One of the last improvements which trigonome try has received from analysis, is in the theory of angular sections before mentioned. Notwithstand ing the attention which has been devoted to this subject by some of the most profound analysts from the time of Euler to the present day, it remained until a very late period in an imperfect state. Formulae, expressing relations between the sine and cosine of an arc, and those of its multiples, were established by Euler, and subsequently con firmed by the searching analysis of Lagrange,which have since been proved inaccurate, or true only un der particular conditions; and it was not until within a few years that the complete exposition of this theory was published, and general formula as signed, expressing those relations. In the year 1811, Poissin detected an error in a formula of Euler, expressing the relation between the power of the sine or cosines of an arc, and the sines and cosines of certain multiples of the same arc. But the most complete discussion of the subject, which has hitherto appeared in a separate form, is contained in a memoir read before the academy of sciences at Paris, by Poinsot, an eminent French mathemati cian, in the year 1823, and further developed by him in another memoir, published in the year 1825.
The reader will also find some interesting discus sions on this subject in the Bulletin Universe!, scat tered in various parts of that work for the last three years.
Works on trigonometry have been so numerous that it would be vain to attempt an enumeration of them here. The English elementary treatises have been, with two exceptions, uniformly geometrical. The first treatise in which the subject was present ed in an analytical form was that of the late Pro fessor Woodhouse of Cambridge. At a more re cent period, a much more detailed treatise has been published by Dr. Lardner. This treatise is perhaps more exclusively analytical than any which has yet appeared. Dr. Lardner has borrowed from geometry no other principle except the proportion ality of the sides of equiangular triangles. In this treatise we find a very complete analysis of angular sections, including all the recent corrections and improvements. The following treatise is abridged from this work by the consent of the author.