Is there any good reason why a protractor starts from right to left, unlike a scale, which starts from left to right?

Question

While studying through the number system, i notice that positive side is from 0 to +ve infinity. The direction is left to right. However, this is opposite in case of angles. The sort of curved number system starts from 0 to 180 ( from right to left). Is their any good reason why unlike a number system, direction of measuring angles is right to left ?

PS: I actually first thought that, it's due to +x +y axis-plane coming on the right side. But then what about -x + y plane coming on the left side. The angles should go negative after 90. ie. -91, -92.... -180.

Logic probably is something different! What is it ?

Thanks V.

Answer 1

Mathematicians always measure angles in the counterclockwise direction; a clockwise angle the same size as one of $30^\circ$ is called a $-30^\circ$ angle. The protractor is consistent with this practice.

Why mathematical practice measures counterclockwise, I do not know. I was going to justify it by reference to the common layout of the cartesian plane ($x$ coordinate increasing left to right, as the language is written, and $y$ coordinate increasing bottom to top in accordance with every linguistic metaphor of increase) but on further thought I saw no reason why the zero angle couldn't have been be on the $y$-axis, with angle increasing clockwise, and I wonder why it wasn't done this way, for consistency with existing nautical practice.

As Daniel McLaury points out, this would have made the vertical axis the real axis and the horizontal axis the imaginary axis.

There is a real question of history here, which may or may not be resolvable; some of these things are just mysteries. I believe nobody knows why, when the first automobile traffic lights were constructed, they had red on the top and green on the bottom, opposite to the design of the railroad signal lights they imitated.

Answer 2

Well, there is going have to be some arbitrary choice somewhere. That said, the choice of angles is compatible with the way we generally choose to lay out the complex plane: real numbers on the horizontal axis increasing from left to right and imaginary numbers on the vertical axis.

In other words, if +1 is going to be to the right and -1 is going to be to the left, then we have to put the zero to the right, since $1 = e^{0 i}$ and $-1 = e^{\pi i}$. If we want to put $+i = e^{i \pi/2}$ pointing up, then moreover we need to measure counterclockwise. (Of course there's no particularly firm distinction between "$+i$" and "$-i$" -- that's an arbitrary convention too.)

If we want to get touchy-feely about it, I imagine that these choices are probably a consequence of the fact that our conventions were developed by people in 18th century France who read left-to-right rather than right-to-left or top-to-bottom.

Answer 3

My guess is that it has to do with the original sundials. While sundials that were placed on the floor had shadows moving clockwise, those on the wall had shadows moving counterclockwise. So I would argue that it's because mathematicians like having their sundials on the wall instead of the floor. The reason for this I cannot explain.

Answer 4

I am surely wrong here, but can there be a relation between quaternions and the anticlockwise rotation they produce when one is multiplied by another one? Sounds far-fetched though -_-"