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Why and when was the hour divided into exactly 60 minutes (and not for example 70 or 80)?

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It comes from the ancient Babylonian numeration system which had base 60. (The reason for the choice of such a base is simplicity of calculation: 60 is divisible by 2,3,4,5,6,10,12,15,20,30. Much more convenient than base 10, whose only justification is the number of fingers on both hands).

It was used mainly in astronomy (ancient people had little need in dividing an hour into smaller parts, and if necessary they used 1/2 or 1/4 of an hour). Then with the spread of clocks, it started to be used in daily life.

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    $\begingroup$ It should be emphasised that measurement of angle was historically far more refined than the measurement of time, and that time has merely inherited approaches that were originally devised for the measurement of angle. $\endgroup$ – Steve Feb 21 at 0:16
  • $\begingroup$ I don't know what is the ground for your assertion that measuring angles precedes measuring time, historically. In early astronomy, these two tings are inseparable. $\endgroup$ – Alexandre Eremenko Feb 21 at 3:34
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    $\begingroup$ Babylonians are over a 1000 years late for this. This was the Sumerian number system. The Babylonians inherited it, just as we have today for measuring time and angles. $\endgroup$ – Paul Sinclair Feb 21 at 4:43
  • $\begingroup$ @Eremenko: The base 60 system is also based on the fingers, but in a different way. See Wikipedia on the sexagesimal number system. $\endgroup$ – Jonathan Rosenne Feb 21 at 10:42
  • $\begingroup$ @AlexandreEremenko, I didn't say measuring angle precedes measuring time, I said the measurement of angle was far more refined, and that it is in angles unrelated to timekeeping (such as latitude and longitude calculations) where the minute has first come into widespread use, only later to be transferred over as a unit of timekeeping also. (1/2) $\endgroup$ – Steve Feb 21 at 12:34

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