### Earth’s Shape and Spin Won’t Make You Thin

Using a simple pendulum we can determine the shape of the Earth. That amazing story is told in this week’s That’s Maths column in The Irish Times (TM057 or search for “thatsmaths” at www.irishtimes.com ).

Is the Earth oblate like an orange (Newton) or prolate like a lemon (the Cassinis)?

Many of us struggle to lose weight, or at least to keep our weight within a manageable range. There is no easy way to do this, but could geophysics provide some assistance? The Earth is approximately spherical, but there is a slight flattening towards the poles. This is a consequence of the rotation when the planet was forming during the early history of the solar system.

The Shape of the Earth

The gravitational attraction of a sphere of uniform density is the same as if all the mass were concentrated at the centre. Thus, it is the same everywhere on the surface. But is the Earth a perfect sphere? Newton believed that rotation would bring about a flattening, with the polar radius smaller than that at the Equator. This would make it an oblate spheroid, flattened like an orange. Dominique Cassini and his son Jacques took a contrary view: their measurements in France indicated that the Earth is elongated at the poles, a shape called a prolate spheroid, more like a lemon.

If the Earth were flattened, the length of a meridional arc of one degree of (geographic) latitude would increase from Equator to poles. If the Earth were elongated, as held by the Cassinis, the length of such a meridional arc would decrease towards the poles. But which was true? There was a lively controversy between the Orangemen and the Elder Lemons.

Ellipsoidal models of the Cassinis (left) and of Newton and Huygens (right). All angles shown are equal, but they are not geocentric[image from NOAA ].

The French Geodetic Survey

To resolve the issue, in 1735 the French Academy of Sciences proposed a geodetic expedition to make measurements of the meridian arc in polar and tropical regions. Two groups set out from Paris, one to Peru and one to Lapland. The north-bound group, led by Pierre Louis Maupertuis, was back in Paris within a year or so.

It is an amazing fact that, by means of a humble pendulum, we can determine the geometric shape of the Earth. We can calculate the relative strength of gravity at two latitudes by measuring the period of the pendulum at the two locations. In 1673, the French astronomer Jean Richer found that his pendulum clock, which kept good time in Paris, lost 2.5 minutes per day in Cayenne. This means that the gravitational attraction in Cayenne is about 0.35% smaller than in Paris. From this we can calculate that the effective force of gravity increases by about 0.5% as we go from Equator to pole.

Among the scientists accompanying Maupertuis was Alexis Claude Clairaut, a notable French mathematician, geophysicist and astronomer. In his Theory of the Figure of the Earth, Clairaut published a formula that allows the ellipticity of the Earth to be calculated from surface measurements of gravity. An analysis of the Lapland measurements confirmed that the Earth is an oblate spheroid, as Newton had predicted. On hearing of the result, Voltaire congratulated Maupertuis “for flattening the Earth and the Cassinis”.

Finnish postage stamp featuring an image of Maupertuis, issued to commemorate the 250th anniversary of the French Geodetic Mission to Lapland.

Variations of gravity have important geophysical consequences, for inertial navigation, GPS location and so on. But, when it comes to weight control, they are no substitute for the old reliables, good diet and regular exercise. A man weighing 100kg can lose about half a kilo by moving from Anchorage to Zanzibar but, of course, his body mass does not vary, only the pull of the Earth upon it.