×

On the “Tetsujutsu Sankei” of Takebe and comparison with the “Arithmetica Infinitorum” of Wallis. (Sur le “Tetsujutsu Sankei” de Takebe et comparaison avec “Arithmetica Infinitorum” de Wallis.) (French) Zbl 0657.01006

Histoire des mathématiques, Journ. SMF, Marseille/France 1983, Cah. Hist. Philos. Sci., Nouv. Sér. 20, 11-23 (1987).
The aim of the paper is to show some particular aspects of Wasan, the traditional Japanese mathematics, which reached its highest stage of development between the 17th century and the end of the 19th century, by comparing some extracts from the Tetsu-jutsu San-kei of Katahiro Takebe with the Arithmetica infinitorum of John Wallis. There are remarkable resemblances in those parts that concern the calculation of \(\pi\), despite the great differences of the scientific traditions within which the two mathematicians worked. Besides summary accounts of the work of Wallis and Takebe, the author includes a brief history of Wasan, whose chief representative in the seventeenth century was Seki Kowa. Both Wallis and Seki employ first-order differences in their respective heuristic methods of interpolation and extrapolation. Takebe’s originality lies in his employment of higher-order differences, which led him finally to a formulation equivalent to the Taylor series. Takebe employs the idea of the potential infinite as an heuristic principle, whereas the idea of an actual infinite of the type envisaged by Cavalieri underlies the Arithmetica infinitorum of Wallis. The most profound difference between the two mathematicians, however, is that, whereas Wallis clearly recognizes the need for deductive reasoning in mathematics (in addition to heuristic techniques), there is no trace of deductive reasoning in the work of Takebe. Indeed this absence of deductive reasoning is characteristic of Wasan.
[For the entire collection see Zbl 0643.00006.]

MSC:

01A27 History of Japanese mathematics
01A50 History of mathematics in the 18th century
01A45 History of mathematics in the 17th century

Biographic References:

Takebe, Katahiro; Wallis, John

Citations:

Zbl 0643.00006