This is a very mediocre history. I shall illustrate its mediocrity by criticising this quotation:
"The formulation of the Aristotelian 'law of motion' metrically as a function [velocity proportional to motive power over resistance], so that it became quantitatively refutable, was an achievement of the greatest importance, even though neither Bradwardine nor any of his contemporaries discovered an expression that fitted the facts or indeed applied any empirical quantitate test." (p. II.70)
This makes no sense. Nothing is added to the verbal expression by turning it into a formula.
In what sense did the law suddenly become "quantitatively refutable" by this transformation? Obviously not because it enabled Galileo-style objections based on joining bodies of different weights, since such objections were raised already in Antiquity (p. II.65). Nor because it drew attention to the case resistance=0 or the possibility that the force caused no motion, since these cases was discussed in detail by Aristotle himself (p. II.62-63).
Crombie's answer is cryptic: "Using his metric formulation, Bradwardine was able to show" various things, most notably "Bradwardine argued that Aristotle's law meant that if a given ratio p/r produced a velocity v, then the ratio that would double this velocity was not 2p/r but (p/r)^2" (p. II.71). Why on earth would (p/r)^2 double the velocity? This claim is nowhere in Aristotle. Apparently Bradwardine "argued" that it is implicit in Aristotle, but we see no trace of this alleged "argument." It seems to me that the mistaken belief that (p/r)^2 doubles the velocity was in fact introduced by mathematics rather than eliminated by it. Crombie has no evidence that any error of this kind was ever committed in the pre-"metrical" period.