chord-product-cubic

Chord-Product Invariant for Third Order Lines

Summary: The product-of-roots formula gives the cubic analogue of the conic chord-rectangle property. If $B,C,D$ are the points where the curve meets the axis $AZ$, then for any chord $LMN$ at abscissa $x$, $PL\cdot PM\cdot PN=\delta PB\cdot PC\cdot PD$. As a consequence, the ratio $\frac{PL\cdot PM\cdot PN}{PB\cdot PC\cdot PD}=\frac{pl\cdot pm\cdot pn}{pB\cdot pC\cdot pD}$ is constant across all parallel chords (§247). Similar properties extend to fourth, fifth, and higher-order lines.

Sources: chapter10 (§247).

Last updated: 2026-05-02.

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From product of roots (§247)

The general cubic with $\alpha =1$, $y^3+(\beta x+ϵ)y^2+(\gamma x^2+\zeta x+\theta )y+\delta x^3+\eta x^2+\iota x+\kappa =0,$ has product of ordinate roots $y_1{y}_2{y}_3=-(\delta x^3+\eta x^2+\iota x+\kappa ).$ With the figure-44 sign convention (two ordinates on one side, one on the other) so that the three ordinates are $PL,PM,-PN$, the product reads $-PL\cdot PM\cdot PN=-(\delta x^3+\eta x^2+\iota x+\kappa )\text{i.e.}PL\cdot PM\cdot PN=\delta x^3+\eta x^2+\iota x+\kappa .$

Setting $y=0$ in the cubic recovers $\delta x^3+\eta x^2+\iota x+\kappa =0,$ whose roots are the abscissas of the points $B,C,D$ where the axis $AZ$ meets the curve. Therefore $\delta x^3+\eta x^2+\iota x+\kappa =\delta (x-AB)(x-AC)(x-AD),$ and substituting back, $PL\cdot PM\cdot PN=\delta \cdot PB\cdot PC\cdot PD.$

Invariance across parallel chords

The factor $\delta$ depends only on the leading coefficient — not on the abscissa $x$. Hence for any parallel chord $lmn$ at abscissa $x^{\prime }$, $\frac{PL\cdot PM\cdot PN}{PB\cdot PC\cdot PD}=\delta =\frac{pl\cdot pm\cdot pn}{pB\cdot pC\cdot pD}.$

This property is completely analogous to the property we found above for second order lines from the product of their ordinates. (source: chapter10, §247)

Comparison with the conic case

In chord-rectangle-property (chapter 5, §§92–100) the product-of-roots formula for the conic $\alpha +\beta x+\gamma y+\delta x^2+ϵxy+\zeta y^2=0$ gave $PM\cdot PN=\frac{\delta x^2+\beta x+\alpha }{\zeta }=\frac{\delta }{\zeta }(x-AE)(x-AF),$ so $PM\cdot PN/(PE\cdot PF)$ is constant for parallel chords. The cubic case adds one more factor ($PN$, $PD$) and one more axis intersection ($D$), but otherwise repeats the construction term-for-term.

Higher orders

Euler notes only that “there are similar properties for fourth, fifth, and higher order lines” (§247). The product-of-ordinates formula for an order-$n$ line gives a degree-$n$ polynomial in $x$ that splits over the axis intersections, so the ratio ${\prod }_{i}P{M}_i/{\prod }_{j}P{B}_j$ is again a constant in $x$. He does not develop the explicit form for $n\ge 4$.

Related pages

• chapter-10-on-the-principal-properties-of-third-order-lines
• chord-rectangle-property — second-order analogue from chapter 5.
• diameter-and-center-of-cubic — the parallel construction from sum-of-roots.
• three-asymptotes-cubic — uses the same sum-of-roots argument but applied to curve and asymptotes simultaneously.