Schema for vertical declining dials
Vertical declining dials are sundials that indicate local apparent time. Vertical south dials are a special case: as are vertical north, vertical east and vertical west dials. The word declining means that the wall is offset from one of these 4 cardinal points. There are dials that are not vertical, and these are called reclining dials. A sundial schema uses a compass and a straight edge to firstly derive the essential angles for that latitude, then to use this to draw the hourlines on the dial plate. In modern terminology this would mean that graphical  techniques were used to derive formula_1 and formula_2 and from it formula_3. Basic calculation. There are four basic angles that are needed to construct a vertical declining dial, Waugh described them such: The four basic calculation have a certain symmetry. At this point only three lines matter, the vertical, the substyle length and substyle height. A circle marked off in 15° angles is needed (circular protractor). Wigham-Richardson's method. Before the protractor became ubiquitous, compasses and the Scale of Chords were used for laying out an angle. This method originally used them. At this point only three lines  matter, the vertical, the substyle length and substyle height. A circle marked off in 15° angles is needed (circular protractor). Use of Dialing rulers. Foster Serles Dialling Scales (1638) Foster is credited for producing a set of scales to assist in the laying out of the hour line on a dial. To use them SH and SD must already be known. The scales are placed on the SD line, and lines are drawn using the calculated SH value rather than the actual latitude. Zarbula method. Zarbula is credited with the design of over a hundred sundials in Hautes Alpes  and Piedmont. This region straddles the 45 ° parallel and as such his dials are a special case. He worked directly on the wall, and didn't require to know the latitude or the declination of the dial, these we found by observation. His dials were examples of frescos, and all gave five-minute accuracy. Laying out the dial. All the rest of the dial was laid out using a 45° square, with a 15° measure at the end.