( 529 ) 



Mathematics. — "The rule of N^eper in the f our dimensional space." 

 Hv Dr. W. A. Wythoff). (Communicated by Prof. P. II. 



Schoute. 



(Communicated in the meeting of December 20, 1906). 



1. The wellknown "rule of Neper" eau in principle be formulated 

 as follows : 



If we regard as elements of a spherical triangle A^A^A^ rectan- 

 gular in A 2 , the two oblique angles A l and A t the hypothenuse a t 

 and the complements of the two other sides h n — a, and \n — a 3 x ) 

 we can apply to each formula generally holding for the rectangular 

 spherical triangle the cyclic transformation 



(A„ \ n — « s , a 2 , \üt — a,, AA 

 without its ceasing to hold. 



Fig. 1. 



We prove this rule by prolonging the sides A x A 3 and A % A s which 

 (Fig. 1) for convenience'sake we shall imagine as <^ ^ .t, through the 

 vertex A^ ■= A' r with segments A\ A' a and A\ A\ so that A x A\ = 

 A^ A' s == a jr. The spherical triangle A'\ A\ A\ then proves to be 

 again rectangular, namely in A' a , whilst furthermore between the 

 elements of both spherical triangles the following relations prove to 

 exist : 



/ 1 1 -r it • 



A' — i- Jt 



ft- J 2 Jl 



rr 



n — a 9 = « 2 ; a 2 



From this is evident that the above mentioned cyclic transformation 



can be applied to the elements of each rectangular spherical triangle 

 without their ceasing to be the elements of a possible rectangular 



] ) These are the complements of what Neper himself calls the "quinque 

 circulares partes" of the rectangular spherical triangle. See N. L. \V. A. Gravelaar, 

 John Naimer's werken, Verh. K. A. v. YV\, First section, vol. VI. X".. 0, page 40. 



:]() 



Proceedings Pioyal Acad. Amsterdam. Vol. IX. 



