# 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. – David H Oct 31 '15 at 7:37
• Related. – HDE 226868 Oct 31 '15 at 17:38

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.