Elephantine
(one of many variants)

No myths so far: this story is all true. By calculating the relative angle of elevation of the midday sun in three cities on the same meridian and at the same season -- Alexandria, Syene (modern Aswan), and Meroë (the chief city of what the Greeks called Aithiopia; modern Bagrawiya, Sudan) -- with figures for the distances between these cities, and with the assumption that the earth's curvature was spherical -- he came up with a figure variously reported as either 250,000 or 252,000 stadia.

In terms of angular precision, the calculation was very exact. In terms of absolute distances, not so much. Eratosthenes' own writings on the subject do not survive. It's most likely that he calculated 250,000 stadia, but that the figure got "rounded" to 252,000 so as to give a tidy figure of 700 stadia per degree of the earth's circumference (360° × 700 = 252,000). More importantly, Ptolemaic methods for surveying long distances were very inexact -- not nearly as good as Roman measures, for example -- and, to boot, there were many variants of the stadion ranging from ca. 157 metres to ca. 262 metres. The problem of Eratosthenes' units is hair-pullingly complicated. (Maybe we'll revisit the subject one day. But then again, maybe not: it really is a messy topic.)

But how he calculated the earth's circumference -- that's where the myths come in. Here's an extract from an especially popular and influential account:
One day, while reading a papyrus book in the library, he came upon a curious account. Far to the south, he read, at the frontier outpost of Syene, something notable could be seen on the longest day of the year. On June 21st, the shadows of a temple column or a vertical stick would grow shorter as noon approached. And as the hours crept towards midday, the sun's rays would slither down the sides of a deep well, which on other days would remain in shadow. And then precisely at noon columns would cast no shadows, and the sun would shine directly down into the water of the well. At that moment the sun was exactly overhead. It was an observation that someone else might easily have ignored -- sticks, shadows, reflections in wells, the position of the sun: simple everyday matters. Of what possible importance might they be?
-- Carl Sagan, Cosmos (1980; episode 1, "The shores of the cosmic ocean")
NOT how Eratosthenes calculated the earth's circumference
The story of the well is repeated in a lesson at the Khan Academy, in an even more colourful (and fictional) account in Julia Diggins' 1965 book String, Straightedge, and Shadow, and in many more places.

And it is mostly false. There was a well like this at Syene: but it certainly had nothing to do with Eratosthenes, and nothing at all to do with his calculation.

Here's Pliny the Elder's account of the well (Natural History 2.183 [§75]):
Similarly they say that in the town of Syene, 5000 stadia south of Alexandria, no shadow is cast at noon on the solstice. A well was made to test this, and it was entirely illuminated. This showed that the sun was directly overhead at that time. Onesicritus states that the same thing happens at that season in India south of the river Hyphasis [i.e. the modern Beas].
(The source Pliny was using is vague: "south of the Hyphasis" sounds like it should indicate Punjab, but the tropic runs considerably south of Punjab. It does run close to Pataliputra, though, further to the south-east, which was the capital of the Maurya Empire until the early 2nd century BCE: that is probably what Pliny's Hellenistic source was talking about.) Strabo also describes the well (17.1.48), and his phrasing tends to suggest that it was moderately famous, though unlike Pliny he doesn't say that its unique feature was the very reason it was built.

No ancient source, at all, anywhere, connects this well to Eratosthenes. In the surrounding context of these passages, however, both Pliny and Strabo do mention the actual method that Ptolemaic surveyors used for measuring latitude: they used an instrument called the gnomon (Pliny NH 2.182; Strabo 17.1.48).

Gnomons were originally for determining the date of the solstice. This practice goes back many thousands of years, in many different civilisations. Meton used a gnomon in Athens in 432 BCE for this purpose; the Shang people in ancient China were using them to measure solstices in the 13th century BCE. In Egypt, one of the reliefs in the jubilee chapel of Senusret I depicts a 9-metre-high gnomon in connection with the festival of Min in the 20th century BCE.

In its simplest form, the gnomon was a vertical rod that cast a shadow. Measuring the ratio between the shadow and the length of the rod would constitute a gnomon reading. Old Egyptian gnomons typically had a bifurcated tip to make the shadow more defined (as in the Senusret relief). The Egyptians could use hand-held ones to mark the passage of time at night. Travellers in the Greco-Roman world had portable gnomons or sundials, in the form of a round vessel with a stylus embedded in the centre and markings on the sides: this set-up was called a skaphion (Greek) or umbilicus (Latin). The Egyptians were well aware that near the tropic it was difficult to measure the solstice with a vertical gnomon, because the shadows are so short at midsummer, and so they adopted the practice of tilting the gnomon to the north and supporting it with struts. (Further reading, for those with JSTOR access.) In later gnomons, plumb bobs were used to ensure it was exactly vertical: to judge from the hieroglyphic for Elephantine mentioned above, it looks like this was the case with the Ptolemaic gnomon. (And incidentally, remember that the symbol for Elephantine included a try square? It so happens that gnomon was also the Greek word for a try square.)

Ancient travellers regularly took gnomon readings as a way of recording their latitude. Pliny, just before the passage quoted above, tells us
Portable timepieces are not used the same way everywhere, because the sun's shadows change every 300 stadia, or at most 500 stadia [i.e. about 0.5° latitude]. So in Egypt at the equinox the shadow is only a little more than half the length of the umbilicus -- what they call a gnomon. In the city of Rome the shadow is 8/9 the length of the gnomon at the same season; in the town of Ancona it is 1/35 longer (i.e. 8/9 + 1/35, or 0.917); and in the region of Italy called Venetia the shadow is the same length.
Vitruvius gives another list of equinoctial gnomon readings in various cities. Ptolemy has an extended account of gnomons and latitude in book 2 of his Almagest. And Martianus Capella indicates that Eratosthenes' measurement was based on gnomon readings taken at the equinox (De nuptiis 6.597).

The practice of taking gnomon readings as a geographical measurement goes back at least to Pytheas of Massalia, a famed traveller of the early mid-4th century BCE. According to Martianus Capella (De nuptiis 6.595), Pytheas took gnomon readings all the way from southern France to "Thoule" (either Iceland, or one of the island groups in the North Sea, or perhaps Scandinavia). Also before Eratosthenes' time, Philon, a surveyor for Ptolemy II, reported gnomon readings at Meroë in his book the Aithiopika or Voyage to Aithiopia. The book itself does not survive, but this report does:
Philon discusses the latitude (κλῖμα) of Meroë in the Voyage to Aithiopia that he wrote. He says the sun is directly overhead 45 days before the summer solstice, and he reports his readings of the ratio between the gnomon and its shadow at the solstices and equinoxes. Eratosthenes agrees very closely (συμφωνεῖν ἔγγιστα) with Philon.
Philon, New Jacoby 670 F 2 (=Strabo 2.1.20)
So, to recapitulate:
  • Eratosthenes didn't use a well to measure anything: the instrument of choice was the gnomon.
  • Taking latitude readings from a gnomon was standard practice for Ptolemaic surveyors decades before Eratosthenes came along.
  • The summer solstice barely came into it; gnomon readings were taken year-round, but it looks like equinoctial readings were the basis of Eratosthenes' calculation.
  • Eratosthenes didn't take the readings himself: he probably used Philon's published work as the basis of his calculation.
The well at Syene was just a striking, large-scale visualisation of the Tropic of Cancer. Pliny's wording suggests it may not even been built until after Eratosthenes' calculation anyway. And native Egyptians had known for centuries that Syene was on the Tropic of Cancer. (In fact Syene was slightly north of the tropic: it's at 24.09° N, and the tropic was at 23.72° N in Eratosthenes' time. The plane of the ecliptic shifts slightly over the millennia: currently it's at 23.44°.)

Monday 16 November 2015

On the ‘losing’ of Troy

In Greek legend, the Trojan War ended with the Greeks using a colossal wooden horse to burn the city, sack it, and raze it to the ground. Men and boys were killed, women and girls were enslaved and transported. There were no survivors and no remains. Troy was utterly destroyed.

Many modern people are under the impression that the same thing happened to the historical city of Troy: that Troy ceased to exist at the end of the Bronze Age; that it was destroyed by the Greeks at the end of the Bronze Age, so that even the location of the city was lost; and that the ruins of the real Troy lay undiscovered until Heinrich Schliemann’s excavations in the 1870s-80s.

Everything in the above paragraph is unequivocally false.

We won’t focus on Schliemann today — his activities at Hisarlık (the modern name for the hill of Troy’s citadel) offer more than enough material for a lengthy debunking all by themselves. Today we’ll focus on the ‘loss’ of Troy. When was Troy ‘lost’?

Early Greek epic certainly gives us ample reason to think of Troy as a city that has been destroyed utterly. At one point in the Iliad, a defeated Trojan begs the Greek Menelaos for mercy. Menelaos is considering taking him captive instead of killing him, but his brother Agamemnon pops out of nowhere and says (Il. 6.55-60):
‘Menelaos, my dear, why do you care so much
about these men? Have you and your house been treated so finely
by the Trojans? Let none of them escape sheer destruction
at our hands, not even any boy that a mother carries
in her womb: let none escape, but let all the people
of Ilios be utterly destroyed, unmourned, wiped to oblivion!’
(Ilios/Ilion is an alternate name for Troy.) And here’s a fragment from the Little Iliad, a poem from the lost Epic Cycle, which uses one family as an emblem for the massacre of the children and enslavement of the women (fragment 21 ed. Bernabé = fr. 29 ed. West):
But Achilleus’ great-hearted shining son
led Hektor’s wife in captivity back to the hollow ships,
and he took her son from the embrace of his lovely-haired nurse,
grabbed him by the foot and threw him from a tower. As he fell,
a bloody death and hard fate snatched him up.
Sack of Troy: Neoptolemus kills king Priam,
bludgeoning him with the corpse of his grandson.
(Attic, ca. 520-10 BCE; Louvre)
Euripides’ play the Trojan Women (415 BCE) gives another angle on the sack of Troy. There the narrative focus is firmly on the survivors, the women of the city, who are about to be carted off into slavery while still mourning for their husbands and sons, in an act of destruction that was not caused by any of them. (Euripides’ picture of the destruction wrought upon Troy is especially thorough because he was using the legend as an allegory for current events: the previous summer, the Athenians had decided to commit genocide on the island of Melos, slaughtering the entire male population and enslaving all the women, rather than allow the Melians to remain neutral in the Peloponnesian War.)

That’s the legend. What about the reality?

Archaeological evidence is the most reliable way of corroborating or disproving the stories. And one piece of archaeological evidence is popularly linked to the legend. There are traces of a large fire on the citadel of Troy dating to the end of the level called ‘Troy VIIa’, that is, ca. 1190 BCE. The archaeological layers are numbered Troy I, II, III, etc. starting from the lowest and earliest level: the higher the number, the shallower and more recent the archaeological remains are. The fire of Troy VIIa is popularly equated with the legendary war especially thanks to Michael Wood's BBC TV documentary series and book In Search of the Trojan War (1985). For what was known at the time, it’s an extremely competent piece of work. The entire series can be watched on YouTube here.

It is perhaps worth pointing out that, among ancient historians who believe the Trojan War actually happened, they gravitate more towards Troy VIh as the best candidate for a historical war. That would put the ‘fall of Troy’ about a century earlier. Very few ancient historians nowadays would opt for Troy VIIa. But that’s neither here nor there. As another incidental by-the-way, some archaeologists involved in excavation at Troy would now refer to that layer as ‘Troy VIi’, not VIIa (for reasons that don’t matter just now).

The important thing, and it really is worth emphasising, is that Troy was not destroyed at that time. On the contrary: after the fire of Troy VIIa, the citadel was promptly rebuilt. It continued to be occupied without any pause for another 250 years or so. The population dwindled — not an exceptional thing: that also happened at many other sites in Greece and Anatolia in the early 12th century BCE — and the site was finally abandoned ca. 950 BCE.

Did I say abandoned? Well, yes... but the story doesn’t end there. Troy was not ‘lost’ in 950 BCE either. In fact, Greek colonists settled the site once again starting in the early 8th century BCE. It became a Greek city, and a part of the Greek world.

There were probably other ethnic groups already living in the region, who would account for references in Homer to Lelegians and other peoples living with and allied to the legendary Trojans. These peoples are not part of the history of Bronze Age Troy: there is no evidence to put these groups there in the 12th century, in spite of large quantities of Hittite textual evidence about the regional and ethnic divisions of Anatolia. In the Greek city of Troy, the main state cult was to Ilian Athena. And again, this Greek cult, dating to the time of Greek colonisation, accounts for the references to a cult of Athena in the legendary Troy: in Homer, the only cult inside the city walls that is mentioned is the shrine to Athena on the citadel (at Iliad 6.269-70 and 6.297-311), even though Athena is vehemently opposed to the legendary city.

Greek Troy continued to be inhabited for another two thousand years.

It went on to have a colourful history. Xerxes visited Troy on his way to invade Greece in 480 BCE, and made offerings to Ilian Athena as a propaganda gesture: it made it look as if he had come to avenge king Priam. When Alexander captured Troy from the Persians in 334 BCE, he too made offerings to Ilian Athena, gave Troy special legal privileges, and ordered the construction of a new temple to Athena. From 306 BCE Troy enjoyed still more status as the capital of a league of cities in the Troad.
Coin of Antiochus III, 197 BCE
The Seleucid king Antiochus III joined Xerxes and Alexander on the list of leaders who honoured Ilian Athena and Troy. So too did the Roman general Cornelius Scipio, when he overthrew Antiochus.

In the Roman era, Troy came to be more important still, as a emblem for Roman-Greek relations. There was already a long-standing legend that Romans had Trojans in their ancestry, so Troy took on great symbolic importance. There may have been one bad hiccup in 85 BCE: there’s a story (not corroborated) that a mutinying Roman commander, Fimbria, sacked the city and boasted that he had done in ten days what Agamemnon had taken ten years to do.

But afterwards Julius Caesar, as dictator of Rome, continued the tradition of honouring the city with tax breaks and other privileges. Caesar’s family claimed descent from the goddess Venus via the Trojan Aeneas, so Caesar tried to shift the emphasis of Trojan religious life away from Athena (Minerva) towards Aphrodite (Venus), and he issued coins showing Venus on one side and Aeneas’ flight from Troy on the other. To some extent this stuck: Aphrodite continued to appear on some later Trojan coins. Suetonius, a gossip-mongering biographer, claims that there was even a rumour floating around just before Caesar’s assassination that he had been planning to abscond with the city’s armies and treasury and set himself up as king of an eastern empire, with his capital in either Troy or Alexandria. That rumour is certainly untrue. But it may just be true that the rumour did exist.

Under the Principate, another new temple to Ilian Athena was built in the reign of Augustus. Many other public works followed, and Troy reached the pinnacle of its historical size and importance. In the 4th century CE, when the emperor Constantine was planning to establish a second capital city for the eastern half of the empire, he was seriously considering Troy as an alternative to Byzantium. It would never have made sense to actually choose Troy ahead of Byzantium — a tourist trap ahead of a major economic power with major strategic significance — but it shows that Troy still had huge symbolic importance.

Its importance only began to fade after around 500 CE, when it was badly damaged by a major earthquake. Increasing urbanisation around Constantinople must also have leeched people and money away from Troy. Even so, in the 10th century it became the seat of a minor Byzantine bishop. But it must have been badly hit by the Byzantine-Turkish wars in the 11th century; by the time the Ottomans finished conquering the region in 1308, it had probably been abandoned for some time.

At that point, and only at that point, does it begin to make any kind of sense to speak of Troy as ‘lost’. And even then, it was only ‘lost’ in the sense that people in the Latin west no longer had any direct knowledge of it because they didn’t travel in the region very much.
Edward Daniel Clarke, 1769-1822
Pretty much as soon as western visitors started writing memoirs of their tours in the area, the location of Troy became ‘known’ again. Five hundred years later, in 1801, the English traveller Edward Daniel Clarke became the first modern westerner to write about the site. In the 1850s, when British forces were stationed in the region during the Crimean War, an engineer named John Brunton carried out some brief excavations and uncovered a Roman mosaic, as can be read in his memoirs. As far as Brunton was concerned, there was no particular doubt or controversy about the identification of the site as Troy. Frank Calvert and Johann Georg von Hahn were the first people with archaeological expertise to visit the site, in 1863-65.

Schliemann was neither the discoverer of Troy nor the first person to identify the site as Troy (in fact he doubted Calvert's word on the matter at first). He was just the first excavator to get down to Bronze Age material, and he liked to pretend that there was an entrenched orthodoxy against him for rhetorical reasons. In reality, Troy didn’t need to be ‘discovered’: it was never lost in any meaningful sense.

Further reading. Trevor Bryce, The Trojans and their neighbours (Routledge, 2006), chapter 7 is an excellent brief summary of the history of Greco-Roman Troy (VIII, IX, and X). On the city’s cultural and political signifiance in the same age, see Andrew Erskine, Troy between Greece and Rome (Oxford, 2001).

Wednesday 4 November 2015

Were the Greeks scared of irrational numbers?

There is a widespread notion that the discovery of irrational numbers was a thing of horror to the ancient Greeks. This went especially for the school of Pythagoras. Pythagoras is best known today for a famous theorem about right-angled triangles -- and we shall look at that theorem another day -- but in antiquity, his significance lay in the fact that he was a semi-legendary guru who founded a philosophico-religious sect in southern Italy.

No writings by Pythagoras himself survive (and it is extremely unlikely he ever wrote any). But the things we hear about the sect make it sound bizarre at times: depending on who you read, the Pythagoreans conveyed their teachings only orally and only in a cave, they had weirdly specific beliefs about reincarnation, and they venerated unexpected plants like fava beans and mallow. The vast majority of this information is reported very late, and is almost certainly false; the bits that are true (whichever ones they are) are difficult to understand out of context.

The legendary Pythagorean veneration of orderly, rational numbers is well exemplified by a passage in William Meissner-Loeb’s graphic story Epicurus the sage, volume II (1991). Here the philosopher-hero Epicurus happens upon a group of Pythagoreans holding a ceremonious gathering to recite the powers of 2 (‘2 ... 4 ... 8 ...’), and Epicurus terrorises them by shouting out random numbers. They lose their concentration and flee, crying out, ‘Unclean numbers! Unclean! Unclean!’ and ‘Ahhhh! It’s happening again!’

In 1972 the mathematician Morris Kline wrote in his book Mathematical thought from ancient to modern times (vol. 1, p. 32):
Numbers to the Pythagoreans meant whole numbers only. ...Actual fractions... were employed in commerce, but such commercial uses of arithmetic were outside the pale of Greek mathematics proper. Hence the Pythagoreans were startled and disturbed by the discovery that some ratios -- for example, the ratio of the hypotenuse of an isosceles right triangle to an arm or the ratio of a diagonal to a side of a square -- cannot be expressed by whole numbers. …The discovery of incommensurable ratios is attributed to Hippasus of Metapontum (5th cent. B.C.). The Pythagoreans were supposed to have thrown Hippasus overboard for having produced an element in the universe which denied the Pythagorean doctrine that all phenomena in the universe can be reduced to whole numbers or their ratios.
Not to put too fine a point on it, but every claim in this paragraph -- apart from the bit about Greek commerce using fractions -- is untrue. In reality,
  1. Fractions were an integral (ha ha) part of Greek mathematics and held an important place in the Pythagorean theory of harmonics.
  2. There was no Pythagorean doctrine about reducing all phenomena to ratios.
  3. There is no evidence that anyone was ‘startled and disturbed’ by irrationals.
  4. The attribution of the discovery to Hippasus is speculative.
  5. No one threw Hippasus off a ship.
Kline is not alone. And worse, an apparently reputable source like Kline can mislead more popular writers. Simon Singh, in his bestseller Fermat’s last theorem (1997), goes seriously overboard -- even more so than Hippasus --
[T]he idea that rational numbers... could explain all natural phenomena... blinded Pythagoras to the existence of irrational numbers and may even have led to the execution of one of his pupils. One story claims that a young student by the name of Hippasus as idly toying with the number √2, attempting to find the equivalent fraction. Eventually he came to realise that no such fraction existed, i.e. that √2 is an irrational number. Hippasus must have been overjoyed by his discovery, but his master was not. Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. ...Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destory Hippasus’ argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning.
The father of logic and the mathematical method had resorted to force rather than admit he was wrong. Pythagoras’ denial of irrational numbers is his most disgraceful act and perhaps the greatest tragedy of Greek mathematics.
(For the record: we know nothing of the circumstances of the discovery, there was no execution, and Hippasus lived in the late 5th century BCE, more than a century after Pythagoras’ death.)

Singh paints Hippasus’ discovery in vivid colours. Does that make up for the fact that it is not only imaginative, but also completely imaginary? Hm.

I do not exactly blame Singh. Half of the relevant primary sources have never been translated into any modern language. But it does go to show how a story that is already distorted can metamorphose into something completely fictional.

So, what does the actual evidence tell us? The surviving testimony is as follows, in chronological order.
  • Late 2nd century CE: Clement of Alexandria, Stromateis 5.9.57. Clement reports that a Pythagorean named "Hipparchus" revealed the teachings of Pythagoras in a book. As a symbol of his expulsion from the sect, the Pythagoreans erected a gravestone as if he were dead.
  • 3rd-4th century CE: Iamblichus, Life of Pythagoras tells us:
    • 88-9 (§18): Hippasus, a Pythagorean, revealed the discovery that the vertices of a regular dodecahedron coincide with the surface of a sphere, and because of his impiety he was lost at sea;
    • 246 (§34): a man who made public the nature of rational and irrational ratios was so hated by the Pythagoreans that they expelled him and erected a tomb as if he were dead;
    • 247 (§34): a man who revealed the construction of the dodecahedron drowned at sea, punished by a divinity; others say that this happened to the man who revealed the nature of rational and irrational ratios.
  • Early 4th century CE: Pappos’ commentary on Euclid’s Elements, book 10 (in the surviving Arabic version, 2.§2, p. 64 Thomson [warning: large PDF file]), and an anonymous ancient commentator on the Elements (scholion on book 10, proposition 1; lines 41-5 and 71-9 in the TLG text). According to these sources the Pythagoreans, to illustrate their reverence for ratios, spread a fable that the man who made public the existence of irrationals died by drowning. And the moral of this fable was that things that are irrational (alogon) prefer to be kept hidden and unspoken (alogon); and that someone who is too greedy for knowledge "gets sunk in the sea of reincarnation, and dashed by its chaotic currents" (εἰς τὸν τῆς γενέσεως ὑποφέρεται πόντον καὶ τοῖς ἀστάτοις ταύτης κλύζεται ῥεύμασιν).
  • Later than the 6th century CE: an interpolation in David of Armenia’s Exegesis of the Categories (Commentaria in Aristotelem Graeca vol. 18.1 [incorrectly attributed to Elias], p. 125). This interpolation, of unknown date, reports that a Pythagorean who wrote a book called On irrational proofs died in a shipwreck for disgracing secret teachings.
And the upshot of this testimony is:
  1. Hippasus did not discover irrationals: he made secret Pythagorean doctrines public.
  2. The nature of these doctrines is unclear. It may have been the nature of rational and irrational numbers; it may have been the existence of the dodecahedron, or the fact that its vertices coincide with a sphere.
  3. He was not executed or thrown off a ship: he died in a shipwreck, and some moralists attributed this to divine agency and made an allegorical fable out of it.
  4. Alternatively, his former comrades built a tomb for him, to represent that he was dead to them.
But the worst of it is that even this honest summary is probably completely untrue as well. The fullest account comes from Iamblichus, and Iamblichus is notoriously untrustworthy. Pappos makes it clear that as far as he was concerned, it was a morality fable, not a sequence of historical events. Most of the late biographical material about Pythagoras is based on one or both of two accounts written in the 1st century CE, six centuries after Pythagoras’ death: one by Nicomachus of Gerasa, the other by Apollonius of Tyana. To judge from Iamblichus, Nicomachus routinely attributed miracles to Pythagoras, and -- no joke -- regarded him as an avatar of the god Apollo. Apollonius came to be regarded as a miracle-worker himself, in a surviving ‘biography’ which dates to the 4th century. Their biographies, and the surviving one by Iamblichus, are more like gospels for a Pythagorean mystic cult than anything historical.

None of them can be trusted an inch.

Trustworthy testimony about Pythagoras and the Pythagoreans is in short supply. Generally speaking, the earlier, the better: and Iamblichus and the others are very late. We get hints about Pythagorean doctrines in Herodotus (5th cenutry BCE), Plato, and Aristotle (4th century BCE). But most of that material relates to the mystical side of Pythagoreanism: in particular, the early sources have nothing to say about Pythagorean teachings about irrational numbers. So we have essentially no corroboration for anything that Iamblichus and other late sources have to tell us. It is all suspect, and it is mostly false.

For what can be recovered about 5th-century-BCE Pythagorean teachings about mathematics, a good starting place would be Reviel Netz’ essay ‘The problem of Pythagorean mathematics’ (C. A. Huffman, ed., A history of Pythagoreanism, Cambridge, 2014, pp. 167-84): Netz argues that the Pythagorean mathematician par excellence of the time was not Hippasus, for whom no early evidence exists, but rather Archytas, about whom Aristotle tells us a good deal.

Did I say Plato has nothing to say about irrational numbers? Well, not in relation to Pythagoreanism, maybe. But one of Plato’s dialogues does have a section devoted to a discovery made by Theaetetus of Athens, that numbers other than exact squares (1, 4, 9, 16, 25...) have irrational square roots (Theaet. 147d-148b). Theaetetus was no slouch: much of book 10 of the Elements may well be his doing. Theaetetus divides the integers into two groups: exact squares, which he called ‘square and equilateral’ (τετράγωνόν τε καὶ ἰσόπλευρον), and numbers that are not squares but are ‘rectangular’ (ἑτερόμηκες). He calls their square roots, respectively, a ‘length’ (μῆκος) and a ‘power’ (δύναμις); and ‘lengths’ and ‘powers’ are incommensurable with one another. ‘And similarly for solids,’ he finishes on a tantalising note.

In the dialogue, what is Socrates’ reaction to the revelation of irrational numbers? Is he horrified? disoriented? ‘startled and disturbed’?

No. He is impressed at a nifty mathematical discovery.

As we all should be. Irrational numbers were not a skeleton in the Pythagoreans' closet: if the Pythagoreans had anything to do with their discovery -- and that’s a big if -- they should instead be regarded as one of the Pythagoreans’ greatest achievements. But in reality, it’s most likely that credit for the achievement belongs to Theaetetus: and he was not executed or ostracised, but was highly respected for his mathematical work.