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38 Historiographic Studies ancient mathematics, were empirical rather than a priori, their considerable ancient development required little refined observation and even less experiment. For a person schooled to find geometry in nature, a few relatively accessible and mostly qualitative observations of shadows, mirrors, levers, and the motions of stars and planets provided an empirical basis sufficient for the elaboration of often powerful theories. Apparent exceptions to this broad generalization (systematic astronomical observation in antiquity as well as experiments and observations on refraction and prismatic colors then and in the Middle Ages) will only reinforce its central point when examined in the next section.

Clearly he participated significantly in aspects of the movement here called Baconian. But, as Leonardo's career also indicates, instrumental and engineering concerns do not make a man an experimentalist, and Galileo's dominant attitude toward that aspect of science remained within the classical mode. On some occasions he proclaimed that the power of his mind made it unnecessary for him to perform the experiments he described. On others, for example when considering the limitations of water pumps, he resorted without comment to apparatus that transcended the capacity of existing technology.

Mathematical versus Experimental Traditions 61 tury was a gradual shift in the perceived identity of mathematics. Until perhaps the middle of the century such topics as celestial mechanics, hydrodynamics, elasticity, and the vibrations of continuous and discontinuous media were at the center of professional mathematical research. Seventy-five years later, they had become "applied mathematics," a concern separate from and usually of lower status than the more abstract questions of "pure mathematics" which had become central to the discipline.