How the Earth Was Weighed

It can be answered. And the method is unexpected: you do not weigh the Earth at all.

You hang two lead balls in a room, measure the very faint pull between them, and the Earth's mass follows by calculation.

How can that work?

Every object attracts every other. There is such a pull between two people sitting across a table — far too weak to feel, but there.

Its size depends on two things: how heavy the objects are, and how far apart. Heavier means stronger; further means weaker.

The obstacle is how weak the force is. Between two people at a table it amounts to less than the weight of a speck of dust. No ordinary scale comes close.

So a new kind of scale had to be built — one that does not press down, but twists.

A little further in

Eratosthenes worked out the Earth's circumference in 240 BC. Its size was settled then. Its weight took another two thousand years.

Size and weight are different kinds of problem. For size you measure a shadow; for weight you must lift the thing. If it cannot be lifted, another route is needed.

The door Newton left open

Newton's law says two bodies attract with a force proportional to the product of their masses and inversely proportional to the square of their distance. But the equation carries a constant of proportionality.

The trouble is how absurdly weak that force is. Between two people sitting across a table, gravity amounts to less than the weight of a speck of dust. Measuring it needs a correspondingly delicate balance.

A balance made from a thread

The apparatus was designed by the geologist John Michell, who died before completing it. It passed to Cavendish.

The principle: hang a rod horizontally from a fine wire, with a small lead ball at each end. Bring a large lead sphere near one small ball and gravity draws it. The rod turns a fraction, twisting the wire.

The smallest disturbance ruins it. A person standing nearby warms the air, and the draught pushes the rod. Cavendish sealed the apparatus in a wooden case, moved to the next room, and read the scale through a telescope poked through a hole in the wall.

So how much?

Knowing the force between the small and large masses, and knowing both masses and their separation, gives the gravitational constant. With the constant in hand, the acceleration of falling bodies at the surface and the Earth's radius yield the Earth's mass.

1. Measure the force between two lead masses in a laboratory
2. With both masses and the distance known, the gravitational constant follows
3. The acceleration of gravity at the surface is already measured
4. The Earth's radius has been known since Eratosthenes
5. Put the three into Newton's equation and the Earth's mass is what remains

What Cavendish published was not a mass but a density: 5.48 times water. The modern figure is 5.51. An experiment from 1798, correct to within one per cent.

As a footnote: the gravitational constant remains among the least precisely known physical constants. Others are pinned to many decimal places; this one still varies slightly between experiments, because the force is so weak.

The question that remainsTo measure the unmeasurable you must convert it into something measurable. What that we now call unmeasurable would yield to such a conversion?