The Sumner LineofPositionIn the first half of the 19th century, Captain Thomas H. Sumner developed a method to construct a LineofPosition from a single Altitude measurement. He "conceived" this method more or less "forced by circumstances" in December 1837 while he was approaching the St. George's Channel between Ireland and Wales with only a deduced reckoning (DR) position and strong SSE winds setting his ship to the Irish lee coast. Heavy clouds had prevented taking any Altitudes for days. In the morning when they were already sailing the St. George Channel, the clouds cleared for a short instance and he was able to take the Altitude of the Sun. He then had to consider how to extract the most information possible of this single Altitude to improve the knowledge on his position. Celestial Navigation in the beginning of the 19th Century
Prior to the development of Celestial Navigation, sailors navigated
by "deduced" (or "dead") reckoning (DR).
In order for this method to work, the navigator needed a way to measure
both course and sailed distance.
Course was measured by a magnetic compass, which had been known in Europe
since at least the end of the 12th century.
Distance was determined by a speed and time calculation.
Speed was measured by "Heaving the Log"
while time was measured with a sand glass. The Noon Sight
The first "Celestial" methods for marine navigation enabled the
determination of Latitude by measuring the Altitude of the Sun
or the Moon at the moment it passed the local Meridian.
For the Sun this happens at Local Apparent Noon ("Noon Sight").
For this method, only the Declination of the sighted celestial
body for that day had to be known.
Another popular method, was to determine the Altitude of Polaris
during twilight and do some correction, because the position of
Polaris is not exactly on the Celestial North Pole. Longitude by Chronometer
The astronomical research performed at the Royal Observatory
in Greenwich which was boosted by the invention of the telescope,
ultimately led to increasingly accurate ephemeral data and in
1767 the first English Nautical Almanac appeared.
The accurate Ephemeris enabled a new method to determine
the mean solar Greenwich time by means of "lunar distances".
Captain Cook managed to obtain an amazingly good charting of
the Australian and New Zealand coasts using this method. The "Time Sight"
The most widely used sightreduction method in the late 18th century
and well into the 20th century was the "time sight".
And the body most frequently used for this was the Sun.
With the Altitude of the Sun the local apparent time at sea could be
calculated, which could further be translated in Longitude.
Originally Latitude was calculated in units of time, which gave the
"time sight" its name. LHA = acos([sin(Hc)  sin(Lat)*sin(Dec)] / [cos(Lat)*cos(Dec)]) The Declination (Dec) of the Sun was taken from the Nautical Almanac. The Sun's Local Hour Angle (expressed in units of "time") is directly related to the observer's local apparent time (LaT): LaT = 12:00:00 GMT  LHA if the Sun is East of the local Meridian LaT = 12:00:00 GMT + LHA if the Sun is West of the local Meridian For calculating Longitude, the local apparent time must be translated into local mean time (LmT): LmT = LaT + EoT
The value EoT is the "EquationofTime" which is the time difference
between Mean Time and Apparent Time on the Prime Meridian.
This value varies from day to day and is listed in the Nautical Almanac. Lon [°] = (LmT  GMT) / 15 [°/hr]
In the times before chronometers became available, GMT had to be
determined by the LunarDistance method. The Sumner LineofPositionThe scheme of determining Latitude by the Noon Sight and Longitude by a "Time Sight" was simple and well established on all seagoing vessels in the beginning of the 19th century. However, it had one basic flaw: an error in the obtained Latitude would also result in an erroneous Longitude. When the Altitude was taken around Noon, the error on Longitude could be very large, because the Parallel of Latitude and the CircleofequalAltitude  obtained by measuring the Altitude  would intersect very unfavourable such that the error on Longitude can be considerably larger that the error on Latitude. If however, the Altitude is taken when the sighted object was due East or West, an error on Latitude would imply only a minor error on Longitude.
Also Captain Thomas H. Sumner was aware of these flaws when he tried
to think how to make the best of his single Altitude observation
at about 2 hours before Noon.
He was worried about his position with the strong SSE winds setting
him towards the Irish lee coast.
His course at that moment was ENE.
The position obtained with his DR Latitude and the "Time Sight",
was East of his DR position (with the "Time Sight" method he used
it could only be due East or West), which was further away from
the coast than his DR Position.
However, as a careful navigator, Captain Sumner was aware that the
obtained Position included a considerable error on Latitude
and an even larger error on Longitude.
So he tried to figure out what would be the consequences of these
inherent errors.
Therefore, he tried to reduce his measured Altitude with an
assumed Latitude 10 Miles further North (an thus closer to the leecoast)
and found a new Position that was even further East than
the previous one.
A third reduction with an assumed Latitude another 10 Miles further
North, yielded a Longitude again further East than the two previous ones. The Constructing of a Sumner LineofPosition
In the beginning of the 19th century Natanial Bowditch developed
and published some simplified calculation schemes for the "Time Sight".
A combination of logarithmic arithmetic and trigonometric functions
was used to to avoid multiplication and division in the required
calculations and to minimize all together the number of
required arithmetic operations.
Besides the classical tables for the logarithmic and trigonometric
functions, Bowditch also made use of special precalculated tables,
which he published and updated in his "American Practical Navigator". cos(LHA) = [sin(Hc)  sin(Lat)*sin(Dec)] / [cos(Lat)*cos(Dec)] 1  cos(LHA) = 1  [sin(Hc)  sin(Lat)*sin(Dec)] / [cos(Lat)*cos(Dec)] = [cos(Lat)*cos(Dec) + sin(Lat)*sin(Dec)  sin(Hc)] / [cos(Lat)*cos(Dec)] = [cos(Lat)*cos(Dec) + sin(Lat)*sin(Dec)  sin(Hc)] / [cos(Lat)*cos(Dec)] = [cos(LatDec)  sin(Hc)] / [cos(Lat)*cos(Dec)] log(1cos(LHA)) = log(cos(LatDec)  sin(Hc)) + log(1/cos(Lat)) + log(1/cos(Dec)) = log(cos(LatDec)  sin(Hc)) + log(sec(Lat)) + log(sec(Dec))
The Sine and Cosine functions were available as tables, but in order to be
able to do the calculations with integer number values, the tabulated values
were log(1cos(LHA)) = log[100000*cos(LatDec)  100000*sin(Hc)] + log(sec(Lat)) + log(sec(Dec))  5 The righthand side of this calculation was performed according to the following scheme (move the mouse over the picture to see the details of the calculation steps):
The LHA expressed as time was then obtained from a precomputed table
implementing the function: For constructing a Sumner Line, two integral Latitude values  one above and one below the DR latitude  were chosen. This avoided the complexity of interpolating between degrees for the Declination values from the Nautical Almanac. For each of the chosen Latitudes the "Time Sight" reduction was performed, eventually yielding two points of the Sumner's LineofPosition. Example
In the following, the Sumner Line is constructed from the data
Captain Sumner obtained with his single Altitude measurement while
sailing the St. George's Channel:

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