Archimedes Sand Reckoner

Greek Learning Greek
1

It might seem strange that someone who is struggling through elementary Greek would want to look at Archimedes, but I have decided to create a little book to take with me on a trip to Sicily in 2020. I intend to include both Sicilian authors and people who wrote about Sicily – mostly easy stuff by writers such as Diodorus and Strabo. When I ran across the intro to the Sand Reckoner, I knew that I had to include it – so that’s where I started.

I’m only including these few sentences from the treatise. I’m prefixing each sentence with my understanding of the gist of it, and I follow the sentence with notes. I’d appreciate comments and corrections.


Archimedes – The Sand Reckoner

Some say that the quantity of sand in the world is infinite.

Οἲονταί τινες, βασιλεῦ Γέλων, τοῦ ψάμμου τὸν ἀριθμὸν ἄπειρον εἶμεν τῷ πλήθει· λέγω δὲ οὐ μόνον τοῦ περὶ Συρακούσας τε και τὰν ἄλλαν Σικελίαν ὑπάρχοντος, ἀλλὰ καὶ τοῦ κατὰ πᾶσαν χώραν τάν τε οἰκημέναν καὶ τὰν ἀοίκητον.

ψάμμου: of sand
ἄπειρον: boundless
εἶμεν: Attic εἴναι
πλήθει: in quantity
τὰν ἄλλαν: Attic τὴν ἄλλην, the rest
ὑπάρχοντος: being
οἰκημέναν: occupied
ἀοίκητον: unoccupied


Others assume that the quantity is not infinite, but that it is not possible to express in numbers/language an upper bound.

Ἐντί τινες δέ, οἳ αὐτὸν ἄπειρον μὲν εἶμεν οὐχ ὑπολαμβάνοντι, μηδένα μέντοι ταλικοῦτον κατωνο­μασμένον ὑπάρχειν[,] ὅστις ὑπερβάλλει τὸ πλῆθος αὑτοῦ.

Ἐντί: Attic εἰσί
ὑπολαμβάνοντι: assuming
μηδένα: none
μέντοι: nevertheless
ταλικοῦτον: so large
κατωνο­μασμένον: being named, being expressed in numbers
ὑπάρχειν: to be, to exist
[,]: J.L.Heiberg has a comma here
ὅστις: I’m taking this to mean ‘which’ because of the preceding μηδένα
ὑπερβάλλει: surpass
πλῆθος: volume

These people would be even more likely to say so if they imagined a pile of sand as big as a pile needed to cover the earth up to the tops of the mountains.

Οἱ δέ οὓτως δοξαζόντες δῆλον ὡς, εἰ νοήσαιεν ἐκ τοῦ ψάμμου ταλικοῦτον ὄγκον συγκείμενον τὰ μὲν ἄλλα[,] ἁλίκος ὁ τᾶς γᾶς ὄγκος ἀναπεπληρωμένων ἐν αὐτῷ τῶν τε πελάγεων πάντων και τῶν κοιλωμάτων τᾶς γᾶς εἰς ἴσον ὕφος τοῖς ὑψηλοτάτοις τῶν ὀρέων, πολλαπλασίως μη γνώσονται μηδένα κα ῥηθήμεν ἀριθμὸν ὑπερ­βάλλοντα τὸ πλῆθος αὐτοῦ.

δοξαζόντες: believing
δῆλον ὡς: clearly
νοήσαιεν: would imagine
ὄγκος: pile, volume
συγκείμενον: composed
τὰ μὲν ἄλλα: otherwise (i.e. it’s a different pile, not the same pile)
[,]: J.L.Heiberg has a comma here
ἁλίκος: Attic ἡλίκος so great
τᾶς γᾶς: Attic τῆς γῆς
ἀναπεπληρωμένων: being filled up
πελάγεων: of the seas
τῶν κοιλωμάτων: the hollowed out (places)
ἴσον: equal to
ὕφος: ???
ὑψηλοτάτοις: the heights
ὀρέων: of the mountains
πολλαπλασίως: so much more
μη γνώσονται: they will not know
κα: Attic ἄν
ῥηθήμεν: would be able to say


Archimedes will attempt to prove that of the numbers named here and shown in writings to Zeuxippus, some exceed not only the volume of sand needed to fill up the earth, but also the volume of the universe.

Ἐγὼ δὲ πειρασούμαι τοι δεικνύειν διʹ ἀποδειξίων γεωμετρικᾶν, αἷς παρακολουθήσεις, ὅτι τῶν ὑφʹ ἁμῶν κατωνομασμένων ἀριθμῶν καὶ ἐνδεδομένων ἐν τοῖς ποτὶ Ζεύξιππον γεγραμμένοις ὑπερβάλλοντί τινες οὐ μόνον τὸν ἀριθμὸν τοῦ ψάμμου τοῦ μέγεθος ἔχοντος ἴσον τᾆ γᾷ πεπληρωμένᾳ, καθάπερ εἴπαμες, ἀλλὰ καὶ τὸν τοῦ μέγεθος ἴσον ἔχοντος τῷ κόσμῳ.

πειρασούμαι: will attempt
ἀποδειξίων: showings, demonstrations
γεωμετρικᾶν: Doric genitive plural
παρακολουθήσεις: you will be able to follow in your head
ἁμῶν: Attic ἡμῶν
κατωνομασμένων: being named, being expressed in numbers
ἐνδεδομένων: ???
ποτὶ: Attic πρός
γεγραμμένοις: written down
ὑπερβάλλοντί: exceeding
τᾆ γᾷ: Attic τῇ γῇ
πεπληρωμένᾳ: filled up
καθάπερ: even as
εἴπαμες: (first aorist?) Attic εἴπομεν

Most astronomers call ‘universe’ the sphere whose center is the center of the earth and which extends to the center of the sun.

Κατέχεις δέ, διότι καλείται κόσμος ὑπὸ μὲν τῶν πλείστων ἀστρολόγων ἁ σφαῖρα, ἇς ἐστι κέντρον μὲν τὸ τᾶς γᾶς κέντρον, ἁ δὲ ἐκ τοῦ κέντρου ἴσα τᾷ εὐθείᾳ τᾷ μεταξὺ τοῦ κέντρου τοῦ ἁλίου καὶ τοῦ κέντρου τᾶς γᾶς. Ταῦτα γάρ ἐντι τὰ γραφόμενα, ὡς παρὰ τῶν ἀστρολόγων διάκουσας.

κατέχεις: you grasp
διότι: since
καλείται: is called
ἁ: Attic ἡ
σφαῖρα: sphere
ἇς: ἥ
κέντρον: center
εὐθείᾳ: straight line
μεταξὺ: space between
ἁλίου: Attic ἡλίου
διάκουσας: you have heard

Aristarchos of Samos has published some hypotheses where the assumptions imply a much bigger universe.
Ἀρίσταρχος δὲ ὁ Σάμιος ὑποθεσίων τινων ἐξέδωκεν γραφάς, ἐν αἷς εκ τῶν ὑποκειμένων συμβαίνει τὸν κόσμον πολλαπλάσιον εἶμεν τοῦ νῦν εἰρημένου.

ἐξέδωκεν: gave out
γραφάς: writings
ὑποκειμένων: assumptions
συμβαίνει: follows as a consequence
πολλαπλάσιον: so much greater
εἶμεν: Attic εἴναι
εἰρημένου: perfect middle-passive paritciple from εἴρω, speak

Archimedes asserts that even if the sphere of sand would be as big as the sphere of fixed stars as Aristarchos hypthesizes, still some numbers named in his treatise Principles will be shown to exceed the count of grains of sand of the said sphere, assuming:

Φαμὲς δή, καὶ εἰ γένοιτο ἐκ τοῦ ψάμμου σφαῖρα ταλικαύτα τὸ μέγεθος, ἁλίκαν Ἀρισταρχος ὑποτιθέται τὰν τῶν ἀπλανέων ἄστρων σφαῖραν εἶμεν, καὶ οὕτως τινὰς δειχθήσειν τῶν ἐν Ἀρχαῖς τὰν κατανομαξίαν ἐχόντων ὑπερβαλλόντας τῷ πλήθει τὸν ἀριθμὸν τὸν τοῦ ψάμμου τοῦ μέγεθος ἔχοντος ἴσον τᾷ εἰρημενᾳ σπαίρᾳ, ὑπο­κειμένων τῶνδε·

φαμὲς: Attic φαμὲν
γένοιτο: aorist optative γίγνομαι
ἁλίκος: Attic ἡλίκος so great
ἀπλανέων: fixed (not wandering)
δειχθήσειν: to be shown - future passive infinitive?
ὑπερβαλλόντας: exceeding

Φαμὲς.. τινὰς δειχθήσειν.. ὑπερβαλλόντας τῷ πλήθει.. we assert… some (numbers) to be shown… to exceed

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Ἐντί τινες etc.:
υπολαμβανοντι is not the participle but Doric for υπολαμβανουσι, 3pl. indic.
εἶμεν is Doric for εἶναι.
πληθος not so much “volume” but “multitude,” the full number.

Οἱ δέ οὕτως δοξαζόντες etc.:
“Those that hold this opinion”, then δηλον ὡς is parenthetical, lit. “(it’s) clear that,” tantamount to “clearly,” as you have it.
ει νοησαιεν … “if they were to conceive …” This if-clause extends all the way down to τοῖς ὑψηλοτάτοις τῶν ὀρέων.
αναπεπληρωμενων perfect not present, “filled up.”
ὕφος just a typo or misprint for ὕψος.
ῥηθημεν baffles me. ῥηθέντα I could understand (aor.pass.participle in acc.&pple construction after γνωσονται). I want it to mean “many times more will they fail to recognize that any number (or “no number,” but surely not?) exceeding its multitude could be expressed.”
—Oh, I suppose ρηθημεν is simply a Doric aor.pass. infinitive, = Attic ρηθῆναι.

ἐγὼ δε …:
αποδειξιων (Attic –έων) ”proofs.”
ενδεδομενων perf.pass. ενδιδωμι, “given, submitted.”
υπερβαλλοντι again 3pl. indic., Attic –ουσιν. “some exceed” “not only the number of sand (i.e. of grains of sand) with a magnitude equal to the earth filled up (with sand), as we said, but also the number with a magnitude equal to the whole f’ing cosmos!!” (But he’s a mathematician, so strike the expletive and the exclamation.)
ἇς = ἧς of course, and τᾶς γᾶς = τῆς γῆς. This is the so-called “doric alpha.”
εντι = εισι.
διακουσας = δι-ήκουσας.

To be concluded after dinner. Perhaps.

3

Isn’t ἐνδεδομένων the perfect mediopassive participle of ἐνδίδωμι?

τοῖς ποτὶ Ζεύξιππον γεγραμμένοις – τα γεγραμμένα “the writings” (again, perfect mediopassive particple, used as a noun)

ποτὶ Ζεύξιππον – to? against? or maybe “addressing” Zeuxippus?

ὅτι τῶν ὑφʹ ἁμῶν κατωνομασμένων ἀριθμῶν καὶ ἐνδεδομένων ἐν τοῖς ποτὶ Ζεύξιππον γεγραμμένοις ὑπερβάλλοντί τινες

". . . that some of the numbers named by us and given in the writings against/to/addressing(?) Zeuxippus . . . "

It’s interesting to see that written Doric prose is not very different from Attic-Ionic, and not essentially different from the Doric-colored choruses in Attic drama, which Athenian audiences must have understood. Written Doric would probably have been quite different from local (“epichoric”) Doric dialects, but written Doric (and, I would imagine, Doric spoken by educated speakers when not speaking their epichoric dialects) could be quite easily understood by anyone who knew Attic-Ionic.

4

I described ἐνδεδομένων as perf.pass., thinking it unnecessary to add “participle.” “Mediopassive” or “middle-passive” is not a term I like to apply to exclusively passive forms (such as γεγραμμενα and ειρημενου below).

The dialectal features go well beyond what we find in the choruses of Attic drama, and are more in line with what we find in the “literary Doric” of Stesichorus Pindar & Bacchylides: e.g. -οντι 3pl.indic. (= –ουσι), ειμεν (ειναι) & εντι (εισι), ειπαμες & φαμες (-μεν), πειρασουμαι (-σομαι), plus ῥηθημεν that I didn’t recognize but should have.

To polish off the remainder:

Ἀρίσταρχος δὲ ὁ Σάμιος etc.:
εξεδωκεν “published”
τοῦ νῦν εἰρημένου “than that stated here” (i.e. in the standard earth-centered model of the cosmos). It’s good to see Archimedes tacitly rejecting that in favor of Aristarchus’ heliocentric model. How he would have been delighted to learn that there’s (almost?) an infinity of universes!

Φαμὲς δή etc.:
ταλικαύτα = τηλικαύτη
“as big as Aristarchus hypothesizes the sphere of the fixed stars to be”
Φαμὲς.. τινὰς δειχθήσειν.. ὑπερβαλλόντας τῷ πλήθει “we claim that some will be shown to exceed in quantity …” (υπερβαλλοντας participle not infinitive after a verb of showing).
υποκειμενων τωνδε genitive absolute, “if the following assumptions are made.”

Archimedes, mathematical genius that he was, had devised a system for expressing extremely high numbers. Before that there was no way they could be “stated” or “named.”

5

Yes, Archimedes’ Doric goes beyond Attic choruses. My point was just that it’s not too difficult to read. There are a number of differences from Attic-Ionic, beyond long alpha (eta of course is an Attic-Ionic innovation), but there aren’t very many differences – they could probably be listed on less than a full page. Once you train yourself to recognize them, literary Doric isn’t too difficult.

6

Thanks, guys! It’s going to take a while for me to digest all this. I take it that no one objects to my rejecting Heiberg’s commas which interfere my interpretation of the sense of the sentences.

7

Heiberg’s commas have syntactical function. His punctuation system is standard in German and Latin and in Greek texts edited in the same fashion. (Heiberg was Danish.) English punctuation follows slightly different principles. The Archimedes manuscripts that Heiberg edited will not have used comma marks as we know them (our comma is a comparatively modern invention), and may have been punctuated quite differently.
So yes you can ignore those commas in translation, but they do help sort out the syntax of the Greek.