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I've always thought that defining conic sections by a locus of points w.r.t the ratio of the distance to the focus and directrix was always "too artificial" - how does one actually discover this mysterious directrix, a line that isn't really lying anywhere when a double-cone is cut by a plane; or a focal point where all the "rays reflecting the conic" would meet (i.e., for a parabola)

Some mathematical thinking/creativity must have gone into "discovering" this fact. I've scoured ancient Greek history but have failed to uncover anything.

The best I have is along these lines (my best interpretation, from History of Greek Math Vol 2 - Heath, Pg 119):

Aristaeus' Solid Loci considered loci which proved to be Conic sections. The theorems that Pappus of Alexandria alludes to are those put forth by the Solid Loci. There's a possibility that the focus-directrix property was known to Euclid but there's no conclusive evidence. However, the focus-directrix was probably "defined" as such by Pappus.

UPDATE: The accepted answer provided a great direction and helped me dig up the Arabic Translation of "On Burning Mirrors by Diocles" that proves the existence of a focus (Proposition 1) as well as provides construction of the parabola via the focus-directrix property (Prop. 4-5). This is probably the "first" extant source that shines light on how this came about and Diocles attributed it to Dositheus, but the latter never furnished a geometric proof it seems.

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    $\begingroup$ I edited the question to make it about history, the educational "rediscovery" side can be addressed on Math Educators SE or Math SE. $\endgroup$ – Conifold Mar 1 '18 at 0:34
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The answer will be disappointing, I am afraid, but typical, "natural" and "artificial" are very relative notions. The "discovery" of the focus-directrix property was likely a technical lemma in investigations of solid loci, which was later also used in the study of curved mirrors. When it did appear Greeks did not make much of it, its modern prominence in the theory of conic sections is a late artifact.

The earliest extant source we have is Diocles' On Burning Mirrors, and he is working off of Euclid's previous work on spherical mirrors. The point was of course to find a shape that would gather sun rays (assumed to be parallel) at a single point, the focus. As Knorr pointed out, reducing ray-focusing property to the focus-directrix property to "discover" that it is possessed by parabola would require something like solving a differential equation, which was beyond the devices of Greeks. He then speculated that someone (perhaps, Archimedes's friend Dositheus) made a "lucky guess", and then proceeded to show that the curve constructed by the focus-directrix property is a parabola (similar demonstration for ellipse is described by Anthemius). Acerbi suggests a different reconstruction in The Geometry of Burning Mirrors in Greek Antiquity, where one can find geometric details of how the property came up as a lemma:

"Focal properties of the parabola were obtained as a by-product of an attempt at proving that the property of the subtangent is a sumptoma of the curve. About the relevance of this property to the eyes of the ancient geometers it is not necessary to spend so many words. Recall in fact that the property of the subtangent was well known to Archimedes... As a consequence, the focal property proves crucial to employ the subtangent as a method to construct a parabola with given vertex and parameter (cf. Con. I.52, that, however, does not lead in an immediate way to a pointwise construction) and makes it possible to solve by reduction a particular case of the problem of tracing a conic section after some of its tangents are known.

"[Assumption] is that a property so important (to us) as the focus-directrix could not be reduced to the status of mere functionality to a pointwise determination of a parabola, as indisputably happens in Diocles’ approach and as Knorr himself has to recognize... Of this property, there is no trace in Apollonius’ Conica as they are now extant (recall that it holds in reference to the axis only, and therefore it is not a sumptoma that can be referred to any diameter).

"Toomer surmised that Diocles himself was the discoverer of the property in the case of the parabola, its validity being extended after him to the other conic sections. The few lemmas Pappus offers, at the very end of book VII of his Collectio... is the only attestation in the ancient corpus of the fact that the focus–directrix property is a sumptoma of the several conic sections (the formulation as a locus bears this out exactly).

Zeuthen (1886, pp. 210–215 and 367–371) observes that, if Pappus had to make such a lemma explicit, then the focus–directrix property should have been taken for granted in the Loci on a Surface. Zeuthen’s remark can be completed by the conjecture that the most “natural” place where the property was established, of course in the above form of a sumptoma formulated as a constraint identifying a locus, is in Aristaeus’ Solid Loci... In sum, if one is not willing to venture in empty conjecturing, then a minimal hypothesis is that the property had been employed in the Loci on a Surface as an intermediate step in the analysis of some locus, but that it was simply considered as a sumptoma of a conic section".

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  • $\begingroup$ Your pointers were insightful. After much digging I found a great reference that provides the arabic translation of Diocles's "On Burning Mirrors" - his very first proposition is the "proof of parallel rays will meet at a focus" and Props 4-5 are reconstructing the parabola via focus-directrix and proving that the result is in fact a parabola. math.harvard.edu/archive/hist_206r_2009/Too_1976.pdf So to your point, I think the entire story is quite interesting and not even a wee bit disappointing! Thanks for the direction ;) $\endgroup$ – PhD Mar 2 '18 at 0:52
  • $\begingroup$ @PhD Vow, that is some serious follow-up, glad it worked out for you! I thought it would be disappointing because it does not give a "natural" way of going from the section-of-a-cone definition to the focus-directrix property. $\endgroup$ – Conifold Mar 2 '18 at 1:02
  • $\begingroup$ You're right. It's not natural "that way". But given a "problem" and finding a solution to it, providing a proof of it seems "natural" enough. It's a great story about the discovery AFAIK and much better than "wondering" how the hell did it come about. I found it really hard to "meditate" and come up with a solution. There are ways to "guess" or reverse engineer this but nothing that would "wow" someone. $\endgroup$ – PhD Mar 2 '18 at 1:11
  • $\begingroup$ Unrelated, but would appreciate your help on this, if possible: math.stackexchange.com/questions/2674177/… $\endgroup$ – PhD Mar 2 '18 at 21:06
  • $\begingroup$ @PhD This is a bit head splitting for me, but I remembered a "natural" way to "discover" the focus-directrix property. It was invented by Dandelin in 1822 and involves inscribing two (one for parabola) so-called focal spheres into the cone, see Dandelin spheres. $\endgroup$ – Conifold Mar 2 '18 at 21:41
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I tend to agree with the sentiment expressed in the question, that the focus-directrix definition of conics seems artificial (and awkward). Yet it has had its adherents. To answer the question 'How could the student discover the focus/directrix properties on her own?' it might be useful to adopt an a posteriori approach, by considering the material in a book of conics that works extensively with these properties.

An example of a book that did use this approach achieved some 19th-century popularity, at least it ran to many editions: Conic sections, treated geometrically by W H Besant. Its proofs may (with meditation) provide some heuristic clues!

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