The Plane or Plain Scale is a navigational device that dates back to the early 1600s but has long since ceased to be used in practice. It could perform the function of a protractor and be used to solve problems in plane trigonometry. In addition, coupled with a suite of remarkable geometric constructions based on stereographic projection, the Plane Scale could efficiently solve problems in spherical trigonometry and hence navigation on a sphere. The methods used seem today to be largely unknown. This paper describes the Plane Scale and how it was used.

In 1915 while the Imperial Trans-Antarctic Expedition's vessel Endurance was icebound in the Weddell Sea, lunar occultation timings were carried out in order to rate the chronometers and thereby find longitude. The original observations have been re-analysed using modern lunar ephemerides and catalogues of star positions. The times derived in this way are found to differ by an average of 20 s from those obtained during the expedition using positions given from the Nautical Almanac and introduces an additional offset of the true positions to the east of those recorded in the log.

During his 1869 expedition down the Green River and through the Grand Canyon, Major John Powell made astronomical observations using a sextant and artificial horizon to fix the locations of key points along the rivers that were only poorly known at the time. Latitude was obtained from the altitude of Polaris or meridian transits of stars or Saturn. Local mean time was determined from equal altitude observations of the Sun. The swamping of one of the expedition's small boats ruined the chronometers, meaning that they could not be used to keep Greenwich mean time and hence find longitude. As a substitute a series of lunar distance observations were undertaken. In this paper observations recorded in Powell's journal are reduced and analysed.

A round of three celestial sights yields three lines of position along which the observer's true position could lie. Due to measurement errors, the lines of position do not intersect at a point but rather form a triangle called the “cocked hat”. The probability that this encloses the observer's true position is well known to be 25% which is the average over all possible cocked hats that could arise when the sights are made. It does not apply to any specific set of sights and in that case the probabilities depend on the statistical distribution of the measurement errors. With fixed azimuths for the observed celestial bodies and assuming a normal distribution for the errors in their measured altitudes, a closed form analytic expression is derived for the probability of the observer's position falling inside the cocked hat and this is related back to the global average. Probabilities for exterior regions bounded by the lines of position are also obtained. General results are given that apply for any number of lines of position.

The location of the wreck of Shackleton's ship the Steam Yacht Endurance is recorded in the expedition log books as 68°39′30′′S 52°26′30′′W. The methods and assumptions that went into obtaining this fix are examined in detail by consulting the original log entries with a view to understanding the size of the errors and uncertainties it may be subject to and providing guidance to possible future searches. It is found that a dearth of navigational sights around the time of the sinking, the inevitable growth of uncertainties in the chronometer time since rating and other factors, introduce the possibility of errors in the position of several nautical miles in both latitude and longitude.

Mapping points on the Riemann sphere to points on the plane of complex numbers by stereographic projection has been shown to offer a number of advantages when applied to problems in navigation traditionally handled using spherical trigonometry. Here it is shown that the same approach can be used for problems involving great circles and/or rhumb lines and it results in simple, compact expressions suitable for efficient computer evaluation. Worked numerical examples are given and the values obtained are compared to standard references.

The running fix or sight-run-sight fix is a classic problem in celestial navigation. Methods employed to obtain the fix traditionally involve advancing a Line of Position (LoP) taken at an earlier time and crossing it with one obtained later. Attempts to generalise the operation of advancing an LoP when the Earth's surface is represented by a plane to the case of the sphere have resulted in proposals that contain poorly constrained approximations or are otherwise fundamentally flawed. A simple rapidly-convergent iterative procedure to obtain a running fix is described that avoids the notion of advanced LoPs and is readily applicable to both the sphere and ellipsoid.