(His thoughts are clearly descendants of Husserl's.) But, if I understand your question properly, Klein's book will be quite helpful in trying to understand "what happened" or "what was different" between the ancients and the moderns; the contention that mathematics was fundamentally different is a very interesting answer to those questions.Of course, the contention assumes that ancients like Archimedes were, in fact, applying mathematics to physics; and thus, the difference between an Archimedes and a Descartes (assuming that Galileo would be a sort of interim character) would be symbolic mathematics and/or time/hindsight.But the use of symbolic mathematics may have been coincident to some fundamental shift in how science was done--or may have been quite important (as it is with modern physics).Of course, a significant change may not have occurred.The preceding era is referred to as before the Common or Current Era (BCE).The Current Era notation system can be used as an alternative to the Dionysian era system, which distinguishes eras as AD ( The year-numbering system as used for the Gregorian calendar is the most widespread civil calendar system used in the world today.
, which contains the first explicit use of what today we'd call infinitesimals, there is no doubt that he applied mathematics to physics.(By Antiquity, I mean a generous time-frame which includes Hellenic Antiquity, Islamic Dar As-Salaam, and Christian Philosophy before the Renaissance, that is before Galileo).The fundamental difference between ancient science and modern science isn't in the use of mathematics, although advances in mathematics since Euclid's day certainly helped fuel the scientific advances of the Enlightenment.For decades, it has been the global standard, recognized by international institutions such as the United Nations and the Universal Postal Union.
The expression has been traced back to Latin usage to 1615, as and became more widely used in the mid-19th century by Jewish academics. One claim I've seen is that Galileo was the first to apply mathematics to mechanics. Surely just as important, if not more, is the development of lens-making technology that allowed him to look at the night sky through a telescope?