upon the Antique Dials. 113 



the second order is commonly of double curvature, and its orthographic pro- 

 jection, like that of the intersections of surfaces of the second order in gene- 

 ral, is of the fourth degree. 



But to return. If Mr CADELL'S equation was less simple and perspicu- 

 ous than it might have been rendered, it was owing to the accident of adopt- 

 ing rectangular instead of polar co-ordinates, and of taking the vertex of the 

 curve instead of the centre of projection for the origin of the co-ordinates. 

 Yet one advantage has resulted from it, that he was driven to pursue his 

 investigations graphically, and thereby exhibit to the eye the general figure 

 of the curve, as perfectly as any equation could exhibit it to the understand- 

 ing. Indeed, they exhibit the course of the curve as clearly as if it had been 

 actually traced on the sphere itself. Indisputably, then, he has the honour 

 of being the first who clearly understood the figure of these curves, as well 

 as the first to illustrate their general character by actual delineation. Had 

 he been equally happy in his analytical investigations, he would have ren- 

 dered this dissertation a work of supererogation ; and it is more than pro- 

 bable that the present investigation would never have been undertaken. He 

 would then, also, whilst he carried conviction to the mind of DELAMBRE, 

 (which had already been made up on the contrary side of the question), have 

 produced a change in the details of the chapter on the " Analemma" in the 

 History of Ancient Astronomy of that illustrious geometer. 



XXVII. 



The equation (\ d ) does not, as appears from (XXV. 1.), include the 

 cases (at least it does not express the relation between the co-ordinates in 

 finite terms) where the point of contact is in the equator of the sphere ; 

 such, for instance, as the east and west vertical dials, or the ZEfcTPOi; 

 AHHAIftTES of " the Tower of Winds" at Athens*. The process is ne- 

 vertheless very simple. 



* STUART, vol. i. pi. 10. 



