method derived from Dr. Halley's formulæ is preferable to all others. After the Algebra was drawn up, we found an investigation of the Binomial Theorem in M. Dubourguet's Algebra by means of the same artifice as that in the work quoted p. 149; the author has extended the demonstration to fractional and negative indices. The different series to which Art. 172-181 are an introduc. tion, may be reckoned among the speculative parts of mathematics. The principal theorem in the Arithmetic of Infinites however, is deduced from the Differential Method, (Art. 179); the application of this formula has been of considerable use in the subsequent part of the volume. In treating of the Conic Sections, the fundamental property or the equation of each curve is derived from the solid: afterwards they are considered in plano; and as the expressions for the ellipse and hyperbola differ in nothing but the signs and the same demonstration frequently answers for both sections by only changing those signs; for which reason the enunciations of some properties of the hyperbola are thought sufficient. That part of Mechanics which relates to the Centre of Gravity is given at some length on account of its extensive use. In Art. 386, 387, 388, different methods of computing the thickness of walls or revetments are compared. The results, as might be expected from different hypotheses, vary considerably. By adopting the method however, in the first work refered to (p.383), we evidently are led to the following conclusion (p. 386) which is correct only in the case of a fluid,-namely, that the lateral pressure of a body of loose earth depends on its height without any regard to thickness.-But as all the computations are founded upon uncertain data, no correction of principle is attempted: and the only alteration is that of giving a more convenient form to M. Belidor's solution, which, as it nearly agrees with the practice of Vauban, seems the least liable to exception. All theories however, respecting the strength of walls, and also that of timber, must necessarily be imperfect. On the latter subject, see an account of the very extensive and laborious experiments of M. de Buffon in Mem. Acad. des Sciences, 1740. The speculative mechanician therefore will scldom find an exact agreement between his conclusions and the results from experiment; particularly in what relates to the working of machinery, because no theory of Friction has yet been discovered which its effects can be calculated; for that reason the subject is not considered in the following pages. As a work of this kind must unavoidably consist of abridgements, considerable care was bestowed in selecting what appeared the most useful to students who have not an opportunity of perusing the separate and more diffuse treatises on the different subjects. Some new solutions are introduced: but the mathematical reader cannot expect much new matter in any form. To conclude. The experience of two or three years proves that it will not be necessary to extend the Course beyond this Volume for the use of the College. Those Officers or Cadets who may gain a thorough knowledge of the principal matters contained in both volumes during their stay, and are inclined to continue the study of mathematics after quitting the Institution, will consult books professedly written on the higher branches, pursue their researches without the assistance of a master. and High Wycomb, Of Evolution or the Extraction of Roots............ Rule for extracting the roots of Numbers ........ Of Dr. Halley's Rational Formulæ for the extraction of Roots..... Questions producing Simple Equations... Demonstration of the Rule of Double Position...................... Page. Of the Differential Method.......... Theorem for the sum of an infinite series of like powers.......... ...... 17S ADDITIONAL EXAMPLES in ALGEBRA...... To find the content of a Paraboloid....... Of the construction of Cubic and Biquadratic Equations........ Of Projectile Motion according to the Parabolic Theory........ Dimensions of Walls for supporting banks of earth Thickness of the Piers for a semicircular Arch.... Curve of equilibrium for the counterpoise to a Draw-bridge.. of finding Areas by means of the Centre of Gravity... Barometer Method of determining the mean velocity of water in a River or PNEUMATICS. |