Naming the days of the week

Ever wondered where the weekday names come from? The answer may go back a bit further than you think. But it’s also incomplete.

The easy part of the answer is that the days are named after the seven planets. Not the modern solar system, though. The planets as they were known to ancient astronomers like Ptolemy. And in English, six of the seven planet names were swapped out, and Old English words or gods’ names put in. But the idea’s the same.

Sunday. Sun; relief of Sol (Museo Nazionale Romano, Rome)

Planet	Roman name	Old English	Modern English
Sun	dies Solis	Sunne	Sunday
Moon	dies Lunae	Mōna	Monday
Mars	dies Martis	Tiw (~Týr)	Tuesday
Mercury	dies Mercurii	Wōden (~Odin)	Wednesday
Jupiter	dies Iovis	Þonar (~Thor)	Thursday
Venus	dies Veneris	Frīg (~Frigg)	Friday
Saturn	dies Saturni	—	Saturday

In antiquity the Sun and Moon were frequently counted among the planets because, like the planets, they travel along the ecliptic relative to the fixed stars. Sometimes ancient writers talk of seven planets (including Sun and Moon), sometimes just five.

That’s the easy bit. I think the more interesting questions are:

1. When did Greco-Roman gods get attached to the seven day cycle?
2. Why are they in that order?

Monday. Moon (CambridgeInColour.com); Luna in chariot (Arch of Constantine, Rome)

Roman weekday names

The weekday names apparently arrived in Rome in the late 1st century BCE. The earliest reference to a day bearing one of the modern names is in the elegiac poet Tibullus, in a poem dating to the early 20s BCE.

aut ego sum causatus aves aut omina dira
      Saturnive sacram me tenuisse diem.

I used birds or bad omens as a pretext,
      or that the day sacred to Saturn detained me.

Tibullus 1.3.17–18

Petronius’ Satyrica alludes to all seven planets being associated with weekdays. However, the date of the Satyrica is in some doubt these days. It used to be dated to the reign of Nero (54–68 CE), but some scholars now think it’s a 2nd century novel.

... two timetables were posted, one on each doorpost. One of them, if I recall correctly, had this note: ‘third day and day before Kalends of January [30–31 December]: our C. dines out.’ The other depicted the course of the moon and paintings of the seven stars.

Petronius, Satyrica 30

The modern sequence of days appears in two graffiti no later than 79 CE, found at Pompeii. One is in Greek; the other, in Latin, omits Wednesday. Both graffiti start the week on Saturday.

θεων ημερας | κρονου | ηλιου | σεληνης | αρεως | ε[ρ]μου | διος | [αφρο]δειτης
Days of the gods: (day of) Kronos, Sun, Moon, Ares, Hermes, Zeus, Aphrodite

Mau 1881: 30

saturni | solis | lunae | martis | iovis | veneris
(day of) Saturn, Sol, Luna, Mars, Jupiter, Venus

Sogliano 1901: 330

Note. These and some further references are helpfully compiled by Schürer 1905: 25–34.

A graffito similar to the second has been found in Tunisia at Thuburbo Maius, not far from Carthage (IL Tun. 710 = Merlin 1944: 126), but that’s probably a couple of centuries later. It too starts the week on Saturday. The canonical sequence of seven appears in many later writers — including a short 4th century poem by Ausonius. In Ausonius, the week now begins on Sunday.

Some of these later writers were puzzled by the order. Plutarch, in the 2nd century, devoted a section of his Table talk to the question ‘Why they name the days after planets but number them differently from their sequence’ (Plutarch, Moralia 672c). Unfortunately that part of the Table talk is lost.

Tuesday. Mars (Fvalk.com); C. F. von Saltza, ‘Týr’ (F. Sander, Edda Sämund den vises, Stockholm, 1893, p. 78)

The order of the seven (or five) planets

The five planets, aside from the Sun and Moon, are the ones that are visible to the naked eye: Mercury, Venus, Mars, Jupiter, and Saturn. (Theoretically Uranus is too — occasionally, and with a keen eye. But only for a rather Simpsons-esque sense of ‘theoretically’.)

Those who are skilled in astronomy say that there are seven bands, on which the seven stars are carried. On the highest is carried the star of Kronos; on the one after that the star of Zeus; on the third the star of Ares; on the fourth the star of the Sun; on the fifth the star of Aphrodite; on the sixth the star of Hermes; and on the seventh the star of the Moon.

Achilles, Eisagoge 16 (Maas 1898: 42,25–30)

So the standard sequence, starting from the outermost planet, is: Saturn, Jupiter, Mars, Sun, Venus, Mercury, and Moon.

This arrangement isn’t really ‘Ptolemaic’, or only incidentally. Ptolemy himself allowed doubt over whether the inner planets are ‘beneath’ or ‘above’ the Sun, since the Sun’s brightness makes it impossible to observe whether Venus and Mercury pass in front or behind. And he’s explicit that he can’t measure the planets’ distance, since he can’t measure any parallax. He just accepts the conventional sequence as, well, a convention.

πιθανωτέρα μᾶλλον ἡ τῶν παλαιοτέρων τάξις καταφαίνεται ...

the order assumed by the older [astronomers] appears the more plausible ..

Ptolemy, Almagest 9.1 (ii.207 Heiberg; tr. Toomer)

Some other features of the ‘Ptolemaic’ system aren’t really Ptolemaic either. Ptolemy has no word for ‘deferent’, the circular orbit on which the epicycle is centred (though he does use the concept); he doesn’t give figures for the distances of the planets. That’s all mediaeval.

Wednesday. Mercury (Cronodon.com); Ian McShane as Wednesday (promotional poster for American gods, 2017–2021)

The standard order took time to become settled. Otto Neugebauer gives a handy run-down of the different sequences seen in different ancient cultures and different ancient authors (1975: 690–693). Some of them omit the Sun and Moon; some are reversed; some omit Jupiter and Saturn. His verdict is that no standard order existed prior to the time of Hipparchos, in the 2nd century BCE.

Egyptian	Jupiter, Saturn, Venus, Mercury, Mars
older Babylonian	Jupiter, Venus, Saturn, Mercury, Mars
Persian/Hellenistic era Babylonian	Jupiter, Venus, Mercury, Saturn, Mars
Archimedes according to ps-Hippolytos Refutatio 4.7–11	Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn, fixed stars
Plutarch On generation 1028b (Pythagorean)	Central fire, Counter-earth, Earth, Moon, Mercury, Venus, Sun
Plutarch On generation 1029b	Mercury, Venus, Mars, Jupiter, Saturn
‘some’ according to Achilles, apparently including Eratosthenes (Maass 1898: 42,30–43,2)	[Mars], Venus, Mercury, Sun, [Moon]
‘others’ according to Achilles (ibid.)	[Mars], Mercury, Sun, Venus, [Moon]
Eudoxos papyrus, inscription of Keskinto	Saturn, Jupiter, Mars, Mercury, Venus
Vitruvius Architecture 9.1.5	Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn
Ptolemy Almagest 9.1, Cicero On divination 2.91–92, Pliny Natural history 2.34–44	Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon
Cicero On the nature of the gods 2.52–53	Saturn, Jupiter, Mars, Venus, Mercury
(Indian) Varāhamihira Pancha-siddhantika 13.39 (i.121 Neugebauer-Pingree)	Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, nakṣatras

(Notice, by the way, that Vitruvius and the 6th century Indian astronomer Varāhamihira have the same sequence as Ptolemy, but reversed; so do Cicero and Plutarch, reversing one another, but both omitting the Sun and Moon.)

In spite of the variation, and the fact that Ptolemy regarded the sequence as purely conventional, the ‘Ptolemaic’ sequence came to be universally regarded as the standard order. Ptolemy calls it ‘the order of previous astronomers’ (ἡ τῶν παλαιοτέρων τάξις), but as mentioned above, it’s inconsistent with pre-2nd century BCE sequences, Neugebauer suggests that no sequence existed until around the time of Hipparchos.

Thursday. Jupiter (Aol.com); Chris Hemsworth as Thor (promotional poster for Thor: the dark world, 2013)

The order of the weekdays

Here’s how the planets get reordered into weekdays, starting with Saturn, as in Ptolemy and the Pompeii graffiti.

We don’t actually have any good evidence on why this reordering happened. It clearly isn’t random, though: notice how for each weekday, you skip two planets — or conversely, for each planet, you skip four weekdays

One modern book on the history of the week, by Eviatar Zerubavel, favours a theory based on the premise that Egyptian astronomers assigned each of the planets to hours of the day (1985: 14–17); however, everything about this theory is hypothetical. It’s claimed by an ancient author, as we’ll see below, but not a very trustworthy author. It can’t be corroborated as anything more than, well, some ancient guy making guesses.

There are three ancient theories on record, including the one Zerubavel prefers. Personally I think all three are pretty tenuous. The lost essay by Plutarch that I mentioned above may have had a fourth explanation, but alas, we’ll never know what it was.

Friday. Venus (Deepsky2000.com); C. E. Doepler, ‘Frigg and her handmaidens’ (W. Wägner, Nordisch-germanische Götter- und Heldensagen, 3rd ed. 1882, p. 109)

The first theory comes from a 6th century Indian astronomer, Varāhamihira.

(Ascending) up from the Moon (each successive planet) is lord of the month, (descending) down from Saturn lord of the hour. (Ascending) up in order (every) fifth (planet) is lord of the day; the lords of the year are clear.

Varāhamihira, Pancha-siddhantika 13.42 (i.121 Neugebauer-Pingree)

This is simply a restatement of the pattern I mentioned: start from Moon = Monday, then for each successive day, move on five planets (counting inclusively; four, counting exclusively). Why anyone would do that, he doesn’t explain. So this is a pretty weak theory.

The second and third are found in Dion Cassius (3rd century CE). He attributes them both to ‘the Egyptians’, which is ... doubtful, to say the least. But let’s hear him out. Here’s theory number two:

For if you apply the so-called ‘principle of the tetrachord’ (which is believed to constitute the basis of music) to these stars, by which the whole universe of heaven is divided into regular intervals, in the order in which each of them revolves, and beginning at the outer orbit assigned to Saturn, then omitting the next two name the lord of the fourth, and after this passing over two others reach the seventh, and you then go back and repeat the process with the orbits and their presiding divinities in this same manner, assigning them to the several days, you will find all the days to be in a kind of musical connection with the arrangement of the heavens.

Dion Cassius 37.18 (tr. Cary)

The tetrachord was indeed the basic element of ancient Greek music: it was a sequence of four notes, spanning an interval of what we would call a perfect fourth (a frequency ratio of 4:3). That is, this theory is that you rotate through the planets in the same way that musical keys modulate through a cycle of fifths.

That isn’t as crazy as it might sound. These are all real things: the tetrachord, the Pythagorean preoccupation with perfect harmonic intervals, and the idea that the planets are tied up with musical theory in some way. Unfortunately we know basically nothing about Pythagoreanism in the 1st century BCE, so we have no way of deciding whether this theory is plausible, or completely daft.

Saturday. Saturn (UniverseToday.com); relief of Saturnus from altar of Malakbel (Palmyra, Syria), Musei Capitolini, Rome (Wikimedia.org)

Here’s theory number three, also from Dion Cassius:

If you begin at the first hour to count the hours of the day and of the night, assigning the first to Saturn, the next to Jupiter, the third to Mars, the fourth to the Sun, the fifth to Venus, the sixth to Mercury, and the seventh to the Moon, according to the order of the cycles which the Egyptians observe, and if you repeat the process, covering thus the whole twenty-four hours, you will find that the first hour of the following day comes to the Sun. And if you carry on the operation throughout the next twenty-four hours in the same manner as with the others, you will dedicate the first hour of the third day to the Moon, and if you proceed similarly through the rest, each day will receive its appropriate god.

Dion Cassius 37.19 (tr. Cary)

This one depends on the premise that each hour of the day is assigned to a planet, and that they’re assigned in the Ptolemaic order.

Day	Day begins at hour	Associated planet
1	1	Saturn
2	25	Sun
3	49	Moon
4	73	Mars
5	97	Mercury
6	121	Jupiter
7	145	Venus
8	169	Saturn

and so on. This sounds kind of plausible. It also has the advantage of predicting Saturday as the first day of the week, which is exactly what we see in early sources like the Pompeii graffiti.

There are problems, however. First, there’s no corroboration in anything we know about Egyptian astronomy for the idea of assigning planets to hours. It could easily be a figment of Dion’s imagination.

Second, Egyptian astronomy is characterised much more by a division of the day (and night) into 12 hours, not 24. According to Robert Hannah, the concept of the 12 hour day is precisely of Egyptian origin (2005: 87). As Herodotos puts it,

But as far as human affairs are concerned, [the priests in Egypt] agreed on this: that the Egyptians were the first to discover the year, and the division of it into twelve seasonal segments; and they discovered this from the stars, as they said. ... They also said the Egyptians were the first to refer to a canon of twelve gods, and that the Greeks adopted this from them ...

Herodotos 2.4 (my emphasis)

(A 12 hour cycle would produce the same result, if we start from the Moon and work our way out. But then we lose the advantage of matching early sources by outputting Saturday as the first day of the week.)

And third, what we do find in actual Egyptian astronomy is the idea of associating hours with specific stars or constellations, not planets.

The hours became associated with certain stars or star groups which rose heliacally at ten-day intervals through the year. Sirius was one of these, and it was joined by 35 other stars ... Collectively they are now known as the ‘decans’ ...

Hannah 2005: 87

This produced a system of ten-day weeks in a seven-week cycle, not a cycle of seven days. Theory number three gives every appearance of being a post hoc rationalisation of the weekday names, not a true explanation.

None of the three theories has any corroboration. Theory 1 is certainly the weakest. But the mismatch between theory 3 and what is actually known about Egyptian astronomy is so glaring that I think it has to be rejected almost as strongly.

The weird result is that the Pythagorean explanation — theory 2, rotating between the planets in musical tetrachords — is the strongest.

Not that it’s a good theory, mind. It sounds quite daft to me. It’s just that, as things stand, we don’t have anything to rule it out.

References

• Hannah, R. 2005. Greek & Roman calendars. Constructions of time in the classical world. London.
• Maass, E. 1898. Commentariorum in Aratum reliquiae. Berlin. [Internet Archive]
• Mau, A. 1881. ‘Scavi di Pompei.’ Bullettino dell’Instituto di corrispondenza archeologica 1881,i–ii: 22–32. [Internet Archive]
• Merlin, A. 1944. Inscriptions latines de la Tunisie. Paris.
• Neugebauer, O. 1975. A history of ancient mathematical astronomy. Berlin/Heidelberg.
• Schürer, E. 1905. ‘Die siebentägige Woche im Gebrauche der christlichen Kirche der ersten Jahrhunderte.’ Zeitschrift für die Neutestamentliche Wissenschaft 6: 1–66. [Zenodo]
• Sogliano, A. 1901. ‘Regione I (Latium et Campania).’ In: Notizie degli scavi di antichità comunicate alla R. Accademia dei Lincei, anno 1901, s.v. ‘Luglio 1901’. Rome. 329–333. [Internet Archive]
• Zerubavel, E. 1985. The seven day circle. The history and meaning of the week. Chicago.

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

Visualisation of a Stoic 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.

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:

• a person can be immersed in pneuma (1.5, 11.16)
• a person can be anointed with pneuma (10.38)
• a person can be filled with pneuma (2.4, 4.8, 6.3–5, 7.55, 9.17, 11.24, 13.9, 13.52)
• pneuma can boil (18.25 ζέων τῷ πνεύματι)
• pneuma can be transmitted by physical contact (8.15–19, 19.6)
• pneuma 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?

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.)

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.

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. Ptolemy treats them solely as a mathematical tool: he makes no claim to their physical reality.

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]

How Eratosthenes measured the earth. Part 4

1. The spherical earth | 2. Eratosthenes’ method | 3. Distance | 4. Angle of the sun

(e) Angle of the sun

Eratosthenes didn’t use gnomons to measure the sun’s angle.

Here once again I have to digress from telling the true story, and dispel a myth. It’s a myth that even many specialists take for granted. I’m just as guilty: in the past I’ve repeated the popular wisdom that Eratosthenes’ measurement was based on gnomon readings. Well, I was wrong.

The sun at midday (source: Stellarium)

The gnomon is the most basic instrument for measuring the sun’s motion. It’s a vertical rod casting a shadow on a horizontal surface. Egyptian observers had been thoroughly familiar with gnomons for many centuries before the Ptolemaic era. Gnomons had several uses:

1. a way of expressing latitude
2. determining exact time of midday, dates of equinoxes and solstices
3. orientation (determining the direction of due north, south, etc.)

The first of these was a comparatively recent innovation: it doesn’t make sense to measure latitude until you know the earth is spherical. The idea is that latitude is expressed as the ratio between a gnomon and the length of its shadow, at midday at the equinox. This ratio is an indirect expression of the sun’s angle.

A gnomon reading measures the ratio of the gnomon to its shadow at midday. This diagram shows a reading taken on the equinox, when the sun’s rays are parallel to the earth’s equator. Using trigonometry, you can use this ratio to calculate the angle θ, since tan θ = o/a.

And this is exactly what Pliny the Elder does (1st century CE). He quotes gnomon readings taken at the equinox.

So in Egypt at midday on the equinox, the shadow of an umbilicus — what they call a ‘gnomon’ — measured a little over half of the gnomon. In the city of Rome, the shadow is one ninth longer than the gnomon. In the town of Ancona, it’s 1/32 more than that. And in the region of Italy called Venetia, the shadow is the same length as the gnomon.

Pliny, Natural history 2.182

Strabo gives the latitudes of Alexandria and Carthage in the same way (2.5.38). Now, remember from Part 2 that at midday on the equinox, the sun’s angle from the vertical is equal to your latitude. This means we can convert Pliny’s and Strabo’s figures to the modern way of expressing latitude — degrees away from the equator — by plugging them into a calculator and using the ‘inverse tan’ function.

	Gnomon reading	Calculated latitude	Actual latitude
Egypt	a little over 0.5	over 27°	24.1° (Aswan) to 31.2° (Alexandria)
Alexandria	0.600 (3/5)	31.0°	31.2°
Carthage	0.636 (7/11)	32.5°	36.9°
Rome	0.889 (8/9)	41.6°	41.9°
Ancona	0.920 (8/9 + 1/32)	42.6°	43.4°
Venice	1.00	45.0°	45.4°

These are reasonably accurate, apart from Carthage.

Note. Ptolemy also puts Carthage too far south, at 32.7°: Geography 4.3.7. And no, it isn’t because Strabo and Ptolemy are confusing Carthage with Leptis Magna — one might imagine that, since the Greek name of Leptis Magna, Neapolis ‘new town’, means the same as Carthage’s Punic name, Qrt ḥdšt ‘new town’. But Ptolemy lists Carthage and ‘Neapolis’ separately: see 4.3.13.

Our earliest evidence for the practice of using gnomon readings as a measure of latitude dates to Pytheas of Massalía in the 300s BCE. Pytheas’ book apparently started out by reporting the latitude of Massalía (modern Marseille).

Hipparchos says that in Byzantion the ratio of a gnomon to its shadow is the same as what Pytheas reports for Massalía ...

Pytheas fr. 6c Mette (Strabo 2.5.8; also fr. 6a, i.e. Strabo 1.4.4)

Elsewhere Strabo reports (2.5.41) that Hípparchos’ summer solstice gnomon ratio for Byzantíon is 120 : 41.8, that is, 2.871. That puts the sun’s angle from the vertical at 19.2°. Taking the earth’s inclination to the ecliptic as 23.8° in that era, that implies a latitude of 43.0° N. The real latitudes of Marseille and Byzantíon are 43.3° N and 41.0° N, respectively.

Pytheas’ book doesn’t survive. We aren’t actually told that he carried on taking gnomon readings on his trip into the North Sea. But it’s fairly strongly implied: we do hear about him reporting on the behaviour of the summer tropic at the Arctic Circle (Strabo 2.5.8).

People wrote books reporting gnomon readings in Africa too. In the early 200s BCE Philon, a Ptolemaic diplomat, reported gnomon readings and other observations of the sun in Meroë.

(Hipparchos states that) Phílon described the parallel of latitude at Meroë, and related it in his Voyage to Aithiopia. He stated that 45 days before the summer solstice the sun is directly overhead; and he reports the ratios of gnomons to their shadows at the solstices and equinoxes. Eratosthenes’ (figures) agree very closely with Philon.

Philon, FGrHist 670 F 2 (Strabo 2.1.20)

‘Agree very closely’ implies their figures weren’t exactly identical: that is, Eratosthenes had another source in addition to Philon. Eratosthenes was well equipped with gnomon readings, it seems.

The second use of gnomons — determining solstices, equinoxes, and the moment of midday — was of much longer standing in Egypt. The Egyptian economy depended on the flooding of the Nile, so the Egyptians were highly motivated to keep track of the solar year. Martin Isler points to reliefs dating back to the 20th century BCE which he explains as gnomons with a forked tip and held vertical by struts. These gnomons are taller than a person — much taller, in the second image below — and the forked tip is designed to improve precision.

Left and centre: reliefs depicting gnomons used in ritual contexts, from a chapel of Senusret I (1900s BCE, left) and the Pylon of Ramesses II at Luxor (1200s BCE, centre). Right: diagram illustrating the movement of the gnomon’s shadow. The view is from overhead, with the gnomon at the bottom of the diagram, casting a shadow upward. The fork on the gnomon is oriented east-west: as the sun transits, the fork improves precision in determining the exact moment of midday. (Source: Isler 1991)

The third use of the gnomon, determining true north, may be even older, though the merkhet was better suited to this purpose. The great pyramid complex at Giza, built in the 2500s BCE, is famously oriented so that the main buildings are aligned with the cardinal directions to a precision of around 0.08°. Gnomons probably weren’t the primary tool for achieving that, but they must certainly have been an important piece of ancillary equipment.

Note. Nell and Ruggles 2014 survey possible orientation methods used at Giza, arguing that the primary method was alignment with circumpolar stars; they allow that gnomons and merkhets served ancillary roles. A stellar method seems inevitable given that some pyramid complexes have slightly different orientations, and precession seems to be responsible for that. They find no suitable pairs of circumpolar stars, however. I suggest Phecta and Megrez, in the Great Bear (Gamma and Delta Ursae Majoris, the two stars on the left side of the ‘scoop’ of the ‘big dipper’). According to Stellarium they were at their closest to being in a north-south alignment in the year –2586, with right ascensions differing by just 1.28 s. That is, in 2587 BCE, when one of these stars was due north, the other would be just 0.005° east or west.

Isler goes on to describe further Egyptian refinements to the gnomon, including (a) the use of plumb bobs to ensure the gnomon is exactly vertical; (b) designs for handheld gnomons to be used by travellers; (c) in southern Egypt, the technique of leaning the gnomon towards due north at a fixed angle to compensate for the sun being directly overhead in summer.

By Eratosthenes’ time, then, the gnomon was fully developed. The idea of using it as a measure of latitude was new-ish, but it worked well — and to judge from Pytheas’ measurement at Massalia, under ideal conditions it could be quite accurate.

But there’s a problem. A gnomon measurement is a ratio between two lengths. How do you convert a ratio to an angle?

You could use the inverse tan function, as I mentioned. Just one snag: trigonometric functions are a modern thing. Aristarchos had started to dabble in trigonometric inequalities, but he was a long way from developing the Taylor series, which is what modern calculators use to compute trigonometric functions. Hipparchos apparently computed a table of chord lengths in a circle, which implicitly draws on the sine function: that sounds good. But Hipparchos himself states that he reckoned the circle in 24 portions, that is, in steps of 7.5°. When Ptolemy quotes latitudes, they’re precise to 1/12 of a degree! Hipparchos’ trigonometric table absolutely didn’t have that kind of precision.

And that’s just one angle. Comparing two angles compounds the inaccuracies.

Note. Hipparchos using steps of 7.5°: commentary on Aratos, 148,26–150,3 Manitius. See also Neugebauer 1975: 299–300, with a reconstruction of Hipparchos’ chord table at 1132 (Table 8).

Moreover, if Eratosthenes had quoted gnomon readings for Alexandria, Syene, and Meroë, then it’s a little remarkable that none of the figures are quoted in any source. It’s not like they’d be hard to express. The real latitudes are 31.2°, 24.1°, and 16.9°. If ancient sources quoted equinoctial ratios of 3/5, 4/9, and 3/10, those would be close enough: they’d translate to latitudes of 31.0°, 24.0°, and 16.7°. But only the first ratio appears in any ancient source, and it isn’t in connection with Eratosthenes (Strabo 2.5.38).

The only numerical figure we get that is actually related to Eratosthenes’ angular measurements is this one.

So the distance from Syene to Alexandria must be 1/50 of a great circle of the earth.

Eratosthenes, Measurement M6 Roller (Kleomedes 1.7, 100,19–21 Ziegler)

Eratosthenes jumped directly to the angular measurement, as a proportion of a circle. He didn’t stop and invent trigonometry on the way. Gnomons must have served only an ancillary role, just as they did at Giza 2300 years earlier. Eratosthenes’ measurement wasn’t done with gnomon readings.

Irina Tupikova (2018) suggests that the instrument he used instead was the skáphē, a bowl with a gnomon sticking up out of the centre and gauge markings along the side of the bowl indicating the number of degrees from the vertical. I’ll accept that that’s possible, but I doubt ancient skáphai were capable of the necessary precision. (The precision of the skáphē isn’t attested or well studied.)

A much likelier candidate is a device described by Ptolemy in the 100s CE. This device has the advantage that there’s evidence of it being used at Meroë in the 200s BCE.

We make a bronze ring of a suitable size ... We use this as a meridian circle [i.e. oriented north-south], by dividing it into the normal 360° of a great circle, and subdividing each degree into as many parts as [the size] allows. Then we take a smaller ring, and fit it inside the first ... the smaller ring can rotate freely inside the larger, with a north-south motion, in the same plane. At two diametrically opposite points on one lateral face of the smaller ring we fix little plates, of equal size, pointing towards each other and the centre of the rings ... [W]e observed the sun’s movement towards the north and south by turning the inner ring at noon until the lower plate was completely enshadowed by the upper one. When this was the case, the tips of the pointers indicated to us the distance of the sun from the zenith in degrees, measured along the meridian.

Ptolemy, Almagest 1.12 (64,12–66,4 Heiberg, tr. Toomer)

Left: the instrument described by Ptolemy (base image: Toomer 1984: 61). The instrument is lined up north-to-south, with the meridian; at midday the inner ring is turned so that A casts its shadow on B, then the angle of the sun is read on the outer ring. Right: a 2nd century BCE graffito from Meroë depicting a seated observer using a similar device, with the sun at top left (source: Garstang 1914, Plate VI No. 1). Garstang (1914: 4) and Depuydt (1998: 173–176) interpret the device as a ‘transit instrument with circle’, i.e. Ptolemy’s device, combined with ‘an azimuth instrument’, i.e. a gnomon, represented by the vertical line extending upward from the ring and touching a sunray.

In 1914 John Garstang described a building in Meróë, dating to the 2nd century BCE, whose outside wall contained graffiti with astronomical calculations and observations. One graffito (above, right) showed something close to Ptolemy’s ring transit device.

Gnomons still had their uses. You’d need one to pinpoint the direction of the meridian, and the exact moment of midday/ And they weren’t made obsolete by Ptolemy’s device: a gnomon doesn’t require as much precision in its engineering.

But when Kleomedes reports that Eratosthenes measured the distance from Syene to Alexandria as 1/50 of a great circle of the earth, we should take it that he means exactly what he says. Eratosthenes didn’t do trigonometry on two gnomon ratios in order to compare them. He directly compared two angular measurements.

(f) The upshot

Most people, if they know anything about Eratosthenes’ measurement, learned about it from Carl Sagan in his 1980 TV series Cosmos.

Eratosthenes’ only tools were sticks, eyes, feet, and brains — plus a zest for experiment. With those tools he correctly deduced the circumference of the earth to high precision, with an error of only a few percent.

Carl Sagan, Cosmos episode 1 (first broadcast 28 Sep. 1980)

This is mostly false. Sagan both diminished and exaggerated the accomplishment.

Eratosthenes was standing on the shoulders of giants. Sagan’s version diminishes the huge amount of work done by previous explorers, researchers, astronomers, and writers; the political infrastructure, the rich publication history, the high precision instrumentation. He glosses over the reasoning and the tools that Eratosthenes himself devised: the most sophisticated geographer of his time, the inventor of lines of latitude and longitude, and the principle that observations at the same meridian can be directly compared.

At the same time, by looking only at Hultsch’s selectively reported result, he also glosses over the blunders and the inaccuracies. The Ptolemaic understanding of the route of the Nile leaves a lot to be desired; the distance measurements are pretty poor; and it turns out that while shadow lengths can tell you your latitude accurately under ideal conditions, they’re not very reliable.

In light of all that, it’s lucky for Eratosthenes’ modern reputation that his measurement was only 16% high. It remained the most accurate measurement of the earth’s size until the modern era. But there was no way at the time for anyone to be sure of that — no one knew Eratosthenes’ measurement was the most accurate one, until modern measurements improved on him!

It’s only in hindsight that we know the techniques and technology he used were the very best that were available. The principle of his measurement, however, was impeccable. Eratosthenes should be given credit for his methodology, more than for his lucky result.

References

• Borchardt 1921. ‘Ein weiterer Versuch zur Längenbestimmung der ägyptische Meilen (itr-w).’ In: Regling, K.; Reich, H. (eds.) Festschrift zu C. F. Lehmann-Haupts sechzigstem Geburtstage. Wien/Leipzig. 119–123.
• Bowen, A. C.; Todd, R. B. 2004. Cleomedes’ lectures on astronomy. Berkeley/Los Angeles.
• Bruins, E. M. 1964. Codex Constantinopolitanus Palatii veteris no. 1, 3 vols. Leiden.
• Carlos Carman, C.; Evans, J. 2015. ‘The two earths of Eratosthenes.’ Isis 106: 1–16. [JSTOR]
• Couprie, D. L. 2011. Heaven and earth in ancient Greek cosmology. New York.
• Depuydt, L. 1998. ‘Gnomons at Meroë and early trigonometry.’ Journal of Egyptian archaeology 84: 171–180. [JSTOR]
• Diels, H.; Kranz, W. 1960. Die Fragmente der Vorsokratiker, 9th ed. Berlin. [Internet Archive: vol. 1, vol. 2] (Note: see Kirk and Raven 1957 for a selection of the fragments, with English translations in the footnotes, and cross-referenced to Diels & Kranz)
• Duncan-Jones, R. P. 1980. ‘Length-units in Roman town planning: the pes monetalis and the pes drusianus.’ Britannia 11: 127-33. [JSTOR]
• Garstang, J. 1914. ‘Fifth interim report on the excavations at Meroë in Ethiopia. Part I. General results.’ Annals of archaeology and anthropology (Liverpool) 7: 1–10. [Google Books]
• Hultsch, F. 1882. Griechische und römische Metrologie, 2nd ed. Berlin. [Internet Archive]
• Isler, M. 1991. ‘The gnomon in Egyptian antiquity.’ Journal of the American Research Center in Egypt 28: 155–185. [JSTOR]
• Kirk, G. S.; Raven, J. E. 1957. The presocratic philosophers. A critical history with a selection of texts. Cambridge. [Internet Archive] (cf. Diels and Kranz 1960)
• Loret, V. 1903. ‘L’átour et la Dodècaschène.’ Sphinx: revue critique embrassant le domaine entier de l’égyptologie 7: 1–24. [Persée]
• Mette, H. J. 1952. Pytheas von Massalia. Berlin.
• Nell, E.; Ruggles, C. 2014. ‘The orientations of the Giza pyramids and associated structures.’ Journal for the history of astronomy 45.3: 304–360. [DOI]
• Neugebauer, O. 1975. A history of mathematical astronomy. Berlin/Heidelberg.
• Priskin, G. 2004. ‘Reconstructing the length and subdivision of the iteru from late Egyptian and Graeco-Roman texts.’ Discussions in Egyptology 60: 57-71. [Academia.edu preprint]
• Roller, D. W. 2010. Eratosthenes’ Geography. Princeton.
• Tupikova, I. 2018. ‘Eratosthenes’ measurements of the earth: astronomical and geographical solutions.’ Orbis terrarum 16: 221–254. [Academia.edu]
• —— 2022. ‘A common-sense approach to the problem of the itinerary stadion.’ Archive for the history of exact sciences 76: 319–361. [DOI]

Further scholarschip on the stadion:

• Engels, D. 1985. ‘The length of Eratosthenes’ stade.’ American journal of philology 106: 298–311. [JSTOR]
• Gulbekián, E. 1987. ‘The origin and value of the stadion unit used by Eratosthenes in the third century B.C.E.’ Archive for history of exact sciences 37: 359–363. [JSTOR]
• Oxé, A. 1963. ‘Die Masstafel des Julianus von Askalon.’ Rheinisches Museum 106: 264–286. [Universität zu Köln | JSTOR]
• Pothecary, S. 1995. ‘Strabo, Polybius, and the stade.’ Phoenix 49: 49–67. [JSTOR]
• Priskin, G. 2004. ‘Herodotus on the extent of Egypt.’ Göttinger Miszellen 201: 63–67. [Academia.edu preprint]

Further scholarschip on the measurement of the earth:

• Diler, A. 1949. ‘The ancient measurements of the earth.’ Isis 40: 6–9. [JSTOR]
• Drabkin, I. E. 1943. ‘Posidonius and the circumference of the earth.’ Isis 34: 509–512. [JSTOR]
• Dutka, J. 1993. ‘Eratosthenes’ measurement of the earth reconsidered.’ Archive for the history of exact sciences 46: 55–66. [JSTOR]
• Nissen, H. 1903. ‘Die Erdmessung des Eratosthenes.’ Rheinisches Museum 58: 231–245. [Universität zu Köln | JSTOR]
• Priskin, G. 2006. ‘The Egyptian heritage in the ancient measurements of the earth.’ Göttinger Miszellen 208: 75–88. [Academia.edu preprint]
• Rawlins, D. 1982. ‘Eratosthenes’ geodesy unraveled: was there a high-accuracy Hellenistic astronomy?’ Isis 73: 259–265. [JSTOR]
• Russo, L. 2013. ‘Ptolemy’s longitudes and Eratosthenes’ measurement of the earth.’ Mathematics and mechanics of complex systems 1.1. 67–79. [DOI]
• Taisbak, C. M. 1974. ‘Posidonius vindicated at all costs? Modern scholarship versus the Stoic earth measurer.’ Centaurus 18: 253–269. [DOI]