Diophantus, Algebra

LCL 362: 514-515

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Theon Alex. in Ptol. Math. Syn. Comm. i. 10, ed. Rome, Studi e Testi, lxxii. (1936), 453. 4–6

Καθ᾿ ἃ καὶ Διόφαντός φησι· “τῆς γὰρ μονάδος ἀμεταθέτου οὔσης καὶ ἑστώσης πάντοτε, τὸ πολλαπλασιαζόμενον εἶδος ἐπ᾿ αὐτὴν αὐτὸ τὸ εἶδος ἔσται.”

Dioph. De polyg. num. [5], Dioph. ed. Tannery i. 470. 27–472. 4

Καὶ ἀπεδείχθη τὸ παρὰ Ὑψικλεῖ ἐν ὅρῳ λεγόμενον, ὅτι, “ἐὰν ὦσιν ἀριθμοὶ ἀπὸ μονάδος ἐν ἴσῃ ὑπεροχῇ ὁποσοιοῦν, μονάδος μενούσης τῆς ὑπεροχῆς, ὁ σύμπας ἐστὶν <τρίγωνος, δυάδος δέ>,1 τετράγωνος, τριάδος δέ, πεντάγωνος· λέγεται δὲ τὸ πλῆθος τῶν γωνιῶν κατὰ τὸν δυάδι μείζονα τῆς ὑπεροχῆς, πλευραὶ δὲ αὐτῶν τὸ πλῆθος τῶν ἐκτεθέντων σὺν τῇ μονάδι.”

Mich. Psell. Epist., Dioph. ed. Tannery ii. 38. 22–39. 1

Περὶ δὲ τῆς Αἰγυπτιακῆς μεθόδου ταύτης Διόφαντος μὲν διέλαβεν ἀκριβέστερον, ὁ δὲ λογιώτατος Ἀνατόλιος τὰ συνεκτικώτατα μέρη τῆς κατ᾿

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Algebra : Diophantus

Theon of Alexandria, Commentary on Ptolemy’s Syntaxis i. 10, ed. Rome, Studi e Testi, lxxii. (1936), 453. 4–6

As Diophantus says: “The unit being without dimensions and everywhere the same, a term that is multiplied by it will remain the same term.”a

Diophantus, On Polygonal Numbers [5], Dioph. ed. Tannery i. 470. 27–472. 4

There has also been proved what was stated by Hypsicles in a definition, namely, that “if there be as many numbers as we please beginning from 1 and increasing by the same common difference, then, when the common difference is 1, the sum of all the numbers is a triangular number; when 2, a square number; when 3, a pentagonal number [; and so on]. The number of angles is called after the number which exceeds the common difference by 2, and the sides after the number of terms including 1.”b

Michael Psellus,c A Letter, Dioph. ed. Tannery ii. 38. 22–39. 1

Diophantus dealt more accurately with this Egyptian method, but the most learned Anatolius collected the most essential parts of the theory as stated by

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DOI: 10.4159/DLCL.diophantus-algebra.1941