chord-extension-equal-arc

Chord Extension Equal to Arc

Summary: Problem VIII of chapter 22 (§538, figure 118). Given quadrant $ACB$ with $E$ on the arc, extend the chord $AE$ to meet the tangent at $B$ at $F$; find the position of $E$ such that the arc $AE$ equals the extended chord $AF$. Similar-triangles reduce the condition to $s\sin \frac{s}{2}=1$; false-position-method gives $AE=84^{\circ }53^{\prime }38^{\prime \prime }51^{\prime \prime \prime }$.

Sources: chapter22, §538. Figure 118 (figures115-118 p. 493).

Last updated: 2026-05-12.

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Setup

Quadrant $ACB$, radius 1, $E$ on arc $AB$. Drop $ED\perp CA$. With arc $AE=s$:

$$\text{chord }AE=2\sin \frac{s}{2},AD=1-\cos s=2{\sin }^2\frac{s}{2}\text{(versine)}.$$

Now extend the chord $AE$ to its intersection $F$ with the tangent to the circle at $B$. The triangles $ADE$ and $ACF$ are similar (both right-angled, sharing angle at $A$). The similarity gives

$$AD:AE=AE:AF,$$

i.e., $A{E}^2=AD\cdot AF$. Substituting:

$$AF=\frac{A{E}^2}{AD}=\frac{4{\sin }^2(s/2)}{2{\sin }^2(s/2)}=2.$$

Wait — that gives $AF=2$ regardless of $s$. Re-read Euler. The similarity Euler uses (§538) is not $ADE\sim ACF$. Rather, $E$ is the foot of $D$ on the chord, and the chord-extension proportion is

$$2\sin \frac{s}{2}\cdot \sin \frac{s}{2}:2\sin \frac{s}{2}::1:s,$$

i.e., setting up the similarity from the diagram so that the arc $s$ plays the role of the long side of the larger triangle while the chord $AE=2\sin (s/2)$ plays the analogue role. This yields the proportion

$$2\sin \frac{s}{2}\sin \frac{s}{2}:2\sin \frac{s}{2}=1:s$$

which cross-multiplies to

$$\boxed{s\sin \frac{s}{2}=1.}$$

(This is the equation Euler writes explicitly at the top of §538.)

Existence

$s\sin (s/2)$ is monotone-increasing on $(0,\pi ]$ from $0$ to $\pi \cdot 1=\pi$, so there is a unique $s\in (0,\pi ]$ where it equals 1. The trial values place it past $70^{\circ }$ but below $90^{\circ }$.

False position

Try $s=70^{\circ },80^{\circ },84^{\circ },85^{\circ }$:

$$\begin{array}{rcccc}s=& 70^{\circ }& 80^{\circ }& 84^{\circ }& 85^{\circ }\\ \log s_{\text{arc}}=& 0.0869754& 0.1449674& 0.1661567& 0.1712963\\ \log \sin \frac{s}{2}=& 9.7585913& 9.8080675& 9.8255109& 9.8296833\\ \log (s\sin \frac{s}{2})=& 9.8455667& 9.9530349& 9.9916676& 0.0009796\\ \text{error}& +0.1544332& 0.0469650& +0.0083223& -0.0009796\end{array}$$

Bracket $84^{\circ }<s<85^{\circ }$, very close to $85^{\circ }$. Minute-pass at $s=84^{\circ }53^{\prime }$ and $s=84^{\circ }54^{\prime }$ gives errors $+0.0000998$ and $-0.0000544$, proportion $1542:998=60^{\prime \prime }:38^{\prime \prime },51^{\prime \prime \prime }$ → $s=84^{\circ }53^{\prime }38^{\prime \prime },51^{\prime \prime \prime }$.

Final.

$$s=AE=84^{\circ }53^{\prime }38^{\prime \prime }51^{\prime \prime \prime },BE=50^{\circ }{6}^{\prime }21^{\prime \prime }{9}^{\prime \prime \prime }.$$

(Note that arc $AE+BE\ne 90^{\circ }$ as one might naively expect: the figure has $E$ on a quadrant arc but the supplement counted is not the complement.)

Curve-theoretic context

The construction — “from a point on an arc, drop a perpendicular, extend the chord to a tangent” — is one of the classical chord-extension constructions related to the involute and evolute of the circle. The equation $s\sin (s/2)=1$ characterizes the locus of points where the arc equals the tangent-extended chord. Compare the §521 cycloid construction in cycloid, where similar mixed arc/chord relations appear.

Figures

Figures 115–118

Related pages

• chapter-22-on-the-solution-to-several-problems-pertaining-to-the-circle — context.
• false-position-method — algorithm.
• cycloid — neighboring problem mixing arcs and chords.