Page:Proceedings of the Royal Society of London Vol 1.djvu/118

Rh 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 in 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 diifers 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 cohstant quantities, by which the ascending series diﬁ'er 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 hyperbole.

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 considcrcd out of the reach of mutual attraction; such as our Sun, Arcturus, Capella, Lyra, Si-