P R E F A CE. This little work has been prepared for the use of the Midshipmen now pursuing their studies at the Naval School, Philadelphia. It was commenced with the intention of using it only “in manuscript;" but as the principal object was to embrace more practical details and a greater variety of examples than are usually given in elementary works upon Algebra, it was found necessary to enlarge it to an extent that would have rendered the use of it in that form impracticable. In preparing it for the press, the original design has been still farther extended, and the work now assumes the form of a distinct, if not a complete, treatise upon the Binomial Theorem and Logarithms. Some articles are in consequence included, which (though essential parts of a treatise upon these subjects) were not called for by the immediate wants of a class whose principal study is Navigation. Those, however, who are acquainted with the extent of the mathematics requisite in a complete theoretical exposition of Navigation, will, perhaps, not regard even these parts as altogether superfluous. As the proper understanding of both the subjects here considered requires a correct knowledge of the nature of “powers and roots," and of “exponents in general,” two short chapters under these heads are prefixed by way of introduction, and for the convenience of reference. A rigid demonstration of the Binomial Theorem, at once simple and sufficiently elementary, has been much sought for by mathematicians. The one here given depends upon a principle which is the foundation of the Differential Calculus, and is in fact little else than a translation of the very simple demonstration afforded by that science into the elementary language of Algebra. It is essentially the same as that given by BOURDON, (Algébre, ed. 1837, Paris,) and by Professor PEIRCE of Harvard University, in his excellent treatise upon Algebra. The latter, however, has not extended the demonstration to the cases in which the exponent is negative or fractional. The mode of deriving the logarithmic formula, in Chapter VII, is that of EULER, with some modification to render it intelligible to those not familiar with the infinitesimal calculus. Under the head “ Transformations of the Logarithmic Formula,” will be found some of the most expeditious formulæ given by CALLET in the “ Précis Elémentaire," prefixed to his “ Tables Portatives,” and by BORDA and DELAMBRE, in their Prefaces to the “ Tables Trigonométriques Décimales.” |