A History of Greek Mathematics and Astronomy

Page 24

In the work On the Measurement of the Earth Eratosthenes is said to have discussed other astronomical matters, the distance of the tropic and polar circles, the sizes and distances of the sun and moon, total and partial eclipses, &c. Besides other works on astronomy and chronology, Eratosthenes wrote a Geographica in three books, in which he first gave a history of geography up to date and then passed on to mathematical geography, the spherical shape of the earth, &c., &c.

Apollonius of Perga was with justice called by his contemporaries the 'Great Geometer', on the strength of his great treatise, the Conics. He is mentioned as a famous astronomer of the reign of Ptolemy Euergetes (247-222 B. C.); and he dedicated the fourth and later Books of the Conics to King Attalus I of Pergamum (241-197 B. C.).

The Conics, a colossal work, originally in eight Books, survives as to the first four Books in Greek and as to three more in Arabic, the eighth being lost. From Apollonius's prefaces we can judge of the relation of his work to Euclid's Conics, the content of which answered to the first three Books of Apollonius. Although Euclid knew that an ellipse could be otherwise produced, e. g. as an oblique section of a right cylinder, there is no doubt that he produced all three conics from right cones like his predecessors. Apollonius, however, obtains them in the most general way by cutting any oblique cone, and his original axes of reference, a diameter and the tangent at its extremity, are in general oblique; the fundamental properties are found with reference to these axes by 'application of areas', the three varieties of which, application (παραβολη {parabolê}), application with an excess (ὑπερβολη {hyperbolê}) and application with a deficiency (ελλειψις {elleipsis}), give the properties of the three curves respectively and account for the names parabola, hyperbola, and ellipse, by which Apollonius called them for the first time. The principal axes only appear, as a particular case, after it has been shown that the curves have a like property when referred to any other diameter and the tangent at its extremity, instead of those arising out of the original construction. The first four Books constitute what Apollonius calls an elementary introduction; the remaining Books are specialized investigations, the most important being Book V (on normals) and Book VII (mainly on conjugate diameters). Normals are treated, not in connexion with tangents, but as minimum or maximum straight lines drawn to the curves from different points or classes of points. Apollonius discusses such questions as the number of normals that can be drawn from one point (according to its position) and the construction of all such normals. Certain propositions of great difficulty enable us to deduce quite easily the Cartesian equations to the evolutes of the three conics.

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T. L. Heath
A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath
A History of Greek Mathematics and Astronomy