= 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 $u''+u=1/p$, and <Binet's equation> gives the radial <acceleration>
$$
a_r=-h^2u^2(u''+u)=-\frac{h^2}{p r^2}.
$$
The <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.<The Mathematical Principles of Natural Philosophy by Isaac Newton (1846 edition)>[Book I, central-force propositions, and Book III, application to planetary and satellite motions]{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.
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