30 6 Sir William Herschel’s observations and experiments 
of the central distances ; and the last gives the difference 
between the cube numbers of any order and the cube of the 
next enclosed order. 
The use to be made of these columns of numbers is by 
inspection to determine how many stars of any particular 
order there ought to be if the stars were equally scattered. 
For instance, let it be required how many stars there should 
be of the 4th order. Then No. 4, in the column of the orders 
points out a sphere of nine times the diameter of the central 
one, and shows that it would contain 729 stars ; but as this 
sphere includes all the stars of the 3d, 2d, and 1st order as 
well as the sun, their number will be the sum of all the stars 
contained in the next inferior sphere amounting to 343 ; which 
being taken from 729 leaves 386 for the space allotted to 
those of the 4th order of distances. 
III. Comparison of the order of magnitudes with the order of 
distances. 
With a view to throw some light upon the question, in what 
manner the stars are scattered in space, we may now compare 
their magnitudes, as we find them assigned in Mr. Bode’s 
extensive catalogue of stars, with the order of their distances 
which has been explained. 
The catalogue I have mentioned contains 17 stars of the 
1st magnitude; but in my figure of the order of the distances 
we find their number to be 2 6. 
The same catalogue has 57 stars of the 2d magnitude; 
but the order of distances admits 98. 
Of the third magnitude the catalogue has 206, and the 
order of distances will admit 218. 
