There's a well known treatise by Apollonius on conic sections, but these objects are absent in Euclid's Elements. Why? If I were to guess, I'd say that conic sections cannot be constructed using a compass and a straightedge, but I cannot be sure.


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While Elements contains no reference to conic sections it does define angled cones, given as definition 18 of Book XI, and it examines some of their properties in Book XII.

History identifies Menaechmus as inventor of conic sections (350-360BC), obtained by considering these angled cones intersecting a fixed plane.

In addition to Elements, Euclid also wrote four books on conic sections, now lost. Indeed, Apollonius makes reference to Euclid's lost work in Conics. Quoting from Conic Sections in Ancient Greece, by Ken Schmarge of Rutgers:

We next come to the (again, lost) works of Aristaeus `the elder' and of the celebrated Euclid on conic sections. Since we do not have the original works by these two men on conic sections, our knowledge of them is derived from the comments of Pappus, whose writings are discussed in Heath, using a translation by Hultsch:

The four books of Euclid's conics were completed by Apollonius, who added four more and produced eight books of conics. Aristaeus, who wrote the still extant five books of solid loci connected with the conics, called one of the conic sections the section of an acute-angled cone, another the section of a right-angled cone and the third the section of an obtuse-angled cone.... Apollonius says in his third book that the `locus with respect to three or four lines' had not been completely investigated by Euclid, and in fact neither Apollonius himself nor any one else could have added in the least to what was written by Euclid with the help of those properties of conics only which had been proved up to Euclid's time; Apollonius himself is evidence for this fact when he says that the theory of that locus could not be completed without the propositions which he had been obliged to work out for himself. Now Euclid-regarding Aristaeus as deserving credit for the discoveries he had already made in conics, and without anticipating him or wishing to construct anew the same system, being moreover in no wise contentious and, though exact, yet no braggart like the other-wrote so much about the locus as was possible by means of the conics of Aristaeus, without claiming completeness for his demonstrations ... (Heath, 1961, pp. xxi-xxii)

The introduction in Schmarge's paper sums things up pretty well :

The knowledge of conic sections can be traced back to Ancient Greece. Menaechmus is credited with the discovery of conic sections around the years 360-350 B.C.; it is reported that he used them in his two solutions to the problem of "doubling the cube". Following the work of Menaechmus, these curves were investigated by Aristaeus and of Euclid. The next major contribution to the growth of conic section theory was made by the great Archimedes. Though he obtained many theorems concerning the conics, it does not appear that he published any work devoted solely to them. Apollonius, on the other hand, is known as the "Great Geometer" on the basis of his text Conic Sections, an eight-"book" (or in modern terms, "chapter") series on the subject. The first four books have come down to us in the original Ancient Greek, but books V-VII are known only from an Arabic translation, while the eighth book has been lost entirely.


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