This method uses the properties of chords to establish distance ${\displaystyle \sin {\frac {\theta }{2}}}$ in the top quadrant, and then transfers this distance onto the latitude radius in the bottom quadrant so that ${\displaystyle \sin \phi \sin \theta }$ is established. Again, a transfer of this measure to the chords in the top quadrant. The final lines establish the formula ${\displaystyle \tan \kappa =}$ ${\displaystyle \sin \theta \sin \phi \over \cos \theta }$ = ${\displaystyle \tan \theta \sin \phi }$

It was used in the Encyclopedia Britannica First Edition 1771- Sixth Edition 1823

• 1: The Gnomon is drawn first against the North South line. In doing so, a diameter at φ degrees to the vertical is drawn- its reflection is needed.
• 2: The circumference is marked off at 15° intervals in the top quadrants. Chords parallel to the horizontal are drawn (the length of these chords will be 2 sin (Θ/2).
• 3: The measurement of each chord is transfered to form a scale along the lower radiuses. When joined these points form a series of parallel lines that are sin θ. sin φ in length.
• 4: These measurements are transferred up to the chord.
• 5: The final hour lines are drawn from the origins through these crossing points.