Pi Day

I'm a bit late for Pi Day, but hey, research takes time. It has become trendy to celebrate the 14th of March as 'Pi Day', because in American notation the date looks like '3/14'. These are the first three digits of the mathematical constant π, or pi, the ratio between a circle's circumference and its diameter. (Assuming we're talking about Euclidean geometry. Which we are.)

Outside the US, some people like to celebrate the 22nd of July instead -- because in everyone else's notation, that looks like '22/7', and 22/7 is a very good approximation for π.

A few incidental bits of trivia about π:

• π is an irrational number. This means it cannot be expressed as a ratio of two integers. Put another way: the circumference and diameter of a circle are incommensurable. Or put yet another way: if you write out π in decimal notation, it will never ever repeat.
• π is also a transcendental number. This means that it cannot be expressed as the solution to a polynomial equation with integer coefficients. That is: given an equation axⁿ + bx^n-1 + cx^n-2 ... + zx⁰= 0, where a, b, c ... are rational, a transcendental number is any number that x cannot be.
• π is widely suspected to be a normal number. This is not known for sure. A normal number is, roughly, one whose decimal expansion shows no patterns, where every digit is equally likely, and every finite sequence of digits is equally likely. This sounds pretty limiting; at present no one really has any idea how to prove that a given number is normal with 100% certainty. But if you look at it statistically, almost all real numbers are irrational; almost all irrational numbers are transcendental; and almost all transcendental numbers are normal. If you randomly pick a number on the real number line, the probability that it will be normal is 1. So, pretty good odds that π is normal, then.
• If you know π to 39 decimal places -- 3.14159 26535 89793 23846 26433 83279 50288 4197 -- then you know it precisely enough to measure a circle the size of the observable universe to a precision finer than the width of an atom.

So much for interesting trivia of the day. What about myths? Give me modern myths about antiquity!

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OK, here's one myth.

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes.

-- Wikipedia, 'Pi'

It is true that Archimedes used this method to calculate π. But it is not true that he devised the method. He just did it with a bit more precision than anyone had done previously. He made an advance, but it was an incremental advance, not something revolutionary. You can find Archimedes' full exposition in a surviving work, the Measurement of the circle.

The 'exhaustion method'. If you draw regular polygons inside and outside the circle, then the more sides the polygons have, the more closely they approximate the actual circumference of the circle. (source: Wikimedia.org)

The illustration shows how the exhaustion method works. Using 96-sided polygons, Archimedes narrowed down the value of π to between 3 10/71 and 3 10/70 -- that is, he found that π is somewhere between 3.1408... and 3.1429...

But the method was already in use 200 years earlier. Antiphon of Athens (ca. 480-411 BCE), Bryson of Heraclea Pontica (ca. 400 to after 340 BCE), and Eudoxus of Cnidus (ca. 391-338 BCE) had all used a similar method to calculate π long before Archimedes came along.

Antiphon, the earliest of the bunch, only used inscribed polygons -- that is, he only drew one shape, inside the circle, but not outside. As a result he only had one bound for the value of π. We don't know much about Eudoxus' effort. We do know that Bryson guessed (wrongly) that π would be given by the arithmetic mean of the inner and outer perimeters; and that Antiphon and Bryson were working on the area of the circle, not its perimeter. It was Eudoxus who showed that the area and perimeter were linked by the square of the radius.

The New Pauly encyclopaedia reports (subscription needed) that it was Eudoxus, not Archimedes, whose influence led to the widespread use of exhaustion for all problems involving infinitesimals. Archimedes' work on π was just a refinement of Eudoxus.

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Here's another myth, from a Time article published on 'Pi Day' this year.

However, not too many generations after [Archimedes'] lifetime, the world experienced a "real decline in math," according to John Conway, mathematics professor emeritus at Princeton University who once won the school's Pi Day pie-eating contest. "Math and science in general went into a great decline from roughly the year zero to the year 1,000, and then the Arabs developed lots of math after that, like trigonometry."

Oooh, do I detect a note of a renowned world expert saying something a little bit silly about another field? I think I do!

What, no love for all those Alexandrian mathematicians of the Roman era? No love for Heron, whose Metrica has recently been published in a new French translation? Or Menelaus, whose work on spherical geometry was foundational for Arabic, Hebrew, and western astronomers for over a thousand years? Not to mention Diophantus, whose work laid down the parameters for the modern study of polynomials, and whose notation foreshadowed the development of algebra?

And then there are many other figures who are, admittedly, lesser, but still made important contributions: Sporus of Poros, who demolished earlier mathematicians' reliance on a curve called the 'quadratrix' in problems to do with squaring the circle; Ptolemy, who in the early 100s CE gained the world record for closest approximation of π (3 + 8/60 + 30/3600, = 3.141666...); and commentators like Pappus, Theon, Hypatia, Proclus, and Eutocius, whose work on Euclid, Ptolemy, and Archimedes were colossally useful in helping later mathematicians to understand the impenetrable language of their predecessors.

I guess it is fair to speak of a decline in Greek mathematics -- but Archimedes was not the be-all and end-all. If there was a decline, it was after the time of Diophantus. Archimedes has a curiously inflated reputation. I suppose that's because there are lots of good stories about him: the story of his death ray; the dramatic story of his death that we find in Plutarch and Valerius Maximus; the story of the bathtub and the running around naked shouting 'eurēka!'; and the story of the Cattle Problem, whose solution involves a number with over 200,000 digits (ca. 7.76 × 10²⁰⁶⁵⁴⁴). Everything about him sounds tremendously exciting. But hey, let's not forget later giants like Hipparchus, Menelaus, and Diophantus, all right?

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.

Hieroglyph
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°.)

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.