- 5. The Role of Mathematics in the Rise of Mechanics
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C H A P T E R 5 THE ROLE OF MATHEMATICS IN THE RISE OF MECHANICS* MECHANICS, as a part of physics, is very mathematical, in imagery, content, and consequences. We will make some observations on the mathematical structure of mechanics during some phases of its development, and we will be concerned with the inward mathematical texture rather than with the outward mathematical setting. But we will not go essentially beyond the beginning of the 19th century. After that time, mechanics and paradigms from mechanics began to penetrate into many other areas of theoretical physics, so that observations on the mathematical nature of mechanics after the early 19th century would have to take large parts of theoretical physics into account. In the 17th and 18th centuries mathematics and mechanics were developing not only in constant interactions but also in close intimacy. But, in the 19th century, relations between mathematics and mechanics, or rather between mathematics and theoretical physics, began to be different. The interactions continued to be no less in importance, but they began to manifest themselves in parallelisms of development rather than in direct interpenetrations , and an analysis of their new relations requires categories of description other than those applicable to previous stages. Modern mechanics came into being soon after the Renaissance, toward the end of the 16th century, and its stages of development, taken by centuries, have been as follows. The 17th century in mechanics extended from * Originally in American Scientist, Vol. 50 (1962), pp. 294311 . Slightly revised. [i79] MATHEMATICS & THE RISE OF MECHANICS Stevin to Newton and culminated in Newton's Principia. The 18th century in mechanics extended from Newton and Leibniz to Lagrange and culminated in his Mecanique analytique. The 19th century in mechanics extended from Lagrange and Laplace up to the first stirrings of quantum theory in 1900, and in a certain qualified sense it culminated first in Maxwell's Electrodynamics, inasmuch as the latter theory imitated hydrodynamic wave theory, and secondly also in leading treatises on statistical mechanics. In the 20th century the leading new theories of mechanics have become, more than before, direct and full representatives of the new physical theories of which they are a part and can hardly be isolated from them for separate analyses. We venture to characterize the past centuries of modern mechanics by the following headings. The 17th century was an age of revelation; the 18th century was an age of patristic organization; and the 19th century was an age of canonical legislation. If we dared to continue we might suggest that the 20th century is an age of reformation, or perhaps an age of reformation and counter-reformation in one. 1. BEFORE THE 17TH CENTURY In Greek antiquity, Hellenic and Hellenistic, if viewed in retrospect and if compared with the 17th century, say, there seemingly was an imbalance between mathematics and rational mechanics. The Greeks had a systematic mathematical astronomy, but outside of this they seemingly did not arrive at a theoretical mechanics or theoretical physics that was equal in status to their mathematics . Their mathematics, whatever its limitations, had the size and organization of a system. Their mechanics however was only an assortment of achievements, whatever their eminence. Some of the difficulties that kept Greek mechanics from developing into a system may have been linked to pe- [180] MATHEMATICS & THE RISE OF MECHANICS culiarities which kept Greek mathematics from advancing beyond the stage at which it was arrested. Anybody who appraises the mechanical work on the balancing of a lever in Archimedes and in his successors, Hellenistic and medieval, cannot fail to notice that in some covert and tacit manner each of them had the concept of the statical moment in his reasoning, but that none of them reached the point of formulating the concept expressly as the mathematical product P · L in which the factor L represents the length of an arm of the lever and the factor P represents the size of the weight which is suspended from it. Archimedes had a pre-notion of the moment in his context, but something in the metaphysical background of his thinking barred him from conceptualizing it overtly and "operationally ." Many physicists after him—including Heron, Pappus (A.D. 320), Jordanus (13th century), and Leonardo da Vinci (15th century)—were groping for some articulation of the notion of the moment; but for over 1,900 years it kept eluding them, untilfinallyIsaac Newton gave a clear-cut formulation of one. It was not the statical moment of rotation, but...

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