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USE OF ACADEMIES
AS WELL AS
IN TWO VOLUMES.
CHARLES HUTTON, LL.D. F.R.S.
LATE PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY ACADEMY.
THE FIFTH AMERICAN, FROM THE NINTH
WITH MANY CORRECTIONS AND IMPROVEMENTS.
BY OLINTHUS GREGORY, LL.D.
Corresponding Associate of the Academy of Dijon, Honorary Member of the Literary
WITH THE ADDITIONS
ROBERT ADRAIN, LL.D. F.A.P.S. F.A.A.S, &e.
CORRECTED AND IMPROVED,
W. E. DEAN, PRINTER.
T. AND J. SWORDS; T. A. RONALDS; COLLINS AND CO.; COLLINS AND HAN
Southern District of New-York, to wit:
BE IT REMEMBERED, That on the 22d day of February, Anno Domini 1831, 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 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 Fifth American from the Ninth London Edition, 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 uitled An Act to amend the several Acts respecting copy-rights."
FRED. J. BETTS,
Clerk of the Southern District of New-York,
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 pro. perties 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 inde. pendent 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