# Greek Mathematics

# VII. Democritus

# Plut. De Comm. Notit. 39. 3, 1079 e

Ἔτι τοίνυν ὅρα τίνα τρόπον ἀπήντησε Δημοκρίτῳ, διαποροῦντι φυσικῶς καὶ ἐπιτυχῶς, εἰ κῶνος τέμνοιτο παρὰ τὴν βάσιν ἐπιπέδῳ, τί χρὴ διανοεῖσθαι τὰς τῶν τμημάτων ἐπιφανείας, ἴσας ἢ ἀνίσους γινομένας; ἄνισοι μὲν γὰρ οὖσαι τὸν κῶνον ἀνώμαλον παρέξουσι, πολλὰς ἀποχαράξεις λαμβάνοντα βαθμοειδεῖς καὶ τραχύτητας· ἴσων δ᾿ οὐσῶν, ἴσα τμήματα ἔσται, καὶ φανεῖται τὸ τοῦ κυλίνδρου πεπονθὼς ὁ κῶνος, ἐξ ἴσων συγκείμενος καὶ οὐκ ἀνίσων κύκλων, ὅπερ ἐστὶν ἀτοπώτατον.

Archim. Meth., Archim. ed. Heiberg ii. 430. 1–9Διόπερ καὶ τῶν θεωρημάτων τούτων, ὧν Εὔδοξος ἐξηύρηκεν πρῶτος τὴν ἀπόδειξιν, περὶ τοῦ κώνου καὶ τῆς πυραμίδος, ὅτι τρίτον μέρος ὁ μὲν κῶνος

# Greek Mathematics

# VII. Democritus

# Plutarch, On the Common Notions 39. 3, 1079 e

Consider further in what manner it occurred to Democritus,^{a} in his happy inquiries in natural science, to ask if a cone were cut by a plane parallel to the base,^{b} what must we think of the surfaces forming the sections, whether they are equal or unequal? For, if they are unequal, they will make the cone irregular, as having many indentations, like steps, and unevennesses; but if they are equal, the sections will be equal, and the cone will appear to have the property of the cylinder, and to be made up of equal, not unequal, circles, which is very absurd.^{c}

This is a reason why, in the case of those theorems concerning the cone and pyramid of which Eudoxus first discovered the proof, the theorems that the cone

^{a}Plutarch tells this on the authority of Chrysippus. Democritus came from Abdera. He was born about the same time as Socrates, and lived to a great age. Plato ignored him in his dialogues, and is said to have wished to burn all his works. The two passages here given contain all that is definitely known of his mathematics, but we are informed that he wrote a book On the Contact of a Circle and a Sphere; another on Geometry; a third entitled Geometrica; a fourth on Numbers; a fifth On Irrational Lines and Solids; and a sixth called Ἐκπετάσματα, which would deal with the projection of the armillary sphere on a plane. As his mathematical abilities were obviously great, it is unfortunate that our information is so meagre.^{b}A plane indefinitely near to the base is clearly indicated by what follows.^{c}This bold inquiry first brought the conception of the indefinitely small into Greek mathematics. The story harmonizes with Archimedes’ statement that Democritus gave expressions for the volume of the cone and pyramid.