It is well recognized and increasingly studied that scientific instruments have transformed the nature of science. The telescope and microscope, for example, literally opened up whole new worlds for scientific examination. Another kind of instrument has also transformed science; but, because of the difficulties of examination of these developments, these instruments are rarely examined with any kind of historical sophistication. These instruments are new techniques of mathematics. While Newton's Principia was not written using the new calculus (so his contemporaries could understand his arguments) Newton's theory could not have been fully developed with out calculus. Similarly, Einstein's theory of general relativity could not have been developed without new mathematical tools which he had to learn in the early years of the second decade of this century. These are the obvious examples, but there are many, many more which simply have not been examined. Professor Truesdell, an award winning mathematician and scientist, has the technical skills and a unique historical sensitivity to carry out the kind of analysis of the role of mathematics in the development of science. Under this grant, he will examine a previously completely ignored area: the history of the calculus of variations for double integrals. Variations of multiple integrals are essential tools of analysis for contemporary theories of fluids, elastic solids and electromagnetism. The use of multiple integrals in the mechanics of continua enables us to comprehend and in part control the world around us today. Studies thus are possible of the oceans, rivers, lakes, tides and winds, acoustic vibrations, flight in the atmosphere, and earthquakes just to give a few examples. The beginnings and the earliest application of variational principles to double and triple integrals are due to Lagrange in his work between 1762 and 1811. Lagrange's studies affected the development of the general theory of elastic deformation in the 1830's by Navier, Poisson, and Cauchy. They had an obvious effect on the research of George Green and, through him, have influenced every branch of mathematical physics. Professor Truesdell will examine this history up through about 1830, tracing the mathematical development in detail and describing its early application to mechanics and differential geometry by Lagrange, Poisson, and Rodrigues. This study will greatly enhance our understanding of the critical interaction of mathematics and science--an interaction which has literally reshaped every one of the sciences.
