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M A T H E M ATICS;

FOR THE
USE OF ACADEMIES
As WELL, As

PRIVATE TUITION.

IN TWO WOLUMES.

By
CHARLES HUTTON, LL.D. F.R.S.

EATE professor or MATHEMATICS IN THE ROYAL Mif, ITARY Academy.

THE FIFTH AMERICAN, FROM THE NINTH
* . LONDON EDITION,

WiTH MANY CORRECTions AND IMPROVEMENTS.
BY OLINTHUS GREGORY, LL.D.

ading Associate of the Academy of Dijon, Honorary Member of the Literary

Cerres
and $. Society of New-York, of the New-York Historical Society, of the
marian societies of Newcastle upon Tyne,

Lite and Philosophical, and the o -
of the Cambridge i.i. society, of the Institution of Civil Engineers, &c. &c.
Secretary to the Astronomical Society of London, and Professor of Mathematics in
the Royal Military Academy.

WITH THE ADDITIONS -
or

ROBERT ADRAIN, LL.D. F.A.P.S.F.A.A.S., &e.
And Professor of Mathematics and Natural Philosophy.

THE WHOLE
CORRECTED AND IMPROVED,

WOL. II.

NEW.YORK :
W. E. D. E.a.W.", PR I.W. T.K. r.
T. AND J. swoops; t. A. RoxALDs; collins AND co.; cott,TNs AND HAN.
NAY; G. AND c. AND H. cAnvill; white, GallaBER, AND white;
O. A. ROORBACH: AND M'ELRATH AND BANGS.

1831.

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Southern District of Yest-York, to wit: BE IT REMEMBER.F.L., That on the 22d day of February, Anno Domini 1881, W. E. DEAN, of the said district, hath deposited in this office the title of a book, the title of which is in the words following, to wit:

“A Course of Mathematics; for the use of Academics 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 Fifth American from the Ninth London Fdition, with many corrections and improvements. By Olinthus Gregory, LL.D. Corresponding Associate of the Academy of Dijon, Honorary Member of the Literary and Philosophical Society of New-York, of the New-York Historical Society, of the Literary and Philosophical, and the Antiquarian Societies of Newcastle upon Tyne, of the Cambridge Philosophical Society, of the Institution of Civil Engineers, &c. &c. Secretary to the Astronomical Society of London, and Professor of Mathematics in the Royal Military Academy. With the Additions of Robert Adrain, LL.D. F.A.P.S. F.A. A.S., &c. and Professor of Mathematics and Natural Philosophy. The whole corrected and improved.”

the right whereof he claims as proprietor. In conformity with an Act of congress, en titled “An Act to amend the several Acts respecting copy-rights.”

FRED. J. B.F.TTS,
Clerk of the Southern District of JYew-York.

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A

COURSE

or

MATHE MATICS, &c. Cl, o | L: 3

PLANE TRIGONOMETRY CONSIDERED ANALYTICALLY.

ART. 1. THERE are two methods which are adopted by mathematicians in investigating the theory of Trigonometry; the one Geometrical, the other Algebraical. In the former, the various relations of the sines, cosines, tangents, &c. of single or multiple arcs or angles, and those of the sides and angles of triangles, are deduced immediately from the figures to which the several inquiries are referred; each individual case requiring its own particular method, and resting on evidence peculiar to itself. In the latter, the nature and properties of the linear-angular quantities (sines, tangents, &c.) being first defined, some general relation of these quantities, or of them in connexion with a triangle, is expressed by one or more algebraical equations; and then every other theorem or precept, of use in this branch of science, is developed by the simple reduction and transformation of the primitive equation. Thus, the rules for the three fundamental cases in Plane Trigonometry, which are deduced by three independent geometrical investigations, in the first volume of this Course of Mathematics, are obtained algebraically, by forming, between the three data and the three unknown quantities, three equations, and obtaining, in expressions of known terms, the value of each of the unknown quantities, the others being exterminated by the usual processes. Each of these general methods has its peculiar advantages. The geometrical method carries conviction at every step; and by

Vol. II. 2

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