Math History

Introduction

The new science that took form during the seventeenth century may be distinguished by both external and internal criteria from the science and the philosophical study or contemplation of nature of the antecedent periods. Such an external criterion is the emergence in the seventeenth century of a scientific "community": individuals linked together by more or less common aims and methods, and dedicated to the finding of new knowledge about the external world of nature and of man that would be consonant with-and, accordingly, testable by-experience in the form of direct experiment and controlled observation. The existence of such a scientific community was characterized by the organization of scientific men into permanent formal societies, chiefly along national lines, with some degree of patronage or support by the state.

Discussion

Mathematics in the new science: a world of numbers

After modern science had emerged from the crucible of the Scientific Revolution, a characteristic expression of one aspect of it was given by Stephen Hales, often called the founder of plant physiology. One of the most famous uses of numerical reasoning in the Scientific Revolution occurs in Harvey's analysis of the movement of the blood. These numbers showed 'that more blood is continually transmitted through the heart, than either the food which we receive can furnish, or is possible in the veins'. Here we may see how numerical calculation provided an argument in support of theory: an excellent example of how numbers entered theoretical discussions in the new science.

Despite the force of the foregoing examples, however, it remains true that the major use of numerical reasoning in the science of the seventeenth century occurred in the exact physical sciences: optics, statics, kinematics and dynamics, astronomy, and parts of chemistry. Numerical relations of a special kind tended to become all the more prominent in seventeenth-century exact science because at that time the laws of science were not yet written in equations. Thus we today would write Galileo's laws of uniformly accelerated motion as x; = At, and S = ] /2 At , but he expressed the essence of naturally accelerated motion (free fall, for example, or motion along an inclined plane) in language that sounds much more like number theory than like algebra: 'the spaces run through in equal times by a moveable descending from rest maintain among them-selves the same rule [rationem] as do the odd numbers following upon unity'.

Galileo's rule, that these first differences (or 'the progression of spaces') accord with the odd numbers, led him to another form of his rule, that the 'spaces run through in any times whatever' by a uniformly accelerated body starting from rest 'are to each other in the doubled ratio of the times [or, as the square of the times]' in which such spaces are traversed. This form of his rule, expressed in the language of ratios, comes closer to our own algebraic expression. Thus while speeds increase with time according to the natural numbers, total distances or spaces traversed increase (depending on the chosen ...