Article ID: CBB000470254

The Importance of Being Equivalent: Newton's Two Models of One-Body Motion (2004)

unapi

As an undergraduate at Cambridge, Newton entered into his Waste Book an assumption that we have named the Equivalence Assumption (The Younger): If a body move progressively in some crooked line [about a center of motion] ..., [then this] crooked line may bee conceived to consist of an infinite number of streight lines. Or else in any point of the croked line the motion may bee conceived to be on in the tangent. In this assumption, Newton somewhat imprecisely describes two mathematical models, a polygonal limit model and a tangent deflected model, for one-body motion, that is, for the motion of a body in orbit about a fixed center, and then claims that these two models are equivalent. In the first part of this paper, we study the Principia to determine how the elder Newton would more carefully describe the polygonal limit and tangent deflected models. From these more careful descriptions, we then create Equivalence Assumption (The Elder), a precise interpretation of Equivalence Assumption (The Younger) as it might have been restated by Newton, after say 1687. We then review certain portions of the Waste Book and the Principia to make the case that, although Newton never restates nor even alludes to the Equivalence Assumption after his youthful Waste Book entry, still the polygonal limit and tangent deflected models, as well as an unspoken belief in their equivalence, infuse Newtons work on orbital motion. In particular, we show that the persuasiveness of the argument for the Area Property in Proposition 1 of the Principia depends crucially on the validity of Equivalence Assumption (The Elder). After this case is made, we present the mathematical analysis required to establish the validity of the Equivalence Assumption (The Elder). Finally, to illustrate the fundamental nature of the resulting theorem, the Equivalence Theorem as we call it, we present three significant applications: we use the Equivalence Theorem first to clarify and resolve questions related to Leibnizs polygonal model of one-body motion; then to repair Newtons argument for the Area Property in Proposition 1; and finally to clarify and resolve questions related to the transition from impulsive to continuous forces in De motu and the Principia.

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Authors & Contributors
Pourciau, Bruce H.
Janiak, Andrew
Dyck, Maarten Van
Verelst, Karin
Takahashi, Ken'ichi
Roux, Sophie
Concepts
Motion (physical)
Physics
Mechanics
Mathematics
Philosophy of science
Space
Time Periods
17th century
18th century
Renaissance
16th century
Early modern
Places
Europe
Italy
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