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Southern District of New-York, ss.

BE IT REMEMBERED, That on the twenty-seventh day of February, in the forty-sixth year of the Independence of the United States of America, George Long, of §: said district, hath *::". in this office the title of a book, the right whereof he claims as proprietor, in the words following, to wit:

“A Course of Mathematics, for the Use of Academies, as well as Private Tuition. In two Volumes. By Charles Hutton, LL.D. F. R. S. Late Professor of Mathematics in the Royal Military Academy. The Third American Edition. From the Fifth, Sixth, and Seventh London Editions. Revised, corrected, and improved. To which is added. An Elementary Essay on Descriptive Geometry, by Robert Adrain, LL.D. F. A. P. S. F. A. A. S. &c, and Professor of Mathematics and Natural Philosophy, in Columbia College, New-York.”

% conformity to the act of Congress of the United States, entitled, “An act for the encouragement of learning, by securing the copies of maps, charts, and boo to the authors and proprietors of such copies, during the time 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 of maps, charts, and books, to the authors § Foot. § ch copies, during the times therein mentioned, and extending the benefits thereos to the arts of designing, engraving, and etching historical

and other prints.” JAMES DILL, Clerk of the Southern District of New-York.

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A SHORT and Easy Course of the Mathematical Sciences
has long been considered as a desideratum 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 be-
nefit 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 Aca-
demy.
oui. 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, dis-
posed in a convenient practical form, demonstrated in a plain
and concise way, and illustrated with suitable examples, re-
jecting whatever seemed to be matters of mere curiosity, and
retaining only such parts and branches, as have a direct ten-
dency and application to some useful purpose in life or profes-
slon.
It is however expected that much that is new will be found
in many parts of these volumes; as well in the matter, 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 simplemanner. 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 Annuities 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 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 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.]

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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 Comic 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 The 2d chapter contains a short geometrical treatise on the elements of Isoperimetry and the marima 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. g 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 the

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