arc, upon the diameter from its origin. C,S, is thus the sine of the arc BDC, or of the angle BAC,.

If the arcs BC, EC, be equal, the anglés CAS, C1AS, must be equal (Book III., prop. xxvii.); and since the sides AC, AC, are also equal, it follows (Book 1. prop. xxvi.) that CS = C, S,. It follows, therefore, that whenever an arc BC is equal to the supplement EC, of another arc BC,, the two have equal sines; that is, an arc or angle, and the supplement thereof, have equal sines.

(4.) THE COSINE. If we refer to the complement of BC viz. CD, and consider D to be the origin of this complement, C will be its termination, and Cs perpendicular to AD will be its sine. But (Book 1. prop. xxxiv.) Cs = SA; hence the part of the diameter EB intercepted between the centre and the sine of the arc is equal to the sine of the complement of that arc; and thus AS is called the cosine of the arc BC, or of the angle BAC. Therefore the cosine of an arc or angle is the portion of the diameter intercepted between the centre and foot of the sine; AS, is thus the cosine of the arc BDC1, or of the angle BAC1.

It follows, from what is shown in last article, that an arc or angle, and the supplement thereof, have equal cosines.

(5.) THE TANGENT. Draw BT, touching the circle at the origin B of the angular measures; and prolong AC till it meets BT in T. BT is called the tangent of the arc BC, or of the angle BAC. Hence the tangent of an arc or of the angle which it measures is the line drawn from the origin of the arc, touching it there, and limited by the prolongation of the diameter drawn through the termination of the arc. BT, is thus the tangent of the arc BDC,, or of the angle BAC1.

E

T

T

When BC, is the supplement of BC, which, as before shown, implies that the angles CAB, C,AE are equal, or, which is the same thing, that the angles TAB, TAB are

U

equal, the tangents BT, BT, will be equal (Book 1. prop. xxvi.); hence an arc or angle and the supplement thereof have equal tangents.

(6.) THE COTANGENT. If we refer to the complement of BC, regarding D as its origin, the touching line Dt will be its tangent. This, therefore, is called the cotangent of the arc BC, or of the angle BAC. Hence the cotangent of an arc or angle is the line drawn from the termination of the quadrant, touching it there, and limited by the prolongation of the diameter through the termination of the are. Dt, is thus the cotangent of the arc BDC1, or of the angle BAC,.

If BC, is the supplement of BC, that is, if the angles CAB, CAE are equal, the angles tAD, t,AD must be equal; hence (Book 1. prop. xxvi.) Dt = Dt1, so that an arc and its supplement have equal cotangents.

(7.) SECANT and COSECANT. The line from the centre which limits the tangent is called the secant; the line from the centre which limits the cotangent is called the cosecant. Thus AT is the secant of the arc BC or of the angle BAC; AT, is the secant of the arc BDC1, or of the angle BAC,. Also At is the cosecant of the arc BC, and At, the cosecant of the arc BDC, ; and, from articles (5) and (6), it follows that an arc and its supplement have equal secants and equal

cosecants.

(8.) Such are the lines which enter into the computations of Trigonometry, instead of the angles to which they refer. These lines all vary with the angles themselves, and their several lengths, through all their variations, as the angle increases minute by minute, are accurately computed and registered in Tables; their numerical values being expressed conformably to the scale AB=1. Whenever, therefore, we know the magnitude of an angle BAC in degrees and minutes, we may, by a reference to a set of Trigonometrical Tables, find correct numerical expressions for the sine, cosine, tangent, &c. of that angle; and vice versa, whenever we know the numerical value of one of these lines, we may, by help of the same Table, find out the angle and the supplementary angle to which it belongs.

(9.) The mutations which the several lines undergo as the angle advances from its initial limit 0 up to its ultimate limit 180° are easily ascertained from contemplating the diagrams. Thus, when the angle is 0 the sine is 0; and it increases with the angle till this becomes 90°, when the sine attains its greatest value, and becomes equal to the radius, or, in numbers, to unity. As the angle, becoming now obtuse, gradually increases, the sine diminishes, and becomes 0 at the other extreme limit. The cosine is greatest at the initial limit, where the arc is 0, and gradually diminishes till the arc or angle becomes 90°, when it vanishes; thus, when the arc or angle is 0, the cosine is 1; when the arc or angle is 90°, the cosine is 0. Beyond 90° the cosines become increasing, and reach the ultimate value 1 when the arc arrives at 180o.

In like manner may the changes which the tangent undergoes be traced. From 0 up to 90° this line increases from O to infinite; and from 90° to 180° it diminishes from infinite to 0. The cotangent, on the contrary, is infinite at the origin of the quadrant, nothing at its termination, and infinite again at the termination of the semicircle.

The secant at the commencement of the arc is the radius, 1; at the extremity of the quadrant, infinite; and at the extremity of the semicircle, radius again. The cosecant at the origin is infinite; at the termination of the quadrant, radius; and at the termination of the semicircle, infinite again.

It is further obvious that the sine of 45o is equal to the cosine; the tangent to the cotangent; and the secant to the cosecant.

(10.) Finally, from a mere inspection of the diagrams at pp. 216, 217, and a reference to the 47th proposition of the first Book of the Elements, the following relations are apparent :

also, by similar triangles, AsC, TBA in the figure at p. 217.

Hence we may always substitute, in any formula for secant, the reciprocal of the cosine; and for cosecant, the reciprocal of the sine; and on this account the secants and cosecant are seldom inserted in the ordinary Trigonometrical Tables, the table of sines and cosines being made to supersede them.

And, in like manner, from the triangles ASC ADt, we have

so that the tangent and cotangent are the reciprocals of each other.

CHAPTER II.

On the Solution of Plane Triangles.

(11.) WE are now in a condition to develop methods for solving the various cases of plane triangles; that is, for determining, by the aid of certain given parts, those which are unknown. Altogether, there are, as before remarked, six parts to a triangle; and, usually, three of these will enable us to discover the others. We say usually, because there are two ways of making a selection of three out of the six parts, so as to leave the remaining parts, to a certain extent, ambiguous; that is, our chosen parts, so far from belonging exclusively to one determinate triangle, may equally well belong to either of two very differently shaped triangles, or, indeed, to an infinite variety of triangles, all different. It will immediately occur to the student that we shall have this latter case of exception when the chosen parts are the three angles, as an infinite number of different triangles may exist, all, nevertheless, equiangular. The other case of exception is that in which the given parts are two sides and an opposite angle; for, as will be presently seen, two very different triangles may exist, having, however, two sides and an opposite angle in one equal, each to each, to two sides and an opposite angle of the other; and accordingly, in the First Book of the Elements, where all the instances of necessary equality of two triangles, having three parts in one respectively equal to three in the other, are considered, we find no mention of the case where the three parts are two sides and an opposite angle. The only instances of necessary equality of all the six parts, from the admitted equality of three, are those brought forward in propositions iv., viii., and xxvi.

Having made these qualifying remarks, we shall commence

with