Time Does Not Run Clockwise. Here’s Proof!

Have you ever wondered why clocks go clockwise?

It is often stated that the first makers of mechanical clocks simply mimicked sundials, the first and most common time-telling device of the ages, with a circular plate for a clock face and a sweeping hand copying the motion of the shadow from a central gnomon. As the shadow on a sundial, allegedly, moves from left to forward to right during the hours of a day, clock makers followed suit and had the clock hands rotate in the direction we now call clockwise.

But is it true that the shadow of a sundial gnomon moves in this clockwise direction? To check, my father, whom I happened to be visiting for the weekend, and I decided to place a stick in the ground and observe the location of its shadow each and every hour of daylight. The following series of photographs shows the setup and the results. (We found that a table-top version of the setup with a light-blue table cloth was easier to read.)

The shadow from a vertical stick moves in a counter-clockwise direction. So then, why do clocks go clockwise? (Experiment conducted March 11, 2018.)
The setups: We marked the four compass points around a vertical stick and used bark chips to mark the location of the shadow each hour. A table-top version turned out to be easier to read.
A time-series of the hours of the day. The rotation of the stick-shadow is clearly counterclockwise.

Our experiment shows that it is patently not true that the shadow on a sundial moves in a clockwise direction! At least one medieval scholar, Paolo Uccello, got it right with his Face with Four Prophets timepiece in the Florence Cathedral dated 1443. The direction of motion on this clock face is counterclockwise.

Paolo Uccello’s clock-face. Photo from https://www.florenceinferno.com/paolo-uccello-clock-florence-cathedral/

What’s going on?

I failed to mention that I was visiting my father in Adelaide, Australia!

Adelaide has a latitude of 34 degrees 92 minutes south, which is south of the Tropic of Capricorn at 23 degrees, 26 minutes south, the southernmost locations at which the Sun ever appears directly overhead at some time of year.

Photo adapted from https://commons.wikimedia.org/wiki/File:World_map_with_tropic_of_capricorn.svg#/media/File:World_map_with_tropic_of_capricorn.svg

This means that the Sun always hangs in the sky a little to the north during the day in my home town and so objects will always cast shadows that point southward, the opposite direction to the Sun. Thus with the Sun rising in the east, the shadow on an Australian sundial starts left (west), moves downwards (south), and ends right (east) as the Sun sets in the west. That is, sundials go counterclockwise in my Australian home locale.

Why do Clocks Go Clockwise?

The answer is because of an (understandable) northern-hemisphere bias.

The Sun never passes directly overhead in the northern hemisphere either (at least north of the Tropic of Cancer at 23 degrees, 26 seconds) but instead hangs in the southern sky. It seems reasonable then to face southwards when contemplating the motion of the Sun. Doing so, one notices that the Sun rises from one’s left each day (east), moves high in the sky, and then sets to one’s right (west). This direction of turning from the left, to forward, to right became known as sunwise.

As all shadows point northward in the northern hemisphere, the shadow of vertical stick moves across the northern arc, not the southern arc, of a crude sundial.

Activity: Do you live in the northern hemisphere? If so, repeat the activity my father and I conducted and see the shadow of the vertical stick move from left (west), to forward (north), to right (east), and thus turn in the opposite direction we observed.

As sundials were the first common time piece, most used in the northern hemisphere, folk naturally came to associate this direction of turning with the passage of time and so sunwise indeed became clockwise with the invention of clocks. And as mechanical clocks were first invented in the northern hemisphere (first China and then in Europe) a northern hemispheric mind-set was inevitable!

Let Aussies Rule the Day! (And let us solve an annoyance in mathematics for you.)

Clocks go clockwise. But have you ever wondered why math does not?

In math class we measure angles in the Cartesian plane from the horizontal axis in a counterclockwise direction and we number the four quadrants of the plane in a curious counterclockwise manner too. This is counter (ha!) to the sense of rotation we use in our everyday lives.

Why the disconnect?

The preferred direction of rotation in mathematics is counterclockwise.

So much of ancient mathematics was motivated by astronomy and the study of the motion of celestial bodies. Sunwise seems like a natural choice of orientation for diagrams in astronomy: draw arcs of motion starting at the left, moving to the top of the page, and then heading to the right. This means we might well expect mathematicians to have set east to the left on a page, south upwards (the sun hangs to the south in the northern hemisphere), and west to the right on a page.

But historical matters of cartography — map making — are muddled and unclear. There are known examples of ancient Egyptian, Chinese, and Middle Eastern maps with south set upwards (though it is not at all clear if this was done for the reason I just suggested). And there are medieval European maps set with east pointing upwards and ones with north pointing upwards. Yet somehow, during the Renaissance period, European scholars uniformly agreed to follow the convention of placing north upwards on maps. (It is suggested that this is a nod to the revered Greek cartographer Ptolemy, ca. 100 A.D., who, for some reason, most often placed north upwards on his maps.) We also naturally came to associate “north” with “up.”

Given this practice of placing north upwards on diagrams — and hence east to the right, west to the left — mathematics followed suit, but with the consequence of now having to imagine the Sun rising from the right and setting to the left in their diagrams. All angles of elevation are now measured from the positive horizontal axis and the motion of the Sun is through the four quadrants of the plane with a curious counterclockwise numbering.

By insisting North be upwards, we are forced to set the motion of the Sun to be counterclockwise in our diagrams.

I shall simply point out that if Australians ever rule the day, “clockwise” would be redefined in terms of Australian sundials, all clocks would be remanufactured to run in the reverse direction, and there would be no disconnect between the directions of rotation in everyday life and in mathematics!

What got me thinking about sundials

Before visiting my father I was visiting schools in south east Asia. At one school in Vietnam I was asked to give a demonstration class on the topic of time to sixth-graders. I really had no idea what I was really being asked to cover mathematically, so I just started the class with a general conversation about matters of time.

We talked about the count of days in a year and the reason for a leap-year, and the count of months in the year. We observed that, despite their names, September, October, November, and December are not the seventh, eighth, ninth, and tenth months of the year and talked about that curiosity while googling on the topic of the months. We chatted about different number bases and why some societies chose to work in base twelve, perhaps because there is a natural way to count to twelve on one hand (see this TedEd talk ) and perhaps because twelve is a much “friendlier” number in matters of trade and measure where one often wants to work with halves, thirds, and quarters of quantities. We conjectured that this might be the origin of 12 being the count of hours in a day. We noticed that the Egyptians seemed to be first to divide the daylight hours into this number.

When folk decided to, or were able to, record time at night as well, we thought it seemed natural to use a count of 12 again, and conjectured this is what led us to say “24 hours in a day.” (More internet research is to be done!)

As “10:20” could now mean a morning hour or a night hour we discussed the two ways to distinguish between the two: the use of “a.m.” and “p.m.” and the use of 24-hour military time. Students looked up the Latin expressions for “a.m.” and “p.m.” and their meaning and we discussed the common practice of writing “12 p.m.” as being meaningless. (Please write “12 noon.”)

We ended the Google-intensive class with an exercise:

a) Tomorrow I am flying from Hanoi, Vietnam to Adelaide, Australia. I depart 1025 hours Hanoi time and arrive 2145 hours Hanoi time. How many hours long is my flight?
b) I forgot to mention I depart 1025 March 7 and arrive 2145 March 8. (It’s a very slow flight.)
c) Did I forget to say March 7, 2018 and March 8, 2020? (It’s an exceptionally slow flight!)

The students cottoned on that there is a leap-year to consider in part c).

It turns out this quirky exercise was along the lines of what I was meant to teach.

But the lesson I would really like to conduct at the school would be a year-long one! Hanoi, at a latitude of 21 degrees, 01 seconds, is just south of the Tropic of Cancer. If we built a large school-yard sundial, that is, inserted a vertical post in the ground and marked the location of its shadow at a few fixed hours each day, we’d might observe something mighty curious! I would love to see it.

Question: What might students in Darwin, north Australia, notice too?
Question: Folk living between the Tropic of Capricorn and the Tropic of Cancer, say, on the equator perhaps, observe the Sun passing directly overhead how many days of each year?

Some Mathematical Questions and Ponderables

Even with just one day of data there are some curious mathematical questions to be asked.

Look again at the photo of our 11-hours of daylight in Adelaide, Australia.

a) Is it surprising that the spacing between the bark chips marking each hour is not uniform?
b) It seems unlikely that the Sun was at its highest point during the day at 12 noon. Is it possible to estimate from this photo the time of day the Sun was at its peak?
c) According to this photo the Sun rose just north of east and set just south of west. Is that possible? Is our data an illusion? Does the Sun not actually rise directly from the east each day of the year?
d) Sun dials don’t use vertical sticks at their centers but instead angled ones. (Set at what angle?) What is the reason for this?
e) The Earth’s axis of rotation is not perpendicular to the plane the Earth sits in when revolving around the Sun. Is it possible to determine its angle of tilt to this plane using shadows?

Here are two twelve-month projects.

a) Suppose we marked the location of the tip of the shadow cast by a fixed vertical stick each day at noon for a year? Do we expect the location of the tip to change from day-to-day? If so, what path should the tip trace?
b) For those living less than 23 degrees from the equator, do conduct the sundial experiment each day for a year. When does the shadow cast by the stick change direction of rotation?

And some questions of history.

a) Have you ever noticed how the number four is presented in Roman numerals on clock faces? Although the subtraction principle is used to express nine as IX, clock-makers don’t use the principle with the number four. Why?
It is the clock-makers tradition to write four as III on clock faces.

There are many dubious theories on the internet about the origin of this tradition and I would personally like to know the definitive story behind the matter. (Also, watch out! Many artists’ depictions of clocks and non-working models of clocks often make the mistake of writing IV for four.)

b) Look at Paolo Uccello’s clock face. He didn’t use the subtraction principle at all in his Roman numerals. But what made him choose counter-clockwise for his ordering of the numerals? Florence, after all, is in the northern hemisphere!
c) What is the story of our 12-month calendar? There used to be only ten months in our calendar year with a “quiet time” over the winter period. What happened to make it a 12-month calendar, with February set as the “tag along” month whose count of days doesn’t match the others?
d) What is detailed history of cartography that led us all to agree to place north upwards on maps?

End Notes

  1. Might you indeed be interested in conducting the same sundial experiment in your part of the world? If so, please email me a photo of your results at tanton.math@gmail.com with a statement of permission to share it in another blog post. Be sure to also include your location, the latitude of your location, and the date you conducted the activity. Thanks!
  2. I have never studied astronomy. This piece, by Neil DeGrasse Tyson, is an informed essay on what one can measure from a stick placed in the ground. (Thank you Philip K. for pointing me to it.)