Reading proofs in Chinese commentaries 449
grouped into a unique division. Th e technical term chosen for this division
refers to the motivation of the eff ected transformation. As a consequence,
algorithm 5
Ci, Cs > (Ci Cs + Ci^2 + Cs^2 )h > (Ci Cs + Ci^2 + Cs^2 )h/9 > [(Ci Cs+ Ci^2 + Cs^2 )h/9]/4
is transformed into the algorithm
Ci, Cs > (Ci Cs + Ci^2 + Cs^2 )h > (Ci Cs + Ci^2 + Cs^2 )h/36
which is equivalent to it and identical to the desired algorithm. Th is was
what was to be obtained: the correctness of the procedure provided by Th e
Nine Chapters is established. Th e way in which the proof was conducted
highlights in the best way possible how the activities of shaping an algo-
rithm and proving the correctness are intertwined.
Such is the type of proof that I suggest designating as ‘algebraic proof in
an algorithmic context’. It is characterized by the articulation of the three
lines of argumentation I distinguished. However, clearly, the second line
of argumentation is the one that is specifi c to it. Several points need to be
made clear to explain the expression by which I suggest referring to this
kind of proof.
First, to justify the fact that I speak here of an ‘algorithmic context’, it will
be useful to compare what we analysed with a translation in modern terms.
Th e reasoning we followed can be rewritten as the following sequence
of steps:
V
CC C C
hCC C C
is i siiss=
+⎛
⎝
⎜
⎞
⎠
⎟ +
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
++
33 3 3
.
3
3
4
9
2222.
⎡⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
⎡⎣ ++⎤⎦
=
⎡⎣ ++⎤⎦
.
3
3
4
.
9
1
4
.
3
2222hCC C C hCC C C his i sis i s.
.
(^66)
Th e fi rst line encapsulates the fi rst line of reasoning, which establishes
an algorithm fulfi lling the task required by the terms of the problem. In
the following lines, corresponding to the second line of argumentation,