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    The Plane Scale and Stereographic Projection in Early Navigation
    From: Robin Stuart
    Date: 2024 May 29, 06:03 -0700

    On a visit to my home town of Dunedin, New Zealand I became aware of some navigational tools in the collection of the Toitu Otago Settlers (formerly Early Settlers) Museum that belonged to Captain Andrew Elles who was master of the Philip Laing that arrived with Scottish settlers in 1848. One of these instruments is made of ivory and had been identified by an "expert" as a Vernier ruler (whatever that might be). To me, for something to be a Vernier anything it should have a Vernier scale which this did not. Further investigation identified it as a Plane Scale. While its principal use is as a protractor, three of the scales on it, the tangent, secant and semi-tangent, were particularly interesting. van Poelje (https://eeuwen.home.xs4all.nl/images/Pics/Kombuispraat/Pleinschaal/200404 Gunter Rules in Navigation V13-1.pdf) states that the semi-tangent scale was used in stereographic projections. I was skeptical of this but it turns out that there is a very rich set of geometric constructions that can be used to solve problems in spherical trigonometryand hence navigation. These can be streamlined  by using the scales. Personally I find it remarkable that great-circle distances can be found by measuring an angle in a construction on the plane. Since these methods appear to be all but unknown today, I have written them up in the attached preprint. Constructive comments are welcome. In many cases geometric arguments serve to prove the constructions. However with regard to the constructions in the Appendix B.5 and B.8, I can prove analytically that they work but don't know of a geometrical argument. If anyone does I'd be interested in hearing it.

    For anyone looking for a gentle introduction to geometric constructions using stereographic projection, I recommend this series of short lectures:

    Stereographic Projection: https://www.youtube.com/watch?v=uRMMfo74hSg
    Small circles: https://www.youtube.com/watch?v=YRhbsPtNgYo
    Pole of a great circle: https://www.youtube.com/watch?v=w3I1NrOkQ8g
    Opposite of a pole: https://www.youtube.com/watch?v=sjf2rxeUU3o
    Great circle through 2 points: https://www.youtube.com/watch?v=YmiTRbYxw6c

    Robin Stuart

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