NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Robin Stuart
Date: 2025 Jul 12, 05:37 -0700
Last year I posted a preprint on the use of the Plane Scale https://navlist.net/Plane-Scale-Stereographic-Projection-Early-Navigation-Stuart-may-2024-g55907 This device allows problems in spherical trigonometry to be solved using graphical constructions that exploit the properties of stereographic projection. I found the techniques for determining great and small circle distances to be truly remarkable. These are described the Appendix B.5 and B.8 of the paper. I have not found mention of them except in
Bion, M. (1723). The Construction and Principal Uses of Mathematical Instruments. London: John Senex & William Taylor. Available at: https://archive.org/details/constructionprin00bion/page/n3/mode/2up
and its nicely reproduced web version
Rougeux, N. (2022). https://www.c82.net/math-instruments/book1-additions-chapter6
I would like to have had a simple geometric argument to explain why these techniques work but perhaps that's expecting too much as other key properties of stereographic projection (circles on the sphere map to circles on the plane; conformal map) are not necessarily intuitively obvious.
I was able to prove analytically that the methods work as described and because this seems to be practically unknown I have written it up for myself and anyone else who might be interested. I'm happy to answer questions if anyone troubles to read it,
Robin Stuart
