Chapter 8: On Transcendental Quantities Which Arise from the Circle

Summary: Euler turns from $a^z$ and $\log y$ to the second great class of transcendentals — circular arcs, sines, and cosines on the unit circle. After fixing notation (radius $=1$, [[pi|$\pi$]] = half the circumference, $\sin z$ and $\cos z$ as functions of arc length), he derives the algebraic apparatus of trigonometry: addition formulas, product-to-sum, half-angle, and a recurrent-series structure for arcs in arithmetic progression. The complex factorization $(\cos z+i\sin z)(\cos z-i\sin z)=1$ leads to De Moivre’s formula. Reapplying the [[infinitesimal-and-infinite-numbers|$\omega$/$j$ device]] of Chapter 7 produces the power series for $\sin v$ and $\cos v$, and combining it with $(1+z/j{)}^j=e^z$ yields the most famous identity in mathematics, [[eulers-formula|$e^{iv}=\cos v+i\sin v$]]. Inverting, the arc-from-tangent series gives [[arctangent-series|$\arctan t=t-t^3/3+t^5/5-\cdots$]], and a small change of variable gives Machin-style fast series for $\pi$.

Sources: chapter8

Last updated: 2026-04-27

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Overview

Chapters 6 and 7 built the first transcendental class — exponentials and logarithms — analytically. Chapter 8 builds the second: the circular transcendentals. Euler’s stated motivation (§126) is twofold: these are an independent genus of transcendental quantity, and they will turn out to coincide with logarithms and exponentials of complex arguments. Both promises are kept by §138.

The chapter has four movements:

1. §126–§131 — synthetic trigonometry on the unit circle. Euler fixes the modern conventions (radius 1, arc as the independent variable, $\sin$ and $\cos$ as named functions) and then catalogs the algebraic identities every reader of the Introductio will need: Pythagorean identity, sum/difference, periodicity, product-to-sum, sum-to-product, half-angle. The arithmetic-progression remark of §129 — sines/cosines of $z,y+z,2y+z,\dots$ form a recurrent progression with denominator $1-2nz+(m^2+n^2)z^2$ — links this material back to Chapter 4.
2. §132–§134 — complex factorization and the trig power series. Recognizing $(\cos z{)}^2+(\sin z{)}^2=1$ as $(\cos z+i\sin z)(\cos z-i\sin z)=1$ unlocks De Moivre’s formula $(\cos z\pm i\sin z{)}^n=\cos nz\pm i\sin nz$, and binomial expansion gives finite-$n$ identities for $\cos nz$ and $\sin nz$ in powers of $\sin z$ and $\cos z$. Then comes the master move: let $z$ be infinitely small (so $\sin z=z$, $\cos z=1$) and $n$ infinitely large with $nz=v$ finite. The same coefficient-collapse $(j-m)/(nj)=1/n$ used in Chapter 7 produces the canonical series $\cos v=1-v^2/2!+v^4/4!-\cdots$ and $\sin v=v-v^3/3!+v^5/5!-\cdots$.
3. §135–§137 — practical computation. Tangent and cotangent series by division. Construction of trigonometric tables: it suffices to know $\sin$ and $\cos$ for arcs up to 30°; everything else follows by addition. Half-arc and double-arc formulas extend the range. Euler’s table-builder pragmatism is on full display — the §134 series for $\sin (m\pi /(2n))$ already runs to 28-digit accuracy.
4. §138–§142 — the bridge to logarithms and applications to $\pi$. §138 returns to De Moivre with $\cos z=1$, $\sin z=z$ infinitesimal, and recognizes $(1\pm iv/j{)}^j=e^{\pm iv}$ from Chapter 7. The result is [[eulers-formula|$e^{iv}=\cos v+i\sin v$]], $\cos v=(e^{iv}+e^{-iv})/2$, $\sin v=(e^{iv}-e^{-iv})/(2i)$ — sines and cosines as complex exponentials. §139 inverts: $z=\frac{1}{2i}\log ((\cos z+i\sin z)/(\cos z-i\sin z))$. §140 substitutes $\tan z$ to get the arctangent series $\arctan t=t-t^3/3+t^5/5-\cdots$, and at $t=1$ recovers Leibniz’s $\pi /4=1-1/3+1/5-\cdots$. §141 uses $t=1/\sqrt{3}$ for a faster series, and §142 the Machin decomposition $\pi /4=\arctan (1/2)+\arctan (1/3)$ for fastest convergence.

See also: pi, sine-and-cosine, trigonometric-addition-formulas, trigonometric-recurrent-progression, de-moivre-formula, sine-and-cosine-series, eulers-formula, arctangent-series, machin-like-formula.

Structure of the chapter

§126 — Setup: the unit circle and the symbol $\pi$

Radius (= “total sine”) is 1. Half the circumference is irrational; Euler reports the value to 113 digits beginning $3.14159265358979323846\dots$ “For the sake of brevity we will use the symbol $\pi$ for this number” (source: chapter8, §126). This sentence is the moment $\pi$ enters mainstream notation. See pi.

§127 — Notation: $\sin z$, $\cos z$, $\tan z$, $\cot z$

The arc $z$ is the independent variable. $\sin z$ and $\cos z$ are functions of arc length, not angle measure (though the two coincide for radius 1 in radians). Special values $\sin 0=0$, $\cos 0=1$, $\sin (\pi /2)=1$, $\cos (\pi /2)=0$, $\sin \pi =0$, $\cos \pi =-1$, etc. The Pythagorean identity $(\sin z{)}^2+(\cos z{)}^2=1$. Co-function relations $\cos z=\sin (\pi /2-z)$. Tangent and cotangent as ratios. See sine-and-cosine.

§128 — Sum/difference formulas and periodicity

The four addition identities

$\sin (y\pm z)=\sin y\cos z\pm \cos y\sin z,\cos (y\pm z)=\cos y\cos z\mp \sin y\sin z$

are taken as known. Substituting $y=\pi /2,\pi ,3\pi /2,2\pi$ produces a 16-row table reducing $\sin ((4n+k)\pi /2\pm z)$ and $\cos$ of the same to $\pm \sin z$ or $\pm \cos z$ for $k=1,2,3,4$ and any integer $n$ (positive or negative). This is the periodicity catalog. See trigonometric-addition-formulas.

§129 — Arcs in arithmetic progression are a recurrent series

Let $\sin z=p$, $\cos z=q$, $\sin y=m$, $\cos y=n$. Then

$\sin (y+z)=mq+np,\cos (y+z)=nq-mp,$

$\sin (2y+z)=2mnq+(n^2-m^2)p,\cos (2y+z)=(n^2-m^2)q-2mnp,$

and so on. The arcs $z,y+z,2y+z,3y+z,\dots$ form an arithmetic progression, and the sines (and cosines) form a recurrent progression with denominator $1-2nz+(m^2+n^2)z^2$. Read off the recurrence:

$\sin ((k+1)y+z)=2\cos y\cdot \sin (ky+z)-\sin ((k-1)y+z),$

with the same identity for cosine. See trigonometric-recurrent-progression.

§130–§131 — Product-to-sum, sum-to-product, half-angle

Adding and subtracting the §128 sum/difference formulas:

$\sin y\cos z=\frac{1}{2}(\sin (y+z)+\sin (y-z)),\cos y\cos z=\frac{1}{2}(\cos (y-z)+\cos (y+z)),$

$\sin y\sin z=\frac{1}{2}(\cos (y-z)-\cos (y+z)).$

Setting $y=z=v/2$ gives the half-angle formulas $\cos (v/2)=\sqrt{(1+\cos v)/2}$, $\sin (v/2)=\sqrt{(1-\cos v)/2}$.

§131 changes variables: let $a=y+z$, $b=y-z$, so $y=(a+b)/2$, $z=(a-b)/2$. The identities above become the four sum-to-product theorems

$\sin a+\sin b=2\sin \frac{a+b}{2}\cos \frac{a-b}{2},$

$\sin a-\sin b=2\cos \frac{a+b}{2}\sin \frac{a-b}{2},$

$\cos a+\cos b=2\cos \frac{a+b}{2}\cos \frac{a-b}{2},$

$\cos a-\cos b=-2\sin \frac{a+b}{2}\sin \frac{a-b}{2},$

from which Euler reads off six ratio identities (e.g. $\frac{\sin a+\sin b}{\sin a-\sin b}=\frac{\tan ((a+b)/2)}{\tan ((a-b)/2)}$). See trigonometric-addition-formulas.

§132 — Complex factorization

The Pythagorean identity factors:

$(\cos z+i\sin z)(\cos z-i\sin z)=(\cos z{)}^2+(\sin z{)}^2=1.$

Multiplying two such factors:

$(\cos y+i\sin y)(\cos z+i\sin z)=\cos y\cos z-\sin y\sin z+i(\sin y\cos z+\cos y\sin z)=\cos (y+z)+i\sin (y+z).$

The conjugate factor multiplies to $\cos (y+z)-i\sin (y+z)$, and the three-factor case to $\cos (x+y+z)\pm i\sin (x+y+z)$ (source: chapter8, §132). Even though the factors are complex, “they are quite useful in combining and multiplying arcs.” See de-moivre-formula.

§133 — De Moivre’s formula and the binomial expansions

Iterating §132 yields

$(\cos z\pm i\sin z{)}^n=\cos nz\pm i\sin nz$

for any integer $n$. Solving for the real and imaginary parts:

$\cos nz=\frac{(\cos z+i\sin z{)}^n+(\cos z-i\sin z{)}^n}{2},\sin nz=\frac{(\cos z+i\sin z{)}^n-(\cos z-i\sin z{)}^n}{2i}.$

Expanding both sides by Newton’s binomial gives the finite-$n$ identities

$\cos nz=(\cos z{)}^n-\frac{n(n-1)}{2!}(\cos z{)}^{n-2}(\sin z{)}^2+\frac{n(n-1)(n-2)(n-3)}{4!}(\cos z{)}^{n-4}(\sin z{)}^4-\cdots ,$

$\sin nz=n(\cos z{)}^{n-1}\sin z-\frac{n(n-1)(n-2)}{3!}(\cos z{)}^{n-3}(\sin z{)}^3+\cdots .$

(source: chapter8, §133). See de-moivre-formula.

§134 — The power series for $\sin v$ and $\cos v$

The infinitesimal/infinite move: let $z$ be infinitely small, so $\sin z=z$ and $\cos z=1$; let $n$ be infinitely large, so that $nz=v$ is finite. Substituting into §133 — and using the Chapter 7 collapse $n(n-1)\cdots (n-k+1)=n^k$ for $n$ infinite, hence each $\left(\genfrac{}{}{0ex}{}{n}{k}\right)(\cos z{)}^{n-k}(\sin z{)}^k=(nz{)}^k/k!=v^k/k!$ — yields

$\cos v=1-\frac{v^2}{2!}+\frac{v^4}{4!}-\frac{v^6}{6!}+\cdots ,$

$\sin v=v-\frac{v^3}{3!}+\frac{v^5}{5!}-\frac{v^7}{7!}+\cdots .$

Euler immediately tabulates $\sin (m\pi /(2n))$ and $\cos (m\pi /(2n))$ as power series in $m/n$, with leading coefficients $\pi /2=1.5707963267948966\dots$ and ${\pi }^3/3!\cdot 2^3$ (correctly $0.6459640975\dots$) (source: chapter8, §134). The coefficients shrink fast enough that 28-digit accuracy is reached in a handful of terms when $m/n<1/2$. See sine-and-cosine-series.

§135 — Tangent and cotangent series

By long division $\tan v=\sin v/\cos v$ and $\cot v=\cos v/\sin v$. Euler writes out 25-digit numerical series in $m/n$ for $\tan (m\pi /(2n))$ and $\cot (m\pi /(2n))$. The closed-form expansions $\tan v=v+v^3/3+2v^5/15+\cdots$ and $\cot v=1/v-v/3-v^3/45-\cdots$ are not derived here — Euler defers their justification to §197.

§136–§137 — Building the table

Once $\sin$ and $\cos$ are known up to 30°, all other values follow by addition. Setting $y=\pi /6$ in the §130 sum-to-product identity, and using $\sin (\pi /6)=1/2$:

$\cos z=\sin (\pi /6+z)+\sin (\pi /6-z),\sin z=\cos (\pi /6-z)-\cos (\pi /6+z).$

So sines/cosines from 30° to 60° follow from those of $z$ and $\pi /6-z$, both below 30°. §137 does the same for tangent (using $\tan 2a=2\tan a/(1-{\tan }^{2}a)$) and notes the secant/cosecant formulas $\csc z=\cot (z/2)-\cot z$, $\sec z=\cot (\pi /4-z/2)-\tan z$.

§138 — Euler’s formula

Apply §133 with the §134 substitutions $z=v/j$ (infinitesimal) and $n=j$ (infinite):

$\cos v=\frac{(1+iv/j{)}^j+(1-iv/j{)}^j}{2},\sin v=\frac{(1+iv/j{)}^j-(1-iv/j{)}^j}{2i}.$

But $(1+z/j{)}^j=e^z$ from Chapter 7. Setting $z=iv$ in one factor and $z=-iv$ in the other:

$\cos v=\frac{e^{iv}+e^{-iv}}{2},\sin v=\frac{e^{iv}-e^{-iv}}{2i}.$

Adding $i\sin v$ to $\cos v$ gives the most famous identity in analysis:

$e^{iv}=\cos v+i\sin v,e^{-iv}=\cos v-i\sin v.$

(source: chapter8, §138). “From these equations we understand how complex exponentials can be expressed by real sines and cosines.” See eulers-formula.

§139 — Logarithms of complex numbers and the arc

Let $n$ be infinitely small, $n=1/j$ with $j$ infinitely large. Then $\cos (z/j)=1$ and $\sin (z/j)=z/j$. From the §125 inverse-binomial identity $\log (1+x)=j((1+x{)}^{1/j}-1)$, substitute $1+x=\cos z+i\sin z$ and $\cos z-i\sin z$; the cosine equation gives a tautology, but the sine equation gives

$\frac{z}{j}=\frac{\frac{1}{j}\log (\cos z+i\sin z)-\frac{1}{j}\log (\cos z-i\sin z)}{2i},$

hence

$z=\frac{1}{2i}\log \frac{\cos z+i\sin z}{\cos z-i\sin z}.$

The arc itself is the imaginary part of a complex logarithm. See arctangent-series.

§140 — The arctangent series

Divide numerator and denominator inside the log by $\cos z$:

$z=\frac{1}{2i}\log \frac{1+i\tan z}{1-i\tan z}.$

But §123 already gave $\log \frac{1+x}{1-x}=2(x+x^3/3+x^5/5+\cdots )$. Substituting $x=i\tan z$ — and noting $i^2=-1$ kills the even powers in the right way — yields

$z=\tan z-\frac{(\tan z{)}^3}{3}+\frac{(\tan z{)}^5}{5}-\frac{(\tan z{)}^7}{7}+\cdots .$

Calling the arc whose tangent is $t$ by $\arctan t$:

$\arctan t=t-\frac{t^3}{3}+\frac{t^5}{5}-\frac{t^7}{7}+\cdots .$

At $t=1$: $z=\pi /4$ and Leibniz’s formula

$\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots .$

(source: chapter8, §140). See arctangent-series.

§141 — Faster convergence via $\arctan (1/\sqrt{3})$

Leibniz’s series converges too slowly for practical $\pi$ computation. Try instead $t=1/\sqrt{3}$, so $z=\pi /6$:

$\frac{\pi }{6}=\frac{1}{\sqrt{3}}-\frac{1}{3\cdot 3\sqrt{3}}+\frac{1}{5\cdot 3^2\sqrt{3}}-\cdots ,$

$\pi =\frac{2\sqrt{3}}{1}-\frac{2\sqrt{3}}{3\cdot 3}+\frac{2\sqrt{3}}{5\cdot 3^2}-\cdots .$

Each term is about a third the previous, but every term is irrational. Euler’s verdict: “By means of this series the value of $\pi$ itself, which was previously exhibited, was determined with incredible labor.”

§142 — Machin’s decomposition

If $a+b=\pi /4$, then $\tan a\tan b=(1-\tan a)/(1+\tan a)$ via $\tan (a+b)=1$. Choose $\tan a=1/2$; then $\tan b=(1-1/2)/(1+1/2)=1/3$, so

$\frac{\pi }{4}=\arctan \frac{1}{2}+\arctan \frac{1}{3},$

$\pi =4\left(\frac{1}{1\cdot 2}-\frac{1}{3\cdot 2^3}+\frac{1}{5\cdot 2^5}-\cdots \right)+4\left(\frac{1}{1\cdot 3}-\frac{1}{3\cdot 3^3}+\frac{1}{5\cdot 3^5}-\cdots \right).$

Both series are rational and converge geometrically — “with much more ease than with the series mentioned before.” (source: chapter8, §142). See machin-like-formula.

Notable points

• The chapter is a mirror of Chapter 7, with a quarter-turn in the complex plane. Chapter 7 took $a^{\omega }=1+k\omega$ for an infinitesimal real $\omega$ and produced $e^z$ and $\log y$. Chapter 8 takes $\sin z=z$, $\cos z=1$ for an infinitesimal real $z$ and produces the trig series; the identical machinery — collapse $(j-m)/(nj)=1/n$, raise to the $j$-th power — works identically. The bridge §138 simply observes that the real and imaginary copies of the same machinery agree once $i$ is allowed.
• Euler’s formula is forced by the Chapter 7 machinery. $e^{iv}=\cos v+i\sin v$ is not posited — it is the only way the §134 trig series and the §122 exponential series can coexist with De Moivre’s formula. Euler does not present the identity as a deep insight but as an unavoidable computation.
• Every value of every trig function is now computable. The §134 series, refined by the §136–§137 reduction-to-30° tricks, gives any $\sin$, $\cos$, $\tan$, $\cot$, $\sec$, $\csc$ to arbitrary precision. The need for laborious geometric constructions in old trig tables disappears, just as the §123 logarithmic series replaced Briggs’s geometric-mean computation.
• $\pi$ enters the canon here, computed by a fast series. Pre-Newtonian computations of $\pi$ relied on Archimedes-style polygon perimeters (slow, error-prone). Leibniz’s $1-1/3+1/5-\cdots$ is conceptually beautiful but practically useless. The Euler version (§141 with $t=1/\sqrt{3}$, §142 Machin-style) makes $\pi$ a routine table-lookup quantity. By this chapter’s end, both $e$ and $\pi$ have been pinned down by rapidly convergent rational series.
• “Logarithms of complex numbers” appears casually. §139 writes $\log (\cos z+i\sin z)$ without flinching. It is well-defined (= $iz$) up to $2\pi ik$ — a multivaluedness Euler will treat properly only later. Here the focus is on extracting real arcs from the formula, and the multivaluedness is invisible.

Why this chapter matters

Chapters 6, 7, 8 form a unit. After Chapter 6 introduces $a^z$ and $\log y$, Chapter 7 makes them analytic, and Chapter 8 does the same for $\sin$, $\cos$, $\tan$. The unifying observation — every “transcendental” Euler considers is the limit of a binomial expansion controlled by the same $(j-m)/j=1$ identity — reduces the entire elementary transcendental world to a single technique.

The byproducts are immense: Euler’s formula $e^{iv}=\cos v+i\sin v$ becomes the reusable instrument of the rest of the Introductio; the arctangent series is the main rapid-computation tool for $\pi$ from this point until the early 20th century; the Machin-style decomposition will be re-applied (with smaller fractions) to push $\pi$‘s digit count into the hundreds within a few decades of the Introductio’s publication.

Related pages

• pi
• sine-and-cosine
• trigonometric-addition-formulas
• trigonometric-recurrent-progression
• de-moivre-formula
• sine-and-cosine-series
• eulers-formula
• arctangent-series
• machin-like-formula
• infinitesimal-and-infinite-numbers
• exponential-series
• logarithmic-series
• eulers-number
• chapter-7-on-exponentials-and-logarithms-expressed-through-series
• chapter-4-on-the-development-of-functions-in-infinite-series