All philosophers of the Enlightenment sought to design theories of knowledge which would justify holding that the new mechanistic science of the day was indeed knowledge in the traditional sense of certainty. Philosophers of this period generally all agreed that this justification required a foundation of starting propositions which were themselves indubitable. The justification of mechanistic science was to be achieved by showing it rested on such a foundation. Furthermore all of the philosophers of this "modern" period followed Descartes in assuming that the one object of knowledge of which the subject is directly certain is (and can only be) his/her own ideas.
However, while virtually all philosophers of the modern period shared this starting point and goal, they disagreed over the what these foundational ideas were and where they came from. The two main approaches have been traditionally labeled "rationalist" and "empiricist." Rationalists regarded the evidence of illusions, dreams, hallucinations, etc., to establish that sensory perceptions could serve as the grounds only for beliefs about the appearances of things. Since (according to the mechanistic world view) reality is held to be very different from its appearance to our senses, rationalists concluded sensory perceptions cannot be the basis for scientific knowledge. Scientific knowledge must therefore rest on ideas supplied not by the senses but by the "mind" or -as they put it- by "Reason." For the typical rationalist, mathematics (particularly geometry) exemplified exactly this characteristic: the objects of mathematics are not perceived but conceived; they are not objects of the senses but thought by the mind. Hence the judgments of mathematics have a universality and necessity lacking in judgments based on sensory perceptions. The obvious dependence of the new mechanistic science on the mathematical description of moving bodies is thus regarded as what gives the new science its certainty; it is the bridge connecting rationalist epistemology to the mechanistic world-view. Descartes claimed the key ideas displayed "clarity" and "distinctness" to the attentive mind, and he claimed his notion of "thinking substances" ("minds") and "extended substances" ("bodies") to have just this clarity and distinctness, Fellow rationalists agreed up to the last step. They complained that Descartes' ideas of "mind" and "body," far from being clear and distinct, were in fact obscure and confused. Thus they built their conceptions of reality on different (allegedly) "innate" ideas and quite naturally produced philosophies radically different from the Cartesian version of mechanism. Aside from Descartes himself (a French Catholic), the brightest philosophical lights of this tradition were Spinoza (a jew living in Holland) and Leibniz (a German Protestant). It provided the philosophical undercurrent for much brilliant mathematical physics done in France, Holland, and Germany throughout the 17th and 18th centuries. Hence we speak of the tradition of "continental rationalists."
Empiricists came to regard this whole approach as radically suspicious. Since every idea is by definition the "private" possession of the mind that thinks it, there is no way to demonstrate that any (allegedly) innate idea does indeed display the characteristics of clarity and distinctness. Indeed the whole hypothesis of innate ideas was regarded by empiricists as unprovable, incoherent, and unnecessary for a proper account of knowledge. Thus empiricists proposed to show how all the ideas requisite for the kind of knowledge afforded by the new science could be derived from sensation. For epistemologists of this persuasion the certainty accorded the new science rests on the claim that its description of nature rests on observational evidence which, in the last analysis reduces to an enormous number of particular sensations of particular subjects.
The mathematical "laws of motion" of the new science are generalizations reached inductively by reasoning from enormous numbers of particular statements to universal conclusions. British philosophers typically regarded this empirical approach as the key to the success of their hero, Isaac Newton (though in fact that claim is certainly disputable). British philosophers had a markedly empirical bent already in the late Middle Ages in the philosophy of William of Ockham, and the Renaissance philosopher Francis Bacon. Descartes contemporary, Thomas Hobbes, also a defender of the new mechanistic science was in many ways an empiricist, but the tradition of "classical British empiricism" is generally regarded as beginning with John Locke because it was he who sought to bring the empiricist approach to doing philosophy in the new "modern" way -a la Descartes- by beginning with the certainty of our own ideas.
Locke undertakes to construct an empiricist theory of knowledge by what he calls his "historical plain method" of showing how all the contents of the mind -which is a "blank slate" (tabula rasa) at birth- enter the mind through experience, either externally through the senses (ideas of sensation) or internally through experience of its own operations (ideas of reflection). But Locke also wants to defend the mechanistic world-view of the new science -in his case Newton's achievement- and in doing so his alleged empiricism includes many rationalistic "impurities." In the subsequent history of British philosophy George Berkeley (an Irish Anglican bishop) and David Hume (a Scot agnostic) purified Locke's empiricism establishing the tradition we call "classical British empiricism" which -with modifications- has continued to wield considerable influence in epistemological discussions right through the present time.