[Math] Where do the arithmetic and geometric series formulas come from? Was their discovery by chance

Question

I was reading the first chapter of Courant's and Robbins's "What is Mathematics", and there's a section on mathematical induction in which the authors go over the proofs of the arithmetic/geometric series formula (as they're proved inductively).

Recognizing that these proofs are more of a verification though (in page 15 it's mentioned that in some cases a "proof gives no indication of how this formula was arrived in the first place"), they show how you can derive these by playing around with the sums and their terms. For example, the arithmetic series formula for the first n integers can be found like this, but there's no mention as to why would anyone try to sum up these "sums" (sorry for being repetitive) with their terms rearranged that way. Same for the geometric series one (see here), where they're subtracted.

In summary, my question is: is there any other explanation to the discovery of these formulas? I don't think it was sheer luck. Who's credited for their creation? Is there any way to know their origin?

Thanks in advance for the help 🙂

Answer 1

Although others suggest this dates back to Euclid papers (which I would let for another answer if someone has more information about that), there are several sources suggesting that arithmetic and geometric series and possibly also some formulas/procedures to sum them up were already known to Egyptians sometime around $1650$ BC. Although it is certainly considered a speculation to a degree, I thought it is worth mentioning (and it is too short for a comment).

According to Expansions and Asymptotics for Statistics (Chapman & Hall/CRC Monographs on Statistics & Applied Probability, Page 1:

For example, the ancient Egyptians worked with geometric series in practical problems of partitions. Evidence for this can be found in the Rhind papyrus, which is dated to $1650$ BCE. Problem $64$ of that papyrus states the following.

Divide ten heqats of barley among ten men so that the common difference is one eighth of heqat of barley.

Put in more modern terms, this problem asks us to partition ten heqats into the arithmetic series $$10=a+\left(a+\frac{1}{8}\right)+\left(a+\frac{2}{8}\right)+\dots+\left(a+\frac{9}{8}\right).$$ That is, to find the value of $a$ in this partition. The easiest way to solve this problem is to use a formula for the sum of a finite arithmetic series.

A student in a modern course in introductory probability has to do much the same sort of thing when asked to compute the normalising constant for a probability function of given form. If we look at the solutions to such problems in the Rhind papyrus, we see that the ancient Egyptians well understood the standard formula for simple finite series.

For the geometric series, there is for example a following problem according to Mathematics in Ancient Egypt: A Contextual History, Page $79$-$80$:

An example of these is number $79$ of papyrus Rhind: \begin{array} \text{}\\ \text{The contents of a house. }& &\text{houses:} 7& \\ .& 2801 & \text{cats:} 49\\ 2& 5602 & \text{mice:} 343\\ 4& 11204 & \text{grain:} 2301\\ \text{Total:}& 19607 & \text{kernels:} 16807 \\ & & \text{Total:} 19607 \\ \end{array}

This is usually interpreted as the following mathematical problem: There are $7$ houses, each house contains $7$ cats, each cat has eaten $7$ mice, each mouse has eaten $7$ halms of grain, each halm of grain contained $7$ kernels. What is the sum of all of these? The complete text of the source consists of the title followed by a calculation (left column of the preceding translation) and a list of the individual items and their numbers (right column of the preceding translation). The calculation is the multiplication of $7$ times $2801$, which can be explained as an alternative method of determining the requested total.$^{41}$

However footnote at the same page adds:

This problem is another example where modern mathematics has obscured the historical judgment of previous scholars. Today, we would interpret this problem as an example for a geometric series, which we calculate using the formula $$S=a\frac{r^n-1}{r-1},$$ which for the numerical values of this problem would be $$S=7\frac{7^5-1}{7-1}=7\frac{16806}{6}=7\times 2801.$$ Based on the possibility that the formula could be calculated in a way that results in the multiplication indicated in the text of the Rhind papyrus, no. $79$, it has been claimed that the Egyptian scribe knew of the geometric series and this formula to calculate it. Using two of the criteria that were set up by Eleanor Robson to evaluate an ancient mathematical text, that is, historical sensitivity and cultural consistency, speculation of this kind can for now be excluded. There is no evidence whatsoever within the entire hieratic corpus of the use of a procedure "equivalent" to this formula.

So we probably will never know for sure, but at least we know that Egyptians played with something what we nowadays call arithmetic and geometric series.