15

Where did the angle convention (in mathematics) come from?

One would imagine that a clockwise direction would be more 'natural' (given sundials & the like, also a magnetic compass dial).

Also, given time and direction conventions, one would imagine that the zero degree line would be vertical.

There are two parts to this question: (1) Why do we measure angles anticlockwise? (2) Why do we take the zero degree line to be along the $x$-axis.

(This was inspired by https://matheducators.stackexchange.com/questions/9874/why-do-we-conventionally-treat-trig-functions-as-going-anti-clockwise-from-the-r.)

Community
  • 1
copper.hat
  • 172,524
  • 3
    I don't know the history but now I'm curious too - good question! Before Descartes, presumably there was already a standard depiction of the number system as a horizontal line, with direction of increase being to the right as that's the direction of writing in Europe. (In versions of software localized for Semitic languages, the direction of increase is R->L, and e.g. the order of "OK Cancel" buttons gets reversed.) Perhaps Descartes built upon the existing 1D portrayal and used the next available dimension. Whether here or in Arabia, the vertical direction of increase is up, not down. (cont) – BrianO Oct 30 '15 at 03:21
  • @BrianO: The cultural aspect never occurred to me! – copper.hat Oct 30 '15 at 03:25
  • 1
    Given that orientation, and sin, cos and tan being what they are, it seems the direction of increase for angles has to go from $\theta=0$ at $(1,0)$ to $0 < \theta < \frac {\pi} 2$ in the first quadrant -- hence, counterclockwise. // This all sounds plausible, anyway to me(!)... but take with a boulder of salt: I've never read Descartes on analytic geometry, nor checked out manuscripts of Nicolas of Cusa, etc. – BrianO Oct 30 '15 at 03:25
  • @BrianO sounds very plausible. – Ben Grossmann Oct 30 '15 at 12:08
  • 1
    This question has essentially been asked and answered on [hsm.se] over here. The post there was inspired by the same ME.SE post. I suggest to merge this post with the HSM post. – Danu Oct 30 '15 at 21:08
  • @Danu: Thanks for the link; I did a fairly extensive search beforehand but HSM never appeared. The linked question has some plausible reasoning, but I am looking for something more definitive, if such a thing exists. – copper.hat Oct 30 '15 at 21:33
  • @Danu. I don't think there is a way to merge questions across sites. – Jyrki Lahtonen Oct 31 '15 at 09:27
  • @JyrkiLahtonen That could be true: Perhaps, then, a migration? What do you think, @copper.hat? – Danu Oct 31 '15 at 17:20
  • @Danu: I have no objection, albeit do not see a problem with it remaining here either. – copper.hat Oct 31 '15 at 17:36
  • Why the downvote? – copper.hat Dec 09 '18 at 17:07
  • Why another downvote? – copper.hat Aug 19 '23 at 16:08

3 Answers3

5

This is by no means answering the question as it is. However, It is just to give a historical piece (taken from "The history of Stokes' Theorem" written by Katz) that might come handy when thinking of the first part of the question (CW vs. CCW or ACW).

enter image description here

As it can be seen, Green's theorem as introduced by Cauchy is undecided about the orientation of the curve.

I repeat that I didn't attempt to answer the question as asked. I just try to extend the question a bit hoping that it helps us to find the answer (if any) to the original question.

Amir Asghari
  • 888
1

We measure the angles with the $x$-axis. So one of the arm of the angle is $x$-axis and the other arm is also on $x$-axis if the angle is zero. This is why we take zero degree line along the $x$-axis.

In rectangular coordinate system we have four quadrants. Now we move the second arm which is fixed to the origin. When we move the second arm in counter clockwise direction then we have pattern of going from quadrant I-II-III-IV.

In history mathematicians worked on height and distance problems in which they were required to find height of a tower(say) without directly measuring it. In those problems they required to find the angle between the line of sight(seeing the highest point on the tower) and the surface of Earth. This means we go counter clockwise for measuring the angle.

Today we have coordinate transformation so we can always define everything in new coordinate system according to are convenience.

In my class text book of mathematics while doing trigonometry I found this problem and I searched for the solution. Above thing is what I got while searching for the solution.

Singh
  • 2,108
  • 3
    I would suspect that the sun and astronomy would have had a greater influence on convention. In any case, the above doesn't show why you would measure angles in an anticlockwise manner. – copper.hat Oct 30 '15 at 03:46
1

It is perhaps "natural" to adopt these two conventions:

  1. The zero angle "should" correspond to the positive $x$-axis.
  2. Small but positive angles "should" be in the quadrant where $x$ and $y$ are positive.

Given that we also adopt the convention that the $x$-axis points rightwards and the $y$-axis points upwards, the anticlockwise convention then also necessarily follows.