论张衡的圆周率

The Ratio of Circumference to Diameter by Zhang Heng

认为张衡的圆周率不但是最粗疏的（比其前的刘歆率还要粗疏），而且其理论也有错误。其实张衡是第一个从理论上求得圆周率的人，他从“为术者”那里继承了丸柱误率，认为立方／丸＝（π／４）２，并把其中的经验值９／１６改为１０／１６，从而求得。他的这种想法是很精彩的，而且除却为术者的说法有误以外，他的全部推导都是正确的，而他所开辟的从理论而求圆周率的道路则是非常有意义的，刘徽正是沿着这条路而获得巨大成绩的，张衡在圆周率上的贡献太被人们忽略了。

it is principally from Liu Hui's Comments on Nine Chapters that we know about ZhangHeng's theory on the ratio of eircumferenee of a circle to its diameter. It is noticed that not only theapproximate error of his ratio is very large, larger than his forerunners, but also his very treatementwas based on an erroneous old argument that the ratio of the volume of a sphere to its circumscribedcylinder is equal to(π/4)2.Hence his theory of the ratio is underestimated by all of us.NonethelessZhang Heng was the first one to evaluate the ratio theoretically.Adopting the old argument, hechanged the experimental value 9/16 in relevant formula to10/16so as to get the result Suchan idea was very excellent,and excep the adopted erroneous old argument, the whole of his deductionis quite correct. It is very important that he opened up the way to evaluate the ratio from theoreticalside; it is just alons this line that Liu Hut got fruitful results.