ConJl ruction of the Heavens , 
The fame otherwife. 
If a different arrangement of the ftars fhould be fckotcd, 
fuch as that in fig. 2. where one ftar is at the vertex of a cone ; 
three in the circumference of the firft feclion, at an equal dii- 
tance from the vertex and from each other fix in the circum- 
ference of the next fedtion, with one in the axis or center; 
and fo on, always placing three fhars in a lower fe&ion in fuch a 
manner as to form an equilateral pyramid with one above them ; 
then we fhall have every ftar, which is fufficiently within 
the cone, furrounded by twelve others at an equal diftance from 
the central ftar 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 2n z 1 §«* + f»; 
where n ftands for the number of terms in each feries. To 
find the angle which a line vx, palling from the vertex v over 
the ftars v, n , h , l, &c. to x, at the outfide of the cone, makes 
with the axis; we have, by conftrudtion, vs in fig. 3. 
reprefenting the planes of the firft and fecond fedlions = 
2 x cof.30 0 ~ to the radius p s, of the firft fedlion = 1. Hence 
it will be %/ <p z — 1 —vp—\vm\ or v m — 2 s/ of — 1 : and, by 
trigonometry, ~==.~T. Where T is the tangent of the 
required angle to the radius R (c) ; and putting / = tangent of 
(c) In finding this angle we have fuppofed the cone to be generated by a 
revolving rectangular 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 furface, wherefore the above fuppofition of the generation of the cone is 
not ftridtly true ; but then thefe waves are fo inconfiderable, that, for the pre- 
fent purpofe, they may fafely be uegledted in this calculation. 
I i 2 
half 
