point. Azimuth is measured in degrees from 0° to 360°. The conventional symbol for azimuth is the letter  A or Z. Astronomical  Triangle The  solutions  of  problems  involving  the  three coordinate systems are made by means of spherical trigonometry.  A  figure  of  prime  importance  is  the spherical triangle that lies on the celestial sphere and whose  vertices  are  the  pole,  the  zenith,  and  the celestial   body   involved.   This   is   known   as   the astronomical or the PZS (pole-zenith-star)   triangle. The astronomical triangle is shown in figure 15-3. As in the case of all spherical triangles, the sides can be  expressed  as  the  angles  subtended  at  the  center  of the  sphere.  In  the  astronomical  triangle,  the  side between the pole and the zenith is the  colatitude (90°  -  0),  between  the  pole  and  the  star  is  the codeclination or polar distance (90° - 6), and between the zenith and the star is the  coaltitude,  or zenith distance  (90°  -  h).  The  angle  at  the  zenith  is  the azimuth angle (A) of the body. The angle at the pole is the hour angle (t). The angle at the star is known as the parallactic  angle and is little used in computa- tions.  If  the  three  elements  of  the  astronomic  triangle are known, the others can be found by means of spheri- cal trigonometry. The fundamental equation is the law of cosines. cos  a  = cos b cos c  + sin  b sin c cos  a, in which a, b, and c are  the  sides  of  a  spherical  triangle, and A is the angle opposite side  a (B and C are the angles   opposite   sides   b and   c,   respectively).   All formulas required for the solution of the astronomic triangle may be derived from this law of cosines. Astronomical  Tables Used  by  Surveyors The declination and Greenwich hour angle of the sun, moon, and selected planets are given for every even hour of GMT for everyday in the year in the daily pages of the Nautical Almanac and the Ephemeris of the Sun, Polaris, and Other Selected Stars. These publications are prepared by the U.S. Naval Observatory and are available  for  sale  at  the  U.S.  Government  Printing Office, Washington, D.C. Condensed tables of data are also available from various manufacturers of surveying equipment. Suppose that you want to determine the GHA and declination of the sun for an observation made at zone time  l&23m18’   on   17   May   1986   in   longitude 79”37’12”W.  The  ZD  is  +5;  therefore,  GMT  Of  the observation  was  15~23m18’. Table 15-1 shows the relevant daily page of the 1986 Nautical Almanac. You can see that for  15~~O&  on 17 May the GHA listed for the sun is 45054.8’.  For the extra 23”’18S   you  turn  to  a  table   of   increments   and corrections in the back of the book. Table 15-2 shows the relevant page of the table. Under  23m and beside 18^s in the Sun column you find an increment of  5049.5’. The GHA of the sun at the time of observation, then, was 45°54.8’  + 5049.5’, or 51044.3’. On  the  daily  page  of  17  May  (table  15-1),  the Nautical Almanac gives a sun declination for 15%0’’’00’ GMT on  N 19°21.3’.  At  the  foot  of  the  column,  you see a small d and the figure 0.6. In the increments and corrections table (table 15-2), you see a column of v or d corrections  for  declination.  You  go  down  this column to the figure 0.6, where you find that the  d correction in this case is 0.2'. Whether you add this correction or subtract it depends upon whether the declination of the sun is increasing or decreasing with time. A glance at the daily page shows that in this case, it is increasing; therefore, the declination of the sun at the  time  of  observation  was   N 19°21.3’ + 0.2’,  or N  19021.5’. On an opposing daily page of the Nautical Almanac (table  5-3),  the  declinations  of  a  select  list  of  57 prominent stars are given. Instead of the GHAs of these stars, however, the  sidereal hour angle  (SHA)  of  each star is given. The sidereal angle of a star is its arc distance westward from the vernal equinox or first point of Aries. The GHA of a star is its arc distance westward from the hour circle of the first point of Aries. For GHA of a star, you first determine GHA of the first point of Aries in the same manner described for the sun. You can see Aries listed in tables 15-2 and 15-3. You then add this to the SHA of the star, as given in the daily  page  of  the  Nautical  Almanac  (table  15-3).  This can be stated as follows: GHA star= GHA Aries + SHA star. If the result is greater than 360°, you subtract 360° from it. For declination of a star, you use the declination listed on the daily page; this is good for a star at anytime of the day. NOTE: The  SHA  and  GHA  of  the  vernal  equinox are factors used in star observations; however, neither is applicable in observation of navigational plaints. 15-7