AB contains 30° or AB=30°. If AB=th of the circumference ABDE, AB = + 360° = 51° + 5 3 = 51° 25′ + 1'=51° 25′ 42"+1". 6 7 The values of 1" and of like quantities are, usually, ex 7 6 pressed by means of decimals; thus, 1", retaining only the two first figures, equals 0.85; and 4th of the circumference would be expressed by 51° 25′ 42′′.85. But it is occasionally useful (see Astron. p. 397.) to extend the division of the circle beyond that of seconds, and to introduce, with their proper symbols, thirds, fourths, &c. In such an extension, 1" would equal 51" 25.7 and the foregoing arc, the seventh of the circumference, would be expressed by 6 7 51° 25′ 42′′ 51" 25.7. 4. The arcs of circles, it has appeared, are proper measures of the angles which they subtend; if the angles be increased, the arcs are also increased, and in the same ratio; and knowing the value of one, is, in fact, knowing the value of the other. But, in Trigonometry, the values of angles are made to depend on the values of certain right lines, drawn according to certain rules, but not varying as the angles vary. The lines just alluded to are called sines, tangents, secants, &c. which it now becomes necessary to define. The Sine of an Arc is a right line drawn, from one extremity of an arc, perpendicularly to a diameter passing through the other extremity. The Cosine of an Arc is a right line intercepted between the centre of the circle and that point in the diameter (the foot of the sine) at which the sine of the same arc drawn perpendicularly to the diameter meets it. The Versed Sine is a part of the diameter intercepted between the common extremity of the arc and diameter, and the foot of the sine. The Tangent of an Arc is a right line, drawn from one extremity of it and perpendicularly to a diameter passing through it, and terminated by its intersection with another diameter passing through the other extremity of the arc and produced beyond it. The Secant of an Arc is a line intercepted between the centre of the circle and that extremity of the tangent of the same arc which lies without the circle. The Chord of an Arc is a straight line joining its two extremities. The Complement of an Arc less than a quadrant is its defect from a quadrant. The Co-tangent, Co-secant of an arc are, respectively, the tangent, and secant of its complement, and, therefore, may be drawn according to the preceding directions, by considering the complement of the arc to be the arc itself. If we now illustrate these definitions, and assume, in the annexed diagram, AB to be the arc: then, see p. 3. 1. 24. BF is the sine of AB, and F is what we have called the foot of the sine. CF is the cosine of AB (see p. 3. 1. 27.) AF is the versed sine of AB (1. 3.) AT is the tangent of AB (1.6.) CT is the secant of AB. A line joining A and B would be the chord of AB: as Bgb, bfb' (see fig. p. 7.) are the chords of the arcs BQb, bab'. The complement of AB is (l. 15.) BQ, since AB+BQ=quadrant: now BQ being considered as the arc of which the sine, tangent, secant, &c. are required, its sine, cosine, tangent, secant, by the preceding definitions, are (see fig. p. 7.) respectively, Bg, Cg, Qt, Ct: and, accordingly, (see pp. 3, 4.) those same lines are respectively, the cosine, sine, co-tangent, co-secant of the arc AB. The lines that have hitherto been drawn expound the sine, cosine, &c. of an arc AB less than a quadrant: but if we take A b greater than a quadrant, then, according to the above definitions, of is the sine of the arc AQb. Cf is the cosine, Af is the versed sine, CS the secant, As is the tangent. In order to determine the co-tangent and co-secant of this arc Ab we must vary and extend the definition of the complement of an arc: now, (see p. 4. 1. 15.) the arc being less than a quadrant, its complement was defined to be its defect from that quantity: but an extended definition which should make the complement of an arc to be the difference between it and a quadrant would suit both arcs greater and less than a quadrant; and, according to such definition, Qb (fig. p. 5.) would be the complement of Ab, and Qs (AQ being a quadrant) the tangent of Qb would be the co-tangent of Ab. Cs the secant of Qb, would be the co-secant of Ab. Let the arc be called A, then when A is less than a quadrant (Q), therefore, by what has preceded, sin. A = cos. (A - Q), and sin. (A - Q) = cos. A, tan. A = co-tan. (A - Q), and co-tan. A = tan. (A — Q), sec. A = co-sec. (A - Q), and co-sec. A = sec. (A --- Q). Let now the arc be greater than two quadrants but less than three: and let AQad represent such an arc, then, by the preceding definitions (fig. p. 5.), Lastly, let the arc be greater than three quadrants but less than four, or less than the circumference of the circle; and let AQak represent such an arc, then We may also use the definitions of p. 3, 4, and draw the sines, cosines, &c. of arcs greater (if we may so call them) than the circumference: for instance, suppose the arc to be equal to the circumference plus the arc AB; then, guided by the definition (see p. 3,) begin from A, one extremity, and go round the circle in the direction AQ a B' till you arrive at B the other extremity of the arc (= AQa A + AB): from B draw BF perpendicularly to A a passing through A, and BF is the required sine of the arc, CF is the cosine, AF the versed sine, AT the tangent, and CT the secant: which are evidently the sine, cosine, &c. of the arc AB. Hence, admitting the existence of arcs greater than a circle, and applying to such the original definitions of p. 3, 4, we have these equalities sin. AB = sin. (circumference + AB), or sin. A = sin. (2ᅲ + A), calling AB, A, and the circumference 2; also cos. A = cos. (2 π + Α), tan. A = tan. (2ᅲ + A), sec. A = sec. (2ᅲ + А), and in a similar manner we might easily obtain more like equalities. The sine, cosine of an arc are thus expressed by means of the sine, cosine, &c. of other arcs: but they may also be expressed |