ON NAPIER'S RULES*. 



To the Editor of the " Quarterly Journal of Mathematics." 



Some time ago, my friend Mr K. L. Ellis sent me the fol- 

 lowing remarks upon Napier's rules. 



NAPIER'S rules for the solution of right-angled spherical tri- 

 angles are generally presented merely as a memoria technica ; 

 and when so presented do not exhibit the principle upon which 

 they depend. To investigate and exhibit that principle is the 

 purpose of this paper. 



LEMMA. 



If the three sides of a tetrahedron are right-angled triangles, 

 no right angle being at the apex, then the base is also a right- 

 angled triangle. 



Let OAB be a right-angled triangle in A, and similarly 

 OAC. Let the third side OCB be right-angled in G. 



Then will the base A CB be also right-angled in 0. 



Since Napier's method of investigating his rules was independently discovered 

 by Mr Ellis, the Editor is of opinion that this Essay, written by the Dean of Ely, 

 will not be without interest to the readers of this volume. 



