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PAPPUS was a contemporary of Theon of Alexandria, and taught mathematics in that city during the reign of Theodosius I. He wrote a commentary on the Great Syntaxis [Almagest] of Ptolemy, which has not come down to us. He is known to us by the work entitled Synagoge or Assemblage; a collection in eight books of mathematical papers having no very distinct connection, and consisting of commentaries on the geometrical work of the previous six centuries, enriched by very fruitful additions of his own. For the history of ancient mathematics this work, of which the last six books and part of the second have been preserved, is invaluable. Already in the third century B.C. the filiation of discovery, so evident in this science, had been traced by Eudemus, a pupil of Aristotle, parts of whose work have been preserved by Proclus. Pappus supplies many details of Apollonius and of later writers who would otherwise have been unknown to us. Special studies on isolated problems occupy the greater part of his attention. Various modes of inserting two mean geometrical proportionals are discussed; new methods of inscribing the five regular solids in a sphere are put forward, (book iii). There are special studies on various curves; as the spirals of Archimedes, the quadratix of Dinostratus, and the conchoid of Nicomedes. Much attention is given to the work of Zenodorus on isoperimetry; and new problems on this subject are solved (book v). In the 6th book the earlier astronomers are spoken of.
The 7th book is the most important, historically speaking. We find here the highest point reached by the geometrical analysis of antiquity; the starting-point of the mathematical revolution instituted by Descartes. Pappus speaks of Euclid, Aristaeus, and Apollonius as the chief cultivators of this department, and as having their attention to what were called solid loci, i.e. to problems which could only be solved by the use of one of the conic sections. One of these problems was of this kind: Let three, four, or more lines be given in position; required the locus of the point from which the same number of lines may be drawn to meet them one to each, at given angles, such taht, in the case of three lines, the rectangle of the first two lines may have a fixed relation to the square of the third; or, in the case of four lines, that the rectangle of the first and second may have a fixed relation to rectangle of the third and fourth, and so on. Pappus was aware that in the case of three, or of four, lines the locus of the point was a conic section. But what the locus was when the lines were more numerous he declared himself unable to determine. This was the problem that Descartes undertook to solve. His success, and the momentous consequences that followed from it, depended not merely on the use of algebraic notation in geometrical problems, but on the broad conception of equation and curve as two correlated aspects of the same problem, and on the entire generality of the methods used in attacking simultaneously problems which till then had been handled separately. "The spirit of the geometry of antiquity," Comte remarks in his treatise of Analytical Geometry (p. 7), "was essentially synthetic: that is to say, the various conditions of each problem were studied for the most part in their entirety. It is true that what was called geometrical analysis had been used in an accessory way; and this may be regarded as a first approach to the modern system; although the absence of algebraic conceptions, by which alone the separation of the various conditions of the problem could be fixed and pursued to its final consequences, deprived this procedure of its main value; so that by the geometers of Greece it was more preached than practiced. The spirit of modern mathematics since Descartes is to isolate the various conditions of a problem, and thus arrive at a perfectly general solution for each. It is thus, in the strictest meaning of the word, Analysis."
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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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