You are here

Extending al-Karaji's Work on Sums of Odd Powers of Integers - Algebraic Justification

Author(s): 
Hakan Kursat Oral (Yildiz Technical University) and Hasan Unal (Yildiz Technical University)

In his article in Mebahis-i İlmiyye, Vidinli offers another method of calculating the area of each gnomon making up the square in Figure 3: he computes the area of each gnomon as the difference of the areas of two squares. Thus, the area of the largest gnomon is:  \[ {(1 + 2 + 3 + \cdots + n)}^2 - {(1 + 2 + 3 + \cdots + (n-1))}^2 .\]  Rewriting the sums as \[ {1 + 2 + 3 + \cdots + n} = {\frac{n(n+1)}{2}}\] and \[ {1 + 2 + 3 + \cdots + (n-1)} = {\frac{(n-1)n}{2}}\] then gives the area of the largest gnomon as \[{\bigg[ {\frac{n(n+1)}{2}}\bigg]}^2 - {\bigg[ {\frac{(n-1)n}{2}}\bigg]}^2 = n^3.\]

This argument using the difference of two squares is the basis of the algebraic justification of al-Karaji’s formula for the sum of the cubes that Vidinli gives next in his article. However, this justification does not appear in the manuscript of al-Karaji that we have seen, nor does it appear in Woepcke's translation of the manuscript he saw. The formula for the sum \(S_n\) of the natural numbers from \(1\) to \(n,\) \[S_n = {1 + 2 + 3 + \cdots + n} = {\frac{n(n+1)}{2}},\] is a formula al-Karaji would have known very well.

In Figure 6, the first three lines of Vidinli's argument read as follows: \[ {1 + 2 + 3 + \cdots + n} = {\frac{1}{2}}n(n+1) = S_n\] \[ {1 + 2 + 3 + \cdots + (n-1)} = {\frac{1}{2}}n(n-1) = S_{n-1}\] \[S_n^{\,2} - S_{n-1}^{\,\,2} = {\frac{1}{4}}{n^2} \bigg({(n+1)}^2 - {(n-1)}^2 \bigg) = n^3 \]

Algebraic justification

Figure 6. Algebraic justification of al-Karaji’s formula for the sum of the cubes (from Mebahis-i İlmiyye, 1867, courtesy of the authors).

The areas of the \(n\)th through the first gnomons are then written successively as follows:  \[ S_n^{\,2} - S_{n-1}^{\,\,2} = n^3 ,\] \[ S_{n-1}^{\,\,2} - S_{n-2}^{\,\,2} = {(n-1)}^3 ,\] \(\phantom{.}\) \[\dots\dots\dots\dots\dots ,\] \[ S_2^{\,2} - S_{1}^{\,2} = 2^3 ,\] \[ S_1^{\,2} - S_{0}^{\,2} = 1^3\,\,\,(S_0=0) \]

If we add both sides of these equations, only \( {S_n^{\,2}}\) remains on the left side and on the right side we have the sum of cubes, giving the equality:   \[ {S_n^{\,2}} = 1^3 + 2^3 + 3^3 + \cdots + n^3 .\]

This final equation in Figure 6, then, is al-Karaji’s formula, \[ {{(1+2+3+\cdots +n)}^2} = 1^3 + 2^3 + 3^3 + \cdots + n^3,\] where \( {1+2+3+\cdots +n}\) is denoted by \(S_n.\)

Hakan Kursat Oral (Yildiz Technical University) and Hasan Unal (Yildiz Technical University), "Extending al-Karaji's Work on Sums of Odd Powers of Integers - Algebraic Justification," Loci (August 2011), DOI:10.4169/loci003725

Dummy View - NOT TO BE DELETED