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So I was having a discussion about angles with a student today and they were given a problem like:

convert 3$^{\circ}$ 15' 24" into degrees which is straightforward enough,

$3 + \frac{15}{60} + \frac{24}{3600}$

Afterward the student asked the natural question "Why do we call these minutes and seconds?"

This led us into a discussion of what a "degree" is in the first place. Basically the idea is that the Earth will travel through very close to 1 degree of its orbit in one day since there are 365 days in a year and 360 degrees in a rotation.

Once that is understood we can easily see the connection between angles and time. The question that occurred (to me not the student) after this discussion was: Wait, what about the hours? If a degree is roughly the angle the Earth travels through in a day, then shouldn't a "minute" the the angle that it travels through in a minute of time? But this is incorrect because we are seemingly missing the "hour" measurement.

I imagine that there is some historical reason for this, does anyone have any insight?

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    $\begingroup$ en.wikipedia.org/wiki/Hour_angle $\endgroup$ – Count Iblis Jun 13 '17 at 22:25
  • $\begingroup$ @CountIblis make that an answer $\endgroup$ – Zubin Mukerjee Jun 13 '17 at 22:27
  • $\begingroup$ Yes i did think to try and google the answer first and I saw that page, it does not seem to pertain to the question because it is about a coordinate system on the Earth by making various angle measurements. What I mean is when I have a very precisely measured angle, say 35.2575 degrees this is sometimes broken down into "minutes" and "seconds" of the angle so 35.2575 = 35 degrees, 15 minutes and 27 seconds. $\endgroup$ – Elliot Jun 13 '17 at 22:31
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    $\begingroup$ I would guess that this also has to do with the etymology of "seconds". Originally "minute" came from the other "minute", meaning "small", and "seconds" came from "second minute", i.e. even smaller. So these two words have a generic sense of subdivision that isn't necessarily tied to time or angles. $\endgroup$ – Erick Wong Jun 13 '17 at 22:43
  • $\begingroup$ See also english.stackexchange.com/questions/272126/… on the origin of minutes and seconds. $\endgroup$ – David K Jun 13 '17 at 22:57
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We really don't know for certain where the $360^\circ$ convention comes from. But, the best guess is that it is Babylonian.

The minutes:seconds convention is definitely Babylonian. The Babylonians used a "Sexagesimal" (base 60) numbering system.

1 minute means $\frac 1{60}$ and it could be applied to time, or distance or degrees. 1 second means $(\frac 1{60})(\frac 1{60})$ Rahul provides some more history below.

24 hours in a day (broken into two 12 hour halves) and 12 months in the year, seems to have come to us from Egypt.

Minutes were not used for time keeping until we had sufficiently accurate clocks (late middle ages). So, the minute was a unit of angular measure, long before it was a unit of time.

And seconds as a unit of time came along in the Enlightenment (1700's).

Base 10 numbers didn't make it into Europe at least until about the turn of the millennium. And base 12, base 20, etc. can still be found in some of our archaic measurements. (dozens, scores, shillings, inches, gallons, etc.)

360 has one more thing going for it. It has a lot of factors. Which if you lived in the world before the invention of the decimal point, is probably a nice characteristic.

It is quite possible the the 1 day $\approx$ 1 degree is a happy accident.

However, beyond high school trigonometry, the degrees convention is out of favor. Measurements in degrees does not play well with calculus. So all "higher math" will use radians, instead.

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    $\begingroup$ Etymological fun fact: The word "minute" comes from pars minuta prima ("first small part"), and "second" from pars minuta secunda ("second small part"). Cf. the word "minute" as an adjective. $\endgroup$ – user856 Jun 13 '17 at 22:42
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    $\begingroup$ And "third" is a word (now obsolete) for the sixtieth part of a second: english.stackexchange.com/questions/51860/… , going back to the Latin pars minuta tertia. $\endgroup$ – Michael Lugo Jun 13 '17 at 22:59
  • $\begingroup$ Very interesting, I didn't realize that "minute" literally means $\frac{1}{60}$ since it is so familiar to us as a unit of time now. Are there any other theories about where the 360 number came from? It made all too much sense to me as the approximate number of days in the year but now you have separated the ideas of time and degrees for me. $\endgroup$ – Elliot Jun 13 '17 at 23:49
  • $\begingroup$ Also I am really intrigued by your comment that "Seconds" wasn't used as a time unit until the 1700's. If you know of any books or sources on the subject..... which i guess is kind of broad but i will go with "understanding the development of time units as we understand them today" or about the early development of the babylonian's mathematics specifically related to the developmental of their base 60 system and why they might have chosen this number as opposed to another one. $\endgroup$ – Elliot Jun 14 '17 at 1:19
  • $\begingroup$ I never understood why people think that the rough correspondence between days per year and angles is a coincidence. The division of a circle into 360 degrees first shows up in Ptolemy's almagest, in which he heavily used Babylonian astronomy and mathematics. Of course Ptolemy knew that a year has roughly 365,25 days, just as today's bankers are aware of this despite the fact that a banker's year has exactly 360 days. $\endgroup$ – franz lemmermeyer Jun 14 '17 at 10:57

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