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 $CB^2=AS.SA'=AS'.S'A$
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.
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.