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T. 1.

which are adopted by mathemati ory of Trigonometry: the one ical. In the former, the varimes, tangents, &c. of single or ose of the sides and angles of ately from the figures to which ed; each individual case requirand resting on evidence peculiature and properties of the linagents, &c.) being first defined, quantities, or of them in conessed by one or more algebraother theorem or precept, of developed by the simple ree primitive equation. Thus, tal cases in Plane Trigonomee independent geometrical in-f this Course of Mathematics, rming, 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 Each of these general methods has its peculiar adprocesses. vantages. 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 formula, 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 1st volume, a few of the principal definitions, there given, are here repeated, as follows:

The SINE 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 the circle and the foot of the 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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