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THE circulation of nearly thirty thousand copies of Hutton's Course, sufficiently attests the estimation in which it has been held by mathematical teachers and students throughout the country.

"Long experience in all seminaries of learning," says the author in his original preface, "has shown that such a work was very much wanted, and would prove a great and general benefit; as for want of it, recourse has always been obliged to be had to a number of other books by different authors; selecting a part from one and a part from another, as seemed most suitable to the purpose in hand, and rejecting the other parts-a practice which occasioned much expense and trouble, in procuring and using such a number of odd volumes of various forms and modes of composition; besides wanting the benefits of uniformity and reference which are found in a regular series of composition."

Dr. Hutton's Course of Mathematics was greatly in advance, as to the manner of treating the subjects contained in it, of all works which had then appeared; and for many years the author continued to improve the successive editions, as new discoveries were made or new methods invented. At the close of his long, laborious, and useful life, he committed the work to the care of Dr. Gregory, who, by continual additions and modifications, endeavoured to assimilate it to the growing spirit of inquiry produced by a long period of general peace. Into the last edition, however, greater changes were introduced than had been made in the work since its first composition; and he did me the honor, soon after my appointment to this Institution, to request me to make the greater part of those alterations, under his editorship. It was, however, a matter of deep regret to both of us, that owing to the haste with which the work was urged through the press, adequate time was not given to complete our contemplated improvements. The same cause, also, gave rise to a great number of errata. I am now, however, not without a hope, that the present edition, whilst as free from errata as any mathematical work extant, will be found

to justify the views under which the alterations were commenced, and to give it that preference as a text-book for mathematical instruction which the original work so long enjoyed.

The state of the health of my lamented friend and coadjutor not allowing him to give the attention essential to the editorship of the work, has committed it wholly to my care, to carry out our joint views to the best of my ability. It was not, however, without some reluctance and much anxiety, that I undertook it: and for more than twelve months the present volume has been the unremitting object of my entire labour. Even yet, I am obliged to defer a few of our contemplated improvements for a future edition.

In the arithmetic very little alteration has been made, except a few occasional notes; and in the early part of the algebra comparatively few essential alterations have been made from the last edition. In the multiplication and division I have given prominence to the use of detached coefficients and the synthetic method of division. An elementary investigation of the latter process is annexed, as that of Mr. Horner is not easily understood except by students whose progress is considerably more advanced but a still simpler and more direct one is given amongst the 66 Additions" at the end of the volume, and which I discovered since sheet K, (p. 128), was printed off. To the simpler operations of algebra, where the reason of the step is not apparent at once, investigations are annexed, to secure to the student a complete understanding of the logic of his pro


In the chapters on simple and quadratic equations, the introductory remarks and suggestions, as well as the examples chosen for illustrating the methods by actual working, have been generally exchanged for others better adapted to show the true character of the operations. In the quadratics, the Hindû method of completing the square is enforced, as being generally superior, in respect of facility, to the Italian or common


The chapter on the general resolution of numerical equations has been wholly recomposed; and I hope it will be found free from those logical defects which are so liable to insinuate themselves into abbreviated treatises on subjects involving so many distinct principles as this does. The theory of equations, is, however, carried no further than is requisite for numerical solution though to this extent, great pains have been taken to render it logically complete. Legitimate proofs, on elementary principles, are given of the criteria of De Gua and Budan, for detecting the imaginary roots of an equation; and as brief a form of investigating Sturm's criterion as I could devise, has also been added. Though I am as fully impressed as any

one can be of the great beauty and importance of Sturm's theorem, I have been led, I confess, to introduce it here more in accordance with the dictum of the mathematical public, than from my own conviction of its practical utility in reference to numerical solution-at least till some method less operose and practically embarrassing than is yet known, shall be discovered, of forming his successive auxiliary functions subsequent to the first derivative. Should such a method, in any way analogous to Horner's process for transformation, ever be invented, Sturm's theorem will become practically useful:-but not till then *.

Upon Horner's method of continuous approximation to the roots of equations, I have dwelt at sufficient length to render it easy of comprehension. As the first attempt ever made to compose an elementary treatise on this subject was made by myself in the previous edition of this work, my attention was naturally directed to it subsequently with sufficient precision. to enable me to separate the essential and the useful part of that composition, from the parts which I found superfluous, and make such additions as experience might suggest during my professional use of the volume.

The chapters on indeterminate coefficients, piling of balls, the binomial and exponential theorems, and on logarithms, it will be seen are all written anew, and with especial reference to the order in which the subjects naturally present themselves in a systematic course of study. The same may be said of the chapters on series and finite differences.

The early part of the geometry is unaltered, though in a future edition I propose to remodel it entirely. The doctrine of ratio is put altogether in a new and, I persuade myself, a perfectly logical form; and the theorems depending on ratio are changed in their manner of demonstration, to be in accordance with the same principles. A few theorems of great practical value are added. The chapters on the geometry of planes and solids have also, for the most part, been modified and rewritten.

The practical geometry has been entirely recomposed, and in especial reference to the circumstances under which the problems themselves occur in practice. A number of constructions of this kind, which are believed to be new, and are adapted to peculiar exigences, have been introduced.

* This page was in type before I met with the elegant and instructive Mathematical Dissertations of Professor Young. In that volume, an important improvement is made in simplifying the numerical process, and especially the initial step of it. A few more such improvements would entirely remove the objections which, in a practical point of view, are urged against that important and beautiful process.

Independently of the discussion of the different points connected with Sturm's criterion, the volume of Mr. Young deserves the attention of the mathematical student, beyond any other work that I could suggest to him as he will be at once led to study the logic of the processes involved in elementary mathematics, with more precision than in any other production with which I am acquainted.

The chapter on practical geometry in the field contains a series of problems of great importance to the military profession, to engineers and surveyors, and which form the substance of a course of lectures just delivered at the Royal Artillery Institution.

In the plane trigonometry nothing besides the examples for exercise, of the last edition, remains in this. To give every thing essential to elementary trigonometry investigated in a direct and simple manner, and entirely to exclude all matters of mere scientific curiosity, has been my guiding principle in the composition of these chapters. Trigonometry, therefore, instead of forming two separate treatises in two successive volumes, is now brought entirely into the first; and the examples that are changed in place have been marked by a quotation of the places in which they previously stood, for the convenience of those who wish to make reference to any works founded on the preceding edition. The few additional examples here given, will offer no difficulty to those who have fully mastered those of that edition.

In the mensuration the investigations are, for the sake of continuity, classed differently, but the problems have the same order as before; and a few easy additional examples are added, as the paucity of these in the former editions has often been a subject of complaint. In the artificers' work and land surveying no changes are made in this edition, but a complete revision of them will be given hereafter,—and, but for practical obstacles, would have been given now, as no parts of the entire course require it more than these.

The figures in this edition are nearly all newly-cut, and every attention has been paid to the arrangement of each page, both for convenience of reading and reference, and of losing no space that could possibly be filled up with useful matter. Much of the phraseology, and the entire notation, of former editions has been modernised, and an attempt has been made to render it, with the exceptions already specified, consistent and systematic throughout.

Royal Military Academy, Woolwich,

10th February, 1841.


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