An amazing paragraph from Euler’s Introductio

Today I’d like to share an amazing paragraph from Euler’s 1748 textbook Introductio in analysin infinitorum (Introduction to analysis of the infinite). This twovolume book is what Carl Boyer calls “The foremost textbook of modern times,” edging out, for example, Descartes’s Géométrie, Gauss’ Disquisitiones, and Newton’s Principia. Boyer writes that “Euler accomplished for analysis what Euclid [in his Elements] and Al-Khowarizmi [in his Al jabr wa’l muquabala] had done for synthetic geometry and elementary algebra, respectively.”

The Introductio was Euler’s precalculus textbook. It is full firsts (perhaps most importantly, his emphasis on the role of functions in analysis). It has masterful treatments of the exponential, logarithmic and trigonometric functions, infinite series, infinite products, and continued fractions.

Here’s the paragraph that I want to share, written in Latin (you can see the full page at Google Books):

And here’s an English translation (from Dirk Struik’s A Source Book in Mathematics 1200–1800):

“Let us therefore take the radius of the circle, or its sinus totus, =1. Then it is obvious that the circumference of this circle cannot be exactly expressed in rational numbers; but it has been found that the semicircumference is by approximation =3.1415926535897932384626433832795028841971693993
751058209749445923078164062862089986280348253421
170679821480865132723066470938446+ for which number I would write $\pi$, so that $\pi$ is the semicircumference of the circle of which the radius =1, or $\pi$ is the length of the arc of 180 degrees.”

The first sentence states that he is taking the “total sine,” or the largest sine, to be 1; that is, from now onward he restricting his attention to the unit circle for trigonometry. Prior to this sine and cosine were lengths of segments in a circle of some radius that need not be 1. In the Introductio Euler, for the first time, defines sine and cosine as functions and assumes that the radius of his circle is always 1.

In the next sentence, before the semicolon, Euler states his belief (which he finds obvious—ha, ha, ha) that $\pi$ is an irrational number—a fact that was proven 13 years later by Lambert.

Then, after giving a long decimal expansion of the semicircumference of the unit circle [Update: Richard Green pointed out to me that there’s an incorrect digit—it should be “513282” not “513272”], Euler calls it $\pi$. Although there are a few known uses of the Greek letter $\pi$ for this number prior to Euler, Fred Rickey writes that Euler’s “use of the first person certainly suggests that he thinks he is introducing the symbol for the first time.” Undoubtedly, it was Euler’s adoption of $\pi$ that cemented it in the mathematical lexicon.

What an amazing paragraph!

In the next few pages he goes on to give various trigonometric identities, the formula that we now call DeMoivre’s formula $((\cos z\pm i\sin z)^n=\cos (nz)\pm i\sin(nz)),$

and the exponential equivalents of sine and cosine ($\sin v=\frac{1}{2i}(e^{iv}-e^{-iv})$ and $\cos v=\frac{1}{2}(e^{iv}+e^{-iv})$).

By the way, notice that Euler puts a period after sin and cos, since they are abbreviations for sine and cosine. Also, he had not yet taken i to be the constant $\sqrt{-1}$ (he introduces “i” about 30 years later).

The Introductio has been translated into several languages (including English). I urge you to check it out. It is a wonderful book.