index

Wiki Index

Table of contents for the Introductio in analysin infinitorum — Book II wiki.

Chapter summaries

• chapter-1-on-curves-in-general — coordinate apparatus and basic classification of curves.
• chapter-2-on-the-change-of-coordinates — transformations of axis, origin, and obliquity; general equation; degree invariance.
• chapter-3-on-the-classification-of-algebraic-curves-by-orders — order of a curve = degree of its equation; general equation of order $n$; simple vs. complex curves.
• chapter-4-on-the-special-properties-of-lines-of-any-order — intersection bound; $\frac{n(n+3)}{2}$ determinations; fitting a curve through points.
• chapter-5-on-second-order-lines — conic sections from the general equation: diameters, center, conjugate diameters, tangent ratios, principal axes, foci.
• chapter-6-on-the-subdivision-of-second-order-lines-into-genera — sign-of-$\gamma$ trichotomy into ellipse, parabola, hyperbola; each species developed from the others via limits and substitutions; asymptotes as tangents at infinity.
• chapter-7-on-the-investigation-of-branches-which-go-to-infinity — branches at infinity from real linear factors of the highest member; straight, parabolic, and curvilinear asymptotes catalogued by multiplicity.
• chapter-8-concerning-asymptotes — refining each straight-line asymptote to a curvilinear one $(u-c)+A/t^k=0$; detailed case analysis for double and triple factors; the §218 hyperbolic/parabolic branch dichotomy.
• chapter-9-on-the-species-of-third-order-lines — sixteen species of cubic by branches at infinity, derived case by case from the principal-member factor structure; comparison with Newton’s 72.
• chapter-10-on-the-principal-properties-of-third-order-lines — cubic analogues of the conic diameter / center / chord-rectangle theorems; three-asymptote sum rule; genuine bisecting diameter criterion; simplest oblique forms.
• chapter-11-on-fourth-order-lines — the chapter-9 method extended to quartics; eight cases on principal-member factor structure yield 146 genera (24 detailed, the rest sketched); §271 closing remark on higher orders.
• chapter-12-on-the-investigation-of-the-configuration-of-curves — bounded-region counterpart to chapters 7–11: discriminant $P^2-QR$ controls where ordinates are real, complex, or doubled; ovals, conjugate ovals, conjugate points, nodes, cusps as collisions of its real roots.
• chapter-13-on-the-dispositions-of-curves — local counterpart to chapter 12: translate origin to $M$ and read tangent / singular-point taxonomy off the lowest non-vanishing homogeneous form in $(t,u)$; subtangent, subnormal, multiple-point equations, order-by-multiplicity ceilings.
• chapter-14-on-the-curvature-of-a-curve — second-order extension of the chapter-13 algorithm: osculating parabola, osculating circle and the radius of curvature $R=(A^2+B^2{)}^{3/2}/(2(A^{2}E-ABD+B^{2}C))$, ellipse worked example, inflection by vanishing curvature, cusps and higher-order branches at multiple points, three-genus classification.
• chapter-15-on-curves-with-one-or-several-diameters — invariant-theoretic classification of curves with reflective and rotational symmetry: parity of exponents in $Z(x,y)=0$ encodes diameters and a center; multiple diameters must concur at $C$ with angle $\pi /n$; polar derivation via $\cos ns$ gives the $n$-diameter family, with $\sin ns$ supplying the “equal parts but no diameter” companion; completeness theorem.
• chapter-16-on-finding-curves-from-properties-of-the-ordinate — inverse problem: given a symmetric property of the ordinates, recover the curve. Master technique is Vieta + Newton’s identities on $y^2-Py+Q=0$ (two-ordinate) and $y^3-P{y}^2+Qy-R=0$ (three-ordinate). Five strands: sum/product basics with §366 Variot prize property, $(P-y{)}^n+y^n=a^n$ family, ellipse conjugate-diameter sum-of-squares, three-ordinate Newton machinery with constant-area triangle, and §388 invariance principle $(z,y)\mapsto (-z,a^2/y)$ unifying equal-sum/equal-product families at symmetric abscissas.
• chapter-17-on-finding-curves-from-other-properties — polar reformulation of chapter 16: with $z=CM$ and $\varphi =\angle ACM$ at a fixed point $C$, characterize curves cut by every line through $C$ in 1, 2, or 3 points. Multiplicity rule: $(n-k)$-fold point at $C$ leaves $k$ free intersections. General equations $L-M+N=0$ (two-point) and $K-L+M-N=0$ (three-point); Vieta + §372 trick handles symmetric-function constraints. Geometric highlight: $(CN-CM{)}^2=$ const yields $z=b\cos \varphi \pm c$ (node/cusp/conjugate-point trichotomy, figures 83–85) and $z=b/\cos \varphi \pm c$ — the conchoid of Nicomedes (figure 86) — with the conchoidal-genus generalization to any single-intersection directrix (figure 87).
• chapter-18-on-the-similarities-and-affinities-of-curves — relations between curves rather than properties of a single curve. Thread-degree homogeneity in $(a,x,y)$ characterizes families of similar curves (one scaling ratio); independent scalings $x\mapsto X/m,y\mapsto Y/n$ give affine curves (every ellipse affine to a circle; similarity = affinity for two-term equations like $y^2=cx$); a varying constant not homogeneous with the variables encodes infinite congruent copies placed along a directrix and/or rotated, packaged in one equation via the rigid-motion substitution $t=(z-a)\cos \alpha -(y-A)\sin \alpha$, $u=(z-a)\sin \alpha +(y-A)\cos \alpha$ (figures 90–93).
• chapter-19-on-the-intersection-of-curves — generalizing the line–curve bound to two arbitrary curves: eliminate $y$ between their equations and read intersections off the real roots of the resulting polynomial in $z$. New phenomenon: complex intersections where real roots of the eliminant give complex ordinates (parabola disjoint from circle, figure 96). Closed-form eliminant tables for $\mathrm{deg}y$ pairs $(1,n)$, $(2,2)$, $(2,3)$, $(3,3)$, $(4,4)$; the $(4,4)$ cascade $A,B,C,D\mid a,b,c,d\mid E,F,G\mid H,I,h,i$. Alternative indeterminate-multiplier method (§§483–485) — essentially the resultant via the Bézout construction.
• chapter-20-on-the-construction-of-equations — inverse of chapter 19: given $\Phi (z)=0$, find two curves whose intersection abscissas realize its real roots. Sample constructions for linear (two lines, figure 97), quadratic (line + circle preferred, figure 98; two circles alternative, figure 99), biquadratic (circle + parabola, figure 100), cubic (limit case $E=0$, Bäcker’s rule). The $mn$ intersection bound (§§496–498) governs factorization: two conics ≤ 4, conic + cubic ≤ 6, two cubics ≤ 9, two quartics ≤ 16, in general $mn$. The general method (§§499–504): pick first curve $P+Qy=0$ to keep $y$ single-valued in $z$ and avoid complex intersections; derive the second curve by substituting $y=-P/Q$.
• chapter-21-on-transcendental-curves — guided tour of curves whose defining equation cannot be brought to polynomial form. Five clusters: definition + intercendental + complex-exponent (§§506–511); logarithmic curve with its parity and multi-log paradoxes (§§512–517, fig 101); exponentials $y=x^x$ and $z^y=y^z$ (§§518–519, figs 102, 103); circular-arc genus (sine/tangent/secant lines, cycloid, epicycloid, $y=\arccos \log z$) (§§520–525, figs 104–107); spirals (Archimedes, hyperbolic, lemniscate, logarithmic) (§§526–528, figs 108–111).
• chapter-22-on-the-solution-to-several-problems-pertaining-to-the-circle — closing chapter of Book II. Nine numerical problems where an arc is sought equal to some line, area, or trig function. Eight use the method of false position; Problem IX uses series inversion. Master constant $s_0=42^{\circ }20^{\prime }47^{\prime \prime }14^{\prime \prime \prime }=0.7390847$ (arc = cosine) anchors Problems I, III, IV via linear substitution. §540 closes Book II with the remark that a single rational sine in any of the nine answers would yield a quadrature of (part of) the circle — none did.

Appendix on Surfaces

• appendix-1-on-the-surfaces-of-solids — coordinate apparatus $(x,y,z)$ lifted from Book II to three dimensions; single/multi-valued and algebraic/transcendental dichotomies; diametral planes by parity of exponents; the eight octants and their conjugate/disjoint/opposite relations; full parity calculus for region-symmetries (§§1–25).
• appendix-2-on-the-intersection-of-a-surface-and-an-arbitrary-plane — sections by principal and parallel planes (sphere as worked example); genera by cross-section shape (cylindrical/prismatic = one variable missing, conical/pyramidal = homogeneous equation, unified $Z(z)$ generalization, surfaces of revolution, wedge-cone and affine-section solids, axial-section $z=T+S$); the §§47–50 general oblique-section formula; §51 every plane section of a quadric is a conic (§§26–51).
• appendix-3-on-sections-of-cylinders-cones-and-spheres — algebraic Apollonius. §§52–67 elliptical cylinder with §64 theorem (product of section semiaxes is to product of base semiaxes as $1$ to $\cos \varphi$); §§68–80 cone with full parabola/ellipse/hyperbola trichotomy by sign of $(\cos \varphi {)}^2-m^2(\sin \varphi {)}^2$, subcontrary-circle condition, equilateral-hyperbola condition; §§81–83 sphere section as circle of radius $\sqrt{a^2-f^2{\sin }^2\varphi }$; §§84–85 closing universal section algorithm.
• appendix-4-on-the-change-of-coordinates — §§86–100: 3D analogue of Book II Chapter 2. Three-stage rigid motion (origin translation §87, in-plane rotation $\zeta$ §§88–89, out-of-plane tilt $\eta$ §90) yielding the §92 master substitution with six arbitrary constants — Euler angles before the name. Corollaries: degree invariance (§94), first-order surfaces are planes (§§96–98), the inclination of $\alpha x+\beta y+\gamma z=a$ to each coordinate plane with tangent $-\sqrt{{\alpha }^2+{\beta }^2}/\gamma$ etc. (§§99–100).
• appendix-5-on-second-order-surfaces — §§101–130: classification by asymptotic cone. Bounded criterion = four inequalities $4\alpha \zeta >{\gamma }^2$, $4\alpha \delta >{\beta }^2$, $4\delta \zeta >{ϵ}^2$, $\alpha {ϵ}^2+\delta {\gamma }^2+\zeta {\beta }^2<\beta \gamma ϵ+4\alpha \delta \zeta$. Canonical form $A{p}^2+B{q}^2+C{r}^2+K=0$ by rotation + translation. Six genera: ellipsoid (figure 143), elliptic / hyperbolic hyperboloid (figures 144, 145), elliptic paraboloid (figure 146), parabolic hyperboloid with two asymptotic planes (figure 147), parabolic cylinder with parallel-planes quasi-species. §§127–128 read-genus-off-the-equation algorithm; §§129–130 brief preview of higher orders.
• appendix-6-on-the-intersection-of-two-surfaces — §§131–152, closing Book II. Non-planar curves need two equations; projections onto coordinate planes by elimination (§§134–136, figure 148); intersection algorithm via elimination (§§137–138). Tangency = coincident intersections read off the eliminant’s factor structure: sphere–plane at $f=a\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}$, cone–sphere along a circle at $f=a\sqrt{1+n}$ (§§139–146). Tangent plane at $M$ from two perpendicular section tangents (§§147–149, figure 149). §150 Bezout: projection of intersection has order $\le mn$. §§151–152 closing example: cone ∩ elliptic hyperboloid intersects in the plane $z=x+a$. Final sentence: “These two books have prepared the way for that science.”

Figures

• figures — index of all 40 figure plates (Figures 1–149) converted from figures*. Each plate is also embedded in the wiki pages that reference its figures.

Concepts

From Chapter 1

• abscissa-and-ordinate — variable as interval on an axis, function-value as perpendicular, orthogonal coordinates.
• continuous-and-discontinuous-curves — Euler’s single-expression sense of “continuous,” not the modern one.
• algebraic-and-transcendental-curves — principal division of continuous curves.
• multi-valued-curves — n-valued curves and the parity principle for ordinate intersections.

From Chapter 2

• coordinate-transformations — translation, axis-swap, rotation, and the general rigid-motion substitution.
• general-equation-of-a-curve — the master equation with free parameters $f,g,m,n$; test for equivalence of two equations.
• degree-invariance — degree is preserved under every change of coordinates, rectangular or oblique.
• straight-line-equation — every first-degree equation represents a straight line, and vice versa.
• oblique-coordinates — non-perpendicular ordinates; most general equation of a curve.

From Chapter 3

• order-of-an-algebraic-curve — order = degree of the equation; why genus and term-count fail as classifying principles.
• general-equation-of-order-n — the triangular-number formula $\frac{(n+1)(n+2)}{2}$ for the term count.
• complex-curves — reducible equations as bundles of simpler curves; sum-of-orders lemma.

From Chapter 4

• line-curve-intersection-bound — a curve of order $n$ meets a straight line in at most $n$ points; the converse fails.
• determinations-of-a-general-equation — $\frac{(n+1)(n+2)}{2}$ constants, $\frac{n(n+3)}{2}$ independent determinations (projective scaling).
• curve-through-given-points — $\frac{n(n+3)}{2}$ points pin down an order-$n$ curve; worked conic-through-5-points example.

From Chapter 5

• diameter-of-conic — straight line bisecting all chords of a fixed direction; sum-of-roots construction.
• chord-rectangle-property — the product-of-roots property: $PM\cdot PN/PE\cdot PF$ constant; inscribed-trapezium projective form.
• center-of-conic — all diameters meet in one obliquity-independent point; midpoint of every diameter.
• conjugate-diameters — reciprocal diameter pairs; equal circumscribed parallelograms; sum of squares constant.
• tangent-properties-conic — $AK\cdot BL=C{E}^2$ and the four-tangent ratio; Newton’s Principia lemmas.
• principal-axes-and-foci — the unique orthogonal conjugate pair; foci, latus rectum, focal polar equation.

From Chapter 6

• classification-of-conics — sign of $\gamma$ in $y^2=\alpha +\beta x+\gamma x^2$ partitions conics into hyperbola, ellipse, parabola by branches at infinity.
• ellipse — $y^2=(b^2/a^2)(a^2-x^2)$; foci, $FM+GM=2a$, equal-angle tangent, conjugate-diameter laws $p^2+q^2=a^2+b^2$, $pq\sin s=ab$; vertex form.
• parabola — $y^2=2cx$ as ellipse with infinite axis; $FM=AP+AF$; oblique diameters parallel to axis; isosceles tangent triangle; constant subnormal.
• hyperbola — $y^2=(b^2/a^2)(x^2-a^2)$ as ellipse with $b^2\to -b^2$; four branches; foci at $\sqrt{a^2+b^2}$; $GM-FM=2a$; tangent bisects focal angle.
• asymptotes-of-hyperbola — tangents at infinity, slope $\pm b/a$; equilateral hyperbola; asymptote-coordinate equation $xy=h^2$; tangent bisected at point of contact; $Mm\cdot Mn=b^2$.

From Chapter 7

• branches-at-infinity — criterion: a curve has infinite branches iff its highest member has a real linear factor; complex factors pair into non-vanishing real quadratics.
• rectilinear-asymptote-from-equation — simple real linear factor $\Rightarrow$ straight-line asymptote $ay-bx=-Q/M{|}_{y=b,x=a}$.
• parabolic-asymptote — double real linear factor $\Rightarrow$ parabolic asymptote $u^2+At/g+B/g^2=0$, with parallel-line / no-branch / hyperbolic-curvilinear degenerations.
• curvilinear-asymptote — triple-and-higher repeated factors $\Rightarrow$ asymptotic curves of order up to $k$; normal form $(u-\alpha {)}^k=K/t^{\mu }$.

From Chapter 8

• curvilinear-asymptote-refinement — §§200–203: for a simple linear factor, refine the straight-line asymptote $u=c$ to $(u-c)+A/t^k=0$; quadrant pattern of $z=C/t^k$ by parity.
• double-factor-asymptote-cases — §§204–209: double factor cases — parabola, parallel lines, double-root refinement $(u-c{)}^2+A(u-c)/t^p+B/t^q=0$; $z^2=C/t^k$ shapes.
• triple-factor-asymptote-cases — §§210–214: four leading-equation types for triple factors; cubic asymptote shapes; equal-root sub-cases.
• example-curve-eight-branches — §§215–217: worked sextic $y^3{x}^2(y-x)-xy(y^2+x^2)+1=0$ with single + double + triple factors → 8 branches.
• hyperbolic-and-parabolic-branches — §218: every infinite branch is hyperbolic ($u^j=A/t^k$, converges to line) or parabolic ($u^j=A{t}^k$, recedes); infinitely many species each.

From Chapter 9

• cubic-species-classification — §§223-235: four-case enumeration (${\beta }^2\lessgtr 4\alpha \gamma$, double, triple) producing all sixteen cubic species; signature table.
• simplest-cubic-equations — §237: simplest representative equation for each of the 16 species, with the 72 → 16 Newton-to-Euler mapping.
• euler-vs-newton-cubic-species — §§236, 238: 16 vs 72 reflects the choice to classify by branches at infinity only, not bounded-region shape; the genus/species/variety nomenclature proposal.

From Chapter 10

• diameter-and-center-of-cubic — §§240–246: lo+mo=no diameter from sum-of-roots; multiplicity per ordinate angle; algebraic condition for a center; per-species inventory.
• chord-product-cubic — §247: $PL\cdot PM\cdot PN/(PB\cdot PC\cdot PD)$ constant for parallel chords; cubic analogue of the chord-rectangle property.
• three-asymptotes-cubic — §§248–252: curve and asymptotes share the principal member; $FL+GM=HN$ rule; figure 47 ruled out; species-rate exclusions.
• bisecting-diameter-cubic — §§253–257: genuine bisecting diameter requires doubled real factor; midpoint locus is a hyperbola in general, a line iff $\gamma {ϵ}^2-\beta ϵ\zeta +{\beta }^2\theta =0$; per-species table; Newton homage.
• simplest-oblique-cubic-forms — §§258–259: simplest oblique-coordinate equation for each of the 16 species.

From Chapter 11

• fourth-order-genera-enumeration — §§260–270: eight cases on the quartic principal member; full per-case derivation of the 146 genera; cumulative-count table.

From Chapter 12

• configuration-from-discriminant — §§273–274, §§278–283: writing $y^2=(2Py-R)/Q$ and reading the bounded-region shape from sign analysis of $P^2-QR$.
• first-species-cubic-configurations — §§273–277: the five varieties of the simplest first-species cubic (asymptotic curl, conjugate oval, conjugate point, node, cusp; figures 50–53).
• multiple-points-on-curves — §§281–282: classification of singular points (node = double point, cusp, conjugate point, triple/multiple points) as collisions of consecutive discriminant roots.
• example-bounded-curve-eight-ordinates — §284 octic worked example $2y=\pm \sqrt{6x-x^2}\pm \sqrt{6x+x^2}\pm \sqrt{36-x^2}$ with two cusps and four self-intersections (figure 54).

From Chapter 13

• tangent-by-translation — §§286–291: substitute $x=p+t$, $y=q+u$; linear truncation $At+Bu=0$ is the tangent; special cases $A=0$, $B=0$.
• subtangent-and-subnormal — §§289, 292: $PT=-Bq/A$, $PN=-Aq/B$, normal length; worked examples (parabola, ellipse, seventh-species cubic).
• singular-points-by-jet — §§293–298: when $A=B=0$, the discriminant $D^2-4CE$ classifies double points (conjugate point, node, tangent contact); cubic and quartic forms classify triple and quadruple points.
• general-equation-multiple-point — §§299–302: synthetic form $\sum P_i(x-p{)}^{k-i}(y-q{)}^i=0$ for a $k$-fold point; per-order ceilings on multiple points; flag on Euler’s $r\le n-2$ vs the modern Plücker bound.

From Chapter 14

• osculating-curves — §304: refinement of the tangent — simpler curves that kiss the given curve at $M$.
• osculating-parabola — §§305–307: derivation of the second-order osculator $s^2=(A^2+B^2{)}^{3/2}r/(A^{2}E-ABD+B^{2}C)$ in the rotated normal/tangent coordinates.
• osculating-circle — §§308–310, §318: replace the parabola with the equally-curved circle; radius of curvature $R=(A^2+B^2{)}^{3/2}/(2(A^{2}E-ABD+B^{2}C))$; curve as a chain of circular arcs.
• convexity-from-osculating-circle — §§311–315: signed analysis; sign of $(A^{2}E-ABD+B^{2}C)/B$; collapsed forms for $B=0$ (figure 57) and $A=0$ (figure 58).
• osculating-radius-of-ellipse — §§316–317: ellipse $a^2{y}^2+b^2{x}^2=a^2{b}^2$ gives $R=a^2{b}^2/M{O}^3$ with $MO$ = central perpendicular to the tangent.
• inflection-by-vanishing-curvature — §§319–322: when $A^{2}E-ABD+B^{2}C=0$, classify by leading exponent in $r\sqrt{A^2+B^2}=\alpha s^2+\beta s^3+\gamma s^4+\cdots$; odd → inflection (figure 61), even → smooth (figure 62).
• curvature-at-multiple-points — §§323–331: per-branch analysis at multiple points; cusps from $r^2=\beta s^3$ (figure 63), tangent arcs from $r^2=f{s}^2$ and $r^2=g{s}^2$ (figures 64, 65); higher-order branches $r^j=\alpha s^n$.
• three-genera-of-local-curvature — §§332–335: every local phenomenon is either continuous curvature, an inflection, or a cusp; finite-angle corners (figure 66) and concave/convex shared-tangent meetings (figure 67) ruled out as algebraic; L’Hospital’s “cusps of the second species” via the example $y^4-2y^{2}x-4y{x}^2-x^3=0$.

From Chapter 15

• diameter-and-center-from-equation — §§337–343: parity of exponents in $Z(x,y)=0$ encodes the orthogonal diameter ($y\mapsto -y$), perpendicular pair ($Z=Z(x^2,y^2)$), and center ($\pm$ all-even or all-odd homogeneous parts); diametrically vs alternately equal.
• concurrence-of-diameters — §§344–346: more than two diameters must concur at $C$ at equal angles $\pi /n$ alternating in two kinds; no two parallel diameters; the circle as the infinite-diameter limit.
• n-diameters-by-cos-ns — §§347–354: polar derivation $z=f(\cos ns)$; equation a non-irrational function of $x^2+y^2$ and $\mathrm{Re}(x+iy{)}^n$; simplest order-$n$ curve has $n$ asymptotes bounding a regular $n$-gon (cubic equilateral, quintic pentagon).
• equal-parts-without-diameter — §§355–360: $n$ equal parts with $n$-fold rotational symmetry but no reflection; equation also involves $\sin ns=\mathrm{Im}(x+iy{)}^n$ ($xy$ for $n=2$, $3x^{2}y-y^3$ for $n=3$); dropping the imaginary part recovers diameters.
• classification-of-equal-parts — §§361–363: completeness theorem; equilateral-triangle (figure 75) and parallel-line (figure 76) bootstrap arguments; the law of continuity forces the $\pi /n$ quantization.

From Chapter 16

• two-ordinate-sum-and-product — §§364–366: Vieta basics for $y^2-Py+Q=0$; constant sum gives a diameter, constant product gives the chord-product property; §365 unified conic family $y^2-nxy+x^2=ax+by$; §366 Pierre Variot’s $\pi /4$-chord prize property.
• sum-of-ordinate-powers-curves — §§367–373: curves with $P{M}^n+P{N}^n=a^n$ at every abscissa; the master equation $(P-y{)}^n+y^n=a^n$; alternate $y^n+(Q/y{)}^n=a^n$; integer, fractional, and negative-exponent tables.
• ellipse-sum-of-squares-conjugate-diameter — §368: $P{M}^2+P{N}^2=2C{E}^2$ for chords of an ellipse parallel to a conjugate diameter (figure 79); algebraically the $n=2$ case of the family.
• three-ordinate-curves — §§374–379: $y^3-P{y}^2+Qy-R=0$ with Newton-identity power sums; cube-root sum via auxiliary variable; the quadratic-resolvent general solution; constant-area triangle from three ordinates via Heron.
• relations-at-symmetric-abscissas — §§380–390: $PM$ at $z$ vs $QN$ at $-z$ (figure 80); equal-sum / equal-$n$th-power / equal-product families; §388 invariance principle $(z,y)\mapsto (-z,a^2/y)$ as a unifying group action; §390 order-1, 2, 3 solutions.

From Chapter 17

• curves-from-polar-coordinates — §§391–398: polar setup $z=CM,\varphi =\angle ACM$; $z=P$ with $P$ odd in $\sin \varphi ,\cos \varphi$ gives one-intersection curves; order-$n$ general form requires multiplicity $n-1$ at $C$; conic-through-$C$, double-point-cubic-at-$C$, triple-point-quartic-at-$C$ table.
• two-point-equation-from-fixed-point — §§399–404: $z^2-Pz+Q=0$ with $P$ odd, $Q$ even reduces to $L-M+N=0$ ($L,M,N$ homogeneous of degrees $n+2,n+1,n$); order-$n$ curves cut in two points iff $C$ has multiplicity $n-2$; conic asymptote directions as the “in one point” exceptions.
• sum-and-product-of-chords-from-point — §§405–409: $CM+CN=$ const has no genuine solution (irrationality $\sqrt{x^2+y^2}$); $CM\cdot CN=a^2$ gives the circle plus an infinite family $M/N=(x^2+y^2+a^2)/a^2$.
• constant-chord-interval-conchoid — §§410–418: $C{M}^2+C{N}^2=2a^2$ and the unified family $C{M}^2+nCM\cdot CN+C{N}^2=a^2$; $(CN-CM{)}^2=4c^2$ yields $z=b\cos \varphi \pm c$ (conjugate-point/cusp/node trichotomy at $C$, figures 83–85) and $z=b/\cos \varphi \pm c$ — the conchoid of Nicomedes (figure 86); conchoidal-genus generalization (figure 87).
• ordinate-powers-from-fixed-point — §§419–425: $C{M}^n+C{N}^n=a^n$ via the §372 trick — $z^n=a^n{L}^n/(L^n+(L-M{)}^n)$; even $n$ gives genuine solutions, odd $n$ does not; §425 mixed-symmetric-function generalization.
• three-point-equation-from-fixed-point — §§426–434: $z^3-P{z}^2+Qz-R=0$ becomes $K-L+M-N=0$ ($K,L,M,N$ homogeneous of degrees $n+3,n+2,n+1,n$); order-$(n+3)$ curve with multiplicity-$n$ point at $C$; $pq+pr+qr$, $p^2+q^2+r^2$, $p^4+q^4+r^4$ examples; §434 closing on four-intersection generalization.

From Chapter 18

• thread-constants — §435: every term of a curve equation must have the same total degree in variables and constants together; thread-degree homogeneity is the algebraic skeleton of similarity, affinity, and equal-copies constructions.
• similar-curves — §§436–441, 447–448: $y$ as first-degree homogeneous function of $x$ and the parameter $a$; substitution $x\mapsto X/n,y\mapsto Y/n$ leaves the equation invariant; tangent angles, arcs, and osculating radii scale by the parameter ratio; areas by its square; pantograph construction.
• affine-curves — §§442–446: independent scalings $x\mapsto X/m,y\mapsto Y/n$; affinity contains similarity; not intrinsic (frame-dependent) but preserves order and genus; every ellipse affine to the circle $y^2=2cx-x^2$; two-term equations like $y^2=cx$ collapse affinity back to similarity.
• infinite-copies-of-curve — §§448–456: writing one equation for infinitely many positioned copies of a curve; directrix translation $(y-A{)}^2=2c(z-a)-(z-a{)}^2$ (figure 90), pure rotation $t=z\cos \alpha +y\sin \alpha -c,u=y\cos \alpha -z\sin \alpha$ (figure 91), full directrix + rotation (figure 92), meridian circles $y^2=c^2-2az-z^2$ (figure 93); §455 equal/similar/dissimilar trichotomy.

From Chapter 19

• intersection-of-two-curves — §§457–464: master procedure (eliminate $y$ from the two equations, real roots of the polynomial in $z$ give the abscissas, recover $y$ from either equation); line–curve case substitutes $y=(\gamma -\alpha z)/\beta$ and recovers line-curve-intersection-bound as the $(1,n)$ case.
• complex-intersections — §§465–473: real roots of the eliminant whose corresponding ordinate is complex; condition (irrational $y$ in $z$ after elimination); examples — parabola disjoint from circle (figure 96), and the cubic-vs-parabola §§471–473 spurious-factor case.
• elimination-of-ordinate — §§474–482: closed-form eliminant tables for $\mathrm{deg}y$ pairs $(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(3,3),(4,4)$; the $(4,4)$ cascade of brevity substitutions $A,B,C,D\mid a,b,c,d\mid E,F,G\mid H,I,h,i$ exploiting $a=D$, $G=e$, $I=h$.
• indeterminate-multiplier-elimination — §§483–485: alternative to cascaded subtraction. Multiply each equation by a polynomial in $y$ with unknown coefficients, match coefficients of every $y$-power; $k=m+n-1$ is forced by counting indeterminates vs. matching equations. Essentially the resultant via the Bézout/Sylvester construction.

From Chapter 20

• construction-of-equations — §§486–487, 505: the master inverse-of-chapter-19 program; given $\Phi (z)=0$, find two curves whose intersection abscissas are the real roots; spurious-roots and missing-roots caveats.
• quadratic-by-line-and-circle — §§488–491: linear via two lines (figure 97), quadratic via line + circle (figure 98); §490 reduction of $A{z}^2+Bz+C=0$; simplest form $b=0$ with $AD=-B/(2A)$, $CD=C/B$, radius $AD+CD$.
• quadratic-by-two-circles — §492: alternative quadratic construction (figure 99); six free constants and no extension beyond quadratics — Euler relegates it to an aside.
• biquadratic-by-circle-and-parabola — §§493–495: quartic via circle + parabola (figure 100); Tschirnhaus reduction to $z^4+B{z}^2-Cz+D=0$; case I ($D=E^2$ positive), case II ($D=-E^2$ negative), cubic $z^3+Bz-C=0$ as the $E=0$ limit (Bäcker’s rule).
• intersection-product-degree-bound — §§496–498: two curves of orders $m,n$ meet in at most $mn$ points; bounds for $(2,2),(2,3),(3,3),(4,4)$; how to factor target degree, with the $39\to 42=6\cdot 7$ example.
• general-construction-by-parabola-and-conic — §§499–504: master systematic method. Pick first curve $P+Qy=0$ (single-valued $y$, no complex intersections), derive second by substituting $y=-P/Q$; quartic via parabola $ay=z^2+bz$ yields conic $a^2{y}^2+a(A-2b)yz+(B-Ab+b^2)z^2+Cz+D=0$; conic-type discriminant $A^2-4B-4ac$; circle subcase $b=A/2$; multiply by $z$ trick for awkward degrees.

From Chapter 21

• transcendental-curves — §§506–508, 511: definition; non-irrational form impossible; complex-exponent example $2y=x^i+x^{-i}=2\cos \log x$.
• intercendental-curves — §§509–510: Leibniz’s term for $y=x^{\sqrt{2}}$ and irrational-exponent kin; algebraic approximations of orders 3, 7, 17, 41, …; infinite-order limit; exact construction by logarithms.
• logarithmic-curve — §§512–515 (figure 101): $y=a{e}^{x/b}$; arithmetic $\to$ geometric correspondence; constant subtangent $PT=b$ as the principal property; axis as asymptote.
• discrete-points-below-asymptote — §§515, 517: parity anomaly. Even-denominator $x/b$ gives $\pm$ values, producing dense scatter of points below the axis with no continuous branch; $y=(-1{)}^x$ as starkest example. Paradox unique to transcendentals.
• infinite-logarithms-paradox — §516: every number has infinitely many logarithms, only one real. From $\log (-1{)}^2=0$ via $-1=1/(-1)$; cube roots of unity give triple branches; $a=e^z$ as an infinite-degree equation in $z$. Algebraic shadow of discrete-points-below-asymptote.
• exponential-curves — §518 (figure 102): $y=x^x$. Logarithmic-genus by $m\log y=x\log x$; minimum at $x=1/e$ with $y=e^{-1/e}\approx 0.692$; discrete-point shadow on negative axis; abrupt termination at $B$.
• implicit-exponential-curve — §519 (figure 103): $z^y=y^z$. Diagonal $y=z$ plus branch $RS$ parametrized $z=(1+1/u{)}^u,y=(1+1/u{)}^{u+1}$, meeting at $(e,e)$; rational solution pairs $(2,4),(9/4,27/8),(64/27,256/81),\dots$
• sine-line — §520 (figure 104): $y/a=\arcsin (x/c)$. Infinitely-multi-valued in $y$; periodic continuous curve with diameters $BC$ and parallels; intervals $E^1{E}^2=F^1{F}^2=2a\pi$. Leibniz’s name. Same curve = cosine line up to translation.
• tangent-and-secant-lines — §521: $y=\arctan x$ (infinitely many parallel horizontal asymptotes); $y=\text{arcsec}x$ (infinitely many infinite branches with $|x|\ge 1$). Brief siblings of the sine line.
• cycloid — §§521–522 (figure 105): rolling-circle path of a point at distance $b$ from center on extended diameter. Master equation $y=\sqrt{b^2-t^2}+a\arccos (t/b)$. Ordinary ($b=a$), curtate ($b>a$), prolate ($b<a$).
• epicycloid-and-hypocycloid — §§523–524 (figure 106): rolling on a circle. Parametric $z=(a+c)\cos (s/c)+b\cos ((a+c)s/(ac))$ and analogue for $y$; algebraic iff $(a+c)/a$ rational; hypocycloid for $a<0$; special case $b=-(a+c),c=2a$ yields the sextic $(z^2+y^2{)}^3-2f^2(z^2+y^2{)}^2+f^4{z}^2=0$.
• arccos-log-curve — §525 (figure 107): $y=\arccos \log z$, equivalently $z=e^{\arccos y}$. Combines logs and arcs. Axis crossings $e^{\pi /2},e^{3\pi /2},\dots$ in geometric progression; tangent points also in geometric progression; asymptote is the finite interval $BC$, a non-algebraic phenomenon.
• spirals — §526 (figure 108): general polar setup $z=CM,s=\angle ACM$; any equation $F(z,s)=0$ defines a spiral; multi-valued substitution $s+2k\pi$ produces infinitely many points on any radius.
• spiral-of-archimedes — §526 (figure 109): $z=as$. Tangent at $C$ along axis; perpendicular axis $BC{B}^{-1}$ is a diameter; right-angle crossings of $AC,BC$.
• hyperbolic-and-lemniscate-spirals — §527 (figure 110): $z=a/s$ (asymptote at $\infty$, John Bernoulli’s name); $z=a\sqrt{n^2-s^2}$ (bounded lemniscate $ACBCA$ tangent to lines at angle $n$). Polar analogues of hyperbola and ellipse.
• logarithmic-spiral — §528 (figure 111): $s=n\log (z/a)$, equivalently $z=a{e}^{s/n}$. Equiangular — every radius from center meets the curve at constant angle $\arctan n$; semi-rectangular ($\pi /4$) for $n=1$. The single transcendental-equation spiral Euler distinguishes.

From Chapter 22

• false-position-method — §§531–538: regula falsi. Bracket, linearly interpolate via the error proportion $|e_1|+|e_2|:|e_1|::s_2-s_1:s-s_1$, iterate; logs converted by the constants $1.7581226$ (deg), $3.5362739$ (min), $5.3144251$ (sec); ~one sexagesimal digit per pass.
• arc-equals-cosine — §531 Problem I. Master $s_0=42^{\circ }20^{\prime }47^{\prime \prime }14^{\prime \prime \prime }=0.7390847$, the Dottie number. Reductions $u=s-45^{\circ }$ (Problem III, doubled) and $u=s-90^{\circ }$ (Problem IV) recover this constant.
• equal-area-bisection-problems — §§532–534, figures 112–114: Problem II $s=\sin 2s$ (chord bisects sector, $s=54^{\circ }18^{\prime }{6}^{\prime \prime }52^{\prime \prime \prime }43^{\prime \prime \prime \prime }33^{\prime \prime \prime \prime \prime }$); Problem III $s-\pi /4=\frac{1}{2}\sin 2s$ (ordinate bisects quadrant, $AE=66^{\circ }10^{\prime }23^{\prime \prime }37^{\prime \prime \prime \prime }$); Problem IV $s-\sin s=\pi /2$ (chord bisects semicircle, $s=132^{\circ }20^{\prime }47^{\prime \prime }14^{\prime \prime \prime }$). III and IV reduce to Problem I.
• circle-trisection-by-chords — §535, figure 115: $s-\sin s=2\pi /3$ with substitution $u=s-120^{\circ }$ giving $u=\sin (60^{\circ }-u)$; $u=29^{\circ }16^{\prime }27^{\prime \prime }{0}^{\prime \prime \prime }$, chord $1.9285340$.
• arc-equals-versine-plus-sine — §536, figure 116: $180^{\circ }-s=2\sqrt{2}\cos \frac{s}{2}\cos (45^{\circ }-\frac{s}{2})$ after $1+\cos +\sin$ product reduction; $AE=138^{\circ }11^{\prime }53^{\prime \prime }$; $DE=0.6665578$ is the near-miss with $\frac{2}{3}$ that §540 spotlights.
• arc-equals-half-tangent — §537, figure 117: $2s=\tan s$, sector $ACD$ = half of tangent-triangle $ACE$; $s=66^{\circ }46^{\prime }54^{\prime \prime }14^{\prime \prime \prime }$, tangent $2.3311220$.
• chord-extension-equal-arc — §538, figure 118: $s\sin \frac{s}{2}=1$ from chord-extended-to-tangent similar triangles; $AE=84^{\circ }53^{\prime }38^{\prime \prime }51^{\prime \prime \prime }$.
• arcs-equal-to-tangents-series — §539: all roots of $s=\tan s$. Solutions $s_n$ near $(2n+1)\pi /2$ for $n\ge 1$; asymptotic expansion $s_n=(2n+1)q-2/((2n+1)\pi )-\cdots$ via arctangent-series inversion; ten-row table. The chapter’s only series-inversion problem.
• quadrature-commensurability-remark — §540: if any of the nine transcendental answers had been commensurable with the radius (e.g., $DE=\frac{2}{3}$ in Problem VI), a quadrature of the circle would have followed. None were. Euler stops short of declaring impossibility; the §540 wager is finally settled by Lindemann (1882). Closes Book II.

From Appendix Chapter 1

• surface-coordinates-and-equation — §§3–12: three orthogonal coordinates $AP=x,PQ=y,QM=z$ (figure 119); single-valued vs. multi-valued $z$; six coordinate orderings; the three-plane reference frame.
• algebraic-and-transcendental-surfaces — §§6–7: principal division of regular surfaces; examples $z=x\log y$, $z=y^x$, $z=y\sin x$ as transcendental.
• diametral-plane — §§13–14: parity criterion (even exponents in $z$ → plane $APQ$ bisects the solid); 3D analogue of diameter-of-conic; sphere has all three planes diametral.
• octants-and-region-symmetries — §§15–25: eight octants $PQR,PQr,\dots ,pqr$; conjugate/disjoint/opposite relations; single-variable and joint-degree parity rules for region-congruences; 8-fold central symmetry condition.

From Appendix Chapter 2

• sections-by-coordinate-planes — §§26–31: sphere sections (great circles), parallel-plane stack, congruent-section criterion (one variable absent).
• cylindrical-and-prismatic-surfaces — §§32–33: one variable absent → translate base curve along the missing axis (figure 122); circle/ellipse/polygon bases.
• conical-and-pyramidal-surfaces — §§34–38: homogeneous equation → similar sections growing proportionally with distance from vertex (figure 123); unified $Z(z)$-generalization makes cone and cylinder one family.
• surfaces-of-revolution — §39: lathed solids with $Z(z{)}^2=x^2+y^2$; right cone, cylinder, sphere as special profiles.
• triangular-section-solids — §§40–44: Wallis’s wedge-cone (figure 124) with vertex on a line; right-triangle sections on a curve (figure 125); affine-section solids (figure 126).
• axial-section-solids — §§45–46: $z=T+S$ surfaces with $T$ linear in $y,z$ (figure 127); §46 axial-section substitution $y=v\cos \varphi ,z=v\sin \varphi$.
• oblique-plane-section-method — §§47–50, 84–85: general substitution $x=t\cos \theta +v\cos \varphi \sin \theta ,y=v\cos \varphi \cos \theta -t\sin \theta -f,z=v\sin \varphi$ for the section by any plane; §50 algebraic-surface invariance.
• general-quadric-surface — §51: every plane section of a quadric in $x,y,z$ is a conic; closing remark of chapter 2.

From Appendix Chapter 3

• cylinder-sections — §§52–67: elliptical-base cylinder $a^2{c}^2=a^2{y}^2+c^2{x}^2$ (figure 130); parallel-to-conjugate-axis sections give ellipses of semiaxes $a/\cos \varphi$ and $c$ (§56) with subcontrary circle at $\cos \varphi =a/c$ (§57); §64 theorem $aD\cdot eD=ac/\cos \varphi$ for any oblique section.
• cone-sections — §§68–80: $m^2{n}^2{z}^2=m^2{y}^2+n^2{x}^2$ (figure 134); sections perpendicular to a base axis are hyperbolas (§69); §§70–71 oblique-to-axes hyperbolic sections; §§72–75 parabola/ellipse/hyperbola trichotomy by sign of $(\cos \varphi {)}^2-m^2(\sin \varphi {)}^2$ (parabola at $\tan \varphi =1/m$); §74 subcontrary circle; §§76–80 general oblique case.
• sphere-sections — §§81–85: $x^2+y^2+z^2=a^2$, every plane section is a circle of radius $\sqrt{a^2-f^2{\sin }^2\varphi }$ (§§82–83); §§84–85 closing universal section algorithm (figure 139).

From Appendix Chapter 4

• change-of-coordinates-3d — §§86–92: master rigid-motion substitution. Origin translation, in-plane rotation (figure 140), out-of-plane tilt (figure 141), full composition with three Euler angles $\zeta ,\eta ,\theta$ and three displacements $f,g,h$; six free constants.
• degree-invariance-surfaces — §§93–96: total degree of surface equation invariant under every change of coordinates; sphere stays second-degree; section by any plane has order $\le n$, recovering oblique-plane-section-method §50 from a higher vantage.
• plane-inclination-angles — §§97–100: first-order surfaces are planes (figure 142); $\tan (\text{inclination to }APQ)=-\sqrt{{\alpha }^2+{\beta }^2}/\gamma$ for $\alpha x+\beta y+\gamma z=a$, plus the cyclic versions — direction cosines two centuries early.

From Appendix Chapter 5

• asymptotic-cone — §§104–107: the homogeneous degree-2 part as the surface’s behaviour at infinity. Cone real ↔ unbounded; cone complex ↔ bounded; cone degenerate (factors into linear pieces) ↔ paraboloid family.
• bounded-quadric-criterion — §§108–112: four inequalities $4\alpha \zeta >{\gamma }^2$, $4\alpha \delta >{\beta }^2$, $4\delta \zeta >{ϵ}^2$, and $\alpha {ϵ}^2+\delta {\gamma }^2+\zeta {\beta }^2<\beta \gamma ϵ+4\alpha \delta \zeta$ characterize boundedness. Equality in the fourth = intermediate case giving Genera 4–6.
• quadric-canonical-form — §§113–115: rotation kills cross-terms ($D=E=F=0$), translation kills linear terms; canonical form $A{p}^2+B{q}^2+C{r}^2+K=0$ with three mutually perpendicular diametral planes through the center.
• ellipsoid — §§116–118, figure 143: Genus 1. $A{p}^2+B{q}^2+C{r}^2=a^2$. Three species: sphere, spheroid (prolate/oblate), triaxial.
• elliptic-and-hyperbolic-hyperboloids — §§119–122, figures 144–145: Genus 2 (circumscribes real cone) and Genus 3 (inscribed inside it); collapse to common asymptotic cone at $a=0$.
• paraboloids-and-parabolic-hyperboloid — §§123–125, figures 146–147: Genera 4 ($A{p}^2+B{q}^2=ar$, elliptic paraboloid with elliptic cylinder species) and 5 ($A{p}^2-B{q}^2=ar$, parabolic hyperboloid / saddle with two asymptotic planes); center at infinity.
• parabolic-cylinder-quadric — §126: Genus 6, $A{p}^2=aq$. Two parallel planes as the $a=0$ quasi-species (analogue of two parallel lines as degenerate parabola).
• quadric-classification-algorithm — §§127–128: decision tree reading the genus directly from the original 10-coefficient equation. Same four expressions as bounded-quadric-criterion partition the coefficient space into the six genera.
• higher-order-surface-cones — §§129–130: brief preview. Third-order surfaces classified by factorization of their degree-3 part; many more genera of third-order cones than second-order; no bounded surface of odd order.

From Appendix Chapter 6

• non-planar-curves-by-projection — §§131–136, figure 148: skew curves in space need two equations in $(x,y,z)$; projection onto a coordinate plane by elimination of one variable; two projections determine the curve.
• intersection-of-two-surfaces — §§137–138: master elimination procedure. Surface ∩ plane is the special case $\alpha z+\beta y+\gamma x=f$, recovering oblique-plane-section-method.
• tangency-of-surfaces — §§139–146: a tangency is the coincidence of two intersections, read off the eliminant’s double-root / perfect-square factor structure. Sphere–plane tangency at $f=a\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}$; cone–sphere tangency along a circle for right cones at $f=a\sqrt{1+n}$, at two points for scalene cones.
• tangent-plane-to-surface — §§147–149, figure 149: build the tangent plane at $M$ from tangents to two perpendicular plane sections; subtangents $s,t$ give the trace slope $t/s$ and inclination $z\sqrt{s^2+t^2}/(st)$; normal direction $PV=z^2/s$, $NW=z^2/t$.
• bezout-bound-surfaces — §150: $mn$ bound on the projection of intersection of surfaces of orders $m,n$. 3D lift of intersection-product-degree-bound.
• planar-intersection-condition — §§151–152: intersection lies in a plane iff a common linear relation $\alpha z+\beta y+\gamma x=f$ follows from both equations. Worked example: right cone $z^2=x^2+y^2$ and elliptic hyperboloid $z^2=x^2+2y^2-2ax-a^2$ meet in the plane $z=x+a$. Closes Book II.