The mathematical form of measurement and the argument for proposition I in Newton's principia

Katherine Dunlop

pp. 191-229

in: John Mumma, Marco Panza, Paul-Gabriel Sandu (eds), Diagrams in mathematics, Synthese 186 (1), 2012.

Abstract

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton’s unpublished texts (and the views of his mentor, Isaac Barrow) shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition—the putting-together in space—of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all quantity (which is what measurement makes known) is ultimately related to spatial extension. I use this conception of Newton’s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia (beginning in Sect. 2 of Book I). The placementof Kepler’s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry.