On Euler’s Phi Function

In which we find that Euler’s phi function was neither phi nor a function.

First of all, a shout-out to all of my math(s) friends who are at (or traveling to) the Joint Mathematics Meetings in Baltimore! Now on to some math.

In my research for the “Evolution of…” series of posts, I came across the word totient in Steven Schwartzman’s The Words of Mathematics, which got me thinking about how Euler’s φ (phi) function—also called the “totient function”—came about. The word itself isn’t that mysterious: totient comes from the Latin word tot, meaning “so many.” In a way, it’s the answer to the question Quot? (“how many”?). Schwartzman notes that the Quo/To pairing is similar to the Wh/Th paring in English (Where? There. What? That. When? Then.). So much for the etymology.

It seems to me, though, that the more interesting questions are: who first defined it? how did the notation change over time? I did some digging, and here’s what I’ve discovered.

The first stop on my investigative tour was Leonard Dickson’s History of the Theory of Numbers (1952). At the beginning of Chapter V, titled “Euler’s Function, Generalizations; Farey Series”, Dickson has two things to say about Leonhard Euler:

“L. Euler… investigated the number φ(n) of positive integers which are relatively prime to n, without then using a functional notation for φ(n).”

“Euler later used πN to denote φ(N)…”

Each of these quotations contains a footnote, the first one to Euler’s paper “Demonstration of a new method in the theory of arithmetic” (written in 1758)  and the second to “Speculations about certain outstanding properties of numbers” (written in 1775). In the first paper, Euler is more interested in proving Fermat’s little theorem, which, true to form, he had already proven twice before. However, Euler does define the phi function (on p. 76, though as Dickson says, he doesn’t use function notation), and proves some basic facts about it, including the facts that φ(pm) = pm-1(p-1) [Theorem 3] and φ(AB) = φ(A)φ(B) when and B are relatively prime [Theorem 5]. This paper is in Latin, and while we do see the use of the words totidem and tot, they don’t seem to hold any special mathematical significance.

In the second paper, Euler returns to the phi function, having decided by this time to use π to represent it. Hard-core nerd that he is, Euler provides us with a table of values of πD for D up to 100, and replicates many of the facts he proved in the first paper. It’s interesting to note that, while Euler wrote this second paper in 1775, it wasn’t published until 1784, a year after his death.

It wasn’t until 1801, in Disquisiones Arithmeticae, that Carl Gauss introduced φN to indicate the value of the totient of N. So why did he pick φ rather than Euler’s π? Well, I checked the English translation by Arthur Clarke (no not, that Arthur Clarke), and I think it’s quite likely that he chose it for no discernible reason. In Clarke’s translation, Gauss introduces φ on page 20—and Gauss loved using Greek letters. In pages 5-19 (the beginning of Section II), he uses α, β, γ, κ, λ, μ, π, δ, ε, ξ, ν, ζ — and only after these does he use φ. As to the use of π, which was Euler’s notation, it’s possible that Gauss knew of Euler’s latter work and chose φ because he had already used π, but there’s no way to know for sure. (Also, π was already used for 3.14159… by this point, but if that was his reasoning, it’s odd that he used the symbol π at all.) Most likely, he just picked another Greek letter off the top of his head. It is important to remember that at no point did Gauss use function notation for the totient—it always appears as φN, never φ(N). (Also: Gauss goes on to use Γ and τ before getting tired of Greek and moving on to the fraktur letters 𝔄, 𝔅, and 𝖅.)

The next significant change came nearly a century later in J. J. Sylvester‘s article “On Certain Ternary Cubic-Form Equations,” published in the American Journal of Mathematics in 1879. On page 361, Sylvester examines the specific case npi, and says

pi-1(p-1) is what is commonly designated as the φ function of pi, the number of numbers less than pi and prime to it (the so-called φ function of any number I shall here and hereafter designate as its τ function and call its Totient).

While Sylvester’s usage of the word totient has become commonplace, mathematicians continue to use φ instead of τ. It just goes to show that a symbol can become entrenched in the mathematical community, even if a notational change would make more sense. Also of note is the fact that while Sylvester refers to the totient as a function, he doesn’t use the modern parenthesis notation, as in τ(n), but continues in Euler and Gauss’s footsteps by using τn.

And this is where our story ends. Sylvester’s use of the word totient, Gauss’s use of the letter φ, and Euler’s original definition all contributed to the modern construct that we call the phi/totient function. Even though Euler’s original definition came in a Latin paper, it wasn’t until Sylvester that the use of totient became commonplace.

However, Euler had proven many of the basic facts about it as early as 1758. So, while the original phi function was neither phi nor a function, it was undoubtedly Euler’s.

The Evolution of Weights and Measures

At long last, I’ve exhausted my curiosity in mathematical etymologies. Many word histories have been explored in the previous three installments:

This time around, I want to look at some of the words we use for measurements. There are a few interesting histories in the metric system (SI), but most of the fun comes from the English Imperial system.

The Roman Empire provided us with the primary pre-SI system of measurement in Europe, from which many of the medieval systems were derived. The Latin word mille gives us two important words today: million (related to “thousand”, as detailed in a previous post), and mile. As Roman legions marched across the Mediterranean world, they measured their distances according to paces, with a thousand paces being milia passuum. A pace is the distance traveled in two full steps, and is about 58-62 inches (depending, obviously, on an individual’s height). Using this reckoning, the Roman definition of a mile clocks in at 4,833-5,167 feet.

When the Roman Empire fractured in the West, their uniform measurement system fractured as well, occasionally with hilarious consequences. Later, by the 18th century, the Roman mile had evolved from one definition to many: there were Scots miles, English miles, German miles, and so on. The German mile was 24,000-some feet (at least according to Wikipedia), compared to the English mile’s comparably-paltry 5,280 feet. (Go check that Wikipedia reference, too—there are many more variants!)

But before I get too distracted by the history of the mile, let’s move on to some other length measurements.

  • Inch — this is a fun one. The word comes from the Latin uncia, which basically means “unit”. The strange thing is that an uncia was a unit of weight rather than length—it was 1/12th of a Roman pound. While the English inch is still 1/12th of its parent measure, the ounce somehow became 1/16th of a pound.
  • Furlong — rather simply, it’s a combination of furrow and long, with a furrow being the length of a ten-acre farm field. This makes it about 1/8th of a mile.
  • Yard and Rod — these two have an intertwined history. Today, a yard is 3 feet long, and a rod is 16.5 feet long. The word yard comes from Old English gierd, meaning “rod” or “stick.” Rod comes from the Old Norse rudda, meaning “club”. According to Schwartzman, the rod and the yard were used somewhat interchangeably during the Medieval period, and only later did they settle on 3 and 16.5 feet (or thereabouts)—the “short” and the “long” yard.   
  • Fathom — originating from the Old English fæðm (“faythm”), meaning “arms” or “grasp”. It was the length of a person’s outstretched arms, and is defined as 6 feet today. Perhaps, given its nautical use, a fathom was the distance you could fall off the boat while still being rescued by someone on board?

While there are lots of other words I could choose from, here are two in particular that have a surprising connection.

  • Pound — comes from the Latin pondus, meaning “a weight.” The abbreviation lb. comes from the Latin word libra, meaning “pound” or “balance.” In most markets, merchants would assess the value of precious metals offered for payment using a balance scale (still with us in the popular imagination today). Indeed, one of the signs of the Zodiac is a balance scale. Of course, you’d need to balance the payment against a set of known weights. Over time, then, the word for the weights themselves came to be the English pound, while the word for the scale itself (libra) evolved into its abbreviation.
  • Liter — comes from the Greek litra, which was a unit of weight. Yes, libra and litra have a common origin! Schwartzman notes that lytre and pound were used interchangeably in England as late as the 17th century. When France adopted a decimal system (the precursor to modern SI units), they borrowed the word litron, changing it from a unit of weight to a unit of volume.

There are many, many more words that I didn’t have the time or energy to write up! But hopefully it’s kept your interest throughout the whole series of posts. Get a copy of Schwartzman’s The Words of Mathematics if you want to learn more.