An angle does not have its own size. It is an opening, and an opening cannot be measured without someone drawing an arc — it is in the choice of this arc that the two ways of reading the same angle appear. One was invented by decree, four thousand years ago, and the other the circle gives for free. Knowing which is which changes everything that comes after.
Do: open the angle with the opening and then move radius of the arc — the angle does not change, the arc changes. Turn on three arcs at the same time to see it all at once, and the division into 360 to see where the degree came from.
| what is measured | what is read | |
|---|---|---|
| the arc | a length — changes with the radius | |
| the radius | what was used as a ruler | |
| radians | arc ÷ radius — does not change with the radius | |
| degrees | the slice of a turn, if the turn is cut into 360 |
1. Move the radius. The arc stretches and shrinks — it is a length, and depends on where you drew it. But the line of radians does not move. Turn on three arcs at the same time: three different lengths, one ratio only. This is the radian: not a unit that someone chose, but the arc measured with the radius itself.
2. Open until the full turn. The arc becomes the circumference, and the reading in radians stops at 6.28318… — which is 2π. And then π is no longer a number that appears out of nowhere: it is the circumference divided by the diameter, or how much the circle measures when measured with its own radius. It falls out of the calculation, not from convention.
3. Turn on the division into 360. Nothing in the drawing asks for 360. It is a partition inherited from Babylonian astronomy, chosen because 360 divides well — by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18… The degree is convenient, and that's all. It carries no information about the circle. Half a turn are 180 degrees because someone divided it this way; half a turn are π radians because half the circumference divided by the radius gives π.
Not a preference of notation. Only in radians does sin′ = cos hold — the derivative
of sine is cosine, clean. In degrees, the same derivative comes with a π/180 attached,
and it reappears in every calculation from there on. The choice of unit makes the entire
calculation clean or dirty.
This is seen in taxa.html: the red sine curve is exactly the cosine
— and only exactly because the input is in radians.
Local, self-contained instrument: no network, no external library. Slice of the same family
of par-vira-ponto, taxa and o-circulo.