ConJlrudiloJi of the Heavens, 242 



Thefatne other wife. 



If a different arrangement of the flars fhould be felecled, 

 fuch as that hi fig. 2. where one ftar Is at the vertex of a cone ; 

 three in the circumference of the firO: fe6lIon, at an equal dif- 

 tance from the vertex and from each other -, fix in the circum- 

 ference of the next fedlion, with one in the axis or center; 

 and fo on, always placing three ftars in a lower fedlon in fuch a 

 manner as to form an equilateral pyramid with one above them: 

 then we fliall have every ftar, which is fufficiently within 

 the cone, furrounded by twelve others at an equal diflance from 

 the central fl:ar and from each other. And by the differential 

 method, the fum of the two feries equally continued, into 

 which this cone may be refolved, will be 2?7^+ 1 1 ^^'^ + li''"; 

 where n ftands for the number of terms in each feries. To 

 find the angle which a line v:c^ paffing from the vertex v over 

 the flars 'U, n^ -6, /, &Co to x, at the outfide of the cone, makes 

 with the axis ; we have, by conflrucSlion, v i in fig. 3. 

 reprefenting the planes of the firfl and fecond fe6lions = 

 2 X cof.30° := cp, to the radius p j, of the firfl fe6lion =: i . Hence 



it will be v^(p' — i—v p = \vm\ or vm = 2 s/ (f — i : and, by 

 trigonometry, ?-^=:T. Where T is the tangent of the 



required angle to the radius R (c) j and putting / = tangent of 



(c) In finding this angle we have fuppofed the cone to be generated by a 

 revolving reftangular triangle of which the line vx, fig. 2. is the hypotenufe ; 

 but the ftars in the fecond feries will occafion the cone to be contained under a 

 waving lurface, wherefore the above fuppofition of the generation of the cone is 

 not ftridly true ; but then thefe waves are fo inconfiderable, that, for the pre- 

 fent purpofe, they may fafely be neglected in this calculation, 



I i 2 half 



