450 karine chemla
equalities are reshaped, whereas, in the commentaries, what is rewritten are
instead algorithms. 31 In correlation with this, in the latter case, intermedi-
ary sequences of operations are provided with an interpretation
Second, why do I speak of an ‘algebraic proof ’? I take it as a typical
element of this kind of proof that it involves transforming lists of operations
as such – the second line of argumentation – and that the validity of these
transformations should be addressed. If we observe the transformations
leading from one line to the next one in the modern version of the reason-
ing, sequences of operations are reshaped, with complete generality, and
this leads to transforming a correct equality in a correct way into an equal-
ity that is equivalent and was desired. I claim that, although in a diff erent
form, the same mathematical work is carried out on the basis of algorithms
in the commentary we analysed. Th is is the element that I recognize to be
present in the ancient Chinese text and for which I retain the expression
under discussion. Th is interpretation implies a use of the term ‘algebraic’ in
relation to operating on the operations themselves.
Let us, at this point, recapitulate the transformations that we identifi ed
by means of our analysis and that were carried out on a list of operations.
We had:- i. Eliminating inverse operations that follow each other
31 In an algebraic proof of a more general type, transformations can be applied to both sides
of the sign of equality in parallel, that is, to two lists of operations simultaneously. Th e
formulas used recall those stated by Li Ye in his Sea-Mirror of the Circle Measurements
(1248), where formulas express the fact that diff erent operations on diff erent entities lead to
the same result.Division by 3 Make communicate Multiplications, sums, etc.C (^) i > D (^) i =
a
b
i
i
3
3ai + bi = Ci (............) >
C (^) s D (^) s =
a
b
s
s
3
3 as + bs = Cs
has been transformed into
Multiplications, sums, etc.
C (^) i (............) >
C (^) s