The first inquiry into nature began in opposition to metaphysics. Aristotle took it on himself to show that a science of nature was possible, contra Plato and Parmenides. In order to found the science, Aristotle had to borrow various terms which belong to metaphysics. When these terms are used in the context of physics, the terms are natural, and not metaphysical (the “act” and “potency” used in the definition of motion are borrowed, but in physics they only extend to mobile being, not to being as such).
The second inquiry into nature began in opposition to Aristotle’s physics. The metaphyscial objections of Parmenides no longer bothered anyone, and so no one particularly cared about borrowing various metaphysical terms to explain natural things. The interest was instead on extending the scope of mathematical physics, which advances by borrowing truths from mathematics. When used in the context of the science of nature, the truths are natural, and not mathematical, as Aristotle himself showed in the second book of his physics.
(in calling these two approached “first” and “second” I am stateing a purely contingent fact. It simply happened that one science developed before the other. The two inquiries into nature are so distict that Galileo could have very well written before Parmenides. There is nothing in mathematical accounts that make them dependent on previous metaphysically-borrowed accounts. One does not need to know about matter and form and causes in order to measure the rate of a free fall.)
In both cases, the benefit and the danger of borrowing is the same: certitude. Mathematics and metaphysics are characterized by a certitude that is simply not possible to attain about natural things as such. Both mathematics and metaphysics deal with immobile things, but nature is essentially mobile. Nature cannot exist apart from a part that is unintelligible to us. The certitude of both goes bad in different ways: on the one hand, we cannot spin out all truths about natural things simply by meditating on their definitions, on the other hand we cannot imagine that nature is really nothing but the clarity of mathematical formulae and laws. Both borrowed sciences confer a benefit that the other one fails to provide: metaphysics allows us to say some really unchangeable and categorical statements, but only about nature in general; mathematics gives us an operable and predictive knowledge of the concrete things in nature, but only a hypothetical knowledge.