More Islamic math

Al Karaji (?- 1019) wrote "The Marvelous" and studied algebra of exponents. So
1:x= x:x^2 =x^3: x^2.....

He also gave a limited algorithm for square roots of a polynomials. Al Karaji also proved
1^3 +2^3 +....+10^3 = (1+...+10)^2 (which you may have seen in some "intro to proof" class) by inverse induction.

His proof as follows: Using the diagram above
Area(red square) =  (1 + 2 +  · · · + 9)²
Area (yellow rectangle) =  10(1 + 2 +  · · · + 9)
Area(blue square) = 10² 

Then,
Area(2 yellow rectangle + blue square) = 2* 10(1 + 2 +  · · · + 9) + 10²
                                                                   = 2*10(9*10/2)  + 10² 
                                                                  = 10 * 10²
                                                                  = 10³      (1)

Area (of Entire gnomon) = (1 + 2 +  · · · + 9)² +2*10(1 + 2 +  · · · + 9) +10²

 Plugging (1) into the above equation yields
Area (of Entire gnomon) = (1 + 2 +  · · · + 9)² + 10³
                                           = (1 + 2 +  · · · + 9+ 10)²


By repeating a similar argument for the rest of the number yields Al karaji's results
 1^3 +2^3 +....+10^3 = (1+...+10)^2

QED

Al Samaw al (1112-1174) wrote "Shining Book of Calculation" and performed long division.

Unanswered Question

1. Was there any practical purpose for Al karaji discovery of  1^3 +2^3 +....+10^3 = (1+...+10)^2 ?

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