Great Circle Definitions
Great Circle - the largest circle that can be placed around a sphere. On earth the primary great circles are the equator and the meridians infinite in number (1 minute of arc being equal to 1 nautical mile). On the celestial sphere there are celestial meridians and the equator, hour circles, and vertical circles that can be measured in arc only.
Declination - the celestial equivalent of latitude on earth.
Vernal Equinox - that point on the celestial equator (equinoctial) that the sun on its south to north transit crosses about March 21 in the establishment of the hour circle of Aries.
Equator - the great circle of the earth that lies midway between the extremities of the earth's axis. It is the baseline of latitude for north and south measurement.
Celestial Equator or Equinoctial - a projection from the earth's equator to the celestial sphere.
Greenwich Meridian - the meridian on earth that passes through the Greenwich Royal Observatory. It is the zero line of east / west direction.
Greenwich Celestial Meridian - a projection to the celestial sphere of the Greenwich meridian. It is the zero line of establishment of westerly direction.
Local Meridian - the meridian the observer on earth is on.
Local Celestial Meridian - a projection of the local meridian to the celestial sphere.
Hour Circle - a great circle on the celestial sphere that passes through the poles and a celestial body. It is in constant movement to circle the earth approximately once a day.
Vertical Circles - a great circle on the celestial sphere that passes through two established points (zenith and nadir) and a celestial body.
Hour Angles - angles measured from reference points of the Greenwich meridian, local meridian, and the hour circle of Aries in a westerly direction through 360° to another reference.
Meridian angle - the angle measured from the local meridian in an easterly or westerly direction through 180°.
Great Circle Sailing
The great circle distance between any two points on the assumed spherical surface of the earth and the initial great circle course angle can be found by relating the problems to the solution of the celestial triangle. By entering the tables with latitude of departure as latitude, latitude of destination as declination, and difference of longitude as LHA, the tabular altitude and azimuth angle can be extracted and converted to distance and course. The tabular azimuth angle becomes the initial great circle course angle, N or S for the latitude of departure, and E or W depending on the destination being east or west of point of departure.
If all the entering arguments are integral degrees, the altitude and azimuth angle are obtained directly from the tables without interpolation. If the latitude of destination is nonintegral, interpolation for the additional minutes of latitude is done as in correcting altitude for any declination increment, if either the latitude of departure or difference of longitude, or both, are nonintegral, the additional interpolation is done graphically. Since the latitude of destination becomes the declination entry, and all declinations appear on every page the great circle solution can always be extracted from the volume which covers the latitude of departure.
Great Circle solutions are in one of the four following cases:
Case 1 - Latitudes of departure and destination of same name and initial great circle distance less than 90°. Enter the tables with latitude of departure as latitude argument (Same Name), latitude of destination as declination argument, and difference of longitude as local hour angle argument. If the respondents as found on a right-hand page do not lie below the C-S Line,
Case III is applicable. Extract the tabular altitude which subtracted from 90° is the desired great circle distance. The tabular azimuth angle is the initial great circle course angle.
Case II - Latitudes of departure and destination of contrary name and great circle distance less than 90°. Enter the tables with latitude of departure as latitude argument (Contrary Name) and latitude of destination as declination argument, and with the difference of longitude as local hour angle argument. If the respondents do not lie above the C-S Line on the right-hand page, Case IV is applicable. Extract the tabular altitude which subtracted from 90° is the desired great-circle distance. The tabular azimuth angle is the initial great circle course angle.
Case III - Latitudes of departure and destination of same name and great circle distance greater than 90°. Enter the tables with latitude of departure as latitude argument (Same Name), latitude of destination as declination argument, and difference of longitude as local hour angle argument. If the respondents as found on a right-hand page do not lie above the C-S Line, Case I is applicable. Extract the tabular altitude which added to 90° gives the desired great-circle distance. The initial great-circle course angle is 180° minus the tabular azimuth angle.
Case IV - Latitudes of departure and destination of contrary name and great circle distance greater than 90°. Enter the tables with latitude of departure as latitude argument (Contrary Name), latitude of destination as declination argument, and difference of longitude as local hour angle argument. If the respondents as found on a right-hand page do not lie below the C-S Line, Case II is applicable. If the DLo is in in excess of 90°, the respondents are found on the facing left-hand page. Extract the tabular altitude which added to 90° gives the desired great circle distance. The initial great circle course angle is 180° minus the tabular azimuth angle.
Points along a Great Circle
If the latitude of the point of departure and the initial great circle course angle are integral degrees, points along the great circle are found by entering the tables with the latitude of departure as the latitude argument (Same Name), the initial great circle course angle as the LHA rgument, and 90° minus distance to a point on the great circle as the declination argument. The latitude the point on the great circle and the difference of longitude between that point and the point of departure are the tabular altitude and azimuth angle respondents.
Spherical Triangle Solutions
Of the six parts of the spherical astronomical triangle, these tables utilize three as entering arguments and tabulate two as respondents. The only remaining part of the triangle is the position angle, which is the angle between a body's hour circle and its vertical circle. The position angle can be found through simple interchange of arguments, effecting a complete solution.
The instructions are:
(a) When latitude and declination are of same name, enter the tables with the appropriate local hour angle, with the declination as latitude argument of the same name and the latitude as declination argument, and extract the tabular azimuth angle as the parallactic angle.
(b) When latitude and declination are of contrary name, enter the tables with the appropriate local hour angle and with the declination as latitude argument of contrary name and the latitude as declination argument, the tabular azimuth angle is then the supplement of the parallactic angle which equals 180° minus the azimuth angle). This method generally requires all the volumes of the series. An approximate value of the parallactic angle X, accurate enough for most navigational requirements, can be calculated from the formula, cos X = d / 60', where d is the difference between successive tabular altitudes for the desired latitude, local hour angle and declination.
Within the limitations of the tabular precision and interval, the tabular data of these tables include the solution of any spherical triangle, given two sides and the included angle. When using the tables for the general solution of the spherical triangle, the use of latitude, declination, and altitude in the tables instead of their corresponding parts of the astronomical triangle should be kept in mind. In general if any three parts of a spherical triangle are given, these tables can be used to find the remaining parts this will sometimes mean searching through the volumes to find, for example, the altitude in a particular latitude and a given LHA in order to find the corresponding azimuth angle and declination.
