Chapter 10: On the Principal Properties of Third Order Lines

Summary: Generalizes the conic-diameter / conic-center / chord-rectangle theorems of chapter 5 to cubics. Vieta on the general cubic gives a “diameter” line of points $O$ for which $LO=MO+NO$, a chord-product invariant $PL\cdot PM\cdot PN/(PB\cdot PC\cdot PD)$, and a three-asymptote sum rule $FL+GM=HN$. A genuine bisecting diameter exists only when the cubic is cut by parallel lines in two points; criteria are derived and applied to the sixteen species. Closes with the simplest oblique-coordinate equation for each species.

Sources: chapter10 (§§239–259); figures 44, 45, 46, 47, 48 in figures44-46 and figures47-50.

Last updated: 2026-05-02.

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Program

Just as chapter 5 read the principal properties of conics off the general second-order equation via sum-of-roots, product-of-roots, and tangent limits, chapter 10 does the same for cubics, working from the general equation $\alpha y^3+\beta y^{2}x+\gamma y{x}^2+\delta x^3+ϵy^2+\zeta yx+\eta x^2+\theta y+\iota x+\kappa =0(\S 239).$ With $\alpha =1$, the three ordinate roots $y_1,y_2,y_3$ at fixed abscissa $x$ satisfy $y_1+y_2+y_3=-(\beta x+ϵ),\sum y_i{y}_j=\gamma x^2+\zeta x+\theta ,y_1{y}_2{y}_3=-(\delta x^3+\eta x^2+\iota x+\kappa ).$

Five strands

§§240–246 — Diameter and center

The sum-of-roots formula yields a generalized diameter: a straight line of points $O$ on each chord such that $LO=MO+NO$ (figure 44). Distinct ordinate angles give distinct diameters; they concur at a single point — the center — only under an algebraic constraint on the coefficients. See diameter-and-center-of-cubic.

§247 — Chord-product invariant

The product-of-roots formula gives the cubic analogue of the conic chord-rectangle property: for parallel chords, the ratio $PL\cdot PM\cdot PN/(PB\cdot PC\cdot PD)$ is constant, where $B,C,D$ are the points where the curve meets the axis. See chord-product-cubic.

§§248–252 — Three rectilinear asymptotes

When a cubic has three rectilinear asymptotes (case 2 of cubic-species-classification), curve and asymptote share the principal cubic member, so the same sum-of-ordinates formula applies to both. The intervals between curve and asymptote on any chord satisfy $FL+GM=HN$ (two on one side, one on the other). Used to exclude certain species combinations. See three-asymptotes-cubic.

§§253–257 — The genuine bisecting diameter

A cubic intersected by a line in only two points (doubled real factor in the principal member) admits the second-order kind of analysis: midpoints of parallel chords. In general they lie on a hyperbola; on a straight line iff a divisibility condition holds. Algebraic criterion derived in §§255–256, applied species by species in §257. Concludes with: “These properties of diameters were well known to Newton, and for that reason it has been a pleasure to commemorate his work in this place.” See bisecting-diameter-cubic.

§§258–259 — Simplest oblique-coordinate forms

The simplest representative equation for each of the 16 species in oblique coordinates, the cubic counterpart of simplest-cubic-equations. See simplest-oblique-cubic-forms.

Figures

Figures 44–46

Figures 47–50

Related pages

• chapter-5-on-second-order-lines — second-order analogue. Diameter, center, chord-rectangle, conjugate diameters all generalize here.
• chapter-9-on-the-species-of-third-order-lines — sixteen-species classification used throughout §§246, 257, 259.
• diameter-and-center-of-cubic
• chord-product-cubic
• three-asymptotes-cubic
• bisecting-diameter-cubic
• simplest-oblique-cubic-forms