Chapter 17 — On Finding Curves from Other Properties

Summary: Inverse-problem chapter parallel to chapter-16-on-finding-curves-from-properties-of-the-ordinate, but with the parallel-ordinate setup replaced by polar coordinates at a fixed point $C$. Vieta’s symmetric functions of the chord-distances $CM,CN$ (and triples $CM,CN,CP$) recover the curve up to a homogeneous-decomposition rule $L-M+N=0$; the chapter’s headline result is the conchoid of Nicomedes and its node/cusp/conjugate-point relatives.

Sources: chapter17 (§§391–434); figure 81 in figures81-85; figures 82–85 in figures81-85; figures 86–87 in figures86-89.

Last updated: 2026-05-11.

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The chapter at a glance

Euler abandons the chapter-16 parallel-ordinate frame because the new properties — length of $CM$, angle $\angle ACM$, product of two chord-distances, sum of squares — are not naturally expressed by ordinates. He sets up polar coordinates at a fixed point $C$: $z=CM$ and $\varphi =\angle ACM$ (figure 81). Rectangular coordinates re-enter through $y/z=\sin \varphi$, $x/z=\cos \varphi$, $z=\sqrt{x^2+y^2}$.

The chapter unfolds in five strands, in order of how many points a straight line through $C$ meets on the curve.

1. Polar setup and the one-point case (§§391–398)

§§391–394 introduce $(z,\varphi )$ and isolate the single-intersection condition: $z=P(\sin \varphi ,\cos \varphi )$ meets each line through $C$ in just one point iff $P$ is odd in both $\sin \varphi$ and $\cos \varphi$. §§395–398 translate the parity condition to rectangular coordinates ($P$ odd in $x/z,y/z$) and tabulate the explicit equations for orders 1–4. The order-$n$ pattern: lines, conics through $C$, cubics with a double point at $C$, quartics with a triple point at $C$ — i.e. a multiplicity-$(n-1)$ point at $C$. See curves-from-polar-coordinates.

2. The two-point case (§§399–404)

§§399–400 set $z^2-Pz+Q=0$ with $P$ odd, $Q$ even in $\sin \varphi ,\cos \varphi$. The non-irrational rectangular form forces $P=Mz/L$ and $Q=N{z}^2/L=N(x^2+y^2)/L$ with $L,M,N$ homogeneous of degrees $n+2,n+1,n$. The general equation collapses to

$L-M+N=0.$

§§401–402 specialize to the conic, where the exceptional “one-intersection” lines coincide with asymptote directions (a hyperbola has two such directions; a parabola has one; an ellipse has none). §403–404 give the order-$n$ rule: a curve of order $n$ is cut by every line through $C$ in 2 points iff $C$ is a multiplicity-$(n-2)$ point of the curve. See two-point-equation-from-fixed-point.

3. Sum and product of $CM,CN$ (§§405–409)

§405 asks for $CM+CN=$ const: would require $P=$ const, but $P=Mz/L$ carries the irrationality $z=\sqrt{x^2+y^2}$, so no curve satisfies this strictly. §406 allows additional intersections and gets concentric-circle-pair families. §§407–409 ask for $CM\cdot CN=$ const: the circle is the obvious solution (Euclid’s Elements), and from each higher order one gets a non-circular family $M/N=(x^2+y^2+a^2)/a^2$. See sum-and-product-of-chords-from-point.

4. Squared-sum, constant interval, and the conchoid (§§410–418)

§§410–411 solve $C{M}^2+C{N}^2=2a^2$ via $P^2-2Q=2a^2$; the general equation is

$2L^2(x^2+y^2)-2LM(x^2+y^2)+M^2(x^2+y^2)-2a^2{L}^2=0.$

§§412–413 unify under the family $C{M}^2+nCM\cdot CN+C{N}^2=a^2$.

§§414–418 are the chapter’s geometric highlight: $(CN-CM{)}^2=$ const — equivalently constant chord interval $MN=2c$ — produces in lowest order $z=b\cos \varphi \pm c$. Three sub-cases by the comparison of $b$ to $c$:

• $b<c$: conjugate point at $C$ (figure 83)
• $b=c$: cusp at $C$ (figure 84)
• $b>c$: node at $C$ (figure 85)

The same problem on the other general family gives $z=b/\cos \varphi \pm c$, the conchoid of Nicomedes (figure 86), with directrix $EF$ as asymptote. The “conchoidal genus” of curves $z=bP\pm c$ for any odd $P$ generalizes the construction to any single-intersection curve as directrix (figure 87). See constant-chord-interval-conchoid.

5. Higher symmetric functions and three-point curves (§§419–434)

§§419–425 take $C{M}^n+C{N}^n=a^n$: even $n$ gives genuine solutions, odd $n$ does not (squaring doubles intersections). The closed form

$z^n=(x^2+y^2{)}^{n/2}=\frac{a^n{L}^n}{L^n+(L-M{)}^n}$

(via the §372 trick from chapter 16) covers all even cases. See ordinate-powers-from-fixed-point.

§§426–434 promote the line $CM$ to a three-intersection chord: $z^3-P{z}^2+Qz-R=0$ with $P,R$ odd and $Q$ even reduces to

$K-L+M-N=0$

where $K,L,M,N$ are homogeneous of degrees $n+3,n+2,n+1,n$. $C$ is a multiplicity-$n$ point of the order-$(n+3)$ curve. Constant $pqr$, constant $pq+pr+qr$, constant $p^2+q^2+r^2$, and constant $p^4+q^4+r^4$ are worked out as examples; §434 closes by noting that the four-intersection problem requires nothing new. See three-point-equation-from-fixed-point.

Connections to earlier chapters

• The parity-of-exponents trick echoes diameter-and-center-from-equation (chapter 15), now applied to the polar variables $\sin \varphi ,\cos \varphi$.
• The $L-M+N=0$ decomposition mirrors §365’s $y^2-nxy+x^2=ax+by$ conic family from two-ordinate-sum-and-product, but in the polar frame.
• The §423 “easier method” explicitly recalls §372 of sum-of-ordinate-powers-curves — the same Newton-identity machinery, redirected from parallel ordinates to chord-distances.
• The $n-2$ multiplicity bound at $C$ (two-point-equation-from-fixed-point) is the polar analogue of the line-curve-intersection-bound.

Figures

Figures 81–85

Figures 86–89

Related pages

• curves-from-polar-coordinates
• two-point-equation-from-fixed-point
• sum-and-product-of-chords-from-point
• constant-chord-interval-conchoid
• ordinate-powers-from-fixed-point
• three-point-equation-from-fixed-point
• chapter-16-on-finding-curves-from-properties-of-the-ordinate