An astonishing equation

But let’s return to our mysterious equation. To find the period of small oscillations of a mathematical pendulum as a function of the length of the suspension, the following formula is used:

 

And here it is—our π! Let’s substitute the parameters of Huygens’ pendulum into this formula. The length of the string l in Huygens’ pendulum equals 1. The T – oscillation equals 2. Plugging these values into the formula, we get π²=g.

So, have we found the answer to our question? Well, not quite. We already saw that the equality is only approximate. It doesn’t feel right to equate 9.87 and 9.81 exactly. Does this mean that the meter has changed since then?

With revolutionary greetings from France

Yes, indeed, it did change! This occurred during the reform of the units of measurement initiated by the French Academy of Sciences in 1791. Intelligent people suggested maintaining the definition of the meter through the pendulum, but with the clarification that it should specifically be a French pendulum—at the latitude of 45° N (approximately between Bordeaux and Grenoble).

However, this did not sit well with the commission in charge of the reform. The problem was that the head of the commission, Jean-Charles de Borda, was a fervent supporter of transitioning to a new (revolutionary) system of angle measurement—using grads (a grad being one-hundredth of a right angle). Each grad was divided into 100 minutes, and each minute into 100 seconds. The method of the seconds pendulum did not fit into this neat concept.

The true and final meter

In the end, they successfully got rid of the seconds and defined the meter as one forty-millionth of the Paris meridian. Or, alternatively, as one ten-millionth of the distance from the North Pole to the equator along the surface of the Earth’s ellipsoid at the longitude of Paris. This measurement slightly differed from the “pendulum” meter. The commission, without false modesty, dubbed the resulting value as the “true and final meter.”

The idea of a universal standard accessible to everyone waved goodbye and faded into the sunset. Need an accurate standard for the meter? No problem! All you have to do is measure the length of a meridian and divide it by a few million. By the way, the French actually did this—they physically measured a portion of the Paris meridian, the arc from Dunkirk to Barcelona. They laid out a chain of 115 triangles across France and part of Spain. Based on these measurements, they created a brass standard. Incidentally, they made a mistake—they didn’t account for the Earth’s polar flattening.

Conclusion

Let’s return to our equation once again. Now we know where the inaccuracy comes from: π² and g differ by about 0.06. If it weren’t for yet another attempt to reform and improve everything, we would now have a slightly different value for the meter and the elegant equation π² = g. Later, scientists did return to defining the meter through unchanging and reproducible natural constants, but the meter standard was no longer the same.