LONGITUDE BY CHRONOMETER - This method uses an assumed latitude and calculates the longitude that a position line crosses it. The position line obtained is actually part of a small circle, as opposed to great circle, where any observer can stand and the heavenly object would have the same altitude in the sky. When plotting the small segment of this circle on a chart it is drawn as a straight line, the resulting tiny errors are too small to be significant.
The assumed latitude is usually obtained from a DR or Dead Reckoning position. This is worked out by applying the distance from the last known position either by log or by the estimated speed over time with the course steered. A sight is taken; that is the distance above the horizon of a heavenly object is measured with a sextant and the exact time noted in UTC. The sextant angle obtained is corrected for dip (the error caused by the observers height above the sea) and refraction to obtain the true altitude of the object above the horizon. This is then subtracted from 90° to obtain the angular distance from the position directly above, the zenith. This is referred to as the True Zenith Distance. The true zenith distance of the object is also the distance (in arc) on the earth's surface from the observer to where that object is overhead, the geographical position of the object.
Using a Nautical Almanac, the declination (celestial latitude), and the Greenwich hour angle (celestial longitude) are obtained of the observed object for the time of observation. Using the haversine formula the local hour angle of the position where the position circle crosses the assumed latitude is calculated. The local hour angle is the difference in longitude from the observer's position and the geographical position of the observed object. Hour angles, unlike longitude which is measured east and west from Greenwich, are always measured west from 0° through to 360°.
The local hour angle is then added to the Greenwich hour angle to obtain the longitude where the position line passes through the assumed latitude.
To draw the position line on a chart the azimuth or bearing of the heavenly object must be known. It is usually calculated but could have been observed. A line at right angles to the azimuth is drawn through the calculated position. The observer is somewhere on this line.
