 Solution

Isaac Newton's achievement was not to introduce mathematics into a previously nonmathematical study of nature. Astronomy, optics and mechanics already had substantial mathematical traditions; Two New Sciences by Galileo Galilei, for example, connects geometrical reasoning with accelerated motion. The distinctive achievement of the Philosophiae Naturalis Principia Mathematica by Isaac Newton was to connect mathematical results about motion and forces with an ambitious, quantitatively constrained account of terrestrial and celestial phenomena.

The method separates a conditional mathematical proposition from its application. Given a specified force law and initial conditions, determine a trajectory. Conversely, given a suitably idealized trajectory and time law, infer what force would produce it. This makes the explanation more demanding than fitting a geometrical curve to positions: the timing and dynamical assumptions matter as well.

A modern calculation illustrates the connection. For a planar orbit under a central force, let $h=r^2\dot{\theta }$ be the conserved specific angular momentum and put $u=1/r$. An ellipse with the attracting point at a focus has $u=(1+e\cos \theta )/p$. Thus $u^{\prime \prime }+u=1/p$, and Binet's equation gives the radial accelerationThe orbit and equal-area timing therefore imply an inverse-square acceleration under the stated assumptions. The modern notation is explanatory; it is not an assertion that Newton wrote this particular differential-equation derivation. The geometrical treatment of central forces in Book I and its astronomical application in Book III supply the historical connection.^[ref]

The programme also joined apparently different domains. Newton's law of universal gravitation connects orbital motion with terrestrial falling motion through the same inverse-square dependence, with distance and time scales accounted for. That unification produced further tests rather than merely attaching one name to two phenomena. Departures from a perfect two-body ellipse, including disturbances by other gravitating bodies, became quantitative problems for the theory instead of automatic refutations of it.

Measurement did not disappear into deduction. Astronomical observations provide imperfectly known inputs; assumptions about mass distribution and neglected interactions restrict an ideal model. A discrepancy can expose a mistaken approximation, an observational error or a deficiency in the dynamical account. The mathematical proof alone does not decide which. The laws and initial data therefore create a disciplined relationship between measurement and explanation, not a proof of all nature from definitions.

Newton also did not claim that successfully describing a force mathematically supplied a complete mechanical account of its ultimate cause. Distinguishing a quantitative law from a speculative mechanism allowed a powerful programme of prediction while leaving explanatory questions open. The achievement was a new integration of mathematical demonstration, empirical inference and common dynamical principles. Its strength lies in the constraints this integration places on physical explanations, rather than in a simple opposition between mathematics and experiment.