## The Greek Mathematicians and Conics

• Menaechmus (~375-325BC) who lived in Macedonia and Greece and was a tutor of Alexander the Great, formalized the notion of conics and found many of their geometrical properties. He solved the famous problem of cube duplication by determining -^2 from the intersection of two conics: a parabola and a hyperbola -with the Cartesian formalism y = x2 and xy = 2, respectively (Fig. 1.1). He suspected that the solution was impossible with the classic method of straightedge and compass since a few decades earlier Hippocrates of Chios (471-410 BC, Athens) has been able to solve the not less difficult problem of the angle trisection by the less restrictive method of "declination" - inclinatio in Latin and vevaiq (neusis) in Greek - which is now called the marked straightedge method.

This method consists of revolving a straight line of a given length through a fixed point until it intersects two fixed straight lines (cf. Arnaudies & Delezoide [6]).

G.R. Lemaitre, Astronomical Optics and Elasticity Theory, Astronomy and Astrophysics Library, DOI 10.1007/978-3-540-68905-8.1, @ Springer-Verlag Berlin Heidelberg 2009

Fig. 1.1 (Left) Angle trisection obtained from the marked straightedge by Hippocrates of Chios in ~430 BC. (Right) Cube duplication obtained from the two-conic intersection method by Menaech-mus in ~340BC. The two-conic intersection method is equivalent to the marked straightedge method

In practice one uses a straightedge on which the given length is copied in two marks by revolving and sliding it at the fixed point. In Fig. 1.1, the angle to trisect is xOA, a length MN = 2xOA is reported on the straightedge that pivots in O, and M,N must be on the straight lines x', y' that are parallel to x, y through A. This method is equivalent to the two-conic intersection method.

• Aristaeus the Elder (~365-300BC, Greece) wrote five books, now lost, entitled Solid Loci and concerning the conic sections. This is known from commentaries by Pappus (290-350 AD, Alexandria), one of the last Greek mathematicians.

• Euclid (325-265 BC, Alexandria) who is the classical reference in the founding of geometry with his 13 books, well known as Elements, also wrote several other works including Optics, Conics, and Surface Loci. Optics solely concerns perspective. According to Pappus, Euclid's Conics, now lost, was mostly a compilation whereas Aristaeus's book gave a more thorough discussion of the discoveries and properties of conics.

• Apollonius of Perga (262-^190BC, born in Perga, now known as Murtana, on the south coast of Turkey) lived in Ephesus and Alexandria. It is to him that we owe the names ellipse, hyperbola, and parabola. He wrote a treatise in eight books, seven of which still exist, entitled Conics, which contains about four hundred propositions. He differentiated the various conic types by the angle of the intersecting plane with respect to the cone angle. Pappus states that "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight books on conics". He also refers to Euclid's last work on conics that is included in the Treasury of Analysis under Surface Loci but states that Apollonius's Conics had become the classical reference on those questions. He gives some indications of the contents of six other works by Apollonius: Cutting of a ratio, Cutting an area, Inclinations (presently lost), On determinate sections, Plane loci, On verging constructions and Tangencies, but none of them seem to mention any optical property of a conic.

• Diocles (~240-^180BC, lived in Arcadia and Karystos near Athens) published On burning mirrors. It is known from comments by Eutocius (480-540 AD) that he solved the problem of angle trisection by inventing a method based on cissoid curves which thus differs from Hippocrates of Chios's method. None of his writings were known in the West before 1920, but more recently 1970) a complete Arabic translation of his work was found in the Shrine Library of Mashhad, Iran. The first translation, by Toomer [162], was published in 1976. It appears that Diocles discovered the property that a parabola can be defined as the locus of points satisfying a constant distance ratio equal to unity from a given point (focus) and a directrix line. Diocles was the first to show that parallel rays are not brought to focus by a spherical mirror - as had been previously thought - but by a paraboloid mirror; he also discusses other geometrical properties at any point of a parabola.

^ The fundamental optical property of stigmatism of a parallel beam reflected by a paraboloid was known by Diocles in ^200 BC as a geometrical property of the parabola.

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