Jade Mirror of the Four Unknowns Chinese mathematics

facsimile of zhu shijie s jade mirror of 4 unknowns

si-yüan yü-jian (《四元玉鑒》), or jade mirror of 4 unknowns, written zhu shijie in 1303 ad , marks peak in development of chinese algebra. 4 elements, called heaven, earth, man , matter, represented 4 unknown quantities in algebraic equations. deals simultaneous equations , equations of degrees high fourteen. author uses method of fan fa, today called horner s method, solve these equations.

the jade mirror opens diagram of arithmetic triangle (pascal s triangle) using round 0 symbol, chu shih-chieh denies credit it. similar triangle appears in yang hui s work, without 0 symbol.

there many summation series equations given without proof in precious mirror. few of summation series are:

1

2


+

2

2


+

3

2


+
⋯
+

n

2


=



n
(
n
+
1
)
(
2
n
+
1
)


3
!





{\displaystyle 1^{2}+2^{2}+3^{2}+\cdots +n^{2}={n(n+1)(2n+1) \over 3!}}






1
+
8
+
30
+
80
+
⋯
+




n

2


(
n
+
1
)
(
n
+
2
)


3
!



=



n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
(
4
n
+
1
)


5
!





{\displaystyle 1+8+30+80+\cdots +{n^{2}(n+1)(n+2) \over 3!}={n(n+1)(n+2)(n+3)(4n+1) \over 5!}}






^ (boyer 1991, china , india p. 203)
^ (boyer 1991, china , india p. 205)