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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 ad chapter contains a short geometrical treatise on the clements 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 matier 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 musi 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 connecied 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 compatation.
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 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 cffects 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 probiems 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 em
ployments 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 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 fluen's in that treatise, let him immediately be conducted through the 10th chapter of the 3d volume. After he has gone over the remainder of the Fluxions with the applications to tangents, radii of curvature, rectifications, quadratures, &c. the Ilth 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 thework.
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 novel. ty 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, I 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 perspicuity and correctness, than in any three volumes of equal size in any language.
BY THE AMERICAN EDITOR.
THE last English edition of Hutton's Course of Niathematics, in three volumes octavo, may be considered as one of the best systems of Mathematics in the English language Its great excellence consists in the judicious selection made by the authors of the work, who have constantly aimed at such things as are most necessary in the useful arts of life. To this may be added the easy and perspicuous manner in which the subject is treated -a quality of primary importance in a treatise intended for beginners, and containing the elements of science.
The third volume of the English edition having been but lately published, is scarcely known at present in this country it is but justice to its excellent authors to state, that they have collected in it a great number of the most interesting subjects in Analytical and Mechanical Science. Analytical Tri. gonometry Plane and Spherical, Trigonometrical Surveying, Maxima and Minima of Geometrical Quantities, Motion of Machines and their Maximum Effects, Practical Gunnery, &c. are among the most important subjects in Mathematics, and are discussed in the volume just mentioned in such a manner as not only to prove highly useful to pupils, but also, to such as are engaged in various departments of Practical Science.
As the work, after the publication of the third volume, embraced most subjects of curiosity or utility in Mathematics, it has been thought unnecessary to enlarge its size by much additional matter.
The present edition however, differs in several respects from the last English one ; and it is presumed, that this difference will be found to consist of improvements. These are principally as follows :
In the first place, it was theuglit adviseable to publish the work in two volumes instead of three; the two volumes being still of a convenient size for the use of students.
Secondly, a new arrangement of various parts of the work has been adopted. Several parts of the third volume of the English edition treated of subjects already discussed in the preceding volumes ; in such cases, when it was practicable, the additions in the third volume have been properly incorporaied with the corresponding subjects that preceded them; VOL. I. A
and, in general, such a disposition of the various departments of the work has been made as seemed best calculated io promote the improvement of the pupil, and exhibit the respective places of the various branches in the scale of science.
And thirdly, several notes have been added ; and numerous corrections have been macie in various places of the work : it were tedious and unnecessary to enumerate all these at present ; it may suffice to remark the few following:
In pages 169, and 263, vol. 1, are given useful notes respecting the degree of accuracy resulting from the application of logarithms ;-these notes will appear the more necessary to beginners, when we observe such oversights commitied by authors of experience.
In page 173, vol. 1, a new definition of surds is given, instead of that by the author of the work.
In the English edition, a surd is defined to be “ that which has not an exact root" In Bonnycastle's Algebra, it is that which has no exaet root.” And in Emerson's Algebra, it is “ a quantity that has not a proper root.” But nowithstanding the weight of authority thus evidently against me, I do not hesitate to assert, that the definition, just stated is altogether erroneous. According to their definition, the integer 2 is a surd, for it“ has not an exact root.”
In the mensuration, page 411, vol. 1, a remark is added respecting the magnitude of the earth. Dr. Hutton has commonly used a diameter of 7957| English milcs, merely because it gives the round number 25,000 for the circumference : in a few places he has used a diameter of 7930. Having some years ago discovered the proper method of ascertaining the most probable magnitude and figure of the earth, from the admeasurement of several degrees of the meridian, I found the ratio of the axis to the equatorial diameter, to be as 320 10 321, and the diameter, when the earth is considered as a globe, to be 7918-7 English miles.
In the additions immediately preceding the Table of Logarithms in the second volume, a new method is given for ascertaining the vibrations of a variable pendulum. This problem was solved by Dr. Hutton, in his Select Exercises, 1787, and he has given the same solution in the present work, see page 537, vol. 2. The method used by the Doctor appears to me to be erroneous; but in order that such as would judge for themselves on this abstruse question, may have a fair opportunity of deciding between us, the Doctor's solution is given as well as my own.
It may be proper to observe, with respect to the new solution, as well as Dr. Hutton's that the resulting formula does