1. In a plane triangle there are six things to be considered, namely, three angles and three sides. But it is sufficient to know a certain number of these in order to determine the rest. It follows, indeed, from what has been proved with respect to equal triangles, that we may always construct a triangle, when we know three of the six parts, which constitute it, provided that one at least of these three parts be a side. In order to render the theory of triangles complete, we must be able to apply the calculus to geometrical figures, the exactness of which is limited by the imperfection of instruments, while there is nothing to prevent the calculus being carried to any degree of precision we choose. Such is the object which we propose to ourselves in Plane Trigonometry.

Those, who first undertook to develop, by a series of numerical operations, or by algebraic formulas, the relations which subsist between the different parts of a triangle, must have found themselves embarrassed by the difficulty of introducing into the calculus the magnitude of angles, which, being measured by the arcs of a circle, cannot be compared with right lines; but they must have soon perceived, that if they could, by any means whatever, calculate a series of triangles, the angles of which should be of all possible values, this series would necessarily Trig.

1

Fig. 1.

contain a triangle similar to the one to be determined, whatever it might be; and that the parts of this last might be deduced from those of the former by a simple proportion. This will be rendered plainer by the following example.

2. I suppose that in the triangle ABC (fig. 1), we know the angle B, the angle C, and the side BC, we find in the series of computed triangles that, which has two angles, b and c, respectively equal to the angles B and C; it will necessarily be similar (Geom. 203) to the proposed triangle ABC; and since all the parts, a b, a c, bc, are known, we have the proportions,

bc: ab:: BC: AB, bc: ac:: BC: AC,

in each of which the first three terms are given, whence we obtain

and since from all other considerations we deduce A = a, all the parts of the triangle ABC are determined.

3. Now that we know the use which may be made of a series of triangles having all possible angles, and the sides of which are calculated, we are led to inquire into the method of constructing such a series. To take the most simple case first, I will suppose that the triangles to be determined are right-angled; it is evident that they may all be formed in the quadrant of a circle by letting Fig. 2. fall from each of the points of the arc AB (fig. 2), perpendiculars MP, M'P', M"P", &c., upon the radius AC, and drawing the radii MC, M'C, MC, &c., the triangles, MPC, M'P' C, M'P'' C, &c., thus formed, are right-angled at P, P', P", &c., and the angles, MCP, M'CP', M" CP", &c., have successively all possible values; the angles, CMP, CM'P', CM"P", &c., which, with the preceding, make a right angle (Geom. 62), will be such, as is required by the nature of right-angled triangles, and there cannot be a right-angled triangle, which is not equiangular with some one of those furnished by a table, constructed as above described. It may be remarked, that these triangles have each the same hypothenuse which is equal to the radius of the arc AB.

4. We may form also a series of right-angled triangles, each having one of the sides comprehending the right angle equal to the radius of the circle; it is sufficient, for this purpose, to raise the indefinite tangent AT from the extremity of the radius AC, and to draw from the centre C, through the points, M, M', M".

&c., the secants CN, CN', CN", &c. It is evident, that the triangles CAN, CAN', CAN", &c. must have all the combinations of angles, which can exist in a right-angled triangle, and among these triangles there will necessarily be found one similar to any right-angled triangle that can be proposed.

5. In the triangles CPM, CP'M', CP"M", &c. the hypothenuse of which does not change, the sides PM, P'M', P'M", &c., which increase with the angles ACM, ACM', ACM", &c., or with the arcs AM, AM', AM", &c. which measure these angles, have received a name on account of this dependence; the line PM is called the sine of the arc ́ AM, the line P'M' is also the sine of the arc AM', and so of the others. It follows from this, that the sine of an arc is the perpendicular let fall, from one extremity of this arc upon the radius, which passes through the other extremity. The lines CP, CP', CP", &c. which diminish, as the arcs AM, AM', AM", &c. increase, are respectively equal, being parallels comprehended between parallels, to the perpendiculars MQ, M'Qʻ, M"Q", &c. let fall from the points M, M', M", &c. upon the radius CB, perpendicular to the radius CA; and it is evident that the lines MQ, M'Q', M"Q", &c., are, with respect to the arcs BM, BM', BM", &c., what PM, P'M', P"M", &c. are, with respect to the arcs AM, AM, AM", &c., and that, consequently, MQ is the sine of BM, M'Q' of BM', and M"Q" of BM", &c.

Two arcs, the sum or difference of which is the fourth part of the circumference of a circle are called complements the one of the other. The arcs BM, BM', BM", &c. are respectively the complements of AM, AM', AM", &c. We designate the lines MQ, M'Qʻ, M"Q", &c., as well as their equals CP, CP', CP", &c. under the name of cosines of the arcs AM, AM', AM", &c. Whence the cosine of an arc is the sine of the complement of this arc, and is equal to that part of the radius comprehended between the centre and the foot of the sine.

The right-angled triangles CPM, CP'M, CP"M", &c., which have all the same hypothenuse, are formed, therefore, by the radius of the circle, and the sine and cosine of the acute angle, which has its vertex at the centre.*

* The part AP of the radius AC, comprehended between the foot

Fig. 3.

6. I pass to the triangles CAN, CAN', CAN", &c. The hypothenuses of these are the secants of the arcs AM, AM', AM", &c., since we call the secant of an arc the radius drawn through one extremity of this arc and produced till it meets the tangent, drawn through the other extremity. The portions AN, AN', AN", &c., taken upon the tangent AT, are the tangents of the arcs AM, AM, AM", &c., since we call the tangent of an arc, that part which is intercepted, on the tangent drawn through one extremity of this arc, by the two radii which terminate it.*

7. If, through the extremity B of the arc AB (fig. 3), we draw the tangent B n, and produce it till it meets the secant CN, the line Cn is the secant of the arc BM, the complement of AM, and is called the cosecant of AM; the line Bn, the tangent of BM, is the cotangent of AM, since we understand by the cotangent and cosecant of an arc the tangent and secant of the complement of this arc. The cotangent and tangent, and the cosecant and secant do not respectively make a part of the same triangle, as we have observed with respect to the sine and cosine.

8. Tangents and secants have with sines and cosines relations, that are very simple, by means of which the one may be found from the other. The triangles CPM and CAN being similar, give CP: PM :: CA: AN; whence AN =

PM X CA

CP

; putting, instead of the lines CP, PM, AN, what they denote, namely, cos AM, sin AM, and tang AM, and expressing the radius CA by R, we have tang AM =

R sin AM
COS AM

From the same triangles CPM and CAN, we deduce also, CP: CM:: CA: CN, whence CN=

CM X CA

; but

CP

of the sine and the extremity of the arc is called the versed sine. This line, however, is not used in trigonometry.

* It will be perceived, that the words secant and tangent are here taken in a different sense, from what they are in the Elements of Geometry, where they are considered, as indefinite right lines, one of which cuts the circle and the other touches it. But in trigonometry these terms are always used to denote lines of a determinate magnitude. Where any doubt might otherwise exist, the latter are called trigonometrical secants and tangents.