98 



Newton, he informs us of the circumstances which led him to the 

 present investigation, namely, the occasion he had some years ago 

 to solve a problem which required the rectification, of an equilateral 

 hyperbola. 



He then enters upon his subject ; and in a first section he investi- 

 gates in nine theorems the several series which apply to this curve, 

 whose different characters, namely, the ratios of their terms, or rather 

 the rates of their convergency and divergency, depend on the relative 

 proportions of their elements. Of these series one only, and that not 

 the best, is all that he has hitherto been able to find hi other works. 

 Two are of the form which is called ascending, and six descending. 

 One of them is of a peculiar form, which can only be understood by 

 turning to the paper. Among these series, he observes, may always 

 be found some which will converge, whether the portion of the hy- 

 perbolic arch taken from the vertex be long or short, or of a mode- 

 rate length ; but the ascending series always differs from the de- 

 scending one by a constant quantity. 



In a second section the author treats of the methods of computing 

 the values of the constant quantities, by which the ascending series 

 differ from the descending ones. Here he has recourse to two methods, 

 of which he has already given an illustration in his Mathematical Es- 

 says : the one by computing the value of both an ascending and de- 

 scending series, taking for the ordinate to the axis some small definite 

 quantity ; and the other by comparing the values of those series to- 

 gether, when the ordinate is taken immensely great. The former 

 method he says is more general ; but the latter, when it can be ap- 

 plied, usually affords the easiest computation. 



In the third section are given five examples, which show the use 

 of these theorems, as well as the manner of choosing such as are best 

 adapted to any particular case. In one of these the author corrects 

 an error in the length of a large arch of an equilateral hyperbola, 

 which was first published in the year 1771, and has been since re- 

 printed by some eminent mathematicians. 



Lastly, he concludes with some remarks on former writers, and 

 takes notice of the defects of two series given by the late Dr. Waring 

 for the rectification of an hyperbola. 



Catalogue of 500 new Nebula, nebulous Stars, planetary Nebula, and 

 Clusters of Stars ; with Remarks on the Construction of the Heavens. 

 By William Herschel, LL.D. F.R.S. Read July 1, 1802. [Phil. 

 Trans. 1802, p. 477.] 



To this catalogue is prefixed a classification of the multitude of 

 sidereal bodies hitherto discovered, not according to their appa- 

 rent magnitudes or appearances on our earth, but according to their 

 peculiar nature and arrangement in the heavens. They are divided 

 into the twelve following classes : 



1 . Insulated stars, or such as may be considered out of the reach 

 uf mutual attraction ; such as our Sun, Arcturus, Capella, Lyra, Si- 



