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THE science of Trigonometry teaches how to calculate the sides and angles of a triangle, from sufficient data. It is founded on the principles of Geometry and Arithmetic.
The sides of every triangle are measured, by referring them to some definite portion of linear extent, fixed by authority—this definite portion is called the unit of measure, such as, one inch, one foot, one yard, &c.
The mensuration of angles is effected by means of that general standard derived from the division of a circle into 360 equal parts, called degrees; each degree being divided into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds, &c.
As angles are proportional to the arcs on which they stand (33.6,) the subdivision of the circumference determines the magnitude of angles; but as the angles of two triangles, whose sides are unequal, may be respectively equal, it follows that the relation of the sides must be considered, in order to determine the magnitude of angles-hence, if one angle of a triangle be given,
the other two cannot be found, without one side at least being given. As there may be an infinite number of equiangular triangles, all differing in magnitude, it obviously follows, that the magnitude of the sides cannot be discovered from the angles, only, unconnected with the consideration of one of the sides at least.
As the portion of the circumference, on which any angle rests, is the measure of that angle, arcs and angles, in calculation, are degrees, minutes, and seconds, though angles and the arcs on which they stand are indifferently expressed by the same number of degrees, minutes, and seconds; yet the arcs only submit to measurement and calculation. Angles being merely inclinations of right lines, do not of themselves present any tangible or definite measure, by which every increase or diminution may be accurately ascertained; they therefore cannot be objects of calculation. Hence the calculation of angles must be effected by the comparison of certain right lines with the arcs on which the angles stand, the circumference of the circle itself being measured by a radius of a given magnitude.
The main object then of Trigonometry, as a science, is to investigate the various relations subsisting between those lines and the circumference of the circle in or about which they are drawn. This relation however cannot be accurately expressed, the radius and circumference of a circle being incommensurable quantities—hence it appears that the exact length of these lines, as compared with the circumference, can only be approximated; but as this approximation can be carried on ad infinitum, the result is sufficiently accurate for all scientific and practical purposes.