traced upon the Surface of the Sphere. 261 



traced upon them both at the same time. Every thing else fol- 

 lowed, and it soon became the universal practice to investigate 

 the properties of curves of double curvature by reference to three 

 co-ordinate planes, and determined by any two equations be- 

 tween the three variables. The method of DESCARTES at best 

 led only to the discussion of curves of double curvature, by 

 means of the equations^zy andf a zo?, instead off'acyz andf'xyz, 

 each equated to zero ; whilst the latter process, which is that of 

 CLAIRAULT, embraced the former as a particular case, granting, 

 however, the resolution of equations of all orders : but that of 

 DESCARTES could in fact only be applied in this particular case, 

 whilst that of CLAIRAULT is independent of any such necessity, 

 and applies with the same success, whether the elimination of 

 any variable between the given equations be possible or not. 



The problem of VIVIANI, which was professedly proposed to 

 bring the powers of the new Geometry to the test, must have 

 awakened attention to this class of inquiries ; and with respect 

 to spherical loci, several problems of considerable difficulty were 

 attempted during the next forty years, with varied degrees 

 of success. The methods almost invariably employed in these 

 inquiries was, to consider the character of the projection of the 

 curve upon the equatorial plane, and, by means of these and the 

 relations between the current vertical ordinates of the sphere, to 

 express all the conditions, relations, and results. A few excep- 

 tions, perhaps, might be made to this statement, the only one of 

 which I can call to mind is that of JAMES BERNOULLI'S solution 

 of VIVIANI'S Florentine Paradox, where he takes only account 

 of arcs of the meridian of the current point, and the angles which 

 those meridians form with a certain given meridian. He does not, 

 however, seem to entertain the slightest notion of its applicability 

 to any other class of problems than that one before him. When 

 CLAIRAULT'S principle, however, became known, all those particu- 

 lar artifices were merged in that general method ; and the sphe- 



VOL. XII. PART I. L 1 



