Chapter 52 Sixth apex
For more than a thousand years after Liu Hui and Zu's father and son, the idea of limit and infinitesimal division has not made significant progress in China, and has not even reached the level of Liu and Zu.In the Yuan Dynasty, Zhao Youqin cut a circle from a regular quadrilateral inside a circle, only to verify that Zu Chongzhi's density ratio was relatively accurate, and made little theoretical contribution.In fact, Liu Hui's thoughts have not attracted enough attention from later generations.At the beginning of the eighteenth century, the French missionary Du Demei (AD 1668-1720) introduced the power series expansions of the three trigonometric functions created by Newton and Gregory, but did not introduce their derivation methods.Mongolian mathematician Ming Antu (AD? - 1766?), Dong Youcheng, Xiang Mingda, Dai Xu, Xu Youren (AD 1800-1860), Li Shanlan, Xia Luanxiang (AD 1823-1864) and others worked with great energy Great achievements have been made in the study of such problems and the power series expansions of logarithmic and exponential functions.They have superhuman intelligence and commendable spirit, which fully demonstrates the courage of the outstanding elements of the Chinese nation not to be left behind by others.However, as the West has entered the era of analytical mathematics, it is impossible to follow the example of using elementary methods to devote years or even decades of hard work to a few formulas without trying to learn other people's advanced mathematical methods.
Among the mathematicians of the Qing Dynasty, Li Shanlan's sharp cone quadrature was the first to surpass Liu Hui and Zu's father and son in terms of infinitesimal division and limit thinking.He pointed out in "Fang Yuan Shan You", "It should be known that all the squares can be turned into planes, and all can be turned into lines", that is, if x is any positive number and n is any positive integer, the value of xn can be expressed as a The plane area can also be expressed as a straight line segment.He further pointed out that "you should know that all the squares have sharp cones", "you should know that all the sharp cones have the principle of accumulation", that is, when x is in the interval [O, h], it means that the plane area of xn is stacked into a sharp cone body.He proposed the algorithm of various pointed cones: the pointed cones stacked by the plane area axn, the height is h, the bottom area is ah, and its volume is (ah×h)/(n+1).This proposition is equivalent to the definite integral ∫axdx=(ah×h)/(n+1)
Figure 40 Spike surgery
He also proposed the proposition equivalent to ∫axdx+∫axdx+...+∫axdx=∫(ax+ax+...+ax)dx.Li Shanlan applied his cone quadrature to the calculation of the area of a circle.For this he considers 1/4 of the unit circle.As shown in Figure 40, OABC is a square with a side length of 1, and 1/4 of its inner circle is OAQC.In order to find the area of OAQC, he first calculates the area of ABCQ outside the inner circle of the square.This is a pointed cone.This cone is the sum of ABD, ADE, AEF, AFG... countless cones.The bottoms of all sharp cones are: BD=BC=1/2, DE=1/4DC=1/(2·4), EF=(1/6)EC=3/(2·4·6), FG= (1/8)FC=(3·5)/(2·4·6·8)...Cone quadrature, the area of the cone ABCQ should be:
Therefore, the area of the unit circle is
In "Exploring the Origin of Logarithm", Li Shanlan also used the cone technique to solve the power series expansion of the logarithmic function.He calculated the product of a pointed cone L(y)=by+by/2h+by/2h+by/4h+…
And it is proved that when y, y, y...Geometric series, the corresponding L(y), L(y), L(y)...Geometric series, so L(y) has logarithm nature.
If by=1, y=(n-1)h/n, then
L=[(n-1)h/n]=(n-1)/n+1/2[(n-1)/n]+(1/3)[(n-1)/n]+...
This is the natural logarithm 1 of n, which is equivalent to the definite integral
These works of Li Shanlan are roughly similar to the work of European mathematicians before Newton and Leibniz completed calculus, and they were completed before he came into contact with Western calculus.Although the preliminary knowledge for completing this work includes Western elementary mathematics introduced since the late Ming and early Qing Dynasties, generally speaking, it is a creative work independently completed on the basis of traditional Chinese mathematics without being influenced by Western calculus ideas. Work.Obviously, the view that Chinese classical mathematics cannot be developed into modern mathematics is untenable.
Xia Luanxiang also has outstanding work in power series expansion, and created a series expansion formula for calculating the integral of the surface area formed by the rotation of a part of an elliptic curve around the major axis (or minor axis), but this is in "Dai Microji Ten Levels" "Completed on the basis of.
Figure 40 Spike surgery
In "Exploring the Origin of Logarithm", Li Shanlan also used the cone technique to solve the power series expansion of the logarithmic function.He calculated the product of a pointed cone L(y)=by+by/2h+by/2h+by/4h+…
And it is proved that when y, y, y...Geometric series, the corresponding L(y), L(y), L(y)...Geometric series, so L(y) has logarithm nature.
If by=1, y=(n-1)h/n, then
L=[(n-1)h/n]=(n-1)/n+1/2[(n-1)/n]+(1/3)[(n-1)/n]+...
This is the natural logarithm 1 of n, which is equivalent to the definite integral
These works of Li Shanlan are roughly similar to the work of European mathematicians before Newton and Leibniz completed calculus, and they were completed before he came into contact with Western calculus.Although the preliminary knowledge for completing this work includes Western elementary mathematics introduced since the late Ming and early Qing Dynasties, generally speaking, it is a creative work independently completed on the basis of traditional Chinese mathematics without being influenced by Western calculus ideas. Work.Obviously, the view that Chinese classical mathematics cannot be developed into modern mathematics is untenable.
Xia Luanxiang also has outstanding work in power series expansion, and created a series expansion formula for calculating the integral of the surface area formed by the rotation of a part of an elliptic curve around the major axis (or minor axis), but this is in "Dai Microji Ten Levels" "Completed on the basis of.