Calculating Longitude

The longitude problem.  The measurement of longitude was a problem  until well into the 18th. century.  Determining latitude was relatively easy because it could be calculated from the altitude of the sun at noon but for longitude, early navigators had to rely on dead reckoning. This was inaccurate on long voyages out of sight of land and a solution to the problem of accurately calculating longitude eluded navigators for many centuries until John Harrison invented the chronometer in the 18th. century.

In July 1714 the British Parliament  passed the Longitude Act and this established a Board of Longitude which offered a £20,000  prize for the person who could invent a means of calculating longitude. Harrison was tempted by the prize and set out to produce an accurate timepiece which could be used to compare local time to Greenwich time (which the timepiece would be set to).  He completed his chronometer in 1759 after several previous attempts and this proved to be so successful that it was used by Captain Cook in his voyages after 1772.

Explanation of the method calculating longitude using a chronometer.

Angular distance between meridians of longitude.  The Earth’s equatorial circumference is 21640.6 n.m.  Since the Equator is a great circle, 1o will subtend an arc of: 21640.6  =  60.113  ≈  60 n.m.   There are 360 meridians of Longitude so it follows that, measuring from the Earth’s centre, the angular distance between adjacent meridians at the Equator is 1o.  Since 1o subtends an arc of 60 n.m. it follows that the distance between adjacent meridians of longitude at the Equator is 60 n.m.   So the angular difference between longitude 10o West and the Greenwich Meridian is 10o; therefore, the distance between them at the Equator is 10 x 60 = 600 n.m.

Time difference between meridians of longitude. We know that the Earth revolves about its axis once every 24 hours.  In other words, the Sun completes its apparent revolution of 360o in 24 hours.  This  means that the Sun crosses each of the 360 meridians of longitude once every 24 hours. So, in 1 hour, the Sun appears to move 15o, in 4 minutes, it appears to move 1o, in 1 minute it appears to move 15′, in 4 seconds it appears to move 1′.

From this, it becomes obvious that there is a direct relationship between arc and time such that 1 minute of time equals 15 minutes of arc.

Calculating longitude by comparing time difference.  If we have two accurate clocks, one calibrated to GMT and the other calibrated to local time, then it is an easy matter to calculate our longitude from the difference between the two times.

For example, if the difference between GMT and local time is three hours, then the difference in longitude must be 3 x 15o.  If local time is ahead of GMT then the local longitude must be East of the Greenwich Meridian and if local time is behind GMT the longitude must be West.

Example:  If it is 18.00 GMT when it is 09.20 local time on the same day, then  local time must be  8 hours and 40 minutes behind GMT.  Therefore Long  = – [(8 x 15o) + (40 ÷ 60 x 15o )]     = – [120o + 10o]     =   -130o   = 130o West

A fuller explanation of this topic is given in the book ‘Astro Navigation Demystified’.

Links:  Calculating Latitude.     Astro Navigation Demystified.

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