There are two entirely different kinds of “minute” or “second”, and you have to be careful not to confuse them. In each case, they are 1/60 of some larger unit. When we measure time, we divide each HOUR into 60 equal parts called “minutes”, and each minute into 60 equal “seconds”.
We also do the same thing when we measure angles, so that each DEGREE (in astrology as well) is divided into 60 minutes, and each minute OF ANGLE is divided into 60 seconds. The units are different; an hour is not the same as a degree in any sense, and their relationship has nothing to do with the angle corresponding to a minute on a clock, or in the earth’s rotation.
The ancient Babylonians liked the number 60 and in fact based their number system on it almost the way we base ours on 10. So any unit would be divided naturally into 60 equal parts. That worked well for various reasons, so early astronomers used that scheme to measure angles, dividing a circle into 360 parts (6 times 60), and then dividing each of those parts into 60 minutes and then into 60 seconds. The word “minute” just meant “little part”, and “second” meant “second division into little parts”!
At some point people started wanting to divide time into smaller parts too; in ancient times it was hard enough to measure hours, but once mechanical clocks were invented it became possible to divide hours up.
How many units should there be in each hour? They might have used 12, as they had long before divided the day and the night each into 12 parts; but someone felt that 60 parts would be nice (probably an astronomer!). So they needed a name for 1/60 of an hour, and since “minute” already meant “1/60 of something”, they called it a minute.
They probably never considered the confusion it would cause when someone looked at a clock and asked how far the hand had moved. If I say it moved 30 minutes, do I mean 30/60 of an hour on the dial, or 30/60 of a degree of angle? That’s unfortunate; but if you remember that the size of a degree has nothing to do with the length of an
hour, and that the two kinds of minutes are just 1/60 of an hour or a degree respectively, then you can see that they just aren’t related.