SoME RELATIONS OF PLANE AND SPHERIC GEOMETRY. 
Davin A. Rorurock. 
Our notions of plane analytic geometry date to the publication by Descartes 
of his philosophical work: ‘‘Discours de la méthode . . . dans les sciences,” 
1637, which contained an appendix on ‘“‘La Geometrie.”’ 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 lines. 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) = 0, between two variables 
represents a plane curve; a single equation, F, (x,y,z) = 0, in three variables 
represents a surface in space; and two equations, F; (x,y,z) = 0, Fs (x,y,z) = 0, 
represent a curve in space. 
In the Cartesian system of coérdinates, 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 codrdinates upon the surface, 
such that any surface-locus may be expressed by a single equation in terms 
of two coérdinates, 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 codrdinates 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-conics in modern notation. The 
correspondence of the spheric equations to the similar equations of plane 
analytics is shown. 
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