chapter-16-on-finding-curves-from-properties-of-the-ordinate

Chapter 16: On Finding Curves from the Properties of the Ordinate

Summary: The inverse problem counterpart to chapters 5–15 (§§364–390). Where the earlier chapters started with a curve equation and read off ordinate properties — diameter, chord-rectangle, conjugate sums of squares — Euler now reverses direction: specify a symmetric property of the ordinates corresponding to each abscissa, and derive the curve equation. The master technique is Vieta’s relations. For a two-valued curve $y^2-Py+Q=0$ the two ordinates have sum $P$ and product $Q$, so any symmetric polynomial of them is a polynomial in $P,Q$; demanding it equal a constant imposes one equation linking $P$ and $Q$ and leaves the other free, generating a whole family of solution curves. The chapter walks this idea through five problem families: (1) Vieta basics — sum gives a diameter, product gives the chord-product property, with Pierre Variot’s prize-winning 45°-line property of conics as an instance (§§364–366); (2) the family $P{M}^n+P{N}^n=a^n$ via $(P-y{)}^n+y^n=a^n$ (§§367–373); (3) the ellipse’s sum-of-squares property on chords parallel to a conjugate diameter (§368); (4) three-valued curves $y^3-P{y}^2+Qy-R=0$ with Newton-identity machinery for power sums, including a constant-perimeter / constant-area triangle from three ordinates (§§374–379); (5) relations between ordinates at symmetric abscissas $\pm z$, with a unified invariance principle “equation unchanged under $z\mapsto -z$ and $y\mapsto a^2/y$” generating equal-sum and equal-product families (§§380–390).

Sources: chapter16 (§§364–390). Referenced figures 19 (in figures19-22) and 77–80 (in figures76-80).

Last updated: 2026-05-11.

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The master idea

For any curve of the form

$y^2-Py+Q=0$

where $P,Q$ are non-irrational functions of the abscissa $x$, the two ordinates $y_1,y_2$ satisfy Vieta’s relations

$y_1+y_2=P,y_1\cdot y_2=Q.$

Every symmetric polynomial in $(y_1,y_2)$ — in particular $y_1^n+y_2^n$, $1/y_1+1/y_2$, the squared-distance sum $y_1^2+y_2^2=P^2-2Q$, etc. — is a polynomial in $P,Q$ via Newton’s identities. So “demand that $f(y_1,y_2)=$ const” becomes “demand that $g(P,Q)=$ const” — a single algebraic relation linking the two coefficients. Either $P$ or $Q$ is then chosen freely as a function of $x$, and the curve is the conic / cubic / higher-order object recovered from the resulting $y^2-Py+Q=0$. Three-valued curves $y^3-P{y}^2+Qy-R=0$ work the same way with one extra free function.

Strand 1 — sum and product of two ordinates (§§364–366)

The two simplest cases recover the chapter 5 conic theorems and immediately extend them:

Condition	Implication	Reference
$P=$ const	sum of ordinates constant → orthogonal diameter	diameter-of-conic
$P=a+nx$	sum is linear in $x$ → oblique diameter	diameter-of-conic
$Q=$ const	product of ordinates constant → curve never meets the axis	chord-rectangle-property
$Q=\alpha +\beta x+\gamma x^2$ factorable	product-of-ordinates / product-of-intervals constant	chord-rectangle-property

§365 then unifies all conics under one family: every curve $y^2-nxy+x^2=ax+by$ is a conic — circle ($n=0$, right-angle ordinates), ellipse ($n^2<4$), parabola ($n^2=4$), hyperbola ($n^2>4$). §366 highlights Pierre Variot’s prize-winning property: for two lines drawn at $\pi /4$ to the principal axes of any central conic (figure 77), the rectangles of segments cut on those lines are equal. See two-ordinate-sum-and-product.

Strand 2 — sum of $n$th powers (§§367–373)

For $P{M}^n+P{N}^n=a^n$, expanding via Newton’s identities gives one relation between $P$ and $Q$:

$n$	Newton expansion	Curve equation
2	$P^2-2Q=a^2$	$y^2-Py+(P^2-a^2)/2=0$
3	$P^3-3PQ=a^3$	$y^2-Py+(P^3-a^3)/(3P)=0$
4	$P^4-4P^{2}Q+2Q^2=a^4$	irrationality forces sign-choice resolution
$n$	(general)	$(P-y{)}^n+y^n=a^n$

The cleanest unified form is $(P-y{)}^n+y^n=a^n$, expressing the obvious fact that if $y$ is one ordinate, the other is $P-y$. An alternate cleanup $y^n+(Q/y{)}^n=a^n$ uses $Q/y$ for the other ordinate. The $n=2$ choice $P=2nx$ gives the ellipse

$y^2-2nxy+2n^2{x}^2-\frac{1}{2}{a}^2=0$

with origin at center. The $n=3$ choice $P=3nx$ gives a second-species cubic. Negative exponents (reciprocal sums) and fractional exponents extend the table; e.g., $\sqrt{PM}+\sqrt{PN}=\sqrt{a}$ rationalizes to $y^2-(a-2\sqrt{Q})y+Q=0$. See sum-of-ordinate-powers-curves.

Strand 3 — the conjugate-diameter sum-of-squares (§368)

A specific elegant case of strand 2 stands alone. For an ellipse with conjugate diameters $AB,EF$ and circumscribed tangent parallelogram $GHIK$ (figure 79), any chord $MN$ parallel to $EF$ has

$P{M}^2+P{N}^2=p{M}^2+p{N}^2=2C{E}^2$

(with $P,p$ where $MN$ meets the parallelogram diagonals). Symmetrically, any chord $RS$ parallel to $AB$ has $P{R}^2+P{S}^2=2C{A}^2$. Algebraically: with $CA=a$, $CE=b$, the ellipse $a^2{u}^2+b^2{t}^2=a^2{b}^2$ in conjugate-axis coordinates rotates back to $y^2-2nxy+2n^2{x}^2=b^2$ exactly matching strand 2’s $n=2$ result. See ellipse-sum-of-squares-conjugate-diameter.

Strand 4 — three ordinates per abscissa (§§374–379)

Same idea with $y^3-P{y}^2+Qy-R=0$, three ordinates $p,q,r$ per abscissa, Vieta giving $P=p+q+r$, $Q=pq+pr+qr$, $R=pqr$. Newton’s identities express any symmetric polynomial in these. Three problem types:

1. Constant power sum $p^n+q^n+r^n=a^n$: solve via Newton’s identity for the chosen $n$, leaving two free coefficients. Negative integer $n$ via $z=1/y$ substitution.
2. Constant root sum $\sqrt{p}+\sqrt{q}+\sqrt{r}=\sqrt{a}$, $p^{1/3}+q^{1/3}+r^{1/3}=a^{1/3}$, etc.: square or cube to chase out the radicals — eventually reducible but tedious.
3. Constant triangle area $\sqrt{2p^2{q}^2+2p^2{r}^2+2q^2{r}^2-p^4-q^4-r^4}/4=a^2$ via Heron: combined with $P=2b$ (constant perimeter) gives the explicit third-order curve

$y^3+m{x}^{2}y-2b{y}^2+nbxy-mb{x}^2+(a^4/b-k{a}^{2}b)y-n{b}^{2}x+a^4/b-k{a}^{2}b+b^3=0$

with constant ordinate sum $2b$ and constant triangle area $a^2$. See three-ordinate-curves.

Strand 5 — ordinates at symmetric abscissas (§§380–390)

The previous strands related ordinates at the same abscissa. Now: relate $PM$ at $AP=z$ to $QN$ at $AQ=-z$ (figure 80). If $y=X(z)$ is the curve equation, then $X(z)=PM$ and $X(-z)=QN$. Decompose $X=(P+Q)/(R+S)$ with $P,R$ even and $Q,S$ odd; then $PM=(P+Q)/(R+S)$ and $QN=(P-Q)/(R-S)$.

Required property	Solution	Geometric meaning
$PM+QN=2a$	$y=a+Q$, $Q$ odd	center at $B$ (rotational symmetry)
$P{M}^n+Q{N}^n=2a^n$	$y^n=a^n+Q$, $Q$ odd	$n$th-power center
$PM\cdot QN=a^2$	$y=Q+\sqrt{a^2+Q^2}$, $Q$ odd	”alternately equal” about $A$
$P{M}^n\cdot Q{N}^n=a^{2n}$	$y=(Q+\sqrt{a^{2/n}+Q^2}{)}^n$, $Q$ odd	nth-power alternating

The §388 master invariance principle: the desired equation should be unchanged under the simultaneous substitution $z\mapsto -z$ and $y\mapsto a^2/y$. From this, Euler builds the general form

$0=M+\left(\frac{y}{a}+\frac{a}{y}\right)P+\left(\frac{y^2}{a^2}+\frac{a^2}{y^2}\right)R+\cdots +\left(\frac{y}{a}-\frac{a}{y}\right)Q+\left(\frac{y^2}{a^2}-\frac{a^2}{y^2}\right)S+\cdots$

with $M,P,R,\dots$ even and $N,Q,S,\dots$ odd functions of $z$. Multiplying through to clear fractions yields polynomial equations of indefinite degree $n$, and tabulating for low orders recovers (§390) the straight-line, conic, and cubic solutions of the equal-product condition. See relations-at-symmetric-abscissas.

Method takeaway

Three reusable moves underlie everything in the chapter:

1. Vieta + Newton’s identities translate any symmetric ordinate condition into one polynomial relation between $P,Q,R,\dots$
2. The ”$y$ is one root, $P-y$ is the other” rewriting $(P-y{)}^n+y^n=a^n$ avoids ever computing the Newton identity explicitly, giving the cleanest equation form.
3. Symmetry-driven invariance (§388): when ordinates at $\pm z$ are related, an explicit substitution captures the condition, and the equation is built from terms invariant under that substitution.

These three moves anticipate the modern algebraic-geometry view that the ring of invariants — for a given group action on $(y_1,\dots ,y_k)$ — pins down the equation.

Connection to other chapters

• Chapter 5 conic theorems (chapter-5-on-second-order-lines): the strand-1 sum-and-product story is the inverse of the §§87–100 diameter and chord-rectangle derivations. The strand-3 sum-of-squares (§368) is the inverse of conjugate-diameters.
• Chapter 10 cubic theorems (chapter-10-on-the-principal-properties-of-third-order-lines): strand 4 is the cubic analogue — Vieta now gives three sums, and chord-product-cubic (§247) appears here as a consequence of demanding $pqr$ = const.
• Chapter 15 symmetry classification (chapter-15-on-curves-with-one-or-several-diameters): strand 5 is exactly the polar-coordinate symmetry analysis from chapter 15 reframed in rectangular coordinates. The “alternately equal” curves of §381–388 are the $\sin ns$ family of equal-parts-without-diameter for $n=1$; the §388 invariance $z\mapsto -z,y\mapsto a^2/y$ is a hidden change-of-variable that makes a curve with center at $B$ into one with center at $A$.
• Newton’s identities: explicitly cited in §375’s power-sum table for three variables — the same identities ground both this chapter and chapter 5’s derivation of the chord-rectangle property.

Figures

Figures 19–22

Figures 76–80

Related pages

• two-ordinate-sum-and-product — §§364–366: Vieta basics; diameter, chord-product, the unified conic family, Variot’s prize property.
• sum-of-ordinate-powers-curves — §§367–373: $P{M}^n+P{N}^n=a^n$ family; the $(P-y{)}^n+y^n=a^n$ master equation; fractional and negative exponents.
• ellipse-sum-of-squares-conjugate-diameter — §368: chord-parallel-to-conjugate-diameter sum-of-squares.
• three-ordinate-curves — §§374–379: cubic-in-$y$ curves; Newton’s identities; constant power sums, root sums, triangle area.
• relations-at-symmetric-abscissas — §§380–390: $PM$ at $z$ vs $QN$ at $-z$; equal-sum, equal-product; §388 invariance principle.
• chord-rectangle-property, diameter-of-conic — chapter 5 forward-direction counterparts.
• chord-product-cubic — chapter 10 cubic analogue.
• equal-parts-without-diameter — chapter 15 link to the “alternately equal” family.