The Stoics and the Holy Spirit

Stoicism, the ancient school of philosophy, had some excellent ideas. Here’s their explanation of how sound works.

We hear when the air between the sound-emitter and the listener is struck. Then it emits a wave in a spherical shape which spreads and arrives at the ears, in the same way that water in a tank forms waves in circles when a stone is thrown in.

Diogenes Laertios 7.158

A Stoic philosopher watches the sunset with a vision cone made of pneuma. (AI generated)

This is terrific. It’s practically what you’ll hear in a modern classroom. But some Stoic ideas were ... well, less excellent. Here’s their theory of vision.

We see when the light between the viewer and the object is stretched tight, in a conical shape. ... The tip of the conical shape is located at the eye, and its base in the direction of the thing seen. Information about the observed object is conveyed by the stretched air, similarly to a cane.

Diogenes Laertios 7.158

That is, they thought there are vision cones extending out of our eyes. Modern readers sometimes call them ‘vision rays’. Ancient theories of vision came in two main varieties, ‘extramissive’ and ‘intramissive’: stuff either comes out of your eye or enters it. The Stoics were firmly in the extramissive camp. As you move your eye around, it moves the vision cone, like a hand moving a cane. You sense the other end of the cone the same way that you can sense things at the end of the cane.

The Stoics thought of the vision cone as physical but intangible, like air, or light. It was made of a substance that they called pneuma: literally, ‘breath’.

In philosophical contexts, the custom is to translate pneuma as ‘spirit’.

Pneuma and logos

Among the concepts that the Stoics used to talk about how the universe works are two influential words that don’t translate straightforwardly: pneuma, πνεῦμα, literally ‘breath’; and logos, λόγος, literally ‘word’ or ‘utterance’.

Pneuma is the physical medium for interactions that aren’t tangible. Light, vision cones, life force, and the soul are all made of pneuma. Pneuma is a material that’s perceptible, and physically real, but incorporeal.

The word has a heritage in Greek mystic and natural philosophy. In Orphic religion, pneuma refers to a physical manifestation of fate. Here’s a schema in the Orphic Derveni treatise (5th century BCE), as tabulated by Gábor Betegh (2004: 202):

Mythological name	Zeus	Moira [‘Fate’]
Intellectual aspect	mind (nous)	wisdom (phronesis)
Physical aspect	air (aer)	breath (pneuma)

In later centuries the Stoics adopted similar terminologies, while dropping the ‘mythological’ names. So did Philo of Alexandria, a 1st century Jewish Bible interpreter who was heavily influenced by the Stoics. Nous becomes logos, phronesis becomes sophia, aer gets sidetracked into the theory of the elements, pneuma holds on to its place.

The traditional English translation ‘spirit’ comes from the fact that the Latin for ‘breath’ is spiritus. For Stoic thought, a better English translation would be ‘force’ — force in the modern sense of an energy field, like magnetism, or classical gravity. Here’s a precisely accurate description of what the Stoics meant by pneuma:

It’s an energy field created by all living things. It surrounds us and penetrates us; it binds the galaxy together.

Obi-Wan Kenobi, Star wars (1977)

χρῶ τῷ πνεύματι, ὦ Λουκᾶ. (Star wars, 1977)

The ancients didn’t think of gravity and magnetism in terms of ‘forces’. Aristotle, for example, imagined gravity as an intrinsic property, not an interaction: objects with the ‘heaviness’ property have a natural motion towards the centre of the cosmos.

But if you could go back in time and teach Zeno or Chrysippos about classical gravitational or magnetic fields, they’d definitely treat them as species of pneuma.

Then there’s logos. To the Stoics, logos was the principle that the world is rational and behaves in an intelligible way. Logic and maths work, sex causes pregnancy, stuff falls downwards, the sun moves around the ecliptic. That’s logos. It’s their word for what the Orphics and Plato called the cosmic nous, ‘mind’: a template for the self-consistency of the physical world.

In the Orphic Derveni treatise, ‘mind’ is a primordial creative force. Elementary matter is made to coalesce into tangible objects by the ‘colliding mind’: in Greek, krou- + nous, a retcon for ‘Kronos’. Zeus absorbs the primordial forces — ‘colliding mind’, time, the forces of life and creation — so that they become his personal attributes. As a result he acquires the persona of Protogonos, ‘first born’.

In Philo, logos is a personal attribute of God, just as Orphic nous is an attribute of the Protogonos.

And if it happens that someone is not yet worthy of being called ‘son of God’, let them hasten to adorn themselves with his protogonos logos, the eldest of angels, an archangel as it were. It possesses many names. For it is called ‘beginning’, and ‘name of God’, and logos, and ‘human in image’, and ‘the one that sees Israel’. ... for logos, the eldest, is an image of God.

Philo, On the confusion of tongues 146–147 (my translation)

The piling-up of alternate names is in much the same vein as the Derveni treatise.

Visualisation of Stoic logos. (AI generated)

Modern interpretations of logos in the Christian New Testament regularly take Philo as their starting point — most notably in John 1.1, where logos is a cosmic primordial force that shapes the tangible world.

In the beginning there was the logos, and the logos was an attribute of God, and the logos was God. This was God’s attribute in the beginning. All things came into existence through it, and nothing that exists came into existence without it.

John 1.1–3 (my translation)

Imagine a translation using Philo’s analogies: ‘In the beginning was divine reason’; ‘In the beginning was the noetic realm of incorporeal Forms’. The text could easily be read that way in the 1st century. It echoes earlier Greek thought too: ‘nothing that exists came into existence without it’ sounds almost like it’s from Parmenides.

But does early Christian thought have more in common with Philo, or with the Stoics? Modern scholars normally opt for Philo. I agree that holds true for Paul’s letters. But in the narrative books of the New Testament, pneuma isn’t just an attribute of God. It pops up in other ways too. There, we’re looking at an adaptation of Stoic language.

New Testament pneuma isn’t supernatural: it’s precisely non-supernatural. It’s an attempt to rationalise the supernatural in naturalistic terms. In the present day, believers sometimes try to rationalise a miraculous healing as a logical medical event; in antiquity, pneuma is exactly the same.

Note. On logos in Philo see Hannah 1999: 77–85; Robertson 2018: 9–28. On reading logos in the New Testament in light of Philo, see Boyarin 2017 (but cast as exclusively Jewish, with no mention of Stoicism). On nous in Plato see especially the Timaios, with discussion by e.g. Menn 1995; Mason 2013. On krou- + nous as Orphic creative force, see Derveni papyrus cols. xiv–xvi (cf. Plato Kratylos 396b), with discussion by Betegh 2004: 185–193.

Pneuma in the Bible

The Hebrew Bible often refers to the rūaḥ or ‘wind’ of God. From around the 3rd century BCE onwards, the Bible used in the Diaspora was in Greek: the Septuagint. And the Septuagint instead talks about God’s pneuma.

כי־כל־עוֹד נשמתי בי ורוח אלוֹה באפי׃

ἦ μὴν ἔτι τῆς πνοῆς μου ἐνούσης πνεῦμα δὲ θεῖον τὸ περιόν μοι ἐν ῥισίν.

... as long as my breath [Hb. nishmā, Gr. pnoë] is in me and the spirit [Hb. rūaḥ, Gr. pneuma] of God is in my nostrils, ...

Job 27.3 (MT, LXX, NRSVue)

Note. πνοή and πνεῦμα could both be translated as ‘breath’, given the right context. Philo, Allegory of the holy laws 1.42, discusses the distinction between them in Genesis 1.2 and 2.7.

So when Hellenistic Jews talked about the ‘wind of God’ in Greek, they were using language that already had a naturalistic sense. The dual senses of pneuma, as the rūaḥ of God and as a Stoic physical ‘force’, could be conflated or distinguished as needed.

Visible pneuma and extramissive vision: ‘there stabbed northward a flame of red, the flicker of a piercing Eye ...’ (Tolkien, The Lord of the Rings, 1955; still from The Return of the King, 2003)

Pneuma appears frequently in the New Testament. The earliest gospel, Mark, envisages it as a physical substance. You can immerse people in pneuma.

ὁ Ἰωάννης ... ἐκήρυσσεν ... ἐγὼ ἐβάπτισα ὑμᾶς ἐν ὕδατι, αὐτὸς δὲ βαπτίσει ὑμᾶς ἐν πνεύματι ἁγίῳ.

John [the Baptist] ... proclaimed ..., ‘I baptised you with water, but he will baptise you with holy pneuma.’

Mark 1.8 (my translation)

Notice that the Greek doesn’t refer to ‘the’ Holy Spirit — there’s no article. This is an uncountable noun, a substance, not a specified entity.

When Jesus casts out demons, he doesn’t do it by praying: he uses pneuma as a physical instrument.

‘If I cast out demons with Beelzebul, what do your own sons use to cast them out? They will be your judges, then. But if it is with God’s pneuma that I cast out demons, then the kingdom of God is already upon you.’

Matthew 12.27–28 (my translation)

The original version of this passage, Mark 3.22–25, doesn’t mention pneuma. The introduction of pneuma makes the statement more physical, more tangible. Luke 11.15–20 makes the imagery more tactile still, writing ‘God’s finger’ instead of ‘God’s pneuma’.

All modern Bibles beg the question in passages like these by personifying pneuma as ‘the (Holy) Spirit’. That choice anticipates the doctrine of the Trinity. It excludes the meanings that the word had when these passages were written.

Occasionally, pneuma is paired with logos. (And notice that pneuma here is emphatically not an attribute of God!)

Once it became evening they brought to [Jesus] many people who were afflicted by demons, and he cast out the pneumata with logos [ἐξέβαλεν τὰ πνεύματα λόγῳ].

Matthew 8.16 (my translation)

The choice of words makes the subtext extremely clear: we’re firmly in Stoic territory. Again, you have to think about what these words meant at the time. A well informed 1st century reader might very well read it like this:

Jesus cast out the intangible physical phenomena by using a noetic template.

You can’t translate this kind of passage neutrally. Because of course these terms aren’t just Stoic terms: pneuma also means rūaḥ, logos also means ‘word’. If you look at Paul’s letters, it’s much harder to see any hint of Stoicism. There, ‘spirit’ usually is going to be the right translation.

Pneuma in Acts

Inspecting every mention of pneuma in the New Testament would be overkill for a short essay like this, but I have checked every occurrence in Acts. Out of the narrative books, Acts mentions pneuma the most — 35 times per 10,000 words, as compared with 10 to 19 in the gospels. (Among Paul’s letters, Galatians and 1 Corinthians both use it over 80 times per 10,000 words.)

Here’s how Acts uses the word pneuma:

• 24× (in 19 distinct episodes) physically, as something visible, a fluid substance (normally without a definite article), or a body part: 1.2, 1.5, 2.2–4, 4.8, 4.25, 6.3–5, 6.10, 7.55, 8.15–19, 9.17, 10.38, 11.16, 11.24, 11.28, 13.9, 13.52, 17.16, 18.25, 20.22.
• 12× (in 11 episodes) as an entity with personhood which speaks: 1.16, 5.32, 8.29, 9.31, 10.19, 11.12, 13.2–4, 15.28, 20.28, 21.11, 28.25.
• 11× ambiguous, with definite article (perhaps as an attribute of God): 1.8, 2.33, 2.38, 4.31, 5.3 (ambiguous syntax), 5.9, 10.44–47, 11.15, 15.8, 16.6, 19.6.
• 9× (in 4 episodes) as a supernatural entity other than God: 5.16, 16.16–18, 19.11–16, 23.8–9.
• 2× in a quotation from the Septuagint: 2.17–18 (≈ Joel 2.28–29).

Pneuma often has the epithet ‘holy’ (ἅγιον πνεῦμα). But even in a text as late as Acts it’s hard to be confident that that isn’t simply meant to be an antithesis to pneumata that are ‘wicked’ or ‘unclean’ (Acts 5.16, 19.11–16). Two passages juxtapose pneuma with logos, both in the third ‘ambiguous’ category (4.31, 16.6).

Some of the passages in the first category — pneuma as a physical substance — clearly envisage it as a fluid: you can be immersed in it (1.5, 11.16), anointed with it (10.38), or filled with it (2.4, 4.8, 6.3–5, 7.55, 9.17, 11.24, 13.9, 13.52), and it can boil (18.25 ζέων τῷ πνεύματι). It can be transmitted by physical contact (8.15–19, 19.6), and it can act as a medium for action at a distance when someone speaks ‘by means of pneuma’ (1.2, 4.25, 11.28).

Modern English-language Bibles can sometimes bring themselves to translate Hebrew rūaḥ as the ‘wind’ of God, instead of ‘Spirit’ with a capital S. And that’s great. It’s reasonable to make an effort with Greek pneuma and logos too!

References

• Betegh, G. 2004. The Derveni papyrus. Cosmology, theology and interpretation. Cambridge.
• Boyarin, D. 2017 [2011]. ‘Logos, a Jewish word. John’s prologue as Midrash.’ In: Levine, A.-J.; Brettler, M. Z. (eds.) The Jewish annotated New Testament, 2nd edition. Oxford. 688–691.
• Hannah, D. D. 1999. Michael and Christ: Michael traditions and angel christology in early Christianity. Tübingen.
• Mason, A. 2013. ‘The nous doctrine in Plato’s thought.’ Apeiron 46: 201–228. [DOI]
• Menn, S. 1995. Plato on God as nous. Carbondale/Edwardsville (IL).
• Robertson, D. 2018. Word and meaning in ancient Alexandria. Theories of language from Philo to Plotinus. Aldershot/Burlington (VT).

Aristarchus and the heliocentric theory

The earth has orbited around the sun since 1609. At least that’s when Kepler’s book on the subject came out, Astronomia nova (‘The new astronomy’). Copernicus had proposed a heliocentric theory in 1543, but with circular orbits it was a lousy model. The geocentric Ptolemaic system continued to be the better model of planetary motion until Kepler came along.

But there was another precedent. Sometime around 280 BCE, in ancient Greece, Aristarchus of Samos proposed a heliocentric model. What exactly did Aristarchus argue? How did he arrive at his theory, what did people think of it, and why did it end up being neglected?

Contemplating the moon (AI generated)

The last question is the simplest: Aristarchus’ theory was neglected because his writings on the subject were lost. Also, other ancient astronomers found that geocentrism, with epicycles, produced a superior model of planetary motion — and they were right. Even though the reason they were right had nothing to do with the planets’ real motion, and everything to do with a form of mathematical analysis that wasn’t fully developed until the 1800s.

Aristarchus of Samos

We don’t know much about Aristarchus’ life. He was born on the island of Samos, probably in the 310s BCE, a decade or two after Alexander’s death. What we know of his dates comes from just three facts:

• We’re told he studied under Straton of Lampsakos, who was the head of the Peripatos in Athens from 287 until 269 BCE.
• Aristarchus observed the summer solstice in 280 BCE.
• His heliocentric model was discussed by Archimedes in the 240s or 230s BCE.

So we know he spent a period in Athens at some point, but nothing else about his movements. We know he developed a heliocentric theory; he measured the sizes and distances of the moon and sun; discovered some trigonometric inequalities; and invented two instruments, something called the ‘disc on a level surface’, and the skaphe, a bowl with a fixed needle and gauge markings for measuring the sun’s position.

Map showing Samos (base image: Google Earth)

Much more precise equipment for measuring the sun’s position was developed pretty soon afterwards. But Aristarchus’ skaphe was straightforward enough to stick around: three centuries later, Pliny reports a bunch of skaphe readings of the sun’s altitude, varying depending on how far north you are.

Note. Pliny, Natural history 2.74, with a description of the skaphe. Aristarchus inventing the skaphe and the discum in planitia: Vitruvius 9.8.1. More precise devices for measuring the sun’s altitude are described by Ptolemy, Almagest 1.12. One of them, a device consisting of two concentric vertical rings, was in use in Meroë, Sudan, by the 2nd century BCE, and is probably also the device used by Eratosthenes: see here for details.

Only one book by Aristarchus survives: On the sizes and distances of sun and moon. It isn’t widely read. In it he measures the moon as being considerably smaller than the earth, and the sun as much bigger. This is also where we find him inventing some bits of trigonometry from first principles.

Note. Edition of Aristarchus’ On the sizes: Heath 1913: 317–414; text and translation at 352–411. Reader beware: Heath’s introduction is seriously marred by relying on Hultsch’s botched reckoning of two ancient distance units, the Egyptian schoinos and Greek stadion. See here. For a more recent and accurate discussion, see Berggren and Sidoli 2007.

The heliocentric theory and On the sizes

The heliocentric theory isn’t mentioned in Aristarchus’ surviving book. We have to rely on other ancient reports — and they aren’t generous with details.

Our main source is Archimedes. He brings up the heliocentric theory in a mathematical exercise, about devising a numerical notation capable of representing very large quantities.

Aristarchus of Samos, however, published writings of certain propositions, where it appears from the premises that the cosmos is many times larger than the standard [i.e. geocentric] cosmos. His suggestion is that the fixed stars and sun remain motionless, and the earth orbits around the sun in a circle, the sun at the centre of its path; and the sphere of fixed stars lies around the sun, with the sun at its centre. And its size [i.e. the sphere of fixed stars] is such that the circle of the earth's orbit has the same proportion to the distance of the fixed stars, as the centre of the sphere has to its surface.

This is obviously impossible: the centre of the sphere has no size, so it has to be understood as having no ratio to the sphere’s surface. So we take Aristarchus’ meaning to be: we suppose that the earth [in the geocentric model] is analogous to the centre of the cosmos [in the heliocentric model]; therefore, the earth’s ratio to the cosmos as we imagine it [i.e. geocentric] is the same as the ratio of the sphere on which the circle of the earth's orbit is inscribed to the sphere of fixed stars [in the heliocentric model].

Archimedes, Sand-reckoner 4–6 (ii.218 Heiberg)

This is obscurely phrased. Essentially, the second paragraph is saying that a lower bound for the size of a heliocentric cosmos has to be vastly larger than that of a geocentric cosmos.

Archimedes doesn’t say why. Presumably because of the parallax problem: as the earth moves around the sun, the fixed stars ought to shift their parallax in a yearly cycle. But they don’t. Therefore, either the earth doesn’t go around the sun, or the fixed stars are enormously more distant than the geocentric model would require.

(Archimedes goes on to work out how many grains of sand it would take to fill a very large cosmos. He calculates a lower bound for the universe’s diameter of a little under 2 light years — or rather, 100 trillion stadia — with room for 10⁶³ grains of sand.)

Still a bit of counting to do (AI generated)

This tells us: (1) Aristarchus proposed a heliocentric model; (2) he appreciated that the distance to the fixed stars has to be treated as effectively infinite. But it doesn’t tell us why Aristarchus thought this was better than the conventional geocentric model.

Two other passages in Plutarch are worth noting, dating to the 2nd century CE. One states that there were only two notable heliocentrists, Aristarchus and Seleucus; and that Aristarchus’ heliocentric model was only a proposal, while Seleucus regarded it as evidently true. The other passage tells a story of Aristarchus having a clash with a Stoic philosopher, Cleanthes. Several other sources discuss whether the earth is in motion rotating on its axis; the most prestigious figures, like Aristotle, Hipparchus, and Ptolemy, conclude it’s the sky that rotates.

Note. Plutarch, Platonic questions 1006c; Plutarch, On the face in the circle of the moon 922f–923a. On the question of Seleucus’ exact contribution, and the meaning of Plutarch’s word ἀποφαινόμενος, see Neugebauer 1975.ii: 697–698. On the earth’s rotation, see Aristotle, On the sky 296a–b (against); Heracleides of Pontus frs. 104–117 Wehrli (in favour); Seneca, Natural questions 7.2.3 (agnostic).

These still don’t tell us why Aristarchus favoured the heliocentric theory. But Aristarchus’ surviving work, On the sizes and distances of sun and moon, gives a pretty broad hint.

Aristarchus calculated the moon to have a diameter equivalent to about one third of an earth diameter, and the sun, about seven earth diameters. He based this on observations of the size of the earth’s shadow on the moon during lunar eclipses, and the angle between the sun and moon when the moon is half full.

His final figures are way off, because his observational tools ... well, sucked. He also wrongly assumed the moon subtends an angle of 2° as seen from earth, when it’s actually 0.5°. (And no, this isn’t a result of a typographical ambiguity.) Also, he couldn’t use trig functions on a modern calculator to convert angle measurements to distances, so he had to discover his trigonometrical inequalities to obtain lower and upper bounds.

An excerpt of Aristarchus’ On the sizes (Heath 1913: 364–365). At the bottom is where he mistakes the angular size of the moon: ‘And since it is assumed that the moon subtends a 15th of a zodiacal sign ...’ (where each zodiacal sign occupies a 12th of a circle, or 30°).

In other respects, his calculations are good. And he got one essential point right: the sun is much bigger than the earth. Sure, his figure for the sun’s size is missing a couple of zeroes. But it may still have been enough for him to infer that the cosmos ought to be imagined as centred on the colossal sun, not the puny earth.

For reference, here are his results, along with the actual figures as determined by modern astronomy.

	Aristarchus	Actual	Actual (km)
lunar diameter	0.3167 to 0.3981 earth diameters	0.2727 earth diameters	3475 km
lunar distance	22.50 to 30.00 lunar diameters	110.6 lunar diameters	384,400 km
solar diameter	6.333 to 7.167 earth diameters	109.2 earth diameters	1,392,000 km
solar distance	18.00 to 20.00 lunar distances	389.2 lunar distances	149,600,000 km

Note. Numbers are given to 4 s.f. The ‘actual’ columns show averages. For Aristarchus’ numbers see Heath 1913: 338. We don’t know what Aristarchus reckoned for the earth’s diameter: he may perhaps have known the 300,000 stadia estimate for the circumference that Archimedes mentions.

On the point of comparative sizes, he was obviously right, and everyone knew it. That may have been enough to push him towards the heliocentric theory. And if he realised that the theory could also explain the retrograde motion of the planets — well, that may have been a nice bonus.

What people thought of Aristarchus’ theory

People didn’t really take to heliocentrism. As I mentioned, we know of only one other ancient heliocentrist by name, Seleucus (mid-2nd century BCE).

Note. Seleucus: from the Erythraean Sea according to Strabo 3.5.9, Diels DG 328.5; from Seleuceia according to Strabo 16.1.6. He argued that Ocean tides are related to the motion of the moon (Strabo 1.1.9, 3.5.9); like Heracleides and Aristarchus he argued that the universe is infinite (Diels DG 328.5; a 10th century report quoted and translated by Pines 1963: 197).

Aristotle offered three separate objections to the idea that the earth orbits the sun — and he did so several decades before Aristarchus came along (On the sky 296a–296b).

1. If the earth were in motion, either that motion must be the result of a force acting on the earth, in which case it’s non-natural and temporary; or it must be a natural motion shared by all parts of the earth, in which case objects should hover relative to the earth’s surface, but they don’t.
2. If the earth orbited around something else we would observe stellar parallax, but we don’t.
3. Weight falls to the ground: that is, it has a natural motion towards the centre of the earth. A natural phenomenon must be universal. Therefore this must actually be motion towards the centre of the cosmos. The fact that the earth’s centre is also in the same place is simply a result of the earth itself gravitating towards the centre.

Points 1 and 3 come from the observed fact that the earth is spherical. Point 2 is more specifically astronomical. We don’t know how Aristarchus would have responded to points 1 and 3; his solution to point 2 was to posit that the universe is infinite, or effectively infinite. Tycho Brahe too, in the 1600s, thought point 2 was heliocentrism’s weak point. Ptolemy thought it was point 3 (Almagest 1.7 = 24-25 Heiberg).

It’s kind of amazing how Aristotle’s points are completely wrong — but to see that they’re wrong, you need another two thousand years of science. Aristotle’s first and third objections weren’t resolved until the publication of Newton’s laws in 1687. Stellar parallax wasn’t measured until the 1830s. He was wrong ... but what a way to be wrong!

There’s no reason to imagine anyone ever thought the heliocentric theory violated any taboos. Writers like Archimedes and Ptolemy were happy to take it seriously and consider its implications, even if they disagreed with it.

The main evidence of active opposition to the heliocentric theory relates to the Stoic philosopher Cleanthes. We know Cleanthes wrote a tract called Against Aristarchus (Diogenes Laertius 7.174). Plutarch has an anecdote of Cleanthes saying that Aristarchus ought to be charged with ‘impiety’ (asebeia): in context it’s clear that that’s just a hyperbolic joke. But his opposition to the theory was genuine.

(Pharnaces said,) ‘You won’t induce me to give an account of what you’re accusing the Stoics of, until I get an account from you for turning the cosmos upside down!’

Then Lucius laughed and said, ‘Just don’t bring a charge of impiety against us! — like when Cleanthes thought the Greeks should accuse Aristarchus of Samos of impiety, because he was disturbing the foundation of the cosmos, trying to preserve observations by suggesting that the sky is immobile, and the earth orbits along the ecliptic, and rotates on its own axis.’

Plutarch, On the face in the circle of the moon 922f–923a

Note. Plutarch’s phrasing ‘preserve observations’ (φαινόμενα σῴζειν) is standard; it also appears in Simplicius’ account of Heracleides’ theory that the earth rotates (519,9–11 ed. Heiberg = Heracleides fr. 108 Wehrli). Russo and Medaglia 1996 prefer the manuscript reading Ἀρίσταρχος ... Κλεάνθη to the usual emendation Ἀρίσταρχον ... Κλεάνθης, so that Aristarchus accuses Cleanthes of impiety rather than the other way round. However, the person in the accusative case here (a) is the one being accused, (b) is from Samos, and (c) is ‘disturbing the foundation of the cosmos’. These are very easily ascribed to Aristarchus; they cannot possibly be ascribed to Cleanthes. Cleanthes (a) wrote a treatise called Against Aristarchus, (b) was from Assos, and (c) was no astronomer. Russo and Medaglia have to make multiple other emendations to get the sentence to make sense, and their emendations rely on speculations about Cleanthes’ teachings. The usual emendation is far more robust.

Cleanthes — a boxer in his youth — ready to throw hands over the heliocentric model (AI generated)

Aristarchus’ theory wasn’t suppressed, it was just abandoned. The weight of opinion was against it. There was Cleanthes’ treatise; and actual astronomers had a more effective model to work with. Heath suggests that it was Hipparchus’ opposition, in the 2nd century BCE, that ‘sealed the fate of the heliocentric hypothesis’ (1913: 308). Certainly the extant Hipparchan-Ptolemaic system, with its eccentric orbits and epicycles, is better at modelling the motion of the planets.

People often scoff at the idea of Ptolemaic epicycles, but they’re missing the point. Epicycles are incredibly effective because each one is a term in a Fourier series.

A simplified form of the Ptolemaic model, ignoring eccentricity (base image: Youtube)

The principle of Fourier analysis is that any periodic function can be modelled as a composite of simple harmonic motions. Each term in a Fourier series represents a circular motion of a given magnitude and frequency. The more terms, the more accurate the model.

So even though Fourier didn’t formalise the idea until 1822, the Ptolemaic system uses the same principle. Ptolemy represents planetary motion as a Fourier series with coefficients determined by trial and error. The first term in the series is the deferent, the second is the epicycle.

No one in the present day objects when an mp3 compresses sound using 1024 epicycles, instead of encoding information about pitches, timbre, and instrumentation. It isn’t physically real, but it’s a very effective model. Epicycles in the Ptolemaic system work exactly the same way.

Approximations of a square wave using Fourier series of 5 terms, 10 terms, and 125 terms (base image: Youtube).

Aristarchus was nearly forgotten by the time Copernicus reintroduced the heliocentric model in the early 1500s. The main source for Aristarchus’ theory, Archimedes’ Sand-reckoner, didn’t appear in print until 1544, the year after Copernicus’ death. Copernicus’ manuscript of De revolutionibus did refer to Aristarchus’ heliocentric theory, but he clearly didn’t know much about it. He removed the reference in the print edition.

Note. First print edition of the Sand-reckoner: Gechauff 1544: 120–127 (current edition: Heiberg 1913, ≈ 1881: 242–291). Copernicus’ manuscript: Biblioteka Jagiellońska, BJ Rkp. 10000 III, at f. 11v: ‘It is feasible that, for these and similar reasons, Philolaus [the Pythagorean] perceived that the earth is mobile; several sources report that Aristarchus of Samos was of the same view, for some reason other than that which Aristotle cites and refutes.’ The print edition mentions Aristarchus only in a separate context (1543: 65v).

References

• Berggren, J. L.; Sidoli, N. 2007. ‘Aristarchus’s On the sizes and distances of the sun and moon: Greek and Arabic texts.’ Archive for history of exact sciences 61: 213–254. [Sci-hub]
• Copernicus, N. 1543. De revolutionibus orbium coelestium. Nürnberg. [Internet Archive]
• Gechauff, Th. (alias Venatorius) 1544. Ἀρχιμήδους τοῦ Συρακουσίου, τὰ μεχρὶ νῦν σωζόμενα, ἅπαντα. Archimedis Syracusani philosophi ac geometrae excellentissimi opera. Basel. [Google Books]
• Heath, T. L. 1913. Aristarchus of Samos. The ancient Copernicus. Oxford. [Internet Archive]
• Heiberg, J. L. 1913. Archimedes opera omnia, 2nd edition (first publ. 1881) vol. 2. Leipzig. [1881 edition: Internet Archive]
• Neugebauer, O. 1975. A history of mathematical astronomy, 2 vols. Berlin/Heidelberg.
• Pines, S. 1963. ‘Un fragment de Séleucus de Séleucie conservé en version arabe.’ Revue d’histoire des sciences 16: 193–209. [JSTOR]
• Russo, L.; Medaglia, S. M. 1996. ‘Sulla presunta accusa di empietà ad Aristarco di Samo.’ Quaderni Urbinati di cultura classica 53: 113–121. [JSTOR]