File:Bedos de Celles method (1790)-(4).svg

Summary

The Dom Francois Bedos de Celles method (1790) otherwise known as the Waugh method (1973)

• Take a large sheet of paper.
• Starting at the bottom, draw a line across, and a vertical one up the centre. Where they cross is important call it O.
• Choose the size of the dial, and draw a line across. Where it crosses the centre line is important call it F
• You know your latitude. Draw a line upwards from O at this angle, this is a construction line.
• Using a square, (drop a line) draw a line from F through the construction line so they cross at right angles. Call that point E, it is important. To be precise it is the line FE that is important as it is length Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): <semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>sin</mi><mo>⁡<!-- &#8289; --></mo><mi>ϕ<!-- &#981; --></mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \sin \phi }</annotation></semantics> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc867c18657f81cf14e9381bb8603ea969d63566" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:4.67ex; height:2.509ex;" alt="{\displaystyle \sin \phi }">.
• Using compasses, or dividers the length FE is copied upwards in the centre line from F. The new point is called G and yes it is important- the construction lines and FE can now be erased.
• From G a series of lines, 15° apart are drawn, long enough so they cross the line through F. These mark the hour points 9, 10, 11, 12, 1, 2, 3 if you take just 3 and represent the points Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): <semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>tan</mi><mo>⁡<!-- &#8289; --></mo><mi>h</mi><mi>sin</mi><mo>⁡<!-- &#8289; --></mo><mi>ϕ<!-- &#981; --></mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \tan h\sin \phi }</annotation></semantics> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c36afcd9804bad917f765b2a27fe8ca8a6c9e2ab" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:10.185ex; height:2.509ex;" alt="{\displaystyle \tan h\sin \phi }">.
• The centre of the dial is at the bottom, point O. The line drawn from each of these hour point to O will be the hour line on the finished dial.
• If the paper is large enough, the method above works from 7 until 12, and 12 until 5 and the values before and after 6 are calculated through symmetry. However, there is another way of marking up 7 and 8, and 4 and 5. Call the point where 3 crosses the line R, and a drop a line at right-angles to the base line. Call that point W. Use a construction line to join W and F. Waugh calls the crossing points with the hours lines K, L, M.
• Using compasses or dividers, add two more points to this line N and P, so that the distances MN equal ML, and MP equal MK. The missing hour lines are drawn from O through N and through P. The construction lines are erased.

Licensing

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