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sections,which are here treated in a manner at once new, easy, and natural ; so much so indeed that all the propositions and their demonstrations in the ellipsis, are the very same,

word for word, as those in the hyperbola, using only, in a very few places, the word sum, for the word difference : also in many of the mechanical and philosophical parts which follow in the second volume. In the conic sections too, it may be observed, that the first theorem of each section only is proved from the cone itself, and all the rest of the theorems are deduced from the first, or from each other, in a very plain and simple manner.

Besides renewing most of the rules, and introducing every where new examples, this edition is much enlarged in several places ; particularly by extending the tables of squares and cubes, square roots and cube roots, to 1000 numbers, which will be found of great use in many calculations ; also by the tables of logarithms, sines, and tangents, at the end of the second volume ; by the addition of Cardan's rules for resolving cubic equations; with tables and rules for annuities ; and many other improvements in different parts of the work.

Though the several parts of this course of mathematics are ranged in the order naturally required by such elements, yet students may omit any of the particulars that may be thought the least necessary to their several purposes ; or they may study and learn various parts in a different order from their present arrangement in the book, at the discretion of the tutor, So, for instance, all the notes at the foot of the pages may be omitted, as well as many of the rules ; particularly the 1st or Common Rule for the Cube Root, p. 85, may well be omitted, being more tedious than useful. Also the chapters on Surds and Infinite Series, in the Algebra ; or these might be learned after Simple Equations. Also Compound Interest and Annui. ties at the end of the Algebra. Also any part of the Geometry, in vol. 1 ; any of the branches in vol. 2, at the discretion of the preceptor. And, in any of the parts, he may omit some of the examples, or he may give more than are printed in the book; or be may very profitably vary or change them by altering the numbers occasionally -- As to the quantity of writing; the author would recommend, that the student copy out into his fair book no more than the chief rules which he is directed to learn off by rote, with the work of one example only to each rule, set down at full length ; omitting to set down the work of all the other examples, how many soever he may be directed to work out upon his slate or waste paper.-In short, a great deal of the business, as to the quantity, and order, and manner, must depend on the judgment of the discreet and prudent tutor or director.

[Dr.

[Dr. Hutton's Preface to the Third Volume of the English Edition,

published in 1811.)

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 a 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 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.

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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 comprehensive view of the trigonometrical analysis, or that in which the chief theorems of Plane and Spherical Trigonometry are deduced algebraically by means of what is commonly denominated 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 to

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the nature and properties of curves, and the construction of of equations. These chapters are manifestly connected, and show how the mutual relations 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 maximum effects of machines in motion ; and developes those principles which should constantly be 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 attending this abstruse enquiry are surmounted by assuming the results of accurate experiments, as to the resistance experienced by bodies moving through the air, as the basis of 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, hydrostatics, hydraulics, projectiles, &c. &c. ; serving at once to exercise the pupil in the various branches of mathematics comprised in the Course, to demonstrate their utility especially to those devoted to the military profession, to excite a thirst for knowledge, and in several important respects, to gratify it. This volume being professedly supplementary to the preceding two volumes of the Course, may best be used in tuition by a kind of mutual incorporation of its contents with those of the second volume. The method of effecting this will, of course, vary according to, circumstances, and the precise employments for which the pupils are destined : but in general it is presumed the following may be advantageously adopted. Let the first seven chapters be taught, immediately after the

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Conic Sections in the 2d volume. Then let the substance of the 2d volume succeed, as far as the Practical Exercises on Natural Philosophy, inclusive. Let the 8th and 9th chapters in this 3d vol. precede the treatise on Fluxions in the 2d; and when the pupil has been taught the part relating to fluents in that treatise, let him immediately be conducted through the 10th chapter of the 2d volume. After he has gone over the remainder of the Fluxions with the applications to tangents, raddii of curvature, rectifications, quadratures, &c. the 11th and 12th chapters of the 3d vol. should be taught. The problems in the 13th and 14th chapters must be blended with the practical exercises at the end of the 2d volume, in such manner as shall be found best suited to the capacity of the student, and best calculated to ensure his thorough comprehension of the several curious problems contained in those portions of the work.

In the composition of this 3d volume, as well as in that of the preceding parts of the Course, the great object kept constantly in view has been utility, especially to gentlemen intended for the Military Profession. To this end, all such investigations as might serve merely to display ingenuity or talent, without any regard to practical benefit, have been carefully excluded. The student has put into his hands the two powerful instruments of the ancient and the modern or sublime geometry; he is taught the use of both, and their relative advantages are so exhibited as to guard him, it is hoped, from any undue and exclusive preference for either. Much novelty of matter is not to be expected in a work like this ; though, considering its magnitude, and the frequency with which several of the subjects have been discussed, a candid reader will not, perhaps, be entirely disappointed in this respect. Perspicuity and condensation have been uniformly aimed at through the performance; and a small clear type, with a full page, have been chosen for the introduction of a large quantity of matter.

A candid public will accept as an apology for any slight disorder or irregularity that may appear in the composition and arrangement of this course, the circumstance of the different volumes having been prepared at widely distant times, and with gradually expanding views. But, on the whole, 1 trust it will be found that, with the assistance of my friend and coadjutor in this supplementary volume, I have now produced a Course of Mathematics, in which a great variety of useful subjects are introduced, and treated with greater perspicuity and correctness, than in any three volumes of equal size in any language.

CHA. HUTTON.

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