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In Katz, V. (1998), A History of Mathematics, p.117, we read that early Greek geometers defined conic section as formed by planes cutting the generating line (the hypothenuse of the generating triangle of the cone) at a right angle. The section was then elliptic, parabolic, or hyperbolic, depending on whether the angle at the vertex of the cone was acute, right, or obtuse, respectively. Appollonius on the other hand, let the conic sections be the result of cutting a cone by a plane at any angle; if it cut both legs of the axial triangle of the cone, then the section was elliptic; if it cut neither of them, and was thus parallel to one of them, parabolic; and if it cut one side and the other side produced beyond the vertex, hyperbolic.

Thus, while the earlier definition has only one variable (the angle at the vertex), the latter definition has two (the angle at the vertex and the angle of the cutting plane to the base plane). I've seen it stated that the curves generated by the former definition are exactly the same as that of the latter, with the exception of the 'degenenerate cases' of a circle and a point. In other words, the more `restrictive' definition can already produce all ellipses (excluding circles), parabolas, and hyperbolas.

Is this true, and was this known to Appollonius?

Ethan Bolker
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    Well, if $\alpha$ is the half-angle at the vertex, then with the first definition, the eccentricity of the resulting conic turns out to be $\tan\alpha$ and its size can be adjusted by translating the plane along the conic’s axis. Can’t comment on Appolonius, though. [To be fleshed out into an answer later if no one else gets to it first.] – amd Sep 12 '19 at 10:07
  • Note that Apollonius cone was not necessarily a right cone: he defined a cone as formed by all lines joining a given point $V$ not lying on plane $\alpha$ with a point of a circle on plane $\alpha$. – Intelligenti pauca Sep 12 '19 at 10:47
  • As for what Apollonius may or may not have known, the History of Science and Mathematics StackExchange might be a better place to ask. – Blue Sep 12 '19 at 15:37

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In the old definition, the conic section depends on two variables: half-angle $\alpha$ at the vertex and distance $VB$ between vertex $V$ and the point $B$ where the perpendicular generatrix meets the plane. One can relate these variables to the parameters of the conic section, for instance semi-axes $a$ and $b$ for an ellipse or hyperbola, and latus rectum for a parabola (in that case $\alpha=\pi/4$ is fixed).

In the general case treated by Apollonius (oblique circular double cone), the intersecting plane is perpendicular to a plane of symmetry of the cone and intersects its generatrices at points $A$ and $B$ (for an ellipse or a hyperbola). Repeating the steps given here (which strictly follow Apollonius' work) we then find: $$ a={1\over2}AB,\quad b={1\over2}\sqrt{BD\cdot AE}, $$ where lines $BD$ and $AE$ are parallel to the base of the cone.

In the special case of the pre-Apollonius definition, those relations become: $$ a=VB{\tan\alpha\over1-\tan^2\alpha},\quad b=VB{\tan\alpha\over\sqrt{1-\tan^2\alpha}}, $$ for an ellipse ($0<\alpha<\pi/4$), and: $$ a=VB{\tan\alpha\over\tan^2\alpha-1},\quad b=VB{\tan\alpha\over\sqrt{\tan^2\alpha-1}}. $$ for a hyperbola ($\pi/4<\alpha<\pi/2$). In both cases eccentricity is found to be $e=\sqrt{1\mp b^2/a^2}=\tan\alpha$.

For a parabola, one finds in the general case that the latus rectum is given by $BD^2/VB$, which becomes $2VB$ with the more restrictive definition.

Hence the old definition could give rise to all possible conic sections, despite being less general, with the exception of circles because $a=b$ is impossible (unless $\alpha=0$) in the case of an ellipse.

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Intelligenti pauca
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