332 ON NAPIERS RULES. 



.'. tan b tan c cos A, 

 or, cos A = tan b cot c, 



which gives the relation between any middle part and the two 

 adjacent ones. 



But in whatever way this relation and the preceding one 

 are established, the point to which I wish to call attention is 

 this, that by considering the system of five associated triangles 

 we immediately generalize any particular result without having 

 to demonstrate it in the separate cases. 



It is clear that an equiangular spherical pentagon would in 

 every case give rise to the system of five equiangular spherical 

 triangles mutually corelated in a manner analogous to the right- 

 angled triangles of which we have been speaking. But in 

 the general case the relation would be too complicated to be 

 useful. 



Having perused the preceding I was anxious to know in 

 what manner the subject had presented itself to Napier's own 

 mind, and having referred to a copy of the Mirifici Logarith- 

 morum Canonis Descriptio in the Cambridge University Li- 

 brary, I found to my astonishment that the mode of treatment 

 invented by Mr Ellis was in reality a re-invention of Napier's 

 own conception of the subject, which owing to some cause or an- 

 other has dropped out from all Cambridge books upon Spherical 

 Trigonometry*. 



The following is translated from Napier. After giving a 

 general explanation of the use of circular parts, he proceeds 

 thus: 



" This uniformity of the circular parts becomes very evident 

 in the case of right-angled triangles formed on the surface of a 

 sphere by five great circles, of which the first cuts the second, 

 the second the third, the third the fourth, the fourth the fifth, 

 and the fifth the first at right angles; the other intersections 

 taking place at oblique angles. 



" Thus, for example, let the meridian DB (fig. 3) cut the hori- 

 zon BE in the point B. Let the horizon BE cut the circle EC of 

 which the sun S is the pole in E. Let EC cut the sun's declina- 



This defect has been recently corrected in Mr Todhunter's Treatise on 

 Spherical Trigonometry. 



