MATHEMATICS AND SCIENCE 395 Third,while the Greeks had employed mathematics freely in their science,there was,as long as the Euclidean basis sufficed for mathematics, a sharp distinction between it and science.Both Plato and Aristotle dis tinguished the two (Chap.3,sec.10 and Chap.7,sec.3),albeit in differ- ent ways;and Archimedes is especially clear about what is established mathematically and what is known physically.However,as the province of mathematics expanded,and mathematicians not only relied upon physical meanings to understand their concepts but accepted mathematical argu ments because they gave sound physical results.the boundary between mathematics and science became blurred.Paradoxically,as science began to rely more and more upon mathematics to produce its physical conclusions, mathematics began to rely more and more upon scientific results to justify its own procedures The upshot of this interdependence was a virtual fusion of mathematics and vast areas of science.The compass of mathematics,as understood in the seventeenth century,may be seen from the Cursus seu Mundus Mathematicus (The Course or the World of Mathematics)by Claude-Francois Milliet Deschales(1621-78),published in 1674 and in an enlarged edition in 1690 Aside from arithmetic,trigonometry,and logarithms,he treats practical geometry,mechanics,statics,geography,magnetism,civil engineering. carpentry,stonecutting,military construction,hydrostatics,fluid flow,hy- draulics,ship construction,optics,perspective,music,the design offirearms and cannons,the astrolabe,sundials,astronomy,calendar-reckoning,and horoscopy.Finally,he includes algebra,the theory of indivisibles,the theory of conics,and special curves such as the quadratrix and the spiral.This work was popular and esteemed.Though in the inclusion of some topics it reflects Renaissance interests,on the whole it presents a reasonable picture of the seventeenth-and even the eighteenth-century world of mathematics One might expect that the mathematicians would have been concerned to preserve the identity of their subject.But beyond the fact that they were obliged to depend upon physical meanings and results to defend their ar- guments,the greatest of the seventeenth-(and eighteenth-)century con tributors to mathematics were either primarily scientists or at least equally concerned with both fields.Descartes,Huygens,and Newton,for example, were far greater physicists than mather aticia ns.Pascal,Fermat,and Leibniz were.active in physics.In fact,it would be difficult to name an outstanding mathematician of the century who did not take a keen interest in science. As a consequence these men did not wish or seek to make any distinctions between the two fields.Descartes says in his Rules for the Direction of the Mind that mathematics is the science of order and measure and includes,besides algebra and geometry,astronomy,music,optics,and mechanics.Newton says in the Principia:"In mathematics we are to investigate the quantities of forces with their proportion consequent upon any conditions supposed;then