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Some Relations of Plane and Spheric Geometry. 



David A. Rothrock. 



Our notions of plane analytic geometry date to the publication by Descartes 

 of his philosophical work: "Discours de la methods . . . dansles sciences,'' 

 1037, which contained an appendix on "La Geometric" In this work Des- 

 cartes devised a method of expressing a plane locus by means of a relation 

 between the distances of any point of the locus from two fixed fines. This 

 discovery of Descartes led to the analytic geometry of the plane, and the 

 extension to three dimensional space gave rise to geometry of space figures 

 by the analytic method. A single equation, f (x.y) = o, between two variables 

 represents a plane curve; a single equation, F 2 (x,y,z) = o, in three variables 

 represents a surface in space; and two equations, Fi (x,y,z) = o, F 2 (x,y,~z) = o, 

 represent a curve in space. 



In the Cartesian system of coordinates, a space curve is determined by 

 the intersection of two surfaces. If we wish to investigate the curves upon 

 a single surface, that is, if we wish to devise a geometry of a given surface, 

 it may be possible to discover a system of coordinates upon the surface, 

 such that any surface-locus may be expressed by a single equation in terms 

 of two coordinates, as in plane geometry. The sphere furnishes a simple 

 example in which a locus upon its surface may be represented by a single 

 equation connecting the coordinates of any point upon the locus. 



Toward the end of the eighteenth century a fragmentary system of 

 analytic geometry of loci upon the surface of the sphere was developed. 

 This early work on Spheric Geometry seems to have originated with Euler 

 (1707-1783), but many of the special cases of spherical loci were investigated 

 by Euler's colleagues and assistants at St. Petersburg. In the present paper 

 are enumerated a number of the early investigations on spherical loci, and a 

 derivation of the equations of sphero-conies in modern notation. The 

 correspondence of the spheric equations to the similar equations of plane 

 analytics is shown. 



50S4 — IS 



