The History of Mathematical Proof in Ancient Traditions

96 bernard vitrac

Let us now consult the indirect medieval tradition, for example the

Arabo-Latin translation by Gerard of Cremona, 71 compared to parts (a)

and (c) of the text edited by Heiberg:

Parts (a) and (c) of Heiberg’s text Gerard of Cremona’s version

Εὐθείας γραμμῆς μέρος μέν τι οὐκ ἒστιν

ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ, μέρος δέ τι

ἐν μετεωροτέρῳ.

Recte linee pars non est una in superfi cie et

pars alia in alto.

Quoniam non est possibile ut ita sit, quod

in exemplo declarabo.

Εἰ γὰρ δυνατόν, εὐθείας γραμμῆς τῆς

ΑΒΓ μέρος μέν τι τὸ ΑΒ ἒστω ἐν τῷ

ὑποκειμένῳ ἐπιπέδῳ, μέρος δέ τι τὸ ΒΓ

ἐν μετεωροτέρῳ.

Si ergo possibile fuerit, sit pars linee ABG

que est AB in superfi cie posita et sit alia

pars que est BG in alto.

Ἔσται δέ τις τῇ ΑΒ συνεχὴς εὐθεῖα ἐπ ̓

εὐθείας ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ.

Protaham ergo a linea AB in data superfi cie

lineam coniunctam linee AB

ἒστω ἡ ΒΔ: δύο ἄρα εὐθειῶν τῶν ΑΒΓ,

ΑΒΔ κοινὸν τμῆμά ἐστιν ἡ ΑΒ ̇

que sit BD. Linea ergo ABG est linea recta

et linea ABD est linea recta, ergo linea AB

duabus lineis BG et BD secundum

rectitudinem coniungitur.

ὅπερ ἐστὶν ἀδύνατόν. Quod est omnino contrarium.

Εὐθείας ἄρα γραμμῆς μέρος μέν τι οὐκ

ἒστιν ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ, μέρος

δέ ἐν μετεωροτέρῳ ̇

Non est ergo linee recte pars in superfi cie et

pars in alto.

ὅπερ ἔδει δεῖξαι. Et illus est quod demonstrare voluimus.

Despite the Arabic intermediary, the reader will easily recognize the

faithfulness of this Latin translation to the Greek, with two exceptions:

• the Latin adds a clause intended to introduce an indirect reasoning
  (a systematic characteristic shared with several manuscripts of the Ishâq–
  Th âbit translation)


• it has neither of the post-factum explanations of the Greek (part b).

(^71) Busard 1984: 338–9.
G
D A
B