pi

$\pi$

Summary: §126 of Chapter 8. With the radius of the circle taken as 1, half the circumference is irrational. Euler reports its decimal expansion to about 113 digits and writes: “For the sake of brevity we will use the symbol $\pi$ for this number.” This is the moment $\pi$ is established in mainstream mathematical notation.

Sources: chapter8 (§126)

Last updated: 2026-04-27

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Setup

Euler fixes the convention that anchors all of trigonometry in the Introductio: the radius of the circle is 1. Under this convention the arc length and the angle measure in radians coincide, so the sine and cosine become functions of arc length rather than of an angular degree. The full circumference of the unit circle is then $2\pi$, the semicircle is $\pi$, and 90° corresponds to the arc $\pi /2$.

The decimal expansion

Euler quotes the value of $\pi$ to about 113 digits (source: chapter8, §126):

$\pi =3.1415926535897932384626433832795028841971693993751058209749445923\dots$

(continuing to ~113 digits before he ends with ”+”, indicating the next digit). He notes that $\pi$ “cannot be expressed exactly as a rational number” — the irrationality is taken as known, not proved here. Lambert’s proof of the irrationality of $\pi$ would not appear until 1761; Euler simply asserts the fact based on the failure of rational candidates.

The digits Euler reports here were computed before the Introductio by laborious classical methods — chiefly Archimedes-style polygon perimeters, refined by Ludolph van Ceulen (35 digits, c. 1610) and others. The chapter’s later sections (§141–§142) replace those laborious methods with rapidly convergent arctangent series.

The symbol $\pi$

Euler writes “we will use the symbol $\pi$ for this number.” He is not the first to use the letter (William Jones, 1706, was earlier), but the Introductio’s circulation made the notation universal. From this section onward in the wiki — and from this section onward in mathematics — $\pi$ denotes the half-circumference of the unit circle.

The convention ”$\pi$ is half the circumference, not the full one” is also Euler’s choice. It survives because most circle and trig identities (e.g. $\sin \pi =0$, $\cos \pi =-1$, $\sin (\pi /2)=1$) read more naturally with this convention.

Status of $\pi$ as a transcendental

By the time Chapter 8 begins, §105 has already established the heuristic that “generic” outputs of transcendental functions are themselves transcendental, but Euler does not invoke that machinery for $\pi$ here. He simply notes the irrationality. The full transcendence of $\pi$ is Lindemann (1882), more than a century after Euler.

How $\pi$ enters the rest of the chapter

• The arcs $\pi /2,\pi ,3\pi /2,2\pi$ generate the periodicity catalog of §128.
• The half-arc $\pi /6$ is the pivot for the table-construction strategy of §136 — sines and cosines from 30° to 60° are derived from those of arcs below 30°.
• The series of §134 for $\sin (m\pi /(2n))$ and $\cos (m\pi /(2n))$ all carry $\pi /2$ as the leading coefficient, in the form $\pi /2=1.5707963267948966192313216916\dots$ (a 28-digit value carved out of Euler’s series).
• The arctangent series of §140 lets $\pi$ itself be computed as $4\arctan 1$ (Leibniz), $6\arctan (1/\sqrt{3})$ (§141), or $4(\arctan (1/2)+\arctan (1/3))$ (§142).

Later appearances in the Introductio

• §167: $\sum 1/k^2={\pi }^2/6$, and the [[zeta-at-even-integers|table of $\zeta (2k)$]] extending through $\zeta (26)$.
• §185: $\pi /2=(2\cdot 2\cdot 4\cdot 4\cdot 6\cdot 6\cdots )/(1\cdot 3\cdot 3\cdot 5\cdot 5\cdot 7\cdots )$, the Wallis product, derived from the redundancy of two §184 product expressions for $\cos (m\pi /2n)$.
• §188–§190: ${\log }_e\pi =1.144729885849400174\dots$ computed by transposing $\sum \log (1-1/(2k+1{)}^2)$ into a doubly-summed series whose columns are the [[zeta-at-even-integers|odd-square sums $A,B,C,\dots$]].
• §369: $4/\pi =1+1^2/(2+3^2/(2+5^2/(2+\cdots )))$, the Brouncker continued fraction obtained by converting the Leibniz series.
• §382: best rational approximations $22/7,333/106,355/113,103993/33102$ from the Euclidean-algorithm continued-fraction expansion of $\pi$.

Related pages

• sine-and-cosine
• trigonometric-addition-formulas
• sine-and-cosine-series
• arctangent-series
• machin-like-formula
• basel-problem
• wallis-product
• log-pi-via-products
• chapter-8-on-transcendental-quantities-which-arise-from-the-circle
• chapter-11-on-other-infinite-expressions-for-arcs-and-sines
• brouncker-formula
• best-rational-approximations
• chapter-18-on-continued-fractions