Page:A History of Mathematics (1893).djvu/130

 Al Kuhi, the second astronomer at the observatory of the emir at BagdadBaghdad [sic], was a close student of Archimedes and Apollonius. He solved the problem, to construct a segment of a sphere equal in volume to a given segment and having a curved surface equal in area to that of another given segment. He, Al Sagani, and Al Biruni made a study of the trisection of angles. Abul Gud, an able geometer, solved the problem by the intersection of a parabola with an equilateral hyperbola.

The Arabs had already discovered the theorem that the sum of two cubes can never be a cube. Abu Mohammed Al Hogendi of Chorassan thought he had proved this, but we are told that the demonstration was defective. Creditable work in theory of numbers and algebra was done by who lived at the beginning of the eleventh century. His treatise on algebra is the greatest algebraic work of the Arabs. In it he appears as a disciple of Diophantus. He was the first to operate with higher roots and to solve equations of the form $$\scriptstyle{x^{2n}+ax^n=b}$$. For the solution of quadratic equations he gives both arithmetical and geometric proofs. He was the first Arabic author to give and prove the theorems on the summation of the series:—

$\begin{align}\scriptstyle{1^2 + 2^2 + 3^2 + \cdots + n^2} & \scriptstyle{= (1 + 2 + \cdots + n)^{{2n+1}\over 3}\text{,}}\\\scriptstyle{1^3 + 2^3 + 3^3 + \cdots + n^3} & \scriptstyle{= (1 + 2 + \cdots + n)^2\text{.}}\\\end{align}$|undefined

Al Karhi also busied himself with indeterminate analysis. He showed skill in handling the methods of Diophantus, but added nothing whatever to the stock of knowledge already on hand. As a subject for original research, indeterminate analysis was too subtle for even the most gifted of Arabian minds. Rather surprising is the fact that Al Karhi's algebra shows no traces whatever of Hindoo indeterminate analysis.