Draft:Alhazen's Summation Formulas

• Comment: This needs to clarify how it relates to Faulhaber's formula and especially the history section of that article, which states that formulas up to the sum of cubes were known much earlier than ibn al-Haytham and which places ibn al-Haytham among a line of other "middle period" researchers including al-Karaji before him from the same milieu. Is it really a separately-notable topic than what is already in that article? —David Eppstein (talk) 22:51, 23 July 2025 (UTC)
  ::Considering its significance to the development of calculus and the amount of literature discussing it, I believe it is good to have a Wikipedia article on this topic. ~~~~

Alhazen's Summation Formulas are formulas for the sum of integral powers. The Arab mathematician and physicist Alhazen (c. 965–1040) derived these formulas by coordinating a geometrical interpretation with a numerical representation. Alhazen was the first to determine the formula for the sum of fourth powers, while the formulas for the sums of squares and cubes were stated earlier by his predecessors. His method can be used to derive the formula for the sum of the powers of the first n integers for any positive integer power. Alhazen's calculation of the volume of a paraboloid, using results he had obtained, is considered a significant breakthrough in the history of integral calculus.

Historical background

Alhazen was a physicist as well as a mathematician. His work was based on the Greek geometrical tradition, but he also pursued practical empirical results in optics and astronomy. The formula for the sum of squares was stated by Archimedes around 250 B.C. in his work on the quadrature of the parabola. The formula for the sum of cubes was first explicitly written down by Aryabhata in India around 500, though it was likely known to the Greeks.

Greek mathematics, with the exception of Diophantus and Heron, does not mention any powers higher than three because they could not be directly interpreted in geometry as lengths, areas, or volumes. Alhazen sought to calculate new results concerning areas and volumes that involved the summation of powers higher than three. He was the first to derive the formula for the sum of fourth powers. His use of areas to represent third and fourth powers was a departure from the strict geometrical interpretations in Greek mathematics. He obtained formulas for the sums of higher powers by coordinating a geometrical interpretation with a numerical representation.

During the early seventeenth century, Faulhaber extended these results up to 17th powers. His formulae, however, did not suggest to him a scheme which might allow for indefinite extension to higher powers.

Significance

Alhazen was the first to determine the formula for the sum of fourth powers. If a method can be found to determine the formula for the sum of fourth powers, then a method can also be found to determine the formula for the sum of any integral power. Ibn al-Haytham demonstrated how to develop the formula for the kth powers from $k=1$ to $k=4$. All of his proofs were similar in nature and could be easily generalized to discover and prove the formulas for the sum of any given powers of integers.

The result obtained by Alhazen is considered significant in the development of integral calculus. Alhazen determined the equations to calculate the area enclosed by the curve represented by $y=x^{k}$ (which translates to the integral $\int x^{k}\,dx$ in contemporary notation), for any given non-negative integer value of $k$. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. In other words, this finding allowed Alhazen to calculate the areas and volumes for curves and surfaces defined by polynomial equations of the form $y=P(x)$.

Derivation

Alhazen's method for deriving the formulas for the sums of higher powers involved coordinating a geometrical interpretation with a numerical representation.

He began by laying out a series of rectangles whose areas represent the terms of the sum. For a power k, a rectangle of area $a^{k}$ is formed with sides of length $a^{k-1}$ and a. He then filled in the rectangle with a series of interlocking strips. By setting the product of the rectangle's dimensions equal to the sum of its rectangular parts, he could derive the formula for the sum. Every formula can be obtained from the preceding one. The top strips, however, require a double summation in every derivation except the first.

The central idea in his proof was the derivation of the equation: $(n+1)\sum _{i=1}^{n}i^{k}=\sum _{i=1}^{n}i^{k+1}+\sum _{p=1}^{n}\left(\sum _{i=1}^{p}i^{k}\right)$ Alhazen himself did not state the result in this general form. He only proved it for specific integers, namely for k = 1, 2, 3, and for n=4. His proofs were by induction on n and could be generalized to any value of k.

Sum of integers

Alhazen's geometric proof for the sum of the first n integers. The total area of the rectangle, n(n+1), is dissected into two sets of smaller rectangles, each representing the sum 1 + 2 + ... + n

To find the formula for the sum of the first n integers, he used a rectangle with dimensions n by n+1. He showed that the area could be represented in two ways: $n(n+1)=\sum _{i=1}^{n}i+\sum _{i=1}^{n}i$ This leads to the formula: ${\frac {1}{2}}n(n+1)={\frac {1}{2}}n^{2}+{\frac {1}{2}}n=\sum _{i=1}^{n}i$

Sum of squares

To find the sum of squares, he used the previously derived formula for the sum of integers. He constructed a rectangle with dimensions $(n+1)$ and the sum of the first n integers. By equating the area of the rectangle with the sum of its parts, he derived the following relationship: $\left(\sum _{i=1}^{n}i\right)(n+1)=\sum _{i=1}^{n}i^{2}+\sum _{i=1}^{n}\left(\sum _{k=1}^{i}k\right)$ Using the known formula for the sum of integers, he solved for the sum of squares: $\sum _{i=1}^{n}i^{2}={\frac {1}{3}}n^{3}+{\frac {1}{2}}n^{2}+{\frac {1}{6}}n$

Sum of cubes and fourth powers

He continued this method for cubes and fourth powers. For the sum of cubes, he used the formulas for the sum of integers and the sum of squares. The equation is: $\left(\sum _{i=1}^{n}i^{2}\right)(n+1)=\sum _{i=1}^{n}i^{3}+\sum _{i=1}^{n}\left(\sum _{k=1}^{i}k^{2}\right)$ This yields the formula: $\sum _{i=1}^{n}i^{3}={\frac {1}{4}}n^{4}+{\frac {1}{2}}n^{3}+{\frac {1}{4}}n^{2}$ He then generalized this method to find the sum of fourth powers. This process requires the use of the three previously derived formulas. For the sum of fourth powers, Alhazen did not provide a full derivation for an arbitrary n. Instead, he derived the result for the specific case of n=4 and then asserted the general formula for any integer n. He stated his result verbally, which can be translated into modern notation in the following factored form: $\sum _{i=1}^{n}i^{4}=\left({\frac {n}{5}}+{\frac {1}{5}}\right)n\left(n+{\frac {1}{2}}\right)\left((n+1)n-{\frac {1}{3}}\right)$ This can be expanded into the more familiar polynomial form: $\sum _{i=1}^{n}i^{4}={\frac {1}{5}}n^{5}+{\frac {1}{2}}n^{4}+{\frac {1}{3}}n^{3}-{\frac {1}{30}}n$

Alhazen's formulas

Here are the formulas Alhazen derived for the sums of the first n integers, squares, cubes, and fourth powers:

• $\sum _{i=1}^{n}i={\frac {1}{2}}n^{2}+{\frac {1}{2}}n$
• $\sum _{i=1}^{n}i^{2}={\frac {1}{3}}n^{3}+{\frac {1}{2}}n^{2}+{\frac {1}{6}}n$
• $\sum _{i=1}^{n}i^{3}={\frac {1}{4}}n^{4}+{\frac {1}{2}}n^{3}+{\frac {1}{4}}n^{2}$
• $\sum _{i=1}^{n}i^{4}={\frac {1}{5}}n^{5}+{\frac {1}{2}}n^{4}+{\frac {1}{3}}n^{3}-{\frac {1}{30}}n$

Alhazen's method can be extended to find a formula for the sum of the powers of the first n integers for any integer power, creating a general recursion scheme.

Application: Volume of a paraboloid

Alhazen applied his summation formulas to calculate the volume of a solid generated by rotating a parabola around a line perpendicular to its axis of symmetry. For a modern student, this would require integrating a fourth-power polynomial.

He used the method of exhaustion, setting up upper and lower sums of cylindrical slices and then refining the slices. The radius of each cylindrical slice follows a square function, so the areas of the slices follow a fourth power. The sum of the areas of these slices involves summing fourth powers.

The specific summation needed for this calculation is: $\sum _{i=1}^{n}(n^{2}-i^{2})^{2}={\frac {8}{15}}n^{5}-{\frac {1}{2}}n^{4}-{\frac {1}{30}}n$ This can be derived from the formulas for the sum of squares and fourth powers.

Following the tradition of Greek mathematics, Alhazen presented his result as a ratio. He stated that the volume of the rotated parabola is 8/15 of the volume of the circumscribed cylinder. This result was a continuation of the tradition of expressing areas and volumes as ratios, such as the area of a triangle being half the area of the parallelogram that contains it. John Wallis later used Alhazen's ratio of 8/15 in his work on fractional and negative exponents.

Legacy

Over the following centuries, Ibn al-Haytham's formula for the sum of fourth powers reappeared in various parts of the Islamic world. It can be found in the work of Abu-l-Hasan ibn Haydur (d. 1413) and Abu Abdallah ibn Ghazi (1437-1514), both of whom lived in modern-day Morocco. Additionally, the formula is present in The Calculator's Key by Ghiyath al-Din Jamshid al-Kashi (d. 1429), a mathematician and astronomer who was most productive in Samarkand (now Uzbekistan) at Ulugh Beg's court. Nevertheless, how these mathematicians became aware of the formula or the purposes for which they used it is unknown.

Further reading

• "Sums of Powers of Positive Integers - Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham (965-1039), Egypt | Mathematical Association of America". old.maa.org. Retrieved 2025-07-25.
• Dennis; Addington, Susan (2009). Alhazen's Summation Formulas. Mathematical intentions.https://www.quadrivium.info/MathInt/Notes/AlHazenSummation.pdf
• "Calculus Before Newton and Leibniz – AP Central | College Board". apcentral.collegeboard.org. Retrieved 2025-07-25.
• Katz, Victor J. (1995). "Ideas of Calculus in Islam and India". Mathematics Magazine. 68 (3): 163–174. doi:10.2307/2691411. ISSN 0025-570X.
• Dennis, David; Kreinovich, Vladik; Rump, Siegfried M. (1998-05-01). "Intervals and the Origins of Calculus". Reliable Computing. 4 (2): 191–197. doi:10.1023/A:1009989211143. ISSN 1573-1340.