The History of Mathematical Proof in Ancient Traditions

166 reviel netz

in a descending order of size. Th e manuscripts have them arranged side by

side, all of equal size.

Th e proposition constructs the circles in a complex way which is then

shown to determine that the circle Λ equals the surface of the cone BAΓ,

circle K equals the surface of the cone BΔE, and Θ the diff erence between

the surfaces, that is the surface of the truncated cone at the lines AΔEΓ.

It is therefore geometrically required that Λ > K, Λ > Θ (the relationship

between K, Θ, though, is not determined by the proposition).

It is clear that Heiberg’s diagram provides more metrical information

than the manuscript diagrams do. In this particular case, indeed, Heiberg

provides more metrical information than is determined by the proposition;

while the manuscripts provide less than is determined by the proposition.

Th is immediately suggests why the manuscripts’ practice is in fact rational.

Let us suppose that the manuscripts would set out to diagram the precise

metrical relations determined by the proposition. It would make sense,

then, to have both Θ and K smaller than Λ. However, how to represent the

relationship between Θ and K? Once Λ appears bigger than both Θ and K,

this is already taken to suggest that diagrams are metrically informative;

and so the reader would look for the diagram relationship between Θ and

K so as to provide him or her with the intended metrical relation. Th us,

a diagram where, say, Λ is greater than both Θ and K, the two, say, equal

to each other, falsely suggests that the intended metrical properties are:

Λ > Θ = K. Th e diffi culty of representing indeterminate metrical relations

inside a metrical diagram is obvious.

Figure 3.1 Heiberg’s diagrams for Sphere and Cylinder i.16 and the reconstruction of

Archimedes’ diagrams.

B

Heiberg Archimedes (reconstruction)

K

Z E

A H Γ

Δ

Θ

Λ

Λ

B

Z E

A H Γ

Δ

K

Θ