The rings. Both rings are the same disc: sixty positions, one every 6°, with the last digits of the Fibonacci numbers at the marks.
The setting. The first ring stays: it is the unit circle. Slide the second along the axis until the rings cross on its mark for the angle’s position, on the clockwise side of the axis, and on its mirror. Under the same crossings the first ring shows 30 minus and 30 plus the step. No protractor or outside tool is needed, because the angle is simply the position, and the ring’s own marks both set it and check it.
The readings. Orange line, centre to centre, ÷ 2 = cosine. Blue line, crossing to crossing, ÷ 2 = sine. Blue ÷ orange = tangent.
The signs. Second ring to the right of the first: cosine positive. To the left: cosine negative. The angle’s position (the thin ray) above the line: sine positive. Below: sine negative. This matches the unit circle.
The tangent line. An upright stands at the first ring’s mark 30, square to the axis. Extend the line from the first centre through the crossing until it meets the upright. The violet length from the foot to that point is the tangent times the radius. It must agree with blue ÷ orange.
The checks. 1: the two crossing digits add to ten (even position) or are equal (odd position). 2: (cos line ÷ 2)² + (sin line ÷ 2)² = 1. 3: the position multiplied back gives the angle.
Inverse. Set the cos line or the sin line instead of the angle and the crossings find the angle (arccos, arcsin).
The menus. 0 mark: turns the whole wheel; the positions always run clockwise. Axis: turns the line of centres to run through another position; the crossings then land on that position plus and minus the step.
Neal’s Net. Solid green chords join the two digits that add to ten; dashed green chords join the two equal digits. Every chord is level or upright. On the axis through 0 and 30, or through 15 and 45, the four readings n, 30 − n, 30 + n and 60 − n are the corners of one rectangle: its four sides light and the digits at its corners are lit. On any other axis no chord joins the readings, so the digits certify nothing there.
Finer angles. Any decimal angle sits on a rung of the ladder (level 1: 3,600 marks of 0.1°; level 2: 216,000 marks of 6 arc-seconds; and so on). The digits for a finer angle are read on that finer ring, which carries the same sixty digits. On the axis through 0 and 30 the same rectangle of the net rides on the setting, so its level and upright lines stay lit across the four readings as the ring slides between the 6° marks, not only on them.