Who invented pi?

bibliophage replies:

Lest anyone think we’re in the homework business, this question arrived March 15, 2000. We figure Soccer Princess has surely flunked out by now.

First, let’s get one thing straight. Being a number, pi (p=3.14 …) isn’t really an invention. If anything, it’s a discovery. You wouldn’t ask who “invented” three-quarters, or the square root of two, or 69, would you? Okay, maybe the last one wasn’t such a great example.

There are really several questions here, and of course I have all the answers. Who was the first person to realize the ratio of any circle’s circumference to its diameter is constant? Who was the first person to realize the fundamental importance of this ratio in finding other quantities, such as areas and volumes? And finally, who was the first person to use the Greek letter p to designate this ratio? The answers, respectively, are “I don’t know,” “I’m not sure,” and “It’s complicated.”

It was probably long before the fact was written down that someone first noticed that the circumference of a circle is about three times its diameter. Sometime after the invention of wheeled vehicles (around 3500 B.C.), she may have noticed that for every revolution of the wheels, the vehicle moves forward about three times the diameter of the wheel. Or it may have been earlier, when she noticed that a leather cord the length of the circumference goes across its diameter three times or a little more. I say “she” because I have noticed that women are more observant — or at least less oblivious — than men (except of course when it comes to noticing when the toilet seat is up). Nobody really knows when this was first noticed, because she probably didn’t write it down, or if she did, it hasn’t come down to us. It is likely that the discovery was made independently several times.

The earliest known written records to throw light on the subject are the Susa mathematical tablets, written in cuneiform about 2000 B.C., and discovered in the 1930s at the site of the ancient city of Susa (now known as Shush, Iran, but try to keep it quiet). At least one Babylonian tablet states that the ratio of the circumference of a circle to the perimeter of an inscribed hexagon is (in modern notation) 1:0.96, implying a value of p=3.125, a value that is too small by about half a percent. But despite this one close approximation, it seems that the usual value in ancient Mesopotamia was the much cruder value p=3, too small by about 4.5%.

But it gets more complicated when it came to calculating the area of a circle. We take it for granted that the ratio of the circumference to the diameter is exactly the same as the ratio of the area to the radius squared (both are equal to pi). But to the ancients, that would not necessarily have been obvious. Their rule for finding the area was to multiply the square of the circumference by one-twelfth. If they (mis)calculated the circumference from the diameter (using p=3), and then used that number to calculate the area, this would imply a value of p=3 in the second calculation as well. However, none of the problems I am aware of are explicitly done this way. In every case, the area is calculated from the circumference, with no clue as to how the circumference was found. A controversial interpretation is that they (correctly) measured the circumference directly instead of incorrectly calculating it from the diameter. This would imply two different values of pi: 3 for calculating circumferences and 3.289 … for calculating areas. The first value is about 4.5% too small, and the second is about 4.7% too large. Note that the 3.289… figure is only implied (if this interpretation is correct), and is not actually found written in the tablets.

Even if the Mesopotamians didn’t know both ratios were the same (which is uncertain), it’s possible (but again uncertain) that the ancient Egyptians did just a couple of centuries later. A remarkable mathematical text was written by an unknown author during the reign of Amenemhet III (around 1820 B.C., give or take a quarter century). About 1650 B.C. the text was copied by a scribe named Ahmes (or Ahmose), after whom it is called the Ahmes papyrus. (It is sometimes called the Rhind papyrus, after A. Henry Rhind, the collector who bought it in the nineteenth century.) We are accustomed to defining pi as the ratio of the circumference to the diameter, with areas and volumes following from that, but the Ahmes papyrus has no circumference problems. It does have several problems that involve a multi-step procedure for finding the area of a circle from the diameter. In modern terminology, it boils down to multiplying the diameter by the square of eight-ninths. The ratio of the area to the diameter squared is what we would call p/4. The Ahmes papyrus thus implies a value of p=(256/81)=3.1605 …, which is too large, but by less than one percent.

Another Egyptian mathematical papyrus, dating from the same era, also uses what we would call the square of eight-ninths. This is the Moscow papyrus (named after the city where it is now kept), or the Golenishchev papyrus (after the collector who bought it). In one problem, the surface area of a “basket” is found to be (in modern terminology) four times the diameter squared times eight-ninths squared. No one really knows what shape the “basket” was supposed to be, but there has been much speculation. If it was an open-ended semicylinder (half of a cylinder cut lengthwise), with its length equal to its diameter, then the answer is right (except for the small error in the value of pi). This implies that the ancient Egyptians knew that the two ratios (area to radius squared and circumference to diameter) were the same. If the basket was supposed to be hemispherical, then the answer is still right (again except for the error in pi). This would imply that the ancient Egyptians knew that the surface area of a sphere was four times the area of its great circle more than a millennium ahead of the earliest known proof of that fact. Not everyone is convinced either interpretation is correct, so it might not mean anything so impressive.

We know with absolute certainty that the Greek mathematician and physicist Archimedes (about 250 B.C.) understood the two ratios (area to radius squared and circumference to diameter) were equivalent. Even if the ancient Mesopotamians and Egyptians didn’t know it (which is uncertain), Archimedes was probably not the first to realize the fact. But he was, as far as we know, the first to prove it formally. He also gave the earliest known proofs for the surface area and volume of the sphere. Archimedes provides us with what is probably the first mathematically rigorous range for pi (as opposed to a practical approximation), correctly stating that it lies between 223/71 and 22/7. The latter is often called the “Archimedean value” of pi, but this approximation was in use long before his time, and continues to be used today. But it is, despite widespread belief to the contrary, only an estimate, since pi is an irrational number and cannot be expressed exactly as the ratio of two whole numbers.

What about the use of the symbol p to represent the ratio? Since it is a Greek letter, it only makes sense that this usage was introduced by the ancient Greeks like Archimedes. Some normally helpful references (such as Klein’s Etymological Dictionary) support this theory, but without evidence. Actually, in writings on the subject before the seventeenth century, the ratio was always expressed rhetorically, not symbolically. Thus every time you wanted to refer to the number, you would have to resort to writing something like (in Latin) quantitas in quam cum multiplicetur diameter proveniet circumferentia (“the quantity by which the diameter is multiplied so the circumference will be produced”). After writing that in longhand a few dozen times, you might get the idea to shorten it up a bit.

That apparently didn’t occur to anyone until 1647 when English clergyman and mathematician William Oughtred in the first English edition of his Clavis mathematicae wrote it p.d, where pi stood for the English word “periphery” (what we would call circumference), the dot was his symbol for division, and the delta stood for the English word “diameter.” The use of p to represent words starting with the letter p, like “periphery,” was not uncommon. Before p=3.14 … caught on, it was variously used to indicate a point, a polygon, a positive number, a power, a proportion, the number of primes in a series, and factorial (which is a product). Oughtred used the same notation in all later English and Latin editions of his book, but not in the earlier first Latin edition. He was a prodigious inventor of mathematical symbols, but most of them have not survived. He did, however, introduce a couple of other symbols we still use: × for multiplication (as distinct from the letter x, which he also used for this purpose) and ± for “plus or minus.”

The first writer to use p alone to stand for 3.14 … was the Welsh mathematician William Jones in 1706 in his Synopsis palmariorum matheseos (the title is Latin, but the text is English). Here it presumably stands for the English word “periphery,” just as in Oughtred, since Jones uses it this way elsewhere in the book. Jones was a friend of Isaac Newton’s (and a publisher of some of his works), but neither Newton nor anyone else followed his lead in using pi this way for many years.

It was left to the great Swiss mathematician Leonhard Euler to popularize p. Before 1736 he used the letter p to represent the ratio, but in that year, he started using p (for Latin peripheria) instead. It is not known if he was influenced by Jones in this usage. Euler introduced a great many other symbols that are still in use, for example, e (the base of the natural system of logarithms), i (for the square root of negative one), S (Greek capital sigma, for the sum of a series), D (Greek capital delta, for a finite difference) and f(x) (for a function of x).

In the early years, many other symbols were used to denote 3.14… and related numbers. In 1655, John Wallis used a square or the Hebrew letter mem (symbols chosen for no apparent reason) to represent what we would call p/4. In 1689, J. Christoph Sturm used e (for ea, a Latin word meaning “it”). In 1739, Johann Bernoulli used c (for Latin circumferentia). After Euler started using p=3.14 …, most other uses for the letter pi gradually died out over about the next half century, thus avoiding much potential confusion.

Pi still has some other uses in math and science, but none of these, in context, is likely to be confused with 3.14… Capital pi (P) is used to indicate a product of a series, a usage introduced by Gauss in 1812. Mathematics also has something called p-modules (which I don’t pretend to understand), particle physics has p-mesons (or pions), and chemists speak of p-bonds. Because pi is the sixteenth letter of the Greek alphabet, astronomers since Johannes Bayer have used the letter to designate the 16th star of any constellation. Lower letters are generally assigned to brighter stars, but a star designated pi is not necessarily the 16th brightest star of the constellation (for example, p-Puppis is number two in brightness in the constellation Puppis). Bayer may have been full of pi-in-the-sky notions, but his system only seems irrational. Changes in the ordering are inevitable given the difficulties presented by the existence of variable stars, imperfect measurements in Bayer’s time, and adjustments to the boundaries of constellations since then. (Puppis, for example, used to be part of a larger constellation, Argo Navis).

So, like I said: I don’t know who first noticed the ratio was constant, I’m not sure who first realized the ratio was good for more than just calculating circumferences, and the story of how the ratio came to be designated by the Greek letter pi is complicated. I told you I had all the answers. And you’re on your own when it comes to learning about 69. Well, send a picture. Maybe we can work something out.

Further reading:

“On the Value Equivalent to p in Ancient Mathematical Texts. A New Interpretation” by A. J. E. M. Smeur in Archive for History of Exact Sciences, volume 6

A History of p by Petr Beckmann

A History of Mathematical Notations by Florian Cajori

bibliophage

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