The History of Mathematical Proof in Ancient Traditions

Reasoning and symbolism in Diophantus 337

Tannery: Let the which is added and taken away from each number <sc.
of the two other given numbers> be set down, ςΑ <:Number 1>. And if it
is added to 20, result: ςΑ Μ ο Κ <:number 1 Monads 20>.

Manuscripts: Let the which is added and taken away from each number
<sc. of the two other given numbers> be set down, One number. And if
it is added to 20, result: One number, 20 Monads.

Here we see Tannery’s most typical treatment of the manuscripts: abbreviat-
ing expressions which, in the manuscripts, are resolved, within the problem
itself. Note the opposite, inside enunciations. For example, the enunciation
to iii .10 which, in Tannery’s form, may be translated:

Tannery: To fi nd three numbers so that the by any two, taken with
a given number, makes a square.

Compare this with, e.g., Par. Gr. 2379:

Manuscript: To fi nd three numbers so that the by any two, taken
with a given ς, makes a .

Tannery, we recall, followed a rational system: inside the proof, all mark-
edly Diophantine symbols were presented in abbreviated form, while in the
enunciation no symbolism was used. We fi nd that the manuscripts some-
times have abbreviated forms where Tannery has fully written words, and
sometimes have fully written words where Tannery has abbreviations. In
other words, Tannery’s rational system does not work. I had systematically
studied the marked Diophantine symbols through the propositions whose
number divide by ten, in Books i to iii , in all the Paris manuscripts. Th ese
are only eight propositions, but the labour, even so, was considerable:
essentially, I was busy recording noise. As a consequence of this, I gave
up on further systematic studies, merely confi rming the overall picture
described here, with other manuscripts.
One notices perhaps a gradual tendency to introduce more and more
abbreviated forms as the treatise progresses (do the scribes become tired,
in time?): Par. Gr. 2378, for instance, has no symbolism in my Book i speci-
mens at all, while they are frequent in Book iii. Th e ordinal numbers, with
their three separate forms (fully spelled out, phonologically abbreviated,
alphabetical numeral based), are especially bewildering. Consider once
again iii .10, once again in Par. Gr. 2378. I plot the sequence of ordinals,
using N for the alphabetic numeral, P for phonological abbreviation and F
for the full version:

NNNNPPFFFFFFFNFFFFPFNNFFFPF.