For any Liouville number $\alpha $, all of the following are transcendental numbers: ${e}^\alpha $, $\log _{e}\alpha $, $\sin \alpha $, $\cos \alpha $, $\tan \alpha $, $\sinh \alpha $, $\cosh \alpha $, $\tanh \alpha $, $\arcsin \alpha $ and the inverse functions evaluated at $\alpha $ of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if ‘Liouville number’ is replaced by ‘U-number’, where U is one of Mahler’s classes of transcendental numbers.

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.

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.

The recent lumber trade war between Canada and the United States deals with Canadian stumpage policies, Canada's log export controls, and U.S. retaliatory duty. This study determines the appropriate level of U.S. countervailing duty (CVD) by employing a vertically interrelated log–lumber model. The theoretical results show that the U.S. CVD can be greater (will be less) than the Canadian subsidy for a vertically related log–lumber market (for lumber market only). Empirical results support the theoretical findings in that the U.S. CVD for the log–lumber market (lumber market alone) is 1.55 (0.91) times the Canadian subsidy.