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APOLLONIUS of Perga, in Pamphylia, so called from the place of his birth, was a younger contemporary of Archimedes, and probably survived him about ten years. Of the details of his life little is known. He studied mathematics at Alexandria under the successors of Euclid. Much of his life was spent in Pergamum. His great work on Conic Sections is dedicated to a certain Eudemus of that city. This work is in eight books, of which, in the time of Descartes, the first four only were known to be extant. Shortly afterwards the fifth, sixth, and seventh books were discovered in their Arabic translation--an important discovery, since they reveal mathematical powers which the students of the first four had hardly suspected. Apollonius wrote many other works on mathematical subjects, of which little more than the title has come down to us. In one of these he endeavours, as Archimedes had done, to enlarge and improve the Greek system of numeration. A fragment of another work, on irrational quantities, has been discovered has been discovered in Arabic: and it is possible that others may yet be restored from a similar source. By contemporaries and successors he was spoken of as the Great Mathematician.
The curves produced by a plane that cuts a cone had been examined by several previous geometers. Allman (pp. 153-163) has shown that they were first observed by Menæchmus, a pupil of Eudoxus, who made use of both the parabola and the hyperbola in solving the famous problem of inserting between two given lines two mean proportionals--in other words, the problem of duplicating the cube. Euclid, the elder Aristaeus, and Archimedes devoted much attention to these curves; the quadrature of the parabola being one of the most remarkable achievements of Archimedes.
Apollonius was accused by some of founding his reputation on unacknowledged debts to these great predecessors. That he had studied their works is obvious: not less so is it that he made the subject his own by exhaustive handling, and by original development of it. The names which the Conic Sections now bear are in all probability due to him. Before his time each of the three curves was regarded as resulting from a plane cutting the side of the cone at right angles; and only right cones, i.e. those in which the axis was perpendicular to the basis, were considered. If the cone were such that the angle at the vertex was a right angle, the section thus made was a parabola: if the vertical angle were acute, the section was an ellipse; if obtuse, an hyperbola. Apollonius showed that all these curves could be produced in every cone, whether right or oblique, by varying the inclination of the cutting plane. His generalized treatment of those three curves, apparently so diverse, was in itself a considerable step. The inevitable tendency of Greek, as contrasted with modern, geometry, was to specialize the study of each curve. In the work of Apollonius we have to recognize the first, though but a slight, advance towards the great conception of Descartes, who applied general methods to the treatment of all curves whatsoever. The names given by Apollonius to these curves illustrate this general treatment. "The rectangle applied to a certain straight line in the section of the acute-angled cone is deficient (ελλείπει) by a square: in the section of the obtuse-angled cone it is excessive (υπερβάλλει) by a square: finally, in the section of the right-angled cone, the rectangle applied (παραβαλλόμενον) is neither deficient nor excessive." (Pappus, quoted by Allman, p. 196).
"The fifth book," says Cantor in his History of Ancient and Medieval Mathematics, "far surpasses the preceeding. Apollonius rises far above his time by a series of propositions on the longest and shortest lines that can be drawn from a given point to the circumference of a conic section. He begins by remarking that previous mathematicians had treated of the theory of shortest lines. But their mode of handling it must have differed substantially from his; their purpose being limited to the determination of the limits(diorismus) within which a given problem was possible. Apollonius gneralized the problem of maxima and minima: regarding it, as he expressly says, as one of those things worthy to be considered on the ground of their intrinsic importance. The way in which he distinguishes special cases in this department, and, by grouping them together, fixes the range of possible cases--the marvellous, indeed almost unnatural, complexity of his demonstrations, excite and deserve our wonder. In determining these longest and shortest lines, Apollonius first treats the cases in which the given point is situated on the acis of the curve. Then follow a series of propositions relating to the modern conception of sub-normals. The constancy of this line in the parabola is shown. Later on, the proof is reached that the greatest and least lines previously spoken of are normals to the curve; thus the problem arises, to draw normals to a conic section from any point in its plane. He now discovers that the number of such normals depends partly on the curve, partly on the position of the point chosen: further, that there are certain points from which only one can be drawn. These points correspond to the centres of the osculating circles, their series forming the curve known as the Evolute. Of the points in question Apollonius was clearly aware, and probably had some conception of the curve resulting from them.
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| This biography is
reprinted from The New Calendar of Great Men. Ed. Frederic
Harrison. London: Macmillan and Co., 1920. |
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