6S4 PHILOSOPHICAL TRANSACTIONS. [ANNO 1785. 



copied ; and I hope the resemblance will not be called a bad one, when it shall 

 be considered how very limited must be the pencil of an inhabitant of so small 

 and retired a portion of an indefinite system in attempting the picture of so un- 

 bounded an extent. 



But to proceed to particulars : Mr. H. begins by giving a table of what he 

 calls gages that have been taken. In the 1st column is the right ascension, and 

 in the 2d the north polar distance, both reduced to the time of Flamsteed's 

 catalogue. In the 3d are the contents of the heavens, being the result of the 

 eages. The 4th shows from how many fields of view the gages were deduced, 

 which have been 10 or more where the number of the stars was not very con- 

 siderable ; but, as it would have taken too much time, in high numbers, to 

 count so many fields, the gages are generally single. Where the stars happened 

 to be uncommonly crowded, no more than half a field was counted, and even 

 sometimes only a quadrant ; but then it was always done with the precaution of 

 fixing on some row of stars that would point out the division of the field, so as to 

 prevent any considerable mistake. When 5, 10, or more fields are gaged, the 

 polar distance in the 2d column of the table is that of the middle of the sweep, 

 which was generally from 2 to 2-l degrees in breadth ; and, in gaging, a regular 

 distribution of the fields, from the bottom of the sweep to the top, was always 

 strictly attended to. The 5th column contains occasional remarks, relating to 

 the gages. As it is not necessary to reprint these tables of numbers, we shall 

 pass them over, and proceed with the remaining part of the paper. 



Problem. — The stars being supposed to be nearly equally scattered, and their 

 number, in a field of view of a known angular diameter, being given, to deter- 

 mine the length of the visual ray. 



Here, the arrangement of the stars not being fixed on, we must endeavour 

 to find which way they may be placed so as to fill a given space most equally. 

 Suppose a rectangular cone cut into frustula by many equidistant planes perpen- 

 dicular to the axis ; then, if one star be placed at the vertex, and another in the 

 axis at the first intersection, 6" stars may be set around it so as to be equally 

 distant from each other and from the central star. These positions being car- 

 ried on in the same manner, we shall have every star within the cone surrounded 

 by 8 others, at an equal distance from that star taken as a centre. Fig. 6, pi. g, 

 contains 4 sections of such a cone distinguished by alternate shades, which will 

 be sufficient to explain what sort of arrangement is here intended. 



The series of the number of stars contained in the several sections will be 1, 

 7, U), 37, 6l, 91, &c. which continued to n terms, the sum of it, by the differ- 

 ential method, will be na -\- n . — - — d! -j- n . — — . —5— d", he. : where a is 

 the first term d ', d", d", &c. the 1st, 2d, and 3d differences. Then, since a = 1, 

 d' = 6, d" = 6, d'" = o, the sum of the series will be ra 8 . Let s be the given 



