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So we've been using summations at least since the dawn of calculus. I'm wondering how the process of summing a function came to be known? Are there events that led to the invention of the summation operation? Can we attribute summations to a particular person, or persons?

How did mathematics evolve to include summation?

Less importantly, but still interesting, is how the summation symbol, $\sum$, came to be used.

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  • $\begingroup$ What do you mean by "the summation operation" exactly? $\endgroup$ Oct 21, 2014 at 23:50
  • $\begingroup$ @EricWofsey: Sorry I was unclear. I'm referring to a generalized notion of something like a finite definite sum of a given function, or even an infinite series. I'm interested in learning about how these concepts evolved, and so I tried to generalize the notion. Since this site is generally interested in specifics, I'm wondering how the finite definite sum of a function came into use. But if possible, in a much broader sense, I'd really like to know how sums came into use. $\endgroup$
    – Matt Groff
    Oct 22, 2014 at 0:28
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    $\begingroup$ Matt, I'd like to draw your attention to an Area 51 site which feels to me like a better fit for your question: area51.stackexchange.com/proposals/65204/… $\endgroup$
    – Todd Trimble
    Oct 22, 2014 at 1:04
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    $\begingroup$ I find this question silly even from a history of mathematics perspective because "summation" goes along with the basic concept of number. It must have prehistoric roots. Here is the earliest reference to some kind of summation in the Bible. Genesis 4:24-- "If Cain shall be avenged sevenfold, truly Lamech seventy and sevenfold." $\endgroup$ Oct 22, 2014 at 7:15
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    $\begingroup$ @usul, I don't get it. You can sum A and B and get C. Then you can sum C and some D. Or you can take seven or seventy seven numbers $A_1,...,A_n$ and sum them by the same process. Wherein lies greater sophistication? (Leave infinity out for now.) $\endgroup$ Oct 22, 2014 at 9:34

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according to this source, the summation symbol $\Sigma$ was first used by Leonhard Euler in 1755:

Quemadmodum ad differentiam denotandam usi sumus signo $\Delta$, ita summam indicabimus signo $\Sigma$.

In the same way that we use the symbol $\Delta$ to denote a difference, we will indicate a sum by the symbol $\Sigma$.

[Institutiones calculi differentialis chapter I, paragraph 26.]

in a similar way one has [...], so that, if for $\sum x^2$, $\sum x$ and $\sum 1$ we substitute the values obtained previously, one finds that [...].

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  • $\begingroup$ Hmm, isn't that question (and that answer as well) more appropriate designed for wikipedia? $\endgroup$ Oct 21, 2014 at 23:24
  • $\begingroup$ As I see it, summation started, when counting was invented, the simplest sequence being repeated additions of one; before that, the relative size of cardinalities can be checked by pairing elements of sets and doesn't need counting or even a notion of numbers (a fact that Cantor utilized in comparing the cardinality of infinite sets). So, alltogether it can be said, that summation must have been invented very long before the invention of calculus; however perceiving sums as "mathematical objects" may indeed have emerged much later. $\endgroup$ Oct 22, 2014 at 0:52
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According to http://en.wikipedia.org/wiki/Binomial_theorem, a special case of the binomial theorem (i.e. for the exponent 2), was already known to Euclid as early as the 4th century B.C. and, as binomial coefficients surface when switching between the product- and the sum-formulation of polynomials, it seems reasonable to claim that summation as a representation of functions is tied to investigations on polynomials.

Other milestones were Newton's 1665 generalization of the binomial theorem to non-integral exponents and later James Gregory's and Brook Taylor's discovery, that certain functions can be converted into a series via a combination of interpreting the function as an infinite polynomial with repeated differentiationg and evaluation at 0.

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We've been adding things up since the dawn of arithmetic. The Pythagoreans knew all about figurate numbers, and how these were built up from gnomons: all they lacked was our current terminology and notation.

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  • $\begingroup$ is there some evidence, that the Pythagoreans actually perceived number as generated by repeated addition and not just as "objects" in a fixed sequence? Little children can also enumerate numbers without being aware that they can be generated by repeated addition. $\endgroup$ Oct 22, 2014 at 2:38
  • $\begingroup$ From en.wikipedia.org/wiki/Gnomon : "Hero of Alexandria defined a gnomon as that which, when added to an entity (number or shape), makes a new entity similar to the starting entity. In this sense Theon of Smyrna used it to describe a number which added to a polygonal number produces the next one of the same type." $\endgroup$ Oct 22, 2014 at 7:08
  • $\begingroup$ thanks for the clarification; that proves that the Pythagoreans actually calculated sums. $\endgroup$ Oct 22, 2014 at 7:19

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