Tower Octagon Size
The size of the tower octagon is determined by its major diagonal, which is the width of the ring space, t, folio 109:01.
The dimension of the tower octagon elements are derived from the tower octagon diagonal t using geometry, folio 109:02.
The side of the tower octagon is labeled s.
The minor diagonal of the tower octagon, labeled u, is a measure of the tower width.
The right-angle triangle formed by t, u, and s is the characteristic triangle of the octagon, with angles of 22.5° and 67.5°, folio 109:02.
This triangle shows up all over the place inside the plant octagon as lines are drawn in the geometric design. The significance of this recurrence is that this triangle facilitates the design and the calculation of dimensions.
Trigonometric relationships are used here to calculate s and u knowing the dimension of t, folio 109:02. This is a trivial operation nowadays, but trigonometry was in its infancy during the Middle Ages.
Because of the prolific presence of the characteristic triangle in this geometric design, the medieval mathematician had to figure out these trigonometric factors once, the sine and cosine of the angles 22.5º and 67.5º, and then use them as needed in the following analyses whenever the characteristic triangle is involved. The similarity property is also useful to calculate the dimension of a new triangle from a known triangle.
What is a trivial and simple calculation nowadays, taking just a few keystrokes on a calculator, was a tedious and more complex calculation process in the Middle Ages, especially considering that the mathematical counting in the Middle Ages was the duodecimal system, as reported by historians.