necessary. Some of the most important theorems in conic sections are also demonstrated in the second book.
The notation of the author has been strictly adhered to, and the double parentheses, which he has used to denote the partial differentials, have been retained, though at present many mathematicians reject them.
For the sake of a more easy method of reference, to any particular part of the work, or to any single formula, the marginal numbers are inserted. These are frequently referred to, in the translation, and in the notes. The introduction of these numbers is the only alteration which has been made in the original work. In other respects it will be found, that the translation has been as nearly literal, as is consistent with a faithful interpretation of the sense of the author. These marginal references might supersede the use of those made by the author, in a few of the most important formulas, but it was thought best to retain them, because they might possibly be referred to, in quoting from the original work. It must be observed that in citing a single formula, the marginal reference will be found on the same line with the formula; but in referring to a particular sentence, or paragraph, it will generally be on the middle line of it.
As the author has supposed the quadrant of a circle to be divided into 100 degrees, each degree into 100 minutes, each minute into 100 seconds, &c., and has applied the usual marks ° ′ ″ &c., to these quantities; it has been found convenient, in the notes, when the sexagesimal division is used, to employ the letters d, m, s, &c., to denote degrees, minutes, seconds, &c., of the common sexagesimal notation; so that 1000″ is equivalent to 324s. This distinction will be adhered to throughout the work.
The notes were written at the time of reading the volumes, as they were successively published. The translation was made between the years 1815 and 1817, at which time the four first volumes, with the several appendices and notes, were ready for publication. Soon afterwards, the American Academy of Arts and Sciences liberally offered to print the work at their expense, but this proposal was not accepted. One of the reasons for not printing it at that time, was the expectation that the author would publish another edition, in which he might modify the first volume, by the introduction of the matter contained in the appendix to the third volume, depending on the improvements made by Mr. Poisson, in the demonstration of the permanency of the mean motions of the planets; and might also correct the second volume, on account of the defects in some parts of the theory of the calculation of the
