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OF

ANALYTICAL GEOMETRY

AND OF THE

DIFFERENTIAL AND INTEGRAL

CALCULUS.

BY ELIAS LOOMIS, A.M.,

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PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY IN THE UNIVERSITY OF
THE CITY OF NEW YORK, AUTHOR OF "A TREATISE ON ALGEBRA,' ELEMENTS
OF GEOMETRY AND CONIC SECTIONS, ""ELEMENTS OF PLANE AND SPHER-
ICAL TRIGONOMETRY, WITH THEIR APPLICATIONS TO MENSURATION,
SURVEYING, AND NAVIGATION," "RECENT PROGRESS OF
ASTRONOMY," ETC., ETC.

NEW YORK:

HARPER & BROTHERS, PUBLISHERS,

82 CLIFF STREET.

1851.

KF7569

UND FRY
LIBRARY

0474475

Entered, according to Act of Congress, in the year one thousand. eight hundred and fifty-one, by

HARPER & BROTHERS,

in the Clerk's Office of the District Court of the Southern District of New York.

PREFACE.

THE following treatise on Analytical Geometry and the Calculus constitutes the fourth volume of a course of Mathematics designed for Colleges and High Schools, and is prepared upon substantially the same model as the preceding volumes. It was written, not for mathematicians, nor for those who have a peculiar talent or fondness for the mathematics, but rather for the mass of college students of average abilities. I have, therefore, labored to smooth down the asperities of the road so as not to discourage travelers of moderate strength and courage; but have purposely left some difficulties, to arouse the energies and strengthen the faculties of the beginner. In a course of liberal education, the primary object in studying the mathematics should be the discipline of the mental powers. This discipline is alike important to the physician and the divine, the jurist and the statesman, and it is more effectually secured by mathematical studies than by any other method hitherto proposed. Hence the mathematics should occupy a prominent place in an education preparatory to either of the learned professions. But, in order to secure the desired advantage, it is indispensable that the student should comprehend the reasons of the processes through which he is conducted. How can he be expected to learn the art of reasoning well, unless he see clearly the foundations of the principles which are taught? This remark applies to every branch of mathematical study, but perhaps to none with the same force as to the Differential and Integral Calculus. The principles of the Calculus are further removed from the elementary conceptions of the mass of mankind than either Algebra, Geometry, or Trigonometry, and they require to be developed with corresponding care. It is quite possible for a student to learn the rules of the Calculus,

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