How Pi Almost Wasn’t

Since you’re reading this essay, you probably already know about the mathematical holiday called Pi Day held on March 14th of each year in honor of the mystical quantity π = 3.14…. Pi isn’t just a universal constant; it’s trans-universal in the sense that, even in an alternate universe with a different geometry than ours, conscious beings who wondered¹ about the integral of sqrt(1–x²) from x = –1 to x = 1 would still get — well, not 3.14…, but exactly half of it, or 1.57…. Therein lies a catch in the universality of pi: why should 3.14… be deemed more fundamental than 1.57… or other naturally-occurring² pi-related quantities?

I suspect that, even if we limit attention to planets in our own universe harboring intelligent beings that divide their years into something like months and their months into something like days, many of those worlds won’t celebrate the number pi on the fourteenth day of the third month. It’s not just because 3.14 is a very decimal-centric approximation to pi (is there a reason to think that intelligent beings tend to have exactly ten fingers or tentacles or pseudopods or whatever?). Nor is it just because interpreting the “3” as a month-count and the “14” as a day-count is a bit sweaty. And it’s not just because holding a holiday to celebrate a number is a weird thing to do in the first place. It’s also because, in our own world, we came close to having a different multiple of pi serve as our fundamental bridge between measuring straight things and measuring round things.

BECOMING A NUMBER

Pi is sometimes called Archimedes’ constant³ because Archimedes was the first mathematician to describe a procedure for estimating pi to any desired level of approximation. Archimedes applied his method to show that pi was between 223/71 and 22/7. Incidentally, Archimedes didn’t think of 22/7, 223/71, or pi as numbers; to him, they were ratios of magnitudes, and the last of them had to be treated geometrically rather than numerically.⁴ But the key thing is that Archimedes did not show that pi lies between 314/100 and 315/100 (corresponding to the modern assertion 3.14 < π < 3.15) or between any other pair of fractions with powers of ten as their denominator. There was no reason for him to do so; the decimal system, with its built-in fixation on power of ten, was far off in the future.

Two millennia after Archimedes, Simon Stevin’s manuals De Thiende (The Tenth) and L’Arithmétique (Arithmetic), published in 1585, brought Europe into the decimal age. Stevin explicitly renounced the old distinction between numbers and ratios, proclaiming emphatically if obscurely “There are no numbers that are not comprehended under number.” In Stevin’s system, integers, fractions, and irrational numbers could all dine together at the same table and participate in the operations of arithmetic by way of their decimal representations.

Stevin didn’t specifically discuss pi, but his framework encouraged mathematicians to think about pi as a number. One such mathematician was Ludolph van Ceulen, who spent decades applying Archimedes’ method to analyze polygons with enormous numbers of sides, obtaining 35 decimal digits of pi. After van Ceulen’s death in 1610, the digits he’d calculated were inscribed on his tombstone and pi was dubbed “the Ludolphine number” in some parts of Germany and the Netherlands.⁵

So now 3.14… was acknowledged as a bona fide number. (Originally I was going to write a blog post called “The Velveteen Ratio; or, How Numbers Become Real”, but that’ll have to wait for some other occasion.)

WRONG NUMBER?

Yes, 3.14… was finally a number, but was it the right number to look at?

William Oughtred, in his 1631 work Clavis Mathematicae (The Key of Mathematics), used the notation “π/δ” where π is the circumference of a circle and δ its diameter; that quotient is our friend 3.14…. On the other hand, James Gregory, in his own 1668 book Geometriae Pars Universalis (The Universal Part of Geometry), focused on “π/ρ” instead, where ρ is the radius of the circle; this quotient is 3.14…’s relative 6.28…. Although the two mathematicians focused on two different ratios, for both Oughtred and Gregory, “π” did not denote a number; it denoted the circumference of a circle of arbitrary size. (I obtained much of the information for this section from Jeff Miller’s document Earliest Uses of Symbols for Constants.)

In 1671, Gregory went on to derive a formula⁶ for one-fourth of the modern pi constant as 1 − 1/3 + 1/5 − 1/7 + …. This formula implies that pi itself is 4 − 4/3 + 4/5 − 4/7 + …. I don’t know if you’ll agree, but I find 1 − 1/3 + 1/5 − 1/7 + … a much prettier sum than 4 − 4/3 + 4/5 − 4/7 + …. Perhaps Gregory felt that way too, and maybe he even wondered from time to time whether 1 − 1/3 + 1/5 − 1/7 + …, or .79…, might be the truly fundamental numerical characteristic of circles.

In 1706 William Jones, in his Synopsis Palmariorum Matheseos (later published in English as A New Introduction to the Mathematics), followed in Oughtred’s footsteps but with a notational twist: he proposed that “π” should denote not the circumference of a circle but the ratio of the circumference to the diameter.

The greatest mathematician of the 18th century, Leonhard Euler, followed Jones in adopting “π” as a dimensionless constant, but he took his time figuring out which constant it should be. In 1729, he used “π” to denote the number 6.28… (using the symbol “p” to denote 3.14…). In the late 1730s, Euler switched to using “π” for 3.14…, but he switched back to using “π” for 6.28… again in 1747. Along the way he wrote the formula for the area of the circle as “A = C r / 2” instead of “A = πr²”, skirting the use of the symbol “π”. Then in 1748, in his deeply influential Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite) he switched again to using “π” to denote 3.14… (and “∆” to denote 6.28…) and he never looked back.

The world took up Euler’s (eventual) choice, and has forever since used “π” to signify 3.14….

So Jones’ convention prevailed. But did it deserve to? Here according to ChatGPT are the ten most widely-used mathematical equations involving π:

Of these, (1), (3), (4), (6), and (7) become simpler when expressed in terms of 2π and (2), (5), (8), (9), and (10) are simpler when expressed in terms of π. I’d call that a dead heat.

Missing from that list is a very important formula involving pi that is not an equation but an approximation:

This formula, due to James Stirling in the 1730s, was expressed by him verbally rather than symbolically, including the Latin words for “the circumference of a circle whose radius is 1”, which is to say, in modern terms, 2π. So Stirling, too, would probably have favored 6.28… over 3.14….

SO WHAT HAPPENED?

Why did Euler eventually stick with 3.14…? We can’t ask him, but it’s natural to think that he wanted to follow precedent. Archimedes wasn’t the only ancient to focus on 3.14… rather than 6.28…. Many civilizations studied circles and found approximate ways to compute their circumferences, but as far as I’m aware, none of them looked at the ratio of circumference to radius. That’s probably because of practical considerations: if you’re holding a disk in your hand or trying to measure a circular plot of land, it’s easier to get an accurate estimate of the diameter than to get an accurate estimate of the radius (other than by estimating the diameter and then dividing by two). And Archimedes was a practical fellow, being not just a mathematician but an engineer as well.

On the other hand, Archimedes was a Greek mathematician following in the tradition of Euclid, and that tradition venerated construction almost as much as it venerated proof. The Greek definition of a circle was in terms of its radius – which makes sense since the classic Greek way to construct a circle was via a compass. So if Archimedes had been feeling more like a disciple of Euclid when he did his work on circles, he might’ve favored measuring the circumference of a circle in terms of its radius rather than its diameter. And then we might well have adopted 6.28… as the circle constant, and even adopted the Greek letter “π” to represent it.

In recent years, it’s been suggested that math would be simpler if we’d all adopted 2π as the fundamental quantity to begin with. The suggestion was first made by Bob Palais in his article π is Wrong!, and others after him have argued for the adoption of the Greek letter tau to represent 6.28…: see Michael Hartl’s Tau Manifesto. As I found above, roughly half of the commonest formulas involving pi become simpler when expressed in terms of tau. Nobody expects τ to prevail over π, but its advocates have a kind of cheerfulness one often finds among the partisans of inconsequential lost causes.⁷ Tau Day is celebrated each June 28, but that date falls between when most school years end and when most math camps start, so the upstart holiday has had trouble gaining traction.

I have two modest proposals of my own. One is that, while we continue to accept 3.14… as the “right” pi, we celebrate Good-Enough Pi Day in honor of the approximation 3.1, which is close enough for most purposes. It could be celebrated on either the 3rd day of the 1st month, which happens to coincide with the perihelion of Earth’s orbit around the Sun, or on the 1st day of the 3rd month, which happens to coincide with the day on which the Earth has traveled one radian past the perihelion. Okay, I’m fudging those dates a bit, but only by a day or two. Could these astronomical coincidences be nudging us in the direction of Good-Enough Pi Day?

But an even more important approximation to pi that deserves wide acclaim is the number three. Three is, after all, the Bible’s approximation to pi – a circumstance which has led one Prof. Beauregard G. Bogusian to contend that the circumference of a circle is exactly equal to thrice its diameter, and that any seeming deviations from equality are due to humanity’s fallen state, as described in his video The Truth About Pi. Three is also the truly trans-universal approximation to pi, not beholden to any particular choice of base and hence likely to appeal to creatures with any number of appendages. As a way of celebrating π ≈ 3, I propose that the third repast of each day be hereafter dubbed Pi Meal, and that we honor the number 3 at each such meal by consuming exactly 3 slices of pie. If we consistently perform this ritual, we will in the course of time approximate the mystical rotundity of the circle to any desired level of approximation.

ENDNOTES

#1. Of course one could argue that in a deeply different universe no one would invent integration or even square roots, let alone try to evaluate that integral. It’s hard to argue with such a position, but it’s also hard to have much fun in a conversation in which any attempt at logical deduction could be countered with “Yes, that’s only logical, but what if the rules of logic themselves were different there?”

#2. 1.57… has fewer fans than 3.14… and 6.28…, but fans do exist. One such is K. W. Regan, who, responding to Lance Fortnow and Bill Gasarch’s blog post on reasons why 6.28… is better than 3.14…, wrote: “I am agreed that the ‘true’ value of ‘pi’ is off by a factor of 2 – but in the opposite direction!” I sympathize with fans of 1.57…. For one thing, 1.57… is the universal coefficient of annoyance that measures how much further you have to walk when there’s a circular obstacle in front of you that you have to walk around instead of through.

#3. Attempts to approximate pi long predate Archimedes so the term is a bit of a misnomer, and you might guess that it was European Grecophiles who introduced the term but you’d be wrong: the term “Archimedes’ constant” was introduced by Japanese Grecophiles in the early 20th century.

#4. The difference in point of view is subtle, and can be hard for modern readers to understand since it requires unlearning familiar ways of thinking about numbers and measurement that are drilled into us from an early age. I explain more about the Greek theory of proportions in my essay Dedekind’s Subtle Knife from last month.

#5. Another term that was used was “van Ceulen’s number”, though this sometimes denoted van Ceulen’s approximation to pi rather than pi itself.

#6. Gregory’s formula was discovered independently two years later by Gottfried Wilhelm Leibniz. Neither one knew it, but the formula had been found by Madhava of Sangamagrama more than two centuries earlier. All three mathematicians derived the formula in the same way: by finding the power series expansion of the function arctan x and plugging in x = 1. To this day, when I want to know the first dozen digits of π, I often type “4*a(1)” into the UNIX program bc, causing my laptop to compute 4 times the arctangent of 1. For more about pi and bc, see John D. Cook’s essay Computing pi with bc.

#7. I understand this kind of fatalistic partisanship; I was a Red Sox fan until they actually won the World Series, at which point being a Red Sox fan lost much of its appeal.