his later editions Yoshidaappended a number of advanced problems
to be solved by competitors. This procedure started among the
Japanese the practice of issuing problems, which was kept up until
1813and helped to stimulate mathematical activity.

Another pupil of Mōri was IMAMURA CHISHŌwho, in 1639, pub-
lished a treatise entitled Jugairoku, written in classical Chinese. He
took up the mensuration of the circle, sphere and cone. Another
author, ISOMURA KITTOKU, in his Kelsugishō, 1660(second edition
1684), when considering problems on mensuration, makes a crude
approach to integration. He gives magic squares, both odd and even
celled, and also magic circles. Such squares and circles became favor-
ite topics among the Japanese. In the 1684edition, Isomuragives
also magic wheels. TANAKA KISSHINarranges the integers 1-96 in
six 42-celled magic squares, such that the sum in each row and column
are 194; placing the six squares upon a cube, he obtains his "magic
cube." Tanaka formed also "magic rectangles." ¹ MURAMATSUin
1663gives a magic square containing as many as 92 cells and a magic
circle involving 129 numbers. Muramatsu gives also the famous
"Josephus Problem" in the following form: 'Once upon a time there
lived a wealthy farmer who had thirty children, half being of his first
wife and half of his second one. The latter wished a favorite son to
inherit all the property, and accordingly she asked him one day, say-
ing: Would it not be well to arrange our 30 children on a circle, calling
one of them the first and counting out every tenth one until there
should remain only one, who should be called the heir. The hus-
band assenting, the wife arranged the children . .; the counting . .
resulted in the elimination of 14 step-children at once, leaving only
one. Thereupon the wife, feeling confident of her success, said,
let us reverse the order. . The husband agreed again, and the
counting proceeded in the reverse order, with the unexpected result
that all of the second wife's children. were stricken out and there re-
mained only the step-child, and accordingly he inherited the property."
The origin of this problem is not known. It is found much earlier in
the Codex Einsidelensis (Einsideln, Switzerland) of the tenth century,
while a Latin work of Roman times attributes it to Flavius Josephus.
It commonly appears as a problem relating to Turks and Christians,
half of whom must be sacrificed to save a sinking ship. It was very
common in early printed European books on arithmetic and in books
on mathematical recreations.

In 1666 SATŌ SEIKŌwrote his Kongenkiwhich, in common with
other works of his day, considers the computation of π( = 3.14).
He is the first Japanese to take up the Chinese "celestial element
method" in algebra. He applies it to equations of as high a degree as
the sixth. His successor, SAWAGUCHI, and a contemporary NOZAWA,
give a crude calculus resembling that of Cavalieri. Sawaguchi rises

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