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HERE are two methods which are adopted by mathematicians in investigating the theory of Trigonometry : the one Geometrical, the other Algebraical. In the former, the vari. ous 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 enquiries 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 conDection with a triangle, is expressed by one or more algebraical equations ; and then every oher theorem or precept, of use in this branch of science, is developed b: 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 Vou. II.


and the three unknown quantities, three equations, and obtaininz, 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 keeping the objects of enquiry constantly before the eye of the student, serves admirably to guard him against the admission of error : the algebraical method, on the contrary, requiring little aid from first principles, but merely at the commencement of its career, is more properly mechanical than mental, and requires frequent checks to prevent any deviation from, truth The geometrical method is direct, and rapid in producing the requisite conclusions at the outset of trigonometrical science; but slow and circuitous in arriving at those results which the modern state of the science requires : while the algebraical method, though sometimes circuitous in the developement of the mere elementary theorems, is very rapid and fertile in producing those curious and interesting formulæ, which are wanted in the higher branches of pure analysis, and in mixed mathematics, especially in Physical Astronomy. This mode of developing the theory of Trigonometry is, consequently, well suited for the use of the more advanced student ; and is therefore introduced here with as much brevity as is consistent with its nature and utility.

2. To save the trouble of turning very frequently to the Ist volume, a few of the principal definitions, there given, are bere repeated, as follows :

The sint of an arc is the perpendicular let fall from one of its extremities upon the diameter of the circle which passes through the other extremity.

The cosine of an arc, is the sine of the complement of that arc, and is equal to the part of the radius comprised between the centre of tbe circle and the foot of ihe sine.

The TANGENT of an arc, is a line which touches the circle in one extremity of that arc, and is continued from thence till it meets a line drawn from or through the centre and through the other extremity of the arc.

The SECANT of an arc, is the radius drawn through one of the extremities of that arc and prolonged till it meets the tangent drawn from the other extremity.

The VERSED SINE of an arc, is that part of the diameter of the circle which lies between the beginning of the arc and the foot of the sine.

The cotANGENT, COSECANT, and COVERSED SINE of an arc, are the tangent, secant, and versed sine, of the complement of such arc.

3. Since

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