Sine Line

Summary: §520. The curve $y/a=\arcsin (x/c)$ — the simplest transcendental curve of the circular-arc genus. Multi-valued: any vertical line cuts it in infinitely many points $s,\pi -s,2\pi +s,3\pi -s,\dots$ Leibniz’s name. Also the cosine line up to a $\pi a/2$ shift. Drawn as figure 104.

Sources: chapter21 §520, figures103-105 (figure 104).

Last updated: 2026-05-12

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The equation (§520)

Working in a unit circle (so arcs are dimensionless angles), the sine line has

$\frac{y}{a}=\arcsin \frac{x}{c},$

i.e. $y$ is proportional to the arc whose sine is $x/c$. Since for any sine value $\sin \varphi =x/c$ there are infinitely many arcs (differing by $2\pi$ or by $\pi -2\varphi$), the ordinate $y$ is an infinite-valued function:

$\frac{y}{a}\in \{s,\pi -s,2\pi +s,3\pi -s,4\pi +s,5\pi -s,\dots \}\cup \{-\pi -s,-2\pi +s,-3\pi -s,\dots \}$

where $s$ is the shortest arc with sine $x/c$. This is the defining feature: not only does the ordinate take infinitely many values, but every line through the curve cuts it in infinitely many points. Algebraic curves can never do this, because line-curve intersection is bounded by the curve’s order (line-curve-intersection-bound).

The figure (§520, figure 104)

Take $CAB$ as the axis with $A$ as origin (figure 104 has it vertical, but it could be either). At $x=0$:

$y\in \{0,\pi a,2\pi a,3\pi a,\dots \}\cup \{-\pi a,-2\pi a,-3\pi a,\dots \}.$

These are the labeled points $A,A^1,A^2,A^{-1},A^{-2},\dots$ on the axis. At $x=AP$:

$P{M}^1=as,P{M}^2=a(\pi -s),P{M}^3=a(2\pi +s),\dots$

— the labeled points $M^1,M^2,M^3,\dots$ on the vertical through $P$.

The curve consists of infinitely many identical pieces $A{E}^1{A}^1,A^1{F}^1{A}^2,A^2{E}^2{A}^3,\dots$, each spanning a height of $\pi a$. Two distinguished parallel lines $BC$ and a mirror, passing through $E$ and $F$ at distances $AC=AB=c$, are diameters (the curve is symmetric across each). The intervals $E^1{E}^2,E^2{E}^3,E^1{E}^{-1},\dots$ all equal $2a\pi$, as do the intervals $F^1{F}^2,F^1{F}^{-1},\dots$

Leibniz called this curve the sine line, because it can be easily used to find the sine of any arc. (source: chapter21, §520)

Sine vs cosine

Setting $y/a=\pi /2-z/a$ in $y/a=\arcsin (x/c)$ gives $z/a=\arccos (x/c)$, so the same curve, with axis shifted by $\pi a/2$, is the cosine line. The two are coordinate translates of each other.

Place in the chapter

The sine line is the simplest case of “transcendental requires circular arcs”. The later §521 tangent-and-secant-lines ($y=\arctan x$ and $y=\text{arcsec}x$) and §§521–522 cycloid are richer examples, and §525 arccos-log-curve mixes logarithms and arcs.

Figures

Figures 103–105

Related pages

• chapter-21-on-transcendental-curves
• transcendental-curves
• tangent-and-secant-lines — same genus, different inverse trig function.
• cycloid — distinguished member of the circular-arc genus.
• logarithmic-curve — analogous “simplest” curve of the logarithmic genus.
• line-curve-intersection-bound — the algebraic bound that this curve definitively violates.