Aryabhata writes the sum of squares and sum of cubes — 1,100 years before Faulhaber

# Aryabhata writes the sum of squares and sum of cubes — 1,100 years before Faulhaber

Add up the first ten positive integers: $1 + 2 + 3 + \cdots + 10 = 55$.

Greek mathematicians knew the closed form: $\Sigma i = n(n+1)/2$. So did Indian mathematicians, separately and earlier.

Add up the first ten *squares*: $1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385$.

Add up the first ten *cubes*: $1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000 = 3025$.

There are clean closed forms for both. The sum of the first $n$ squares is $n(n+1)(2n+1)/6$ — for $n=10$, that's $10 \cdot 11 \cdot 21 / 6 = 385$. The sum of the first $n$ cubes is even cleaner: it equals the square of the sum of the original series, $(n(n+1)/2)^2$ — for $n=10$, that's $(10 \cdot 11 / 2)^2 = 55^2 = 3025$.

That second identity is surprising. The sum of cubes equals the square of a triangular number. There's no obvious reason in the arithmetic itself for this to be true. It's a piece of mathematical beauty that has to be proven.

Aryabhata stated both identities — without proof — in 499 CE.

> The sixth part of the product of three quantities consisting of > the number of terms, the number of terms plus one, and twice the > number of terms plus one is the sum of the squares. The square of > the sum of the (original) series is the sum of the cubes. > > — *Aryabhatiya Ganitapada* II.22, trans. W. E. Clark (1930)

Two formulas, one Sanskrit verse, 1,500 years ago.

What the verse is saying

Modern translation, line by line:

- "Three quantities consisting of the number of terms, the number of terms plus one, and twice the number of terms plus one" $\Rightarrow$ $n$, $n+1$, $2n+1$. - "The product of" those three $\Rightarrow$ $n(n+1)(2n+1)$. - "The sixth part of" $\Rightarrow$ divide by 6. - "Is the sum of the squares" $\Rightarrow$ $\sum_{i=1}^{n} i^2 = n(n+1)(2n+1)/6$.

Then:

- "The square of the sum of the (original) series" $\Rightarrow$ $\left(\sum_{i=1}^{n} i\right)^2 = (n(n+1)/2)^2$. - "Is the sum of the cubes" $\Rightarrow$ $\sum_{i=1}^{n} i^3 = (n(n+1)/2)^2$.

The verse is dense — compressed Sanskrit *sūtra* style — and contains no derivation. It's a *statement* of two correct closed forms. The proofs (induction, or geometric decomposition into square arrangements of cubes) were developed by later Indian commentators and, independently, in the European tradition centuries later.

A small piece of mathematical beauty

The sum-of-cubes identity is the more striking one:

$$1^3 + 2^3 + 3^3 + \cdots + n^3 \;=\; (1 + 2 + 3 + \cdots + n)^2.$$

You can verify it for small $n$ and the equality keeps holding:

| $n$ | $\sum i$ | $(\sum i)^2$ | $\sum i^3$ | |-----|----------|--------------|------------| | 1 | 1 | 1 | 1 | | 2 | 3 | 9 | 9 | | 3 | 6 | 36 | 36 | | 4 | 10 | 100 | 100 | | 5 | 15 | 225 | 225 |

There's a now-classic geometric proof: arrange $n$ squares side by side along the diagonal of an $(n(n+1)/2) \times (n(n+1)/2)$ grid, where the $k$-th square is $k$ on a side and is tiled by $k^2$ unit squares stacked $k$-deep. The total area of these layered squares is $\sum k^3$. But it also tiles a square of side $\sum k$, hence area $(\sum k)^2$. So they're equal.

Aryabhata almost certainly knew a version of that argument — Indian mathematical tradition was geometric-constructive in style — but he didn't write the proof down. The verse states the result. The proof was a teaching exercise.

The European recovery

The European mathematical tradition rediscovered both identities independently, but not for over a millennium.

**Diophantus** (~250 CE) gives $\sum i$ in *Arithmetica*. He does not give $\sum i^2$ or $\sum i^3$ in the surviving books.

**Nicomachus** (~100 CE) notes that consecutive odd numbers sum to cubes — $1 = 1^3$, $3+5 = 8 = 2^3$, $7+9+11 = 27 = 3^3$ — which is a special case of the sum-of-cubes identity, but Nicomachus does not generalise it.

**Alhazen** (~1000 CE, in the Arabic tradition) derives $\sum i^2$ and $\sum i^3$ — independently of the Indian sources, as far as the historical record shows.

**Pierre de Fermat** (~1636) computes $\sum i^p$ for $p = 2, 3, 4$ in letters to Mersenne — independently again.

**Johann Faulhaber** (1631, *Academia Algebrae*) gives polynomial formulas for $\sum i^p$ up to $p = 17$. This is the systematic European treatment that gets named-after-its-author.

**Jakob Bernoulli** (*Ars Conjectandi*, posthumous 1713) introduces the Bernoulli numbers $B_k$ and gives the general formula for $\sum i^p$ as a polynomial in $n$ of degree $p+1$ — the modern combinatorial-analytic generalisation that subsumes Aryabhata's two cases as the $p=2$ and $p=3$ instances.

So Faulhaber's 1631 *p*=2 and *p*=3 formulas are 1,132 years after Aryabhata's verse.

What this leaves us with

Two closed-form summation identities, correct, in one Sanskrit verse, 499 CE. The geometric beauty of the sum-of-cubes identity ($1^3 + 2^3 + \cdots + n^3$ = $(1 + 2 + \cdots + n)^2$) is not an obvious arithmetic fact; it's a piece of structural mathematical beauty that has to be discovered.

Aryabhata discovered it. He stated it. He moved on, in the next verse, to other arithmetic of pyramid-pile geometry. The *Aryabhatiya* doesn't pause on the beauty of the result; it presents it as a calculation tool, in the way modern textbooks present the formula for the area of a circle.

Indian mathematical tradition kept the identities continuously in the curriculum — Brahmagupta restates them (628 CE), Mahavira extends them (850 CE), Bhaskara II teaches them in the *Lilavati* (1150 CE), the Kerala school uses them in its infinite-series work (~1500 CE).

The European tradition rediscovered the same two closed forms, through Alhazen and Fermat and Faulhaber, over the course of roughly a millennium. The Bernoulli generalisation in 1713 is arguably the deeper result — it gives the *pattern*, not just two specific cases. But the two specific cases were already written down in Sanskrit, in 499 CE, in two clauses of a single verse.

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Sources

- [Aryabhatiya, W. E. Clark trans., 1930](https://archive.org/details/The_Aryabhatiya_of_Aryabhata_Clark_1930) — verse II.22 cited above. - Plofker, K. (2009). *Mathematics in India*. Princeton University Press, ch. 4. — the Sanskrit summation-identity tradition from Aryabhata through the Kerala school. - Faulhaber, J. (1631). *Academia Algebrae, Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden*. Augsburg. — the eponymous European polynomial-summation treatment. - Bernoulli, J. (1713). *Ars Conjectandi*. Basel: Thurnisius. — Bernoulli numbers and the general $\sum i^p$ polynomial.

Related claims

- [Aryabhata's pi approximation](/c/0b862684-d325-5002-b054-169bd2253ef9) — same chapter, same author, same year. - [Aryabhata's kuttaka](/c/a0ac1d5a-b8a4-5228-bb4e-e69636d8d613) — same chapter, the same compact-sutra style applied to integer congruences instead of summations.