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the equation with full generality. Th is distinction between cases relates to a
concern about the validity of the operations in the proof and the generality
of the proven statement. Li Rui distinguished cases in such a way that the
proof carried out through polynomial computations would be valid for all
triangles of the same case. Th is step ensured the correctness of the kind of
algebraic proof which he conducted in a way that Li Ye’s proofs before him
did not.
As a consequence, Li Rui established general equations through poly-
nomial computations – the proof of their correctness – that were valid for
the particular category of triangles delineated and he probably developed
this structure of proof intentionally. Otherwise, there would be no reason
for him to diff erentiate diff erent cases of a given type. Yet, even though this
feature reveals that Li Rui was interested in the generality of procedures,
like the ancient Chinese mathematical texts, he expressed this property for
each case within the context of a particular problem, which he thus used as
a paradigm. We see here again that the search for generality and the ways
in which generality is expressed both account for specifi c features of the
practice of proof that was constructed.
Several other elements manifest how, through his mathematical practice,
Li Rui simultaneously presented himself as continuing the tradition of his
Chinese predecessors of the past and yet changed it. His deployment of
geometrical diagrams to provide yet another (geometric) proof of the cor-
rectness of the equation is one of these elements. However, although the
diagrams clearly call to mind Li Ye’s own diagrams in his Yigu yanduan ,
completed in 1259, or Yang Hui’s 1261 commentary on Th e Nine Chapters ,
they betray diff erences, due not least of all to an infl uence of Western
practices with geometrical fi gures. Further, like Xu Guangqi before him, Li
Rui seems to be using the concept of ‘meaning ( yi ′)’ in a way that displays
affi nity with how the commentators on Th e Nine Chapters used the same
term. Th is reveals a continuity of mathematical theory that has not yet been
addressed adequately.
In addition, Tian surmises that Li Rui was also interested in showing
the power of the ‘procedure of the right-angled triangle ( gougushu )’ – the
ancient name and formulation for Pythagoras’ theorem – to solve any
problem in a uniform way. Li’s book can be interpreted as having explicitly
developed the system covered by this older procedure, even if it had been
presented in the past in relation to a particular problem.
In conclusion, the Detailed Outline of Mathematical Procedures for the
Right-Angled Triangle demonstrates a synthesis of goals for and techniques
of proof, which take their origins from both East and West. Th e book