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Geoffrey,

Close's Textbook of Topographical and Geographical Surveying https://books.google.com/books?id=IDBRAAAAYAAJ, which we know Worsley used in the reduction of the occultations, has Hints To Travellers as its source for Raper's Method.

As editor of Hints to Travellers, in my opinion, E.A.Reeves did an appalling job. Unfortunately in turning his occultation reduction method into a recipe in words for surveyors to apply Raper had introduced an ambiguity or error that was widely misinterpreted and propagated into Close's textbook. It will produce incorrect results when the observer's latitude and stars declination are of opposite names (signs). Fortunately for an observer in the polar regions, latitude and declination will generally have the same name and the problem and did not arise for Worsley.

We give a derivation of Raper's Method in a form more palatable to a modern reader starting  on page 76 of our paper https://www.canterburymuseum.com/assets/DownloadFiles/Navigation-of-the-Shackleton-Expedition-on-the-Weddell-Sea-pack-ice.pdf. Reeves also gives a derivation in his Trigonometry: Plane and Spherical but it's hard to follow and looks more like a regurgitation from other sources. He fails to notice the the problem. Also under his editorship the description of the Raper's Method was changed from 4 figure to 5 figure logarithms. Unfortunately by that time the actual meaning of the constants, things like log₁₀(120/π), that go into the calculation seems to have been forgotten. These were extended by just adding a zero on the end! If you look at Close's occultation reduction form all the fixed constants have a zero in their 5th decimal place. The reduction method in Hints to Travellers employs prologs (Proportional Logarithms) but they offer no obvious advantages and probably complicate the calculation,

Robin Stuart