THE beneficial improvements lately made, and still making, in the plan of the scientific education of the Cadets, in the Royal Military Academy at Woolwich, having rendered å further extension of the Mathematical Course adviseable, I was honoured with the orders of his Lordship the Master General of the Ordnance, to prepare a third volume, in addition to the two former volumes of the Course, to contain such additions to some of the subjects before treated of in those two volumes, with such other new branches of military science, as might appear best adapted to promote the ends of this important institution. From my advanced age, and the precarious state of my health, I was desirous of declining such a task, and pleaded my doubts of being able, in such a state, to answer satisfactorily his lordship's wishes. This difficulty however was obviated by the reply, that, to preserve a uniformity between the former and the additional parts of the Course, it was requisite that I should undertake the direction of the arrangement, and compose such parts of the work as might be found convenient, or as related to topics in which I had made experiments or improvements; and for the rest, I might 1t LI A 2 I might take to my assistance the aid of any other person I might think proper. With this kind indulgence being encouraged to exert my best endeavours, I immediately announced my wish to request the assistance of Dr. Gregory of the Royal Military Academy, than whom, both for his extensive scientific knowledge, and his long experience, I know of no person more fit to be associated in the due performance of such a task. Accordingly, this volume is to be considered as the joint composition of that gentleman and myself, having each of us taken and prepared, in nearly equal portions, separate chapters and branches of the work, being such as, in the compass of this volume, with the advice and assistance of the Lieut. Governor, were deemed among the most useful additional subjects for the purposes of the education established in the Academy. The several parts of the work, and their arrangement, are as follow.-In the first chapter are contained all the propositions of the course of Conic Sections, first printed for the use of the Academy in the year 1787, which remained, after those that were selected for the second volume of this Course: to which is added a tract on the algebraic equations of the several conic sections, serving as a brief introduction to the algebraic properties of curve lines. The 2d chapter contains a short geometrical treatise on the elements of Isoperimetry and the maxima and minima of surfaces and solids ; in which several propositions usually investigated by fluxionary processes are effected geometrically; and in which, indeed, the principal results deduced by Thos. Simpson, Horsley, Legendre, and Lhuillier are thrown into the compass of one short tract. The 3d and 4th chapters exhibit a concise but compre, hensive view of the trigonometrical analysis, or that in which the chief theorms of Plane and Spherical Trigonometry are deduced algebraically by means of what is commonly deno minated a a minated the Arithmetic of Sines. A comparison of the modes of investigation adopted in these chapters, and those pursued in that part of the second volume of this course which is devoted to Trigonometry, will enable a student to trace the relative advantages of the algebraical and geometrical methods of treating this useful branch of science. The · fourth chapter includes also a disquisition on the nature and measure of solid angles, in which the theory of that peculiar class of geometrical magnitudes is so represented, as to render their mutual comparison (a thing hitherto supposed impossible except in one or two very obvious cases) a matter of perfect ease and simplicity. Chapter the fifth relates to Geodesic Operations, and that more extensive kind of Trigonometrical Surveying which is employed with a view to determine the geographical situation of places, the magnitude of kingdoms, and the figure of the earth. This chapter is divided into two sections; in the first of which is presented a general account of this kind of surveying; and in the second, solutions of the most important problems connected with these operations. This portion of the volume it is hoped will be found highly useful ; 'as there is no work which contains a concise and connected account of this kind of surveying and its' dependent problems ; and it cannot fail to be interesting to those who know how much honour redounds to this country from the great skill, accuracy, and judgment, with which the trigonometrical survey of England has long been carried on. In the 6th and 7th chapters are developed the principles of Polygonometry, and those which relate to the Division of lands and other surfaces, both by geometrical construction and by computation. • The 8th chapter contains a view of the nature and solution of equations in general, with a selection of the best rules for equations of different degrees. Chapter the 9th is devoted a to to the nature and properties of curves, and the construction of equations. These chapters are manifestly connected, and, show how the mutual relation subsisting between equations, of different degrees, and curves of various orders, serve for the reciprocal illustration of the properties of both. In the 10th chapter the subjects of Fluents and Fluxional equations are concisely treated. The various forms of Fluents, comprised in the useful table of them in the 2d volume, are, investigated : and several other rules are given; such as it is. believed will tend much to facilitate the progress of students in this interesting department of science, especially those which relate to the mode of finding fluents by continuation. The 11th chapter contains solutions, of the most useful problems concerning the niaximum effects of machines in motion; and developes those principles which should con.. stantly, bę kept in view by those who would labour beneficially for the improvement of machines. In the 12th chapter will be found the theory of the pres... sure of earth and fluids against walls and fortifications; and the theory which leads to the best construction of powder, magazines, with equilibrated roofs. The 13th chapter is devoted to that highly interesting subject, as well to the philosopher .as to military men, the theory and practice of gunnery. Many of the difficulties ats. tending this abstruse enquiry are surmounted by assuming a the results of accurate experiments, as to the resistance exper.. rienced by bodies, moving through the air, as the basis ofia the computations. Several of the most useful problems are solved by means of this expedient, with a facility scarcely to be expected, and with an accuracy, far beyond our most sanguine expectations. The 14th and last chapter contains a promiscuous but extensive collection of problems in statics,, dynamics, hydroen, statics, hydraulics, projectiles, &c, &c; serving at once to exercise |