TRIGONIA, in botany, a genus of the Diadelphia Decandria class and order: Natural order of Malpighiw, Jussieu. Es sential character : five-parted ; pe tals five, unequal, uppermost fhveolate at the base within ; nectary, two scales at the base of the germ ; filaments, some barren ; capsule leguminose, three-cor nered, three•celled, three-valved. There are two species : viz. T. villosa, and T. lxvis ; both natives of South America. TRIGONOMETRY. The business of this important science is to find the angles where the sides are given; and the sides of their respective ratios, when the an gles are given ; and to find sides and an gles, when sides and angles are partly given. To effeet this, it is necessary not only that the peripheries of circles, but also certain right lines in and about cir cles, be supposed divided into certain numbers of parts. The ancients, feeling the necessity of such a pre-division, por tioned the circle into 360 equal parts, which they called ,degrees; each degree was again divided into 60 equal parts, called minutes ; and each minute com prised 60 equal parts, called seconds. The moderns have improved upon this division by the addition of a nonius, or vernier, which may he carried to any ex tent, but is usually limited to decimating the seconds ; noting each tenth part thereof It would have been found a con siderable convenience in mathematics, if the circle had been divided into centisi mal parts, particularly in trigonometriN operations ; thus making every quadrant to consist of 100 degrees, each degree of 100 minutes, and each minute of 100 se conds : there can, indeed, be no doubt but all the arithmetical calculations relating to the periphery, as well as to the se cants, sines, tangents, radii, chords, and complements, would by this reformation have been simplified.
\Ve shall be brief on this head, because it would require more spare than Could be allotted to any one branch of science, were we to follow the whole extent of trigonometry in this place. The foltoulng definitions will be found useful : 1. The
complement of an am is the difference thereof from a quadrant ; thus, if an arc measures 60°, the complement is 2. A chord, or subtense, is a right line drawn from one to the other end of an arc. 3. The sine, or right sine, of an arc, is a perpendicular falling from one end of an arc to the radius drawn, at right an gles thereto, towards the other end of the arc. Hence it is clear that an arc of 60° must hive its secant, its radius, and hi chord, all of the same length; forming an equilateral triangle. The secant and ra dius both proceed from the centre ; but all sines are a vertical line passing through the centre, and invaria bly fall upon a diameter, drawn perpendi cular to that right line. See DIALLING, GEOMETRY, and MATIIEMATICAL hi struments ; under which various explana tions will be found, whereby the student may perceive the necessity for such yefer ence.
The solution of the several cases in plane trigonometry upon four propositions, called axioms, which cannot he too perfbctly understood, and ought. ever to be adverted to.
axiom. I. In any right-lined plane trian gle, if the hypothenuse (or longest side) be made the a circle, the other two sides, or legs, will be the sines of their opposite angles ; but if either of the legs, including the right angle, be made the other leg becomes the tangent of its opposite angle, and the hypothenuse the secant of the same angle. For in the tri angle A B C, (fig. 21. Plate XV. Trigo nometry,) let A B be made the radius of a circle , and with one foot of the com passes on A or B describe a circle : it is plain that the leg B C will be the sine of the angle A, and A C the sine of the an. gle B: but if A C becomes radius, B C will be the tangent to the angle A, and B A the second thereto. Again, by mak ing B C radius, A C will be tangent, and A B the secant of the angle B. Hence it is plain that the different sides take their names according to that side which is made radius.