SPHERICAL TRIGONOMETRY. T CHAPTER I. DEFINITIONS. 1. Spherical Trigonometry treats of the solution of spherical triangles. A Spherical Triangle is a portion of the surface of a sphere included between three arcs of great circles. In the present treatise those spherical triangles only are treated of, in which the sides and angles are less than 180°. 2. The angle, formed by two sides of a spherical triangle, is the same as the angle formed by their planes. 3. Besides the usual method of denoting sides and angles by degrees, minutes, &c.; another method of denoting them is so often used in Spherical Astronomy, that it will be found convenient to explain it here. The circumference is supposed to be divided into 24 equal arcs, called hours ; each hour is divided into 60 minutes of time, each minute into 60 seconds of time, and so on. Hours, minutes, seconds, &c. of time are denoted hy h, m, s, &c. Arcs expressed in time. 4. Problem. To convert degrees, minutes, fc. into hours, minutes, &c. of time. Solution. Since we have 15' = 360° = 24" 15o = 1", and 1° = 1 = 4", and 1", and l' = 4", 1', and l" = 4. Hence ao = 4 a", a' = 4 a', a' = 4 a'; so that to convert degrees, minutes, fc. into time, multiply by 4, and change the marks 01 "respectively, into 15" = m st 5. Corollary. To convert time into degrees, minutes, foc., multiply the hours by 15 for degrees, and divide the minutes, seconds, &'c. of time by 4, changing the marks 'into 01". O The turning of degrees, minutes, &c, into time, and the reverse, may be at once performed by table XXI of the Navigator. 38" = 152 2: 32 28 32 225° 47' 38" = 15" 3m 1032 15" 3m 10% 32 Arcs expressed in time. 2. Convert 17* 19m 13° into degrees, minutes, &c. 3. Convert 12° 34' 56'' into time. Ans. 50m 19° 44, 4. Convert 99° 59' 59'' into time. Ans. 61 39m 59' 56'. 5. Convert 31 2m 12° into degrees, minutes, &c. Ans. 45° 33'. 6. Convert 11° 59m 59 into degrees, minutes, &c. Ans. 179° 59' 45'. 7. When an arc is given in time, its log., sine, &c. can be found directly from table XXVII, by means of the column headed Hour P. M., in which twice the time is given, so that the double of the angle must be found in this column. The use of the table of proportional parts for these columns is explained upon page 35 of the Navigator. When the time exceeds 6", the difference between it and 12" or 24" must be used. |