The History of Mathematical Proof in Ancient Traditions

108 bernard vitrac

two alternative proofs for Propositions x .105–106 are inserted at diff erent

places in the manuscripts. 100 Called here ‘superfi cial’ as opposed to the origi-

nal ‘linear’ Greek proofs, they apply to and argue about rectangular areas.

Let us explain this diff erence by an example, Proposition (Heib.) x .105:

A [straight line] commensurable with a minor straight line is a minor.

First proof in Greek 101

Aliter in Greek = fi rst proof in

medieval tradition

Let AB be a minor straight line and CD

commensurable with AB; I say that CD

i s a l s o m i n o r.

Let A be a minor straight line and B

[be] commensurable with A; I say

that B is minor.

A

B

E

D

C F

C

D

A B

E G

FH

We will consider the two components

(AE, EB) of AB and let DF be constructed

so that (AB, BE, CD, DF) are in

proportion. By vi .22, their squares

will also be in proportion and, thence by

x .11, x .23 Por. it will be shown (CD, DF)

have the same properties as (AB, BE).

Th us, by defi nition, CD will be a minor.

Let CD be a commensurate straight line.

Let the rectangles be constructed:

CE = square on A, width: CF,

FG = square on B, width: FH.

CE is the square on minor A so CE is

the fourth apotome ( x .100).

We have Comm. (A, B).

Th us: Comm. (CE, FG) and Comm.

(CF, FH).

FH is the fourth apotome ( x .103).

Th e square on B = Rect. (EF, FH), thus B

is a minor ( x .94)

• In each of the linear proofs, the argument concerns the two parts of an
  irrational straight line. Th e same type of argument is repeated ten times.
  Th ough repetitive, the approach has the advantage of not employing
  anything other than the Defi nitions of diff erent types and the theory of

100 In the Greek manuscripts the proof aliter to x .105–106 is inserted at the end of Book x , aft er

the alternative proof to x .115, which without a doubt implies that they had been compiled in

this place, aft er the transcription of Book x , in a limited space. Th us, they are in the margins

of manuscripts B and b. In one of the prototypes of the tradition, x .107 aliter has been lost or

omitted, probably for reasons of length, or because it was confused with x .117 vulgo which

follows immediately (but which is mathematically unrelated).

101 My diagrams are derived from those found in the edition of Heiberg ( EHS : iii 191 and 229,

respectively). Th ose of the manuscripts are less general. Th e segments AE, CF are very nearly

equal (the same goes for A, B in aliter ) and divided similarly.