Is there anything known about what drew the attention of ancient greek mathematicians to conic sections and, what were the models they used to study conic sections? 
What I would like to know, is whether ancient mathematician actually became aware of conic sections by looking at sections of circular cones or, whether they noticed them elsewhere, e.g. as the shadow of a coin or (highly speculative) even in the way that gardeners created interesting shapes with two sticks and a rope? 


I know that Apollonius of Perga wrote a thorough treatise about conic sections (cf e.g. <http://en.wikipedia.org/wiki/Apollonius_of_Perga>), but did he also discover them and how did *he* study them? 



Discovery and Study of Conic Sections in Ancient Greece
**Discovery:** [Menaechmus](https://en.wikipedia.org/wiki/Menaechmus) is credited with the discovery of conic sections around the years 360-350 B.C.; he used them to solve the problem of "doubling the cube". The construction required a parabola, which he called "a section of a right-angled cone", and a hyperbola, "a section of an obtuse-angled cone". (The names parabola and hyperbola themselves are due to Apollonius.) [1]


**Origin:** Euclid notes in his Phaenomena that a cylinder cut by a plane not parallel to the base results in a section which resembles a "shield". [Eratosthenes](https://en.wikipedia.org/wiki/Eratosthenes) implied that Menaechmus arrived at his sections by cutting a cone "in triads".


In connection with the suggestion of the OP that "the shadow of a coin" might have led to an early observation of conic sections: It is quite possible that the ancient Greeks would have been aware of conic sections when constructing [sundials](https://en.wikipedia.org/wiki/Sundial), since a sheaf of light rays is a cone which is cut by the plane of the horizon in a hyperbola, and a portion of that hyperbola is then marked out on the sundial [2].


![](https://ilorentz.org/beenakker/MO/sundial.png)



> 
> One may speculate about the circumstances that might have led
>  Menaechmus to discover the curves. Conceivably he could have developed
>  the idea from observing a volcano, or an anthill scuffed off by his
>  sandal, or some artefact like a sharpened stake. However, there is
>  much to recommend the conjecture [going back to Philippe de la Hire,
>  1682] that the sundial was the most probable basis for discovery of
>  the conics: A dial traced on an appropriate oblique plane could show 
>  all three conics at any latitude.
> 
> 
> 


*Sources:*


[1] [Conic Sections in Ancient Greece](http://www.math.rutgers.edu/~cherlin/History/Papers1999/schmarge.html), K. Schmarge (1999).


[2] [Early sundials and the discovery of conic sections](http://www.jstor.org/stable/2688372?seq=1#page_scan_tab_contents), W.W. Dolan (1972).


