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It might be famous that $\pi^2$ is a good approximation to the gravitational acceleration in the unit "meter per second squared".

My explanation for this is the seconds pendulum, which was proposed as the definition of "meter" back to the birth of the unit, competing with the meridional definition.

However, I don't see why the two competing definitions of meter have to be close to each other. I don't know any pre-existing unit of length for their reference. And in the meridional definition, the "ten-millonth" seems to be chosen just for convenience, doesn't it?

So why the "meter" according to the seconds pendulum and the "meter" according to the meridian turn out to be close? Is it purely a coincidence?

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  • $\begingroup$ So there were two groups of people arguing for totally unrelated definitions of the meter... and they just turn out to be almost exactly the same? This is so mindboggling that I don't fully trust the standard historical argument (which is presented in the accepted answer below). $\endgroup$ – Marc May 15 '18 at 19:01
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It is a pure coincidence. And the agreement is not so good. Meter was introduced in connection with decimal system. They wanted all units to be based on decimal system, including angular and time units.

So it was decided to have 100 decimal degrees in the right angle, and 20 hours in a day. Each hour was divided into 100 decimal minutes, a decimal (time) minute into 100 decimal time-seconds. Similarly with angular measures: one degrees is 100 angular minutes.

The kilometer was supposed to be one (angular) decimal minute of the (Paris) meridian. Similarly to the nautical mile which is defined as one angular (sexagesimal) minute of the meridian. And meter is 1/10 of one angular decimal second of the meridian.

During the rule of the French revolutionary government, they actually made clocks, watches and angle measuring instruments with this decimal division. It was abandoned during the rule of Napoleon I.

That acceleration of gravity is roughly 10 meters/$sec^2$, where $sec$ is the Babilonian (sexagesimal) second, is a pure coincidence. A coincidence of the same sort as $\pi^2\approx10$.

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  • $\begingroup$ I knew that the pole to equator meridian was to be 10,000km, but I hadn't realised that is was also meant to be part of the decimalisation of the day. Were there any new names created for those new 'hours/minutes/seconds' ? $\endgroup$ – Philip Oakley Aug 10 '16 at 20:38
  • $\begingroup$ @Philip Oakley: I don't think any new names were invented. They were called hours/minutes/seconds, and when needed to avoid a confusion called decimal hours/minutes/seconds etc. $\endgroup$ – Alexandre Eremenko Aug 15 '16 at 18:05
  • $\begingroup$ Oh shame, I was hoping they might have had some good words for them! $\endgroup$ – Philip Oakley Aug 15 '16 at 23:13

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