Corjlruclion of the Heavens. .241 



PROBLEM. 



Thejlars heirig fuppofed to he nearly equally fcaltcredy ana thai* 

 number, in a fclci of view of a knoivn angular diameter , being 

 given, to determine the length of the vifiial ra\\ 



Here, the arrangement of the ftars not being hxed upon, wc 

 mufl endeavour to find which way they niay be placed lo as to 

 fill a given fpace moft equally. Suppofe a rectangular cone 

 cut into fruftula by many equidiftant planes perpendicular to 

 the axis ; then, if oneitar be placed at the vertex, and another 

 in the axis at the firfl: interfc6lion, fix ftars may be fet around it: 

 fo as to be equally diftant from one another and from the cen- 

 tral liar. Thefe pofitions being carried on in the fame manner, 

 we fliall have every ftar within the cone furrounded by eight 

 others, at an equal diflance from that ftar taken as a center. 

 Fig. I. (tab. VIII.) contains four fedions of fuch a cone diilin- 

 guiihed by alternate Ihades, which will be fufficlent to explain 

 what fort of arrangement I would point out. 



The feries of the number of flars contained in the fe- 

 veral fedions will be 1.7. 19. 37. 61. 91. &c. which 

 continued to n terms, the fum of it, by the differential method, 



will be n a-\-n , "—— d' -{-n , ^— • - — d^\ &c. ; where a is 



2*23' 



the firfl term d\ d^\ d''\ &c. the ifl, 2d, and 3d differences. 

 Then, fince tf = i, d' — 6^ d"—(i, d'^' — o, the fum of the 

 feries will be n^. Let S be the given number of ftars; i, the 

 diameter of the bafe of the field of view ; and B, the 

 diameter of the bafe of the great re(5langular cone ; and, by 



trigonometry, we fhall have Br:: ^ Tfigid ' Now, fince the 

 Vol. LXXV. I i ' field 



