Revisiting Riccioli’s free-fall calculations

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Christopher Graney provides a fascinating description of Giovanni Battista Riccioli’s meticulous 17th-century experiments on free fall (Physics Today, September 2012, page 36). He notes that Riccioli’s results, converted to modern units, provide an estimate of g that is systematically 5% smaller than the current accepted value. The discrepancy might well be attributed to uncertainty over the modern equivalent of Riccioli’s unit of length, the Roman foot. However, one more insight can perhaps be wrung from the data.As Christiaan Huygens reported in his classic 1673 Horologium Oscillatorium, published 22 years after Riccioli’s Almagestum, the period of a pendulum of length l is T = 2π√l/g. Thereafter it is an easy exercise to show that a plot of the distance fallen versus the square of the fall time in units of the pendulum half-period T/2 is a straight line of slope π2l/2. That figure is independent of g and of the conversion to modern units.Plotting Riccioli’s data in that way reveals an accurate straight line from which one can deduce the pendulum length. The answer—with a generous error estimate—is l = 1.00 ± 0.05 Roman inches. As Graney notes, Riccioli himself reported the length to the center of the pendulum bob to be 1.15 inches. The 15% discrepancy—less than 4 mm—is plausibly excusable, though one might infer that Riccioli supposed his measurement was accurate to at least 0.05 inches (about 1 mm).Unfortunately, the discrepancy is in the wrong direction to be attributed either to the distinction between the center of mass and the center of oscillation of a compound pendulum or to the fact, known to Riccioli, that large-amplitude pendulum swings are not quite isochronous. Nevertheless, the discrepancy is intrinsic to Riccioli’s presented data and is not dependent on the disputed length of the Roman foot. One can only presume it reflects on the difficulty of subdividing a Roman foot into such fine equispaced divisions.© 2013 American Institute of Physics.