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CHARLES IIUTTON, LL.D. F.R.S.
LATE PROFESSOR OF MATHEMATICS IN THE ROYAL' MILITARY

ACADEMY.

FROM THE FIFTH AND SIXTH LONDON EDITIONS.

REVISED AND CORRECTED BY

ROBERT ADRAIN, A. M.
FELLOW OF THE AMERICAN PHILOSOPHICAL SOCIETY,

AND
PROFESSOR OF MATHEMATICS IN QUEEN'S COLLEGE, NEW JERSET.

VOL. I.

NEW YORK:
PUBLISHED BY SAMUEL CAMPBELL, WILLIAM FALCONER,
T. & J. SWORDS, JAMES EASTBURN, (Executor to E. Sarjeant)
PETER A. MESIER, EVERT DUYCKINCK, INSKEEP &
BRADFORD, R. MÅDERMUT & D. D. ARDEN, WHIT.
ING & WATSON, THOMAS A. RONALDS, JOHN
TIEBOUT, SAMUEL WOOD, AND

SAMUEL A. BURTUS.
George Long, Printer.

1812.

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THE publishers of this first American edition of Dr. Hutton's Course of its utility to private Students as well as to Colleges and other Seminaries in which Mathematical Science constitutes a branch of education. They also had in view the furnishing of the Military School of their country with a Text Book of high standing, and long in use in the British Military Academy. And in order that this edition might derive advantage from the progress of the Science, and thereby become more worthy of the public patronage, they engaged a gentleman of acknowledged eminence to revise its pages and superintend the printing ; and they confidently trust this duty has been performed with some profit to the work generally. To gentlemen, therefore, who study this delighưful science in private, and to the literary and military institutions of their country, the publish and proprietors look for remuneration--and they feel as though they should not look in vain. An increasing taste for Mathematical Studies will produce a cofrespondent increase of purchasers; while the preference which an honourable patriotism gives to American editions when well executed, will receive additional activity from the super-eminence of the work itself.

Orders for this publication will be thankfully received by any of the proprietors ; all whose names are printed at the foot of the title-page.

New-York, August, 1812.

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District of New-York, ss. BE IT REMEMBERED, that on the eleventh day of August, in the thirty-seventh year of the Independence of the United States of America, Samuel Campbell, of the said district, hath deposited in this office the title of a book, the riglu whereof he claims as proprietor, in the wonls following, to wit:

** A Course of Mathematics. In two volumes. For the use of Academies, as well as Private Tuition. By Charles Hutton, L. L.D. F. R. S. late Professor of Mathematies in the Royal Military Academy. From the fifth and sixth London editions, revised and corrected by Robert Adrain, A. M. Fellow of the American Philosophical Society, and Professor of Mathematics in Queen's College, New Jersey."

In conformity to the act oi the Congress of the United States, entitled "An act for the encouragement of learning, by securing the copies oi maps, charts, and books to the authors and proprietors of such during the times therein mentioned."

And also to an act, entitled * An act, supplementary to an act, entitled an act for the encouragement of learning, by securing the copies pt maps, charts, and books to the authors and proprietors ot' snch copies, during the tinies therein mentioned, and extending the benefits thereof to the art of designiug, engraving, and etching historical and other prints.”

CHARLES CLINTON,
Clerk of the district of Neur-?ork.

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PREFACE.

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A SHORT and Easy Course of the Mathematical Sciences
has long been considered as a desideratuin for the use of Stu-
dents in the different schools of education : one that should
hold a middle rank between the more voluminous and bulky
collections of this kind, and the mere abstract and brief com.
mon-place forms, of principles and memorandums.

For long experience, in all Seminaries of Learning, 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 benefit of uniformity and reference, which
are found in a regular series of composition.

To remove these inconveniences, the Author of the present work has been induced, from time to time, to compose various parts of this Course of Mathematics; which the experience of many years' use in the Academy has enabled him to adapt and improve to the most useful form and quantity, for the benefit of instruction there. And, to render that benefit more eminent and lasting, the Master General of the Ordnance has been pleased to give it its present form, by ordering it to be enlarged and printed, for the use of the Royal Military Academy.

As this work has been composed expressly with the inten. tion of adapting it to the purposes of academical education, it is not designed to hold out the expectation of an entire new mass of inventions and discoveries : but rather to collect and arrange the most useful known principles of mathematics, disposed in a convenient practical form, demonstrated in a plain and concise way, and illustrated with suitable examples, rejecting whatever seemed to be matters of mere curiosity, and retaining only such parts and branches, as have a direct tendency and application to some useful purpose in life or profession.

It is however expected that much that is new will be found in many parts of these volumes; as well in the inatter, as in the arrangement and manner of demonstration, throughout the whole work, especially in the geometry, which is rendered much more easy and simple than heretofore; and in the conic

sections,

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 everywhere new examples, this ednion 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 inher 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 Annuities at the end of the Algebra. Also any part of the Gcometry, 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 he 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 wasze 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. 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 Licut. Governor, were deemed among the most useful additional subjects for the purposes of the education estabJished 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 :

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