The acceleration due to gravity, at the surface of the Earth, is about 9.81 m/s2. (If you are some of my students, you think it's 10, which is confusing for a moment. Fortunately none of my students thought it was 32.)
π2 = 9.87.
The approximate numerical equality of these numbers is not a coincidence.
I was reminded of this by Mark Dominus' post John Wilkins invents the meter, which I found via his post The Wilkins pendulum mystery resolved. In 1668, John Wilkins defined the meter to be essentially the length of the "seconds pendulum", i. e. the pendulum whose period is two seconds (the "mystery" there is that since one wants a pendulum with a light cord and a heavy bob, one has to take into account the moment of inertia of the heavy spherical bob to define this correctly). This turns out to be about 39.1 inches.
A meter is 39.37 inches.
Indeed, originally the length of the seconds pendulum was going to be the length of the meter, but the slightly different suggestion was adopted instead that the meter would be one part in 107 of the distance from the North Pole to the Equator via Paris. (Now, in fact, the meter is defined by assuming the second known and stating the speed of light.) Wikipedia tells me that these definitions were adopted in 1790, 1791, and 1983 respectively; there are others intermediate between the last two.)
The period of a pendulum is approximately T = 2π(L/g)1/2, where L is its length and g is the acceleration due to gravity. If we set T = 2 and L = 1, and solve for g, we get g = π2.
I suspect that the reason the seconds pendulum wasn't adopted is that the formula for the period involves the small-angle approximation; in reality the period of a pendulum depends not only on its length and on the acceleration due to gravity but also on the angle through which it swings. The period is given by
and so the fractional magnitude of the error is about 1/4 sin2 (θ/2); if θ is, say, one degree this is one part in about fifty thousand, or nearly two seconds a day. That error would become an error in the definition of the meter, and perhaps the Powers that Were thought that was a bit too much. (I'm not sure how accurate surveying was at the time, though, so I can't comment on the accuracy of their eventual definition based on the size of the Earth.)