Dom Bédos de Celles

The 26 images below are taken from this work, kept in the St.Bernard's abbey library in Bornem.

• 
  Title page
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 
• 

The Dom Francois Bedos de Celles method (1790) otherwise known as the Waugh method (1973), enables a dial to be constructed on a narrower piece of paper or velum, than using Dürers (1525) method- though it is essentially the same for the hourlines 9 to 3. It relies on a theorem proved on 1682 by P. de la Hire.[1]

• Starting at the bottom of the paper, a line is drawn across, and a vertical one up the centre. Where they cross is the point O.
• Choose the size of the dial, and draw a line across. Where it crosses the centre line is F
• Using the selected latitude. a line is drawn from O at this angle, this is a construction line.
• Using a square, a line is dropped from F through the construction line so they cross at right angles. That point E is important. To be precise it is the line FE that is important as it is length ${\displaystyle \sin \phi }$.
• 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 may 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 and represent the points ${\displaystyle \tan h\sin \phi }$.
• 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.[2]
• However, Dom Bédos de Celles had a unique 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, in his book, 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 = ML, and MP = MK. The missing hour lines are drawn from O through N and through P. The construction lines are erased.[2] The P. de la Hire theorem established that if a line is parallel to the 9 hourline, that is line WF then all the hourlines will be symmetrical around the hourline 6 hours later (i.e. 3). [1]

• 
  Setting out a dial for 52°N. The three initial lines.
• 
  Marking the latitude, laying out length ${\displaystyle \sin \phi }$, and copying to G on the vertical.
• 
  From G casting ${\displaystyle \tan h\sin \phi }$ on the horizontal.
• 
  The actual hour lines 9, 10, 11, 12, 1, 2, 3.
• 
  Construction lines removed.
• 
  Constructing the 7, 8, 4, 5 lines
• 
  Marking the 7, 8, 4, 5 lines
• 
  The completed dial plate for 52°N. Bedos de Celles (1790)

The method became well-known when it was adopted by Waugh, as the construction method to be used for horizontal dials by Albert Waugh, in his 1973 book Sundials: their theory and construction.[2]