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The following text is presented courtesy of:
	The Navigation Foundation
	P.O. Box 1126
	Rockville MD 20850
	Tel: 301-622-6448

It is from The Navigators Newsletter, issue Thrity-Five, Fall 1991. The
almanac references are for the 1991 Nautucal Almanac.

Any errors are mine.


Dan Hogan
    dhhogan@earthlink.net     dhhogan@delphi.com
************************************************

Celestial Navigation by Chronometer Time Sight
by John M. Luykx

     Prior to the development of position line navigation in the mid-
nineteenth century (i.e., after the Sumner Line-1837 and the Marq St,
Hilaire altitude intercept method-1875) the most common procedure for
finding position at sea was was a) to observe the altitude of Polaris
for latitude at both evening and morning twilight; b) to observe the
altitude of the sun for longitude by chromometer time sight both in the
morning and afternoon when the sun was on or near the prime vertical and
c) to observe the altitude of the sun at meridian passage for latitude
at noon.

     In addition to the sextant, this procedure basically required the use
of
a chronometer with an accurately established daily rate, an alamanac and
a set of trig tables. No plotting was required.

     In South latitudes where Polaris was not available for observation,
some
other star was used when available such as the constallation of the
Southern Cross for which as early as 1505 rules had already been
provided for obtaining the latitude.

The day's work consisted basically of the following steps.

1. Advancing the morning twilight observation of Polaris for latitude to
the morning observation of the sun for longitude using mid-latitude or
mercator sailing procedures in accordance with the course steered and
the distance made good by log between the two observations.

2. Advancing the morning sun observation for longitude by time sight to
the noon latitude obtained at the time of meridian passage using mid-
latitude or mercator sailing procedures in accordance with the course
steered and the distance made good good by log between the noon and
afternoon sun observations.

3. Advancing the noon latitude observation to the afternoon observation
for longitude by time sight using mid-latitude or mercator sailing
procedures in accordance with the course steered and the distance made
good by log between the noon and afternoon sun observations.

4. Advancing the afternoon observation of the sun for longitude to the
evening twlight Polaris observation for latitude using mid-latitude or
mercator sailing procedures in accordance with the course steered and
the distance made good by log between the two observations.

NOTE: The longitude by chronometer time sight is particularly dependant
on an accurate latitude if the sun is not due East or West (on the prime
vertical) at the time of the observation.

For the chronometer time sight solution, the observer's latitude(L) the
body declination(d) and the observed altitude(Ho) are required. Given
three sides of the navigation triangle, the meridian angle(t) is
then computed using this formula:
	cos t = (sin Ho - sin L * sin d) / (cos L * cos d)
When computed, the meridian angle (t) is applied to the GHA to obtain
the longitude.

When advancing the latitude and longitude using traverse and mid-
latitude sailing procedures, the following definitions and formula
apply:
	1. Traverse Sailing (see Traverse Table 3 Bowidtch Vol. II.) The
Traverse Tables lists  the East/West and North/South components of any
course from 0d to 359d and distance from 0 to 600 miles. The East/West
component in nautical miles is known as the departure (p)  and the
North/South component as as latitude difference (l) in nautical miles.
For example:
	Departure(p) = 18.1 mi.
	Latitude difference(l) = 8.5 mi.

	2. Mid-Latitude Sailing. Mid-Lat. proceduresare used to convert
departure(p) from miles to difference of longitude(DLo) in degrees and
minutes of arc for any value of the mid-latitude; i.e. the latitude
midway between the of the point of departure and the point of arrival.
Definitions:
	Lm = Mid-Latitude
	DLo = Difference of longitude in degrtees and minutes of arc
	p = Departure in nautical miles
Formulae:
	Dlo = p * sec Lm
	p = DLo * cos Lm.

SAMPLE PROBLEM
	In order to better understand what is involved in the day's work
using the chrometer time sight, the following sample problem with
solutioons is presented: On 24 June 1991, a sailing vessel is underway
from Chesapeake Bay Entrance to Bermuda on course 115 T at a speed
estimated at from 3 1/2 to 7 knots. The 08-00 GMT DR position is N 33d
28.0' W 66d 56.0'
	At GMT 08-50-6 the altitude of polaris was observed Ho 33d 51.9".
The log read: 494.2
       GHA:   44D 39.9'
      LONG:   66D 51.0'
         t:   22d 11.1'
       LHA:  337d 48.9'
        Ho:   33d 51.9'
Correction:      -24.6 (p. 276 Naut. Alm.)
       Lat:   33d 27.3'

	At 13-01-20 the suns altitude was observed near the prime vertical:
Ho 43d 51.9'. Log reading: 513.3
	1) Advance the Polaris latitude: 115d T / 19.1 miles
          l =  8.1 miles
          p = 17.3 miles
     2) Latitude = 33d 19.3'
                       -8.1'
            Lat: = 33d 19.2'
     3) Time Sight
     cos t = (sin Ho - sin L * sin d) / (cos L * cos d)
     where Ho = 43d 51.9'
            L = 33d 19.2'
            D = 23d 25.0'
        cos t = .61901
              = 51d 45.4'
     4) Longitude
          GHA: 14d 45.6'
          + t: 51d 45.4'
       = Long: W 66d 31.0'

     At meridian passage, the sun's altitude was observed:
Ho 80d 10.7',  Log reading: 523.9.
     1) Advance morning longitude: 115d / 10.6 miles
          l =  4.3 miles
          p =  9.3 miles
        DLo = 11.1'
     2) Noon Longitude
          Long: W 66d 31.0' (morning)
         - DLo:       11.1'
          Long: W 66d 19.9' (Noon)
     3) Noon Latitude
                  90d 00.0'
          - Ho    80d 10.7'
    = zenith Dist: 9d 49.3'
         + Dec:   23d 24.9' ( Nautical Almanac p. 127)
           Lat: N 33d 14.2'

     At GMT 19-52-40 the sun's altitude was observed near the prime
vertical: Ho 43d 59.5'.  Log reading: 542.9.
     1) Advance Noon Latitude: 115d / 19.0 miles
          l = 8.0 miles
          p = 17.2 miles
     2) Latitude = 33d 14.2'
                      - 8.0'
	        Lat = 33d 06.2'
     3) Time Sight
cos t = (sin Ho - sin L * sin d) / (cos L * cos d)
    where Ho =  43d 59.5'
           L =  33d 06.2'
           d =  23d 24.7'
       cos t =  .62122
           t =  51d 35.7'
     4) Longitude
          GHA:  117d 34.8'
          - t:   51d 35.7'
       = Long: W 65d 59.1'

     At GMT 00-05-13 the altitude of Polaris was observed: Ho 32d 11.8'.
The log read: 563.1.
     1) Advance afternoon longitude: 115d T / 120.2 miles
          l =  8.5 miles
          p = 18.2 miles
        DLo = 21.7'
     2) Longitude
          Long:   W 65d 59.1'
           DLo:       - 21.7'
        = Long:   W 65d 37.4'
     3) Latitude
           GHA:    273d 56.7'
        - Long:     65d 37.4'
           LHA:    208d 19.3'
            Ho:     32d 11.8'
          Corr:       + 46.2' (Nautical Almanac p. 275)
           Lat:   N 32d 58.0'

     The sample problem shows that the typical day's work in the early
days of chronometer navigation could be accomplished without plotting
lines of position on a chart. Such a procedure may find favor among
yachtsman today. It is simple and forthright and only requires an
almanac and trigonometry table.

     This procedure is ideally suited for the navigation of slow moving
vessels. It is faster and even more reliable and accurate when a
scientific mathematics calculator is substituted for trigonometry
tables.

   

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