VOL. LXXXVI.] PHILOSOPHICAL TRANSACTIONS. ggy 



is double, very unequal, the smaller of the 2 being much smaller than the larger. 

 With a power of 287 I can see the smaller of the 2 stars perfectly well : this 

 shows how little density there is in the comet, which is evidently nothing but 

 what may be called a collection of vapours. 



VII. Mr. Jones's Computation of the Hyperbolic Logarithm of \0 improved: 

 being a Transformation of the Series which he used in that Computation to 

 others which converge by the Powers of 80. To which is added a Postscript, 

 containing an Improvement of Mr. Emersons Computation of the same 

 Logarithm. By the Rev. John Hellins, Ficar of Potters Pury, in North- 

 amptonshire, p. 135. 



1. The method of computing by series is so extensive and useful a part of the 

 mathematics, that any device which facilitates the operation by them will doubt- 

 less be acceptable to those who are proper judges of these matters. In this per- 

 suasion I have employed an hour of that little leisure which my present situation 

 affords me, in improving a calculation of Napier's, for finding the hyperbolic 

 logarithm of 10, which was given by the justly celebrated William Jones, f.r.s. 

 in p. 180 of his Synopsis Palmariorum Matheseos. The same computation, de- 

 scribed in a manner better suited to thei capacities of beginners, was also pub- 

 lished many years afterwards by the learned Dr. Saunderson, in the 2d volume of 

 his Elements of Algebra, p. 633 and 634. Since Dr. Saunderson's time the 

 doctrine of series has been much improved. My present intention is, to exhibit 

 a transformation of the series by which Mr. Jones computed the hyperbolic 

 logarithm of JO to others, the terms of which decrease by the powers of 80 ; so 

 that their convergency is swift, and the divisions by 80 are easily made. 



2. Mr. Jones considered the number 10 as composed of 2 X 2 X 2 X * ; 

 and consequently obtained the log. of 10 by adding 3 times the log. of 2 to 

 the log. of »-. The algebraic series which he used on this occasion was 



1 7- ^ ; + ir^> &c. and the numerical value of - was -v for the lojr. of 



2, and i for the log. of i ; so that he has 



Sum of ^ 5 2 , 2 , 2 . 2 „ (-^-^O- 



3. Now the series j -|- g-^ + j-^ -|- ^, &c. ( = 3l. 2) is evidently 



= 1 X -■ ^ +3^'+ I^^+ 7^'^^-= ^ X •• ^ + J^+ ^+ 7-^' &<^- 

 And if the 1st, 3d, 5th, &c. term of this series be written in ontf line, and thd 



2d, 4th, 6th, &c. in another, we shall have 



4 u2 



