184 reviel netz
Th is should be read in full to get a sense of the manuscript evidence Heiberg
had to contend with (I quote together with my numbering of claims in the
argument. It should be clear that this is something of an extreme case,
though not at all a unique one):
(16) But that ratio which T Δ has to H in square – T Δ has this ratio to PZ in length [(17)
for H is a mean proportional between TΔ, PZ (18) through <its being a mean propor-
tional> between Γ Δ, EZ, too; how is this? (19) For since ΔT is equal to TΓ, (20) while
PE <is equal> to EZ, (21) therefore Γ T is twice T Δ, (22) and PZ <is twice> PE; (23)
therefore it is: as ΔΓ to ΔT, so PZ to ZE. (24) Th erefore the <rectangle contained>
by Γ Δ, EZ is equal to the <rectangle contained> by T Δ PZ. (25) But the <rectangle
contained> by Γ Δ, EZ is equal to the <square> on H; (26) therefore the <rectangle
contained> by T Δ, PZ, too, is equal to the <square> on H; (27) therefore it is: as T Δ to
H, so H to PZ; (28) therefore it is: as T Δ to PZ, the <square> on T Δ to the <square>
on H; (29) for if three lines are proportional, it is: as the fi rst to the third, the fi gure
on the fi rst to the fi gure on the second which is similar and similarly set up]
Th e expression ‘how is this?’ inside claim 18 is without parallel in the
corpus, and seems like a didactic order to a pupil (or, perhaps, an autodi-
dact’s cri de coeur ?). Th e passage from 19 to 21 is indeed extraordinarily
simple (from A = B to A + B being twice A). Th e fi nal explicit quotation from
Euclid’s Elements is natural coming from a didactic context. And overall the
argument is very simple, strikingly so given its length. It is therefore quite
likely that the entire passage from ‘how is this?’ in the end of claim 18 down
to the end of claim 29 is a scholion inserted into the manuscript tradition.
Heiberg’s choice, however, was to bracket starting from step 17 itself – this,
apparently, merely because step 17 begins with a gar.
It would be easy for us to condemn Heiberg’s use of square brackets as
disrespectful to the manuscripts’ evidence, or as involving massive circular
reasoning. But Heiberg’s practice is not unreasonable and is likely to be
correct at least in part. I doubt any editor could have come up with a single
system better than Heiberg – short, that is, of the confession of editorial
ignorance which might have been best of all (and which Heiberg, in a
sense, did fi nally follow – by allowing the bracketed words to be printed
inside the main text). I stand by my judgement of Heiberg as a superb, and
superbly tactful, philologer. Having said that, however, the fact remains that
we cannot really say how correct he was. Th ere are three texts at play here:
(A) Heiberg’s text with the bracketed segments inserted, i.e. the manu-
scripts’ reading.
(B) Heiberg’s text with the bracketed segments removed.
(C) Archimedes’ original text.