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I'm interested in knowing how was the first closed form solution of the sum of squares derived (for historical context/curiosity). I stumbled across an MAA Page that provides a visual proof, without words, of how Archimedes may have thought about it, in the sand.

However, I'm not able to make much sense of it. How exactly can this be interpreted?

Here's Archimedes' proof statement from the above link:

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.

Visual Proof for Archimedes's sum of squares

Here's the explanation on Pg. 8 of the associated PDF:

Archimedes provides a quite elaborate proof of his claim, with no picture. Student Kathe Kanim recently found in Archimedes a rather algebraic proof the insight and inspiration for a geometric picture proof, as displayed in Figure 1 [4]. It seems that she may have rediscovered the picture that was in Archimedes' mind as he was drawing in the sand more than two thousand years ago. Mark up the displayed picture to explain why it proves what Archimedes claims. Notice that it only proves the claim for a particular number of lines. How many? The picture nonetheless should convince you that the claim is true for any number of lines. Explain why it does that. Could you draw the picture necessary to prove it for nine lines? nineteen thousand lines? Explain why you are sure of that. The fact that you are convinced means that this has the nature of a proof by generalizable example, which was a common method of proof in mathematics until perhaps one or two hundred years ago. However, as mathematics became more highly developed, elaborate proofs by this method became less acceptable, because they rely on an intuitive sense that the example generalizes, but intuition can lead one astray or not be the same to everyone when things are complicated. Thus today we require proofs that do not rely just on the readerís acceptance of the intuitive generalization of an example.

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  • $\begingroup$ @DietrichBurde - I'm asking this question after a month or so of stumbling on this article. I have given it fair attention and rumination in that time, but I can't seem to look at the picture and explain it to anyone and "walk them through the proof". Hence the request for help on Math.SE. I'd hope no one really needs to "read the PDF" since the proof is claimed to be self contained. $\endgroup$ – PhD Jan 31 '18 at 20:24
  • $\begingroup$ The illustration is of the fact that $(1+2+\cdots+n) + (n+1)n^2 = 3(1^2+2^2+\cdots+n^2)$ with $n=5$, demonstrated by moving smaller rectangles and squares. It uses $n^2=k^2+(n-k)^2 + 2k(n-k)$ on the left and $(1+2+\cdots+k)+(1+2+\cdots+(k-1))=k^2$ on the right $\endgroup$ – Henry Jan 31 '18 at 20:41

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