Cavendish mis-reported again?

[revised 2008-10-13]
Falconer (1999) has written an arresting account of Cavendish and his famous gravitation experiment (Cavendish, 1798) and highlights three important aspects of his achievement:

1. Cavendish set out to measure the density of the earth as a ratio of the density of water, not the universal gravitation constant G, a concept which lay about 80 years in the future.

2. In the same way that Newton’s theory of Universal Gravitation linked astronomical motion to local motion (i.e. falling objects), the experiment set an example for investigating natural phenomena in a laboratory.

3. It is a seminal paper in measurement science, focussing more on the estimation and elimination of errors than on the experimental results.

That article is a welcome counter to the widespread misreporting in physics texts, course notes and commentaries, that Cavendish set out to measure G (Gonzalez, 2001). However, the article contains two technical errors, which are discussed below, in the hope that these too don’t get repeated by others.

1. Falconer writes (p.474) that “the essential feature of the experiment consisted of using a torsion balance to find the attraction between a lead sphere 8 inches in diameter and another lead sphere 2 inches in diameter”. (Cavendish calls these spheres the “weight” and the “ball” respectively.) In fact, nowhere does Cavendish mention the diameter of the weight, but in an acknowledgment of the balance’s inventor, states (p.470) “the weights which Mr. MICHELL intended to use were 8 inches diameter”. We can estimate the diameter of the “weight” that Cavendish actually used, from his scale diagram (approximately 11 inches), or calculate it from the given mass of 2439000 grains and the density of cast lead as approximately 11.7 inches.

2. A feature of Falconer’s paper is that it expands some missing steps in Cavendish’s calculation of the ratio of the attraction of the weight on the ball to the attraction of the Earth on the ball. The ball and weight had a centre-centre separation of 8.85 inches. Cavendish introduced (p.510) a fictitious particle (what we now call a “point mass”) and a fictitious sphere of water one foot in diameter, which has mass 1/10.64 that of the weight. He writes “...and therefore [the weight’s] attraction on a particle placed at the centre of the ball, is to the attraction of a spherical foot of water on an equal particle placed on its surface, as 10.64 x (6/8.85)² to 1”. Unfortunately, Falconer confuses the ball for Cavendish’s particle, with the result that her rephrasing (p.475) "Then the attraction, F, of the weight on the ball was F=10.64 x (6/8.85)² times the attraction of the sphere of water on the ball if the ball were on the surface of the sphere", is geometrically incorrect and confusing because the ball has a finite radius of 1 inch, which would require the ratio to be (7/8.85)² if it were touching the sphere of water.

References
Cavendish, H. 1798. Experiments to determine the density of the Earth. Phil. Trans of the Royal Society of London 88, 469-526
Falconer, I. 1999. Henry Cavendish – the man and the measurement. J. Measurement Science and Technology 10, 470-477
Gonzalez, A. 2001. "Weighing'' the Earth: a Newtonian Test and the Origin of an Anachronism. Science and Education, 10, 6, 515-543.