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: ${1 + 2 + 3 + \cdots + n} = {\frac{1}{2}}n(n+1) = S_n$ ${1 + 2 + 3 + \cdots + (n-1)} = {\frac{1}{2}}n(n-1) = S_{n-1}$ $S_n^{\,2} - S_{n-1}^{\,\,2} = {\frac{1}{4}}{n^2} \bigg({(n+1)}^2 - {(n-1)}^2 \bigg) = n^3$

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," Convergence (August 2011), DOI:10.4169/loci003725