ference to the surface of a sphere. Hence mathemati. cians have been led to attempt the solution of two general problems. *I. Knowing three of the six things, whether angles or sides, which enter the constitution of a rectilinear triangle, to determine the other three; when it is possible. II. Knowing three of the six things which compose a triangle formed on the surface of a sphere, by the intersections of three planes, which also meet in the centre of that sphere, to determine the other three, when possible. The resolution of the first of these general problems appertains to plane or rectilinear trigonometry; that of the second, to spherical trigonometry. 4. Lines and angles, being magnitudes of different kinds, do not admit of comparison. It becomes necessary, therefore, to have recourse to quantities of an intermediate kind, akin to the one, yet having an obvious dependence upon the other, and serving as a common vinculum. Such are the lineo-angular quantities denominated sines, tangents, &c. which we are about to define. They are lines, but lines which admit of being measured only by parts of an assigned line, the radius of a certain circle; and lines which at the same time depend altogether, for their value, upon arcs of that circle, which arcs are, themselves, adequate measures of the angles included between the radii which limit such arcs. i 5. By means of this happy invention of intermediate quantities, the business of trigonometry is greatly facilitated. For, by imagining a perpendicular let fall from the vertical angle of an oblique angled plane triangle upon the base, or base prolonged, it will at once be manifest that the resolution of triangles generally, may be referred to that of right angled triangles. Thus, assuming for a term of comparison, the hypothenuse of a right angled triangle, equal to unity, for example, com. puting the bases and perpendiculars of all possible right angled triangles having the assigned hypothenuse, and arranging them in different columns of a table, the magnitudes of the different parts of any proposed triangle, would become determinable upon the known principles of similar triangles. Such a table as this, would, as will soon be seen, be no other than a table of natural sines. PLANE TRIGONOMETRY. 6. Plane trigonometry is that branch of mathematics, by which we learn how to determine or compute three of the six parts of a plane, or rectilinear triangle, from the other three ; when that is possible. This limitation is necessary, although there is only one case in which it can occur, namely, that in which the three angles of a rectilinear triangle are given. For, it is plain from Euc. vi. 4, that while the three angles of a triangle remain the same, the sides, though retaining the same mutual relation, may be greater or less, in all conceivable proportions. 7. Lemma I. Let AcB be a rectilinear angle: if, about the point c as a centre, and with any distance, or radius, CA, a circle be described, intersecting CA, CB, the right lines that include the angle AcB, in A and B; the angle AcB will be D to four right angles, as the arc AB to the whole circumference of the circle ADFE, F A Produce Ac to meet the circle in F, and through the centre c draw pe, another diameter, to meet the circle E in D, E. Then, Euc. vi. 33, ang. Acs: right ang. AcD :: arc As : arc AD, and, Euc. v. 4, cor. quadrupling the consequents, we have, angle AcB:4 right angles: arc AB:4AD, that is, to the whole circumference. 8. Lemma II. Let Ace be a rectilinear angle: if, about c as a centre with any two distances ca", CA, two circles be described, meeting cA, cB, in A', B', A, B ; the arc AB will be to the whole circumference of which it is a part, as the arc A'B' to the whole circumference of which it is a part. By, lem. 1, arc AB : whole circum. :: angle AcB : 4 right angles, and, arc A'B' : its whole circum. :: angle A"cB':4 right angles. A Therefore, arc AB : whole circum. :: arc A'B' : whole circum. of its respective circle. Definitions. 9. Let AcB be a rectilinear angle, if about c as a centre, with any radius CA, a circle be described, intersecting cA, CB, in A, B, the arc AB is called the measure of the angle AcB. 10. The circumference of a circle is supposed to be divided, or to be divisible, into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; each of these into 60 equal parts, called seconds ; and so on, to the minutest possible subdivisions. Of these, the first is indicated by a small circle, the second by a single accent, the third by a double accent, &c. Thus, 47° 18' 34” 45", denotes 47 degrees, 18 minutes, 34 seconds, and 45 thirds. So many degrees, minutes, seconds, &c. as are contained in any arc, of so many degrees, minutes, seconds, &c. is the angle of which that arc is the measure said to be. Thus, since a quadrant, or quarter of a circle, contains 90 degrees, and a quadrantal arc is the measure of a right angle, a right angle is said to be one of 90 degrees. 11. The complement of an arc is its difference from a quadrant; and the complement of an angle is its difference from a right angle. 12. The supplement of an arc is its difference from a semicircle ; and the supplement of an angle is its difference from two right angles. 13. The sine of an arc is a perpendicular let fall from o: extremity upon a diameter passing through the Other. 14. The versed sine of an arc is that part of the diameter which is intercepted between the foot of the sine and the arc. 15. The tangent of an arc is a right line which touches it in one extremity, and is limited by a right line drawn from the centre of the circle through the other extreImity. #. The secant of an arc is the sloping line which thus limits the tangent. 17. These are also, by way of accommodation, said to be the sine, tangent, &c. of the angle measured by the aforesaid arc, to its determinate radius. 18. The cosine of an arc or angle, is the sine of the complement of that arc or angle: the cotangent of an arc or angle is the tangent of the complement of that arc or angle. The co-versed sine, and co-secant are defined similarly. To exemplify these definitions by the annexed diagram: , let AB be an assumed arc of a circle described with the radius Ac, and let AE be a quadrantal arc; let BD be demitted perpendicularly from the extre- A' P mity B upon the diameter AA’; \ a parallel to it let AT be drawn. To 2F and limited by CT: let GB and #T EM be drawn parallel to AA’, the latter being limited by c't or ct produced. Then BE is the complement of BA, and angle Bce the complement of angle BCA; BEA' is the supplement of BA, and angle BCA' the supplement of BCA; BD is the sine, DA the versed sine, At the tangent, ct, the secant, GB the cosine, GE the coversed sine, EM the cotangent, and csi the cosecant, of the arc AB, or, by convention, of the angle AcB. Note. These terms are indicated by obvious contractions: Thus, for sine of the arc AB we use sin AB, tangent. . . . ditto . . . . . . tan AB, secant . . . . ditto . . . . . . sec AB, versed sine ditto...... versin AB, cosine...... ditto...... cos AB, cotangent ... ditto...... cot AB, cosecant. . . . ditto...... cosec AB, coversed sine ditto . . . . . . coversin As. Corollaries from the Definitions. 19. (A). Of any arc, less than a quadrant, the arc is less than its corresponding tangent; and of any arc whatever, the chord is less than the arc, and the sine less than the chord. - For, in the preceding diagram, the circular sector cAB is less than the triangle cAT, the former being contained within the latter. That is, by the rules for mensuration of surfaces, #CA x arc AB is less than }cA × tan AT; whence, dividing by cA, there results arc AB less than tangent AT. In a similar way it may be seen that chord AB, is less than arc AB, and sine BD, less than chord AB. The same is also evident from the consideration that a right line AB is less than any curve line terminated by the same points, and the perpendicular BD less than the hypothenuse AB, of a right angled triangle ADB. (B). The sine BD of an arc AB, is half the chord BF of the double arc B.A.F. (c). An arc and its supplement have the same sine, tangent, and secant. (The two latter, however, are affected by different signs, + or -, according as they appertain to arcs less or greater than a quadrant: the |