b + c But a + b + c 2 a; and therefore if S be put for b + c + a the above equation becomes to radius unity; a cos? A sin S. sin S-a. cosec b . cosec c Whence cos 2 rad And by a like process, similar formulæ may be deduced for B с A 2 COS COS cos b. COS C sin b. sin c sin b. sin c A cos b c COS a 2 sin and as cos bocc ; 2 - cos a s2 sin sin b. sinc a + a + b sin we have 2 a + c 6 a + b sin 2 sin b. sin c с sin s b. sin s cosec b . cosecc rad A Hence sino sin s - b. sin S - C. cosec b . cosec c 2 rad? And by a like process, similar formulæ may be deduced for B C sin - and sin 2 %* SCHOLIUM. A A As sin A=2 sin (Form. 7. Trig.) therefore sino A = 4 COS 12 승 A A 2 Hence from the above formulæ we have с 4 sin S. sin S. a. sin S - b. sin S sin? A= sin? b. sinc to radius unity, or с sin? A= 4 rad? . sin S. sin S - a . sin S - b.sin S sin? b. sinoc to a given We may now, for the convenience of reference, collect and arrange the formulæ which have been demonstrated in this proposition and the preceding one. If A, B, C measure the angles of a spherical triangle, and a, b, c be the sides respectively opposite those angles, then if S be put for a + b + c we have 2 cos b. cos c cos A = sin b. sin c 4 rado, sin S. sin S - a. sin S - b. sin S Formulæ 2, 5, and 6, may be thuš expressed logarithmically; the first only of each class is put down, as the others, being perfectly analogous, will present no difficulty, log sin A = 20 + log 4 + log sin S + log sin S-a + log sinS-b+ log sin S-C 2 sin b + sin c А log cos 2 log sin S + log sin S - a + log cosec b + log cosecc 2 20 log sin s 20 b + log sin s MC it log cosec b + log cosec 2 PROPOSITION XX. In any spherical triangle A B C (see figure to Prop. 16.) if CD be a perpendicular drawn from the angle C to the opposite side A B, or A B BD + AD BD-AD produced, then the rectangle of tan and tan is 2 2 BC + AC BC - AC equal to the rectangle of tan and tan 2 2 Let B C = a, A C = b, B D = m, and A D=n; then (Prop. 17.) cos a : cos b : : cos m : cos n, therefore (Theo. 72. Geo.) cos a + eos 0 :: cos m + cos n : cosm But COS : COS a -COS n. a mn a + b : tan 6 mtn Hence cot :: cot And 2 because rectangles of equal altitudes are to each other as their bases, a + b a + 6 a + b 6 tan cot me tn : tan tan :: tan a cot a + 6 to Definitions Plane Trig.) hence rad? : tan 2 tan : rad? 2 mtn : tan m -n tan > and as the first and third terms of this 2 min Stan . : tan proposition are equal, the second and fourth are also equal, or tan mt n ä 6 BD + AD tan tan ; that is tan 2 2 2 2 2 BD AD BC+ AC BC - AC tan = tan tan 2 2 2 Cor. 1. The above equation, converted into an analogy, gives us BD + AD BC + AC BC - AC BD - AD tan : tan ::tan : tan 2 Cor. 2. When the perpendicular C D falls within the triangle BD + AD = A B, and when C D falls without the triangle B D – A D AB BC + AC = A B, therefore in the first case we have tan 2 2 BC - AC BD - A D :: tan : tan -; and in the second case tan 2 BD + AD B C + AC BC - AC AB : tan or by 2 2 2 A B BC+ AC BC - AC invertion and alternation tan : tan 2 2 Cor 3. By this proposition, an oblique angled spherical triangle, whose sides are given, may be divided into two right angled spherical triangles, in each of which the hypothenuse and a side will be given ; and hence the angles of the triangles may be determined without reference to the formulæ in the scholium to the last proposition. : tan CONSTRUCTION AND USE OF THE MARINER'S SCALE. The scale commonly used by mariners is two feet in length, and on one side of it are drawn several scales of equal parts, lines of natural sines, tangents, chords, &c.; and on the other side scales of the logarithmic relations of the numbers representing these lines. The line marked Run, or Rhumbs, exhibits the relative lengths of the chords to every point and quarter-point of the mariner's compass, or to every thirty-second part of the quadrant. |