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According to the MacTutor essay "D'Arcy Thompson on Greek irrationals" (which I take to be a version of Thompson's original essay whose only liberty with the original text is giving English translations where Thompson gives Greek words and phrases), Proclus, following Adrastus, asserted that $2+8+50+288+\dots$ (the sum of twice the squares of the "side numbers" $1,2,5,12,\dots$) equals $1+9+49+289+\dots$ (the sum of the squares of the "diagonal numbers" $1,3,7,17,\dots$). Here is the relevant excerpt:

"The table of side and diagonal numbers has many other properties. For instance, as Proclus tells us, the sum of the squares of two adjacent diagonals = twice the sum of the squares on the two corresponding sides: e.g. $3^2 + 7^2 = 2(2^2 + 5^2)$. And, in Chapter xxiii he shows, following Adrastus, that the sum of the squares of 'all' the diagonals is equal to twice the sum of the squares of 'all' the sides."

I haven't been able to find the text that Thompson is referring to here. In any case, I'm curious whether the original text seems to be an early attempt at manipulating divergent series.

(Is there anything like MathOverflow for questions involving the history of mathematics?)

Interestingly, under modern approaches to regularizing divergent series the equality asserted by Proclus fails: the two divergent sums differ by $\frac12$.

[In response to Sam Hopkins' suggestion, I've posted this question to HSM.Stackexchange.]

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    $\begingroup$ Regarding the parenthetical question: there is a History of Science and Mathematics stack exchange site at hsm.stackexchange.com. $\endgroup$ Apr 1, 2022 at 14:37
  • $\begingroup$ To add to @SamHopkins comment: I usually find history-based questions suitable for MathOverflow if they contains some aspect of mathematical research (e.g. "how did Euler originally prove Euler's theorem?"). I find them more suitable for HSM if they are more properly about historical topics, even if they involve mathematicians (e.g. "what did Euler eat for breakfast on September 19, 1783?", though that one has an easy answer). $\endgroup$ Apr 1, 2022 at 15:12
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    $\begingroup$ Oof, that's a bad joke for September 19, 1783. But you made me look! $\endgroup$
    – Lucia
    Apr 1, 2022 at 17:20
  • $\begingroup$ can someone explain the joke? Euler was not alive on that date, so what did he "eat for breakfast" ? $\endgroup$ Apr 1, 2022 at 19:29
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    $\begingroup$ @Carlo it's the day after Euler died, so he didn't have breakfast. It is a very bad joke... $\endgroup$
    – David Roberts
    Apr 2, 2022 at 2:12

2 Answers 2

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To my knowledge, there is no infinite series in Proclus work. In modern language, the identity is simply $$ \|u+v\|^2+\|u-v\|^2 = 2(\|u\|^2+\|v\|^2) $$ which in its geometric form, is nowadays often attributed to Ptolemy.

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Regarding the reference request, you have been redirected to HSM.SE, so I will address another aspect of your question. Actually, the difference between these two series can be any, depending on what exactly series you have in mind.

Your notation is ambiguous.

For instance, $1+1+1+1+1+...$ can mean either $\sum_{k=0}^\infty 1$, whose regularized value is $1/2$, or $\sum_{k=1}^\infty 1$, whose regularized value is $-1/2$. Also, if we consider the sum $\sum_{k=1/2}^\infty 1$ (with step $1$ between the terms), the regularized value is zero.

Similarly, $(2+8+50+288+\dots)|_{\text{from } 0}-(1+9+49+289+\dots)|_{\text{from } 0}=\sum_{k=0}^\infty 1$, its regularized value is $1/2$. (I do not know the formulas for the general terms of your series, so I use this awkward notation without sum sign)

At the same time, $(2+8+50+288+\dots)|_{\text{from } 1}-(1+9+49+289+\dots)|_{\text{from } 1}=\sum_{k=1}^\infty 1$, its regularized value is $-1/2$.

Ath the same time, the regularized value of $(2+8+50+288+\dots)|_{\text{from } 1/2}-(1+9+49+289+\dots)|_{\text{from } 1/2}$ is zero, so in one sense the authors you are referring to were correct.

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