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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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It is not really a coincidence that the different proposals had similar lengths, because the metre (as it is spelled in French, and in British English) was created in a world which already contained many units of measurement. These varied in two ways:

  • Based on where they were used, and who decreed their correct size; for instance, an "inch" in one country would not be the same as an "inch" in another, as this 19th-century conversion chart demonstrates.
  • Based on what they were used for; for instance, a length of string might be measured in inches, cloth in yards, but a farm in furlongs. These different measures were originally independent, not related by the exact multiples that have been standardised today.

The creators of the metre were aiming to come up with a measure that had some scientific basis (rather than the decree of a monarch), but which would be useful for the same purposes as existing measures.

The closest unit in France at the time was apparently the toise, which was equivalent to about 1.9 metres, slightly longer than an English fathom; the more common unit in England was the yard, roughly half as long. Other historical units of similar length include the klafter, cubit, and ell.

More fundamentally, these units are based on the size of the human body - the span of outstretched arms, a pace forward, and the height of a grown man are all in this range. Those sizes are easy to picture, and useful to proportion things based on.

The two proposed standards were therefore looking for a value somewhere in the range of these existing units, so that it could be used for the same purposes. Two ways to define it were suggested:

  • The length of a pendulum with a particular period; choosing a period of two seconds happens to give a close value to the yard, ell, etc.
  • A fraction of the Earth's circumference; taking one ten-millionth of the distance from the North Pole to the equator gives the right sized unit.

The proposal at the time was to replace all units with decimal systems, including a new decimal second, which made the pendulum definition less appealing. The divisor of 10 million fitted better into this decimal system, but required significant research to define.

In practice, the origin of the definition was soon irrelevant, as neither is reproducible with sufficient accuracy - the Earth is hard to measure, and irregularly shaped; and a pendulum swings slightly differently in different locations. So the actual definition of the metre, as with most previous units, was a specimen against which other measuring devices could be checked.

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