The History of Mathematical Proof in Ancient Traditions

Th e Elements and uncertainties in Heiberg’s edition 101

a property likewise shown for the parallelepipeds ( xi .34) and pyramids
( xii .9).
In the fi rst part of the proof, let us suppose the cones or cylinders on
bases ABCD and EFGH, with heights KL and MN, are equal. If KL is not
equal to MN, NP equal to KL is introduced and the cone (or cylinder) on
base EFGH with height NP is considered (see Figure 1.3).
Schematically, in abbreviated notation, we have (by v .7) a trivial
proportion:

cylinder AQ = cylinder EO ⇒ cylinder AQ: cylinder ES:: cylinder EO:

cylinder ES

in which a substitution is made for each of the two ratios:

cylinder AQ: cylinder ES:: base ABCD: base EFGH (which is justifi ed
by xii.11)
cylinder EO: cylinder ES:: height MN: height PN (S).
From which: base ABCD: base EFGH:: height MN: height PN (CQFD)
However, the proportion ( S ) is an ‘implicit presumption’ in the Arabo-
Latin versions. Admittedly, it may be easily deduced by those who under-
stand Propositions vi .1 and 33, as well as xi .25, that is the way one employs
the celebrated Defi nition v .5. In the Greek manuscripts, though, the
situation is diff erent. Proportion ( S ) is justifi ed on the basis of previous
knowledge: xii .13 in P and Th , xii .14 in b. 81 Th ese Propositions xii .13–14
do not exist in the indirect medieval tradition and thus it may be inferred

Figure 1.3 Euclid’s Elements , Proposition xii.15.

O

Q

L

D

A
K

B

C

R

P

U E

M H N F

S G

81 Here, the indirect medieval tradition is not in accord with ms b which presents the most
satisfying textual state from the deductive point of view! For details, see Euclid/Vitrac 2001: iv
334–44.