# Counterclockwise vs. clockwise

It is common for mathematicians to use counterclockwise (ccw) as positive, and clockwise (cw) as negative. For example, trigonometric functions increase from $0^\circ$ along the positive $x$-axis (East) to $90^\circ$ along the positive $y$-axis (North), sweeping out the first (NE) quadrant of the Cartesian coordinate system. The right-hand-rule rules such coordinate systems.

On the other hand, non-mathematicians are exposed to cw advancement as increasing time in all(?) analog clocks, and so for many of them, cw is the natural sense of positive angular advance. I myself, as a mathematician, view the annual calendar as advancing ccw—summer @$(0,-1)$ South, fall @$(1,0)$ East, Christmas @$(0,1)$ North—but I learned when making a presentation to random people (faculty) that many (most?) view the annual calendar as advancing cw.

My question is:

Q1. Where did the mathematician's ccw convention originate, and where did the clockwise convention originate, and why are they opposite one another?

And secondarily,

Q2. Are there data on how we mentally view the annual calendar advancing—linearly: L/R, R/L, circularly: cw, ccw?

Likely the answer to Q2 depends on nationality/culture.

(Although I've wondered this a long time, this posting was triggered by the MESE posting, "Why do we conventionally treat trig functions as going anti-clockwise from the right?.")

• The origin of the clockwise convention most likely derives from sundials, which by and large were the only practical and reliable clock available until the Middle Ages. Oct 31, 2015 at 7:37
• Related. Oct 31, 2015 at 17:38
• The old Townhall in Prague has a famous clock moving... anticlockwise origami.watch/prague Mar 24, 2022 at 21:32

I suppose that all these notions and terminology come from the early astronomy. It so happened that in the ancient times astronomy was mostly practiced in the Northern hemisphere. If you observe from Northern hemisphere, the sky rotates about you clockwise (East to West). Sun in particular. This determines the clockwise direction, as the time was always measured by the Sun.

However interesting astronomy begins when you start to appreciate and describe the slower motions than the daily rotation: the progress of the planets, Sun and Moon with respect to the "fixed" stars. As all they move generally in the same direction (opposite to the direction of the daily rotation), this direction was traditionally called "positive" or "forward" in astronomy. (Planets sometimes "retrograde" but they retrograde with respect to this positive, natural direction).

So positive direction is counter-clockwise.

As mathematics was always closely connected to astronomy (well, until 20s century), it was natural for mathematicians to take the same direction of rotation as positive. In particular trigonometry was invented and developed for the needs of astronomy. Choosing the other direction in trigonometry as positive would be extremely inconvenient for the main purpose of it.

EDIT All pictures illustrating Ptolemy are made with Sun, Moon and planets moving counterclockwise (as they actually move with respect to the fixed stars, as seen from the Northern hemisphere). Of course I understand that these are not original pictures, but it is hard to imagine that the original pictures looked differently. So for the inventors of trigonometry there was probably no choice but to count argument counterclockwise.

• "the progress of the planets, ... in the same direction (opposite to the direction of the daily rotation), this direction was traditionally called "positive" or "forward"." This is a beautiful hypothesis! I believe it in absence of the detailed knowledge to support it definitively. Oct 30, 2015 at 21:42

This isn't a proper answer, but just a speculation: if I think about how the polar angle is defined in polar coordinates, it only makes sense that it's increasing in the counter clockwise direction: you want to start on the positive half of the $x$ axis, and then increase the angle staying in the $x>0, y>0$ quarter for the first $\pi/2$ radians. This of course was true before the formalization of polar coordinates, for instance in the geometric definitions of the trigonometric functions on the unitary circle: you want the angle to be defined that way so that $\sin\theta$ is positive for $0<\theta<\frac{\pi}{2}$. So my instinctive answer would be: because western writing is done left to right, so the axis of our graphs are oriented that way as we see the right direction as positive, thus we define angles a certain way.