334 ON NAPIERS RULES. 



Woodhouse. " There is no separate and independent proof 

 of these rules ; but the rules will be manifestly just, if it can be 

 shewn that they comprehend every one of the ten results, (1), 

 (2), (3), &c." 



Airy, Encyclopaedia Metropolitana. " These rules are proved 

 to be true only by shewing that they comprehend all the equa- 

 tions which we have just formed." 



The same view has been adopted (I believe) in all our Cam- 

 bridge books. 



Soon after receiving Mr Ellis' paper, given above, I received 

 from him the following : 



Addition to the foregoing. 



In any spherical rectangular polygon all the sides but three 

 may have any assigned length, the length of the other three 

 being functions of them. In the case of the pentagon there are 

 thus two elements arbitrary, the same number as in a right- 

 angled spherical triangle, between which and the pentagon 

 exists a close analogy, which may be developed from the follow- 

 ing lemmas. 



LEMMA 1. 



Of three consecutive sides of a right-angled spherical poly- 

 gon, the extremes cut off quadrants of one another. 



LEMMA 2. 



The angle between the said extremes at their point of inter- 

 section (measured by the deflection of the direction of motion of 

 a point which travels along the boundary of the polygon) is 

 equal to the supplement of the next ; in other words, the com- 

 plement of the angle added to the complement of the side is 

 equal to zero. 



Consider any side of a pentagon and the two sides which 

 are opposite to it and which meet one another at a right angle : 

 they form with that side a right-angled spherical triangle. We 

 will for distinction call that side (1), so that the other sides will 

 similarly be called (3) and (4). Then as (4), (5) and (1) are 

 consecutive, the angle between (4) and (1) is equal to the sup- 

 plement of (5), and similarly the angle between (3) and (1) is 

 the supplement of (2). Also the lengths of the sides of the tri- 



