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Notes
Here, Harper is following Smith (2004). Smith discusses the use of “successive approximations” and “theory-mediated phenomena” in Newton’s method, though he restricts the use of approximation to the deductive step in a staged-process version of Newton’s method.
Newton knows that the deduced relations will only hold approximately, due to confounding perturbations. He provides a generic theory of such perturbations in Books 1 and 2. However, it seems that this is a supplement to the deductions from phenomena, not an intervening inductive step. Newton is explaining how concepts are to be applied to perturbed phenomena—he is not generalizing the perturbation theory from the observations.
I did not claim that Newton did not ever use the mean. Indeed, as George Smith has pointed out to me, Newton actually endorses using the mean. In the Scholium to Problem 3 in De Motu, Newton says “The advantage of this method is that in order to reach a single conclusion many observations can be brought together and readily compared.” Smith also points out that he found that in Proposition 20 of Book 3 in the Principia Newton’s indication that Richer's pendulum clock was slow in Cayenne by 2 min and 28 s per day is an example where he explicitly uses the mean. Smith says:
The number is not in Richer’s report, nor anywhere else I have been able to find before the Principia. What Richer reports is time of crossing the meridian of various stars night after night for months. These differed from the nominal 23 h 56 min 4 s by differing amounts from day to day—ranging from more than 2 min 40 s to a little more than 2 min. So, I personally averaged all of them, discovering that indeed their mean is 2 min 28 s!
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Huggett, N., Smith, G.E., Miller, D.M. et al. On Newton’s method. Metascience 22, 215–246 (2013). https://doi.org/10.1007/s11016-013-9745-y
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DOI: https://doi.org/10.1007/s11016-013-9745-y