operations, carried on by the aid of the Trigonometrical Analysis, of frequent and considerable use. It is desirable, therefore, to extend them, which may be done without much difficulty. Thus with regard to the equality which has just been stated: if 2† represent the circumference of a circle of which the radius is 1, and A be the arc, then sin. A = sin. (π – A), and since A may represent any arc, if instead of A we substitute A n-m π A sin. (m - 1) = sin. (m + 1). n n We may also obtain other general expressions; thus by extending the definition of a sine (see pp. 4, 7.) the subjoined equations are true sin. A = sin. (2 π + A) = sin. (4 π + A), and generally, n being a whole number. For like reasons, sin. (π - A) = sin. (3 π - A) = sin. (5 п — А), and generally, A sin. (ᅲ - A) = sin. { (2 n + 1) π - Α}, n being any number in the progression, 0, 1, 2, 3, &c. Hence, since by p. 12, sin. A = sin. (ᅲ - A) there results this general equation, * The numerical value of w, that of the circumference of a circle of which the diameter is 1, is 3.14159, &c.: 2π, or 2×3.14159, &c. expresses, then, the value of the circumference of a circle, of which the radius is 1. sin. (2n + A) = sin. { (2n + 1) π − A }, or, what amounts to the same thing, if s be the sine of any arc A, it is also the sine of all arcs comprehended under the two formulæ, (2ηπ + Α), { (2n + 1) π - A}, in which n may be any term of the progression, 0, 1, 2, 3, &c. 11. The definition of a cosine being (see p. 7.) like that of a sine, extended to designate the cosines of arcs, that are greater than the circumference, we may, in like manner, obtain general expressions for it. Thus, CF, which is the cosine of AB, may be considered as the cosine of the (circumference + AB) &c. Hence, as before the following equations will be true; cos. A = cos. (2 π + A) = cos. (4 π + A) = cos. (2n + A), n being any term of the progression, 0, 1, 2, 3, 4, 5, &c. But, since the same CF is also the cosine of the arc AB a B', we have CF=cos. (2 π - A) = (for reasons just alledged) cos. (4 π - A) = cos. (2ηπ-Α), n being any term of the progression, 1, 2, 3, &c. or, CF, generally, = cos. {(2n + 2) π - Α }, n being any term of the progression, 0, 1, 2, 3, &c. Hence CF is the cosine of all arcs comprehended within the two formulæ 12. In a similar manner we may investigate the general formulæ of arcs that have the same negative cosine, Cf=cos. (π -A)=cos. (3-A)=, generally, cos. { (2n+1)-4}, n being any term of the progression, 0, 1, 2, 3, &c. and since the same Cf is the cosine also of ABbab' or ᅲ + A Cf=cos. (π + A) = cos. (3ᅲ + A) = cos. { (2n + 1) π + A } . Hence, Cf is the cosine of all arcs comprehended within the two formulæ, π { (2n + 1) π – A}, { (2n + 1) ᅲ + 4 } . The arcs of which fl' is the sine, are π + Α, 3 π + A, and generally { (2n + 1) π + Α } . The arcs of which FB' = fb is the sine, are 2 π - Α, 4 π - A, and generally (2 n + 2) π A. Hence FB' is the sine of all arcs comprehended within the two formulæ, { (2n + 1) + A} and {(2n + 2) π - Α}, n being in each case any term of the progression, 0, 1, 2, 3, &c. In calculations, where FB, FB', and other quantities are involved, if s be the symbol for FB, - s must be the symbol for FB'. For, conceive a line to be drawn a tangent to the circle, at the point opposite to Q in QC produced, and let the distance of any point in the circunference from this line be called z, then FB (s) = z r, and FB' = r - z, or FB' = (z-r). Hence, if in any equation subsisting between trigonometrical lines we wish to pass from the consideration of the point B to that of the point B', we must in such equation substitute -(z-r) instead of z - r, or s, instead of s.* 13. The preceding results may be conveniently represented in a Table, s and c representing the sine and cosine of an arc A. * This hinges on the general doctrine of negative quantities: the scrupulous Student, who is not satisfied with what is here said, is referred to Carnot's Geometrie de position, and his subsequent work on the Theory of Transversals, &c. It is easy from this Table and the expressions for tan. A, co-tan. A, sec. A, &c. namely, to determine the values of the tangent, co-tangent, secant, &c. 18. In some of the preceding expressions, a radius = 1 has been used, and, solely, for the purpose of lessening the number of symbols in the Trigonometrical formule. For, 1, or any power or root of it, used as a multiplier or divisor of an expression, may be expungsin.2 A ed from such expression; thus, instead of , we may more 12 simply write sin. A. Still, however, it is, on many occasions, necessary to use, for the radius, a general symbol such as r, or, an arithmetical value such as 10,000. For this reason, it is desirable to be possessed of some easy and expeditious rule, for converting formulæ constructed with a radius=1 into other formulæ that shall have a different radius. Such a rule may be obtained from the following simple considerations : If (see fig. of next page) BF, bf be drawn similarly inclined to CA, then by similar triangles, : C.B BF = bf. b 1 or BF = bf. (if CB = 1, Сb=r). r 1 In like manner, CF = Cf., AF=af. Now, sines, co r sines, tangents, &c. are drawn after the manner that these lines are; if the angle BFC be a right angle, BF is the sine of the angle BCF to radius 1, bf is the sine to radius r; and, CF, Cf, C |