The History of Mathematical Proof in Ancient Traditions

94 bernard vitrac

remain, the diff erence being especially apparent in the arithmetical Books

vii – viii , as a matter of fact more ‘salvaged’ by these variants than the geo-

metric portions, in particular Book x and the stereometric Books.

An example of a local variant

Th e rather simple example which I propose is that of Proposition xi .1. It

shows how accounting for the indirect medieval tradition allows us to go

beyond the confrontation between P and Th to which Heiberg was con-

fi ned. Th e codicological primacy which he accords to the Vatican manu-

script is not inevitable because all Greek manuscripts, including P , have

been subjected to various late enrichments. It also probably indicates the

intention of these specifi c additions.

As with several other initial proofs in the stereometric books, in xi .1

Euclid tries to demonstrate a property he probably would have been better

off accepting (i.e. as a postulate) – namely, the fact that a line which has some

part in a plane is contained in the plane. 65 Here, the philological aspect inter-

ests me, even though the changes in the text were probably the result of the

perception of an insuffi ciency in the proof. Th e text is as follows:

(a)

Εὐθείας γραμμῆς μέρος μέν τι οὐκ ἒστιν ἐν τῷ

ὑποκειμένῳ ἐπιπέδῳ, μέρος δέ ἐν μετεωροτέρῳ.

Some part of a straight line is not

in a subjacent plane and another

part is in a higher plane.

D

C

B

A

Εἰ γὰρ δυνατόν, εὐθείας

γραμμῆς τῆς ΑΒΓ μέρος

μέν τι τὸ ΑΒ ἒστω ἐν τῷ

ὑποκειμένῳ ἐπιπέδῳ,

μέρος δέ τι τὸ ΒΓ ἐν

μετεωροτέρῳ.

Ἔσται δέ τις τῇ ΑΒ

συνεχὴς εὐθεῖα ἐπ’

εὐθείας ἐν τῶν

ὑποκειμένῳ ἐπιπέδῳ.

ἒστω ἡ ΒΔ ̇ δύο ἄρα

εὐθειῶν τῶν ΑΒΓ, ΑΒΔ

κοινὸν τμῆμά ἐστιν ἡ ΑΒ ̇

ὅπερ ἐστὶν ἀδύνατόν,

For, if possible, let some part AB

of the straight line ABC be in

the subjacent plane, another

part, BC, in a higher plane.

Th ere will then exist in the

subjacent plane some straight

line continuous with AB in a

straight line.

Let it be BD; therefore, of the

two straight lines ABC and ABD,

the common part is AB; which is

impossible,

(^65) On the weaknesses of the foundations of the Euclidean stereometry, see Euclid/Vitrac, 4, 2001:
31 and my commentary to Prop. xi.1, 2, 3, 7.