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I am trying to understand this part of apollonius conics by heath, page 112

The foci are not spoken of by Apollonius under any equivalent of that name, but they are determined as the two points on the axis of a central conic (lying in the case of the ellipse between the vertices, and in the case of the hyperbola within each branch, or on the axis produced) such that the rectangles $AS.SA', AS' .S'A'$ are each equal to "one-fourth part of the figure of the conic," i.e. $\frac{1}{4}p_a.AA'$ or $CB^2$. The shortened expression by which $S, S'$ are denoted is (some greek text) "the points arising out of the application." The meaning of this appear from the fiill description of the method by which they are arr ived at, which is as f}ollows : (more greek text) , " if there be applied along the axis in each direction [a rectangle] equal to one-fourth part of the figure, in the case of the hyperbola and opposite branches exceeding, and in the case of the ellipse falling short, by a square figure."

So basically the focus are the points $S, S'$ such that $CB^2=AS.SA'=AS'.S'A$

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Apollonius defines the foci in this way and goes ahead to prove that it indeed works, but I'm baffled, how could he come to this definition? where does this come from? In proposition 69 he just straight up uses that definition to start proving things, but it just feels like it comes out of the blue.

I have searched far and wide,This site also mentions this concept by the original name that apollonius gave them, "points of application". But I have not been able to find where does it come from.

Intelligenti pauca
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Joaquin Brandan
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The more common way of defining an ellipse (and analogously a hyperbola) is as the locus of points $P$ for which the sum of its distances from two fixed points $S$, $S'$ (the foci) is a constant $2a$. If you then set up a convenient coordinate frame such that $S=(-c,0)$, $S'=(c,0)$, then you get for the locus the equation $x^2/a^2+y^2/b^2=1$, where $b^2=a^2-c^2$. This is the prevalent approach in high-school textbooks.

But it is of course possible to define the ellipse in another way, as the locus of points satisfying the equation $x^2/a^2+y^2/b^2=1$ in a suitable coordinate frame. Then you can show that there are two points: $S=(-c,0)$, $S'=(c,0)$ (where $c^2=a^2-b^2$) such that the sum of the distances from any point of the ellipse to them is $2a$. This is basically what Apollonius did and I don't see why you feel it is not satisfying: it is a definition as fine as the other one, albeit less popular nowadays.

Intelligenti pauca
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  • I understand all of this. I just dont understand how the man came about thinking that the relationship $CB^2=AS.SA'=AS'.S'A$ fits into this. It clearly works as a proof, as seen in his book, but it offers very little insight into how it relates to what I already know of "high school conics", where does it come from or how apollonius imagined it in a way it makes sense other than that "it just works". – Joaquin Brandan Aug 17 '20 at 17:35
  • @JoaquinBrandan One could say the same thing of the high-school definition: why on earth should it be related to sections of a cone? – Intelligenti pauca Aug 17 '20 at 17:41
  • That's the underlying context for my reaserch. I'm working on how both definitions are equivalent without using dandelin spheres. I want to get a more natural way of thinking about both definitions for all conics – Joaquin Brandan Aug 17 '20 at 17:47
  • @JoaquinBrandan Well, both definitions lead to the same cartesian equation: isn't that enough? By the way: I wrote here an Apollonius-like derivation of the ellipse equation. – Intelligenti pauca Aug 17 '20 at 19:52
  • Thanks for the link. I have already worked over the apollonius derivation of the ellipse, this one looks similar but I will look into it nontheless. The thing is that the focus do not show up in this kind of derivations, and to get to them in geometric terms (not analitical geometry) then some arbitrary looking assumptions or postulates (like the one this question is about) are made with no explaination of where they come from. That's why I'm looking for clarification here. I'm not questioning the proof, I'm looking for insight. – Joaquin Brandan Aug 17 '20 at 20:03
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    @JoaquinBrandan Your question is then more about history than mathematics. You might find some insights on page xxxix of Heath's translation, where he compares Apollonius' definition of focus with a theorem by Euclid. Looks like the focus-directrix property was well known in Apollonius' time, but it was not investigated in his treatise. – Intelligenti pauca Aug 17 '20 at 22:10
  • Thank you. That looks like something really worth looking into!. By any chance do you have any relatively modern sources about what you mention in your second paragraph? (that the foci can be derived from the equation). As you said it seems to be a less popular definition and I actually could not find a proof or articles about it, I checked out your other answers and you really seem to know about this stuff so I thought maybe you have one handy or maybe nudge me in the right direction. – Joaquin Brandan Aug 18 '20 at 19:40
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    I read that on page xxxvi of Heath's translation, where he cites a lemma by Pappus to Euclid's (lost) treatise "Surface loci": "the locus of a point whose distance from a given point is in a given ratio to its distance from a fixed line is a conic section, and is an ellipse, a parabola, or a hyperbola according as the given ratio is less than, equal to, or greater than, unity". – Intelligenti pauca Aug 18 '20 at 20:22
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    Anyway I'm not an expert in history of mathematics: I'd suggest you to ask on https://hsm.stackexchange.com – Intelligenti pauca Aug 18 '20 at 20:25