Art. II. A Course of Mathematics. Composed for the Use of the Royal Military Academy, by Order of his Lordship, the Master General of the Ordnance. Vol. III. By Charles Hutton, LL.D. F.R.S. Late Professor of Mathematics in the Royal Military Academy. 8vo. pp. x. 379. Price 12s. bound. Rivingtons. 1810.

MOST of our mathematical readers, and we suppose all of

those who have been engaged in teaching mathematics, are acquainted with the first two volumes of the Course, published for the use of the Woolwich Academy. They made their first appearance in the year 1798; and comprise, in moderate space, concise but comprehensive treatises on arithmetic, algebra, geometry, application of algebra to geometry, plane trigonometry, mensuration of planes and solids, land surveying, artificers' works, conic sections, mechanics, hydrostatics, hydraulics, pneumatics, and fluxions; with an extensive and very interesting collection of exercises, in which the fluxional analysis is applied to the solution of various problems in natural philosophy and military science.

Those volumes were, for some years, found fully adequate to the purposes of that important institution. But the numerous improvements introduced into the academy, by the present active and scientific Lieutenant-Governor, Colonel Mudge, seem to have rendered an improvement of the Course necessary; and Dr. Hutton was therefore requested, by the Master-General of the Ordnance, to prepare a third and supplementary volume. This mathematical veteran, who has been known as an able writer and tutor for full half a century, and who has stood at the head of his profession for nearly forty years, tells us in the preface to this work, written with his characteristic simplicity and modesty, that, from his advanced age, and the precarious state of his health, he was desirous of declining such a task, and pleaded his doubts of being able, in such a state, to answer satisfactorily his Lordship's wishes.'

This difficulty however, [says he,] 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 and 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.' pp. iii,iv.

The work before us, then, is the joint production of two authors and, as the parts actually prepared by each are not specified in the preface, we are left to conjecture, or to determine from internal evidence, how they are apportioned. But the task is by no means difficult. The volume is divided into fourteen chapters; of which the first is evidently the production of Dr. Hutton. The 3d, 4th, 5th, 6th, 8th and 9th, are manifestly drawn up by one hand; and the manner exhibited in them of referring to some of Dr. H.'s performances, proves that they were written by Dr. Gregory. The 2nd chapter may, we think, be safely ascribed to the same hand. The 7th is not sufficiently characteristic of the manner of either author to enable us to decide. The 10th is doubtless Dr. Hutton's; and the 11th as obviously Dr. Gregory's. The three remaining chapters are too strongly marked by some of the peculiar excellencies of Dr. Hutton's manner, to allow any hesitation in imputing them to him."

We cannot give a more concise and fair account of the contents of these various chapters, than is furnished in the preface to the book,-from which, therefore, we shall make another quotation.

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 compass of one short tract.

the

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 geomerical 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 king. doms, 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 Polygo nometry, 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 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 second 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 pressure 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, projec tiles, &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.' pp. iy-vii.

In the composition of this volume, the authors seem to have aimed, in a remarkable manner, at conciseness and utility. Every thing is delivered in the shortest possible space, compatible with perspicuity; and nothing will be found that has not a tendency to some beneficial practical purpose, especially amongst civil and military engineers. Every part of the volume, and indeed every part of the Woolwich Mathematical Course, abounds with useful practical examples. This, indeed, has always appeared to us one of the great excellencies of the work, and what very admirably fits it for the purposes of tuition.-But our readers will not be satisfied, if we do not pass beyond these general remarks.

We proceed then to observe, that the 2nd chapter contains a very neat and simple Essay on the Elements of Isoperimetry. It is purely geometrical: and, though it only occupies 22 pages, exhibits, and clearly demonstrates, several of the most interesting properties relative to isoperimetrical, and equal surfaced, figures. We do not perceive more than three or four properties that we have not met with before, in some or other of the books mentioned by Dr. Hutton in his preface; yet we know not where else to point to a summary, so well suited as the present, to lead the way to the abstruser inquiries in this department of science, and, at the same time, so easy to be understood. Euclid's first book is not simpler-though this chapter contains the demonstration of several properties, which, in no other English work have been demonstrated without the use of fluxions. We think the "reading men" at Cambridge would find it an agreeable introduction to Mr. Woodhouse's treatise on Isoperimetrical Problems, noticed by us some months ago.

The 3d chapter exhibits a brief, but elegant view of the Trigonometrical Analysis, so far as relates to plane trigonometry it contains, also, some curious formula, which we do not remember to have seen elsewhere, and some interesting and useful problems. After the solution of one of these, however, á remark is added which we do not quite understand. The question is this: "There is a plane triangle, whose sides are three consecutive terms, in the natural series of integer numbers, and whose largest angle is just double the smallest : Required the sides and angle of that triangle?' A simple solution is given by means of the trigonometrical formula, from which the sides are found to be 4, 5, and 6; and the angles 41° 24′ 34" 34", 55° 46′ 16" 18", and 82° 491 9′ 8′′, respectively. Then follows this observation: Any direct solution of this curious problem, except by means of the analytical formulæ employed above, would be exceedingly tedious and operose.' Now this is not correct, unless the author meant a direct algebraical solution. That a direct geo

metrical process will soon lead to the same result, we shall endeavour to shew in very few lines. The figures may be readily constructed from our mode of deducing the

Geometrical Analysis. Suppose the thing done, and ABC the triangle whose vertex is C, and whose sides CB, BA, AC, are respectively as three consecutive terms in the increasing series of integer numbers, and the greatest angle ABC equal to twice the least angle BAC. From C upon AB let fall the perpendicular CD; towards A set off upon the base Db = Db, and join bC. Then, because the angle CbB (equal to ABC) is equal to twice the angle CAB, the points A and C are in the circumference of the circle, whose radius is bA or bC, and centre b. But by a well known theorem, AB: AC+ CB (=2 AB) :: AC-CB = 2); AD - DB Ab = 4CB. Hence CB is given; and because AB CB + 1 = AC — 1, (by hypothesis,) the other two sides are given; that is, all the three sides are given to construct the triangle.

Construction. Let the right line AB be set off equal to 5 : from the centres A and B with radius 6 and 4 describe arcs to intersect each other in C; draw AC, BC; and ACB is the required triangle.

Demonstration. The sides CB, BA, AC, are three consecutive terms in the series of natural numbers, by construction. From C upon AB demit the perpendicular CD; set off Db = DB, and draw bC. Then AB (5) AC+ BC (= b + 4) :: ACBC (b-4): AD-DB= Ab = 4: Therefore Ab bC BC. Hence the points A and C are in the circumference whose centre is b and radius bA or bC; and consequently the angle CBb- CBA twice the angle CAB, by Euc. iii. 20. Q.E.D.

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The angles deducible from this solution, will manifestly agree with those given bDr. Gregory.

In the 4th chapter, which contains, first, an enumeration of the General Properties of Spherical triangles, and next, the solution of all the Cases of Spherical Trigonometry, with tables to facilitate the practice, we were much pleased with the perspicuous manner of treating the theorem which relates to the supplementary triangle, and with the scholium which contains a summary of the various cases in which the spherical triangle is susceptible of one or of two forms and solutions. But the most striking article in this chapter, is that relating to solid angles.

These have, in all ages, been regarded as geometrical quantities of a very peculiar kind, the mutual relations of which it is by no means easy to establish. Some geometers have called in question the possibility of the thing altoge ther and others, as Professor Playfair, in the notes to his Elements of Geometry, have affirmed, that no multiple or