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AFAIK, Archimedes is credited with discovering the following formula for computing the sum of squares:

$$1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$$

This seems to have come up in his quest for finding the area of a parabolic segment. I believe the sum of the first $n$ numbers was already known since it's easily derivable by observing the arrangement of stones as triangular numbers (although I haven't been able to find a reference yet, but remember reading about it in my trip down math history).

However, I wasn't able to find any credible reference to Archimedes' derivation of this formula. All derivations taught in school today have some "modern" variation attached to it. How was this derived by Archimedes given the state of math at his time?

The closest reference is this MAA Page but it doesn't give any insight into this (or even state things clearly). Here's a gist on this Math.SE Question.

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  • $\begingroup$ Complete surviving work of Archimedes is easily available online. $\endgroup$
    – Alexandre Eremenko
    Mar 23, 2018 at 2:13
  • $\begingroup$ It’s been hard to find this exact solution. Would you have a reference/link? $\endgroup$
    – PhD
    Mar 23, 2018 at 2:20
  • $\begingroup$ Do you have reason to believe it was a "quest" rather than simply trying a few formulas and finding that this one worked? $\endgroup$
    – Carl Witthoft
    Mar 23, 2018 at 11:11
  • $\begingroup$ @CarlWitthoft - the reason for that word was my understanding of it as a crucial element to his proofs. He had to have spent sometime with a non-algebraic view of math to come up with it. It’s quite hard to if you try it. Perhaps I was projecting an emotion I felt? :) $\endgroup$
    – PhD
    Mar 23, 2018 at 15:15

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Although it is commonly said that Archimedes summed the geometric series in Quadrature of the Parabola, and summed squares of integers in On Conoids and Spheroids and On Spirals, that is not what he did. He "sums" not even areas of figures (as in numbers associated to them) but figures themselves, and he certainly does not relate any of that to stacking pebbles into triangular numbers, let alone to any algebraic manipulations. Those date to neo-Pythagoreans at least three centuries after Archimedes.

To see how different what Archimedes did is from a formula here is the relevant passage from On Spirals in Dijksterhuis's translation:

"If a series of any number of lines be given, which exceed one another by an equal amount, and the difference be equal to the least, and if other lines be given equal in number to these and in quantity to the greatest, the squares on the lines equal to the greatest, plus the square on the greatest and the rectangle contained by the least and the sum of all those exceeding one another by an equal amount will be the triplicate of all the squares on the lines exceeding one another by an equal amount."

"Translated" into a formula, nothing close to which could have been written until Vieta almost two millenia later, this becomes $$(n+1)L_n^2+L_1(L_1+L_2+\cdots+L_n)=3(L_1^2+L_2^2+\cdots+L_n^2),$$ where $L_{i+1}-L_i$ is constant.

How it was derived is discussed at length in section 1.2 of Mathematical Masterpieces by Knoebel et al., who remark that "while Archimedes expressed this as an equality of areas, today we tend to interpret it as just a formula about numbers, ignoring the dimensionality involved", and refer to Kanim's conjectural reconstruction of Archimedes’s geometric reasoning. The most detailed commentary on it with a step by step "discovery" guide is in Pengelley's Sums of Numerical Powers in Discrete Mathematics (pp. 8-10). Kanim, whose idea was published as a short note How Did Archimedes Sum Squares in the Sand? (freely available from MAA) accompanied by a "proof without words" (below), writes for her part:

"The modern transcriptions of his proof (Dijksterhuis, Heath, Heiberg) are completely algebraic and hard to follow. The geometry of Archimedes' proof is depicted visually below. Each step of his written proof is transparent in the geometry of the picture. One wonders if this is the picture Archimedes drew in the sand."

enter image description here

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    $\begingroup$ I just wish I could understand the picture proof. I just dont "see" it 🤔 $\endgroup$
    – PhD
    Mar 25, 2018 at 7:10
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    $\begingroup$ @PhD This is not the case when one can expect to see anything by looking. The picture proof is supposed to be looked at next to the text of Archimedes's demonstration, whose steps it illustrates. The idea is telescoping, and Pengelley's paper prepares for it by walking through "sum of first powers" via squares and gnomons (p.7). $\endgroup$
    – Conifold
    Mar 25, 2018 at 20:15
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You are right about the person (Archimedes), but not about the work. He stated it as proposition 10 of his text On spirals. In modern language, the statement is$$(\forall n\in\mathbb{N}):(n+1)n^2+(1+2+\cdots+n)=3(1^2+2^2+\cdots+n^2).$$See page 162 of this edition of the works of Archimedes.

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  • $\begingroup$ presumably he also had the well-known (now) formula for $\sum^n_1(j)$ ? $\endgroup$
    – Carl Witthoft
    Mar 23, 2018 at 18:32
  • $\begingroup$ @CarlWitthoft My guess is that that formula was trivial for him. $\endgroup$
    – José Carlos Santos
    Mar 23, 2018 at 18:33

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