since the ratio of an arc to its sine is ultimately that of equality, and the sine of A+ dA may be considered as the same with the sine of A; it follows, that I. d sin A= +cos AdA. 2. d cos A=–sinAdA. 13. Since, by No. 12. d sina-cosAdA, or the variation of the sine of an arc is proportional to its cosine; it follows that, near the termination of the quadrant, the slightest alteration in the value of a sine would occasion a material change in the , or the arc itself. Again, from the same Note, d tanA= dA co variation of the tangent is inversely as the square of the cosine, and must therefore increase with extreme rapidity as the arc approaches to a quadrant. 14. It is convenient to reduce the solution of triangles to algebraic formulae. Let a, b and c denote the sides of any plane triangle, and A, B, and C their opposite angles. The various relations which connect these quantities may all be derived from the application of Prop. 11. bo-Eco–a’. 2bc 2. But, since (art. 16. NO. 3.) sino A*=#(1—cos A), it follows, by substitution, that in A-o-o-o-o: a’—(b–c)*_(a+b-c)(a—b-H c) 1. – 4bc , and therefore, s denoting the 4. The second expression being now divided by the third, gives tan #A* =olo, corresponding to Prop. 12. These are the formulae wanted for the solution of the first case of oblique-angled triangles. To obtain the rest, another transformation is required. sin A = 2T. For the same reason, sinE = #, and hence 6. Again, by composition, sin A-F sinBTao , and therefore, The preceding formulae will solve all the cases in plane trigonometry; but, by certain modifications, they may be sometimes better adapted for logarithmic calculation. 2 C. angle, its sine is the same as that of A+B, and consequently -: aginAtoosasino). By a similar transformation, sin A 12. From art. 5, c=a If the angle A be assumed equal to 90°, the preceding formulae will become restricted to the solution of right-angled triangles. 14. From art. 1., cosA= o- *::= whence, a”=b”-- co, which expresses the radical property of the right-angled triangle. #= +. and consequently sin B =+. which corresponds with Prop. 7. 15. From art. 5., * - 16. Again, from the same article, _b__sinh sinE and Besides the regular cases in the solution of triangles, other combinations of a more intricate kind sometimes occur in practice. It will suffice here to notioe the most remarkable of these varieties. 20. Thus, suppose a side, with its opposite angle and the sum or difference of the containing sides, were given, to de |