File:Principia Book 2 Section 6 Proposition 27.jpg

English: In Section 6 of Book 2 of the Principia, Newton examined the motion of a cycloidal pendulum under the effect of various resistances as a function of the instantaneous speed. Compared with an unresisted pendulum, the amplitude is reduced, and the time is increased, in passing between the extremes of motion. He showed how the reduction of distance varies as a function of the initial amplitude for different resistances. In the General Scholium to this section, he performed various experiments with a simple (not cycloidal) pendulum to measure the reduction in amplitude in order to determine the resistance law. However, the small size of the effect, made it impossible to obtain accurate results.

He had argued that the resistance should be proportional to the square of the speed, and assumed, erroneously, that in this case, the increase in the time would be proportional to the initial amplitude. He claimed that by measuring the times of oscillations with different initial amplitudes it would be possible to calculate the time of an oscillation without any resistance. However, as well as his assumption being incorrect, the time differences are so small that he was unable to measure it, lacking any means of determining short periods of time accurately.

See: Principia, A New Translation, by I. Bernard Cohen and Anne Whitman – University of California Press, 1999