Why 10?
Base-10 is a human artefact.
Computers like base-2 for obvious reasons.
There has been base-8, 12, 16, 20, 24, etc, in human history.

For the exact reason it IS a human artefact, and we have 10 fingers to count on, I guess. This is what happens to me when I'm on break... the increased oxygen density on the surface makes my giddy. Or, the lack of density at altitude has done irreparable damage.

 

Wouldn't it be cool if we could have 10-state circuits, reliably?

Base ten devices, both mechanical and electronic, were very common decades ago. Mechanical devices tended to be intrinsically base-10 while electronic devices tended to be binary-coded-decimal at their core.

There were a number of tight parallels, too. For instance, we use two's complement for representing signed integers because we can use the same hardware to do both addition and subtraction with only minor changes. But some systems use one's complement because of it's implementation advantages. The Curta mechanical calculator used nine's complement due to those same advantages.

 

Why 10?
Base-10 is a human artefact.
Computers like base-2 for obvious reasons.
There has been base-8, 12, 16, 20, 24, etc, in human history.

We still have vestiges of the Babylonian base 60 system in our time and angular measures. Base 60 was "nice" because it has a rich set of divisors, namely {2,3,4,5,6,10,12,15,20,30} with the five consecutive small ones being the most valuable in practice. Similarly, base 12 for time was nice since it has {2,3,4,6}.

It would be interesting to know how we ended up with 12 hours in a day but 60 minutes in an hour and 60 seconds in a minute. We have a good understanding of the history of these happening, but not so much the underlying rationale used. I've always thought that the length of a second was probably tied to the period of a human heart beat at rest, which tends to average around 60 beats per second. From there it seems plausible that the thinking went something like this: using base-60 upwards we have 10 heartbeats in a minute and 100 heartbeats in an hour. But the next step up we would naturally want to correspond to a day or a clear fraction of a day or perhaps a small multiple of a day and 1000 heartbeats would be two and a half days, which isn't very convenient. But twelve hours would be very close to half a day and twelve is one of those "nice" numbers with lots of divisors.

I certainly don't know that this was a driving factor, but it seems plausible to me.

Shifting gears, it seems like a bit of a toss op whether a human-driven base-8 (using the eight fingers) or base-10 (using the eight fingers and both thumbs) could have won out. Imagine how much easier a lot of things would be in the modern age of computation if we naturally thought in base-8.

 

It would be interesting to know how we ended up with 12 hours in a day

On a planet in a galaxy far, far away?

 

On a planet in a galaxy far, far away?

 

"The seasonal, temporal, or unequal hour was established in the ancient Near East as 1⁄12 of the night or daytime. Such hours varied by season, latitude, and weather. It was subsequently divided into 60 minutes, each of 60 seconds. Its East Asian equivalent was the shi, which was 1⁄12 of the apparent solar day; a similar system was eventually developed in Europe which measured its equal or equinoctial hour as 1⁄24 of such days measured from noon to noon. The minor variations of this unit were eventually smoothed by making it 1⁄24 of the mean solar day, based on the measure of the sun's transit along the celestial equator rather than along the ecliptic."

Why DID everyone in the world do this?

 

My introduction to the minicomputers was predominantly based on base-8.
The DEC PDP-8 used a 12-bit instruction set. Hence instructions were four octal digits.
DG and HP continued the practice of using octal.

 

Shifting gears, it seems like a bit of a toss op whether a human-driven base-8 (using the eight fingers) or base-10 (using the eight fingers and both thumbs) could have won out.

I find Base-5 the easiest when shifting gears in my car, though I have done it with base-2. The only Base-10 application I remember is when learning to drive a dump truck, and the tranny got stuck in reverse low-range. Definitely a driving factor.

 

The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves.

 

Why did we decide that a radian is 180 / π, so that a full rotation would add up to 360°?

We didn't decide that that was a radian.

The radian is a natural unit of angular measure. It is defined as the ratio of the length of a circular arc to the length of the radius of that arc. The radian is therefore the angle such that the arclength is equal to the radius and it just turns out that the angle of a full circle turns out to be 6.2831853... radians. At some point we defined the value of π to be the ratio of the circumference of a circle to its radius, making the angular measure of a circle expressible as 2π radians.

Completely separate from this we defined a degree to be 1/360 th of the angle of a complete circle.

These two independent measures of angular measure then result in the conversion factor being 180° / π radian.

 

We didn't decide that that was a radian.

Oops, what I meant was 1 rad (in degrees) is equal to 180 / π, or about 57.3°. "Why 360°" was the question I was trying to ask... and without me cheating and looking it up, I can see from your previous answer it must be attributed to the Babylonians also... with Base 60 counting. It would seem logical that they would have needed to come up with the idea of the "year" first.

 

Oops, what I meant was 1 rad (in degrees) is equal to 180 / π, or about 57.3°. "Why 360°" was the question I was trying to ask... and without me cheating and looking it up, I can see from your previous answer it must be attributed to the Babylonians also... with Base 60 counting. It would seem logical that they would have needed to come up with the idea of the "year" first.

I don't see how they would need the concept of a "year" in order to come up with the notion angular measure. Long before I heard about the Babylonians and their base-60 system, I had reasoned that the 360° circle had come from an estimation of the travel of the Earth about the sun in one day. It turns out that this is just one of a handful of possible explanations. Several ancient calendars did use a 360 day year, so I still think it's the most plausible explanation. But even without it, they might well still have chosen 360° for a circle but might have also chosen something else, such as 240° (based on 60° being a right angle, similar to the gradian having 100 grads in a right angle, or 400 grads in a full circle. The scale chosen would likely have taken into consideration the resolution with which practical measurements could be made and trying to make it so that 1° was useful. But I have no idea what kind of resolution they typically achieved in their efforts in various things such as geometry or architecture. I have seen that somewhere in this time frame (~2500 BC) that astronomers were noting that the sun moved about this amount each day in its path across the background constellations.

 

I don't see how they would need the concept of a "year" in order to come up with the notion angular measure. Long before I heard about the Babylonians and their base-60 system, I had reasoned that the 360° circle had come from an estimation of the travel of the Earth about the sun in one day.

You wouldn't need the concept of a year just to establish angular measure, but... in the example of a sundial, they'd obviously have needed to experience a full "day" before they could derive any meaningful concept of "time of day", and subsequently slice it up into 24 hours, 1440 minutes, 86,400 seconds, 108,000 heartbeats, or whatever. Silly comparison maybe- after all even basic animal instinct and circadian rhythms "know" how long a day lasts, and everyone knew what a day was long before the sundial anyway. But similarly, they could measure changes in the sun's position in the sky with each passing day as you said, but without first observing at least one complete journey of the Sun's annual path across the sky (which resembles a figure 8 and not even a circle), how did they divide that figure 8 path into 360 degrees to track each day's passage?

 

The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves.

That's actually how I grudgingly count... money slips out of my hands between my fingers all the time.

 

So, to a creature other than man, how do they measure the time of day or do they live in a binary world of light, no light.

Angular measurements of the sun occurred before the sun dial.

 

You wouldn't need the concept of a year just to establish angular measure, but... in the example of a sundial, they'd obviously have needed to experience a full "day" before they could derive any meaningful concept of "time of day", and subsequently slice it up into 24 hours, 1440 minutes, 86,400 seconds, 108,000 heartbeats, or whatever. Silly comparison maybe- after all even basic animal instinct and circadian rhythms "know" how long a day lasts, and everyone knew what a day was long before the sundial anyway. But similarly, they could measure changes in the sun's position in the sky with each passing day as you said, but without first observing at least one complete journey of the Sun's annual path across the sky (which resembles a figure 8 and not even a circle), how did they divide that figure 8 path into 360 degrees to track each day's passage?

Humans obviously had a long history of interacting with various natural concepts of time long before any system of quantifying it was formalized. The first one almost certain was the day. But we would have also long before realized that there was the thing we now call a year and we would have seen obvious utility in the ability to predict annual changes in weather patterns, vegetation cycles, and animal populations and migrations. Probably the first thing practical thing that we used to measure this was the lunar cycle since twelve such cycles is close enough to one year for most of those kinds of uses. All of this would have probably been well established long before humans learned to write.

I don't think that "figure 8" (more properly known as an analemma) was divided into 360°. I think the analemma came along well after the number of degrees in a circle was established and I don't think that the link between angular measure and the analemma would have been at all obvious or apparent for quite some time.

One thing that developed quite early on -- again, probably well established before we knew how to write -- were an understanding of the celestial night skies and how they could be used to tell time in terms of the seasons. It would have been noted that the constellations shift in the sky over the course of the year and repeat on an schedule that was synced to the seasons. Constellations were defined that were more or less equally spaced around the ecliptic and the full moon's presence in one of these "houses" told the time of year. Also, if you are at least some distance away from the equator such that the pole star (either actual in the case of Polaris or virtual in the case of the southern hemisphere) is well above the horizon, you would have long since realized that the stars rotated about that point every day.

A long time after that -- and long after we knew how to write and had advanced in many other areas -- we had the ability to determine local noon to within a fraction of a minute and to measure the time throughout the day ratiometrically. Imagine adjusting a water clock or candle clock or some other device such that it was calibrated between successive local noons. You can then use it to find the midway point between the two in order to find local midnight. At that point you observe what point in the celestial tapestry is directly overhead using your fixed observation post, probably aligned with the pole star. At this point, if it hadn't already been known from deductions centuries or even millennia earlier, you would realize that the map rotates a relatively constant amount every day and that makes a natural unit of angular measure. But you would also quickly note that it is not quite uniform -- bigger at some times of the year and smaller at others -- nor is it an integer number of increments in a year. So instead of defining a degree to be 1/365.24...th of a full circle, it just seems better from a practical standpoint to define it as 1/360th of a full circle.

 

So, to a creature other than man, how do they measure the time of day or do they live in a binary world of light, no light.

Angular measurements of the sun occurred before the sun dial.

Certainly. I was just using the example of a Mesopotamian sundial since WBahn brought up the subject of the Babylonian's sexagesimal method of time and angle measurements. Indeed, time and angle measurement didn't start with the sundial.

My dog can't "measure" time, but he knows when he's supposed to be fed, (and he'll bark if I'm as little as 5 minutes late), walked, taken to the groomer, and he can predict when my wife comes home from work. If she's rarely late or early he gets confused, but not at all when it's me since he's used to my irregular schedule, particularly earlier in my career. Thus he "knows" time probably a lot more due to behavior association vs instinct, yet I still don't know how he knows it has gone 5 minutes past his feeding time... can he gauge his hunger level that accurately? For less intelligent animals, it's entirely regulated by biological factors such as the release of melatonin and other hormones... which is tied to time of day also. I took up a team of wildlife biologists last August 21 during the total solar eclipse to attempt to observe animal behavior during totality. Ground teams reported things like bird songs ceasing and bats waking up, which could suggest these animals do in fact respond to light vs dark (or maybe warm/cold) binarily as you mention. But higher order mammals responded much more curiously. Here's a lucky shot of a grey wolf from that mission. This was maybe 2 minutes after totality, and how I wish we had more luck in capturing him during 1st and 2nd contact. Even with the return of light, he's just standing there dumbfounded, not even paying attention to his prey. He doesn't know what just happened, but he's smart enough to know something isn't right. One of my biologists suggested he probably just killed his prey since grey wolves usually hunt at night, and the carcass is fresh. But without being able to track him earlier we don't know... did his hunting instinct kick on at 1st contact and did he manage to find and kill it in the space of only a few minutes, and where is the rest of the pack? We did use GPS trackers, but could not detect any other signal, so this could be a lone wolf, who will hunt at any time if it has to.

Now, if day-night progression wasn't an analog phenomenon and delineated by none other than "light vs no light", well I imagine that would have a profound impact on instinct and the perception (if that's even an appropriate term) of time for anything that lives. Not to mention drastically affecting meterological things like the absorbtion rate of solar radiation and the spawning of weather systems. I'm sure Wbahn can add some details here, as he's been three steps ahead of me all along, and at this point all I can do is ride on his coat tails .

 

...celestial night skies and how they could be used to tell time in terms of the seasons...

Would a 20th century BC Norte Chican of the Andes, if he packed his suitcase and moved to Babylonia, have to get a new watch when he got there?

 

Would a 20th century BC Norte Chican of the Andes, if he packed his suitcase and moved to Babylonia, have to get a new watch when he got there?

Yep. Completely different celestial map. But if he understood what his prior map was and how it worked, he could make his new map in one night (though the actual data collection would probably require a lot longer than that due to the limit on how much of the map he could complete in one night). It might also take several nights to identify the location of the pole star, but this could possibly be done in a single night.

Once the pole star location is established, it's a pretty simple matter of dividing up the sky from down to the horizon and then sectioning it (say into twelves). Next you identify a constellation in each section that you believe you can readily identify again. In order to use constellations that are behind the sun at sunrise, you need to map below the horizon and so that will require six months or so to do (I think, potentially, you could do it in a bit over three). I would imagine you would first pick constellations that are always above the horizon initially, and then pick constellations that are further away as them become visible in the night sky.

Of course, as you get close to the equator or to the poles, things change. Some become easier and others become harder.