By Hathaway A.S.

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IBM Corporation and by George Stiblitz of Bell Telephone) were intended mostly for the US government and for commercial use and only in some isolated cases, were they exploited for specific scientific calculations, as in the case of astronomy. Only during World War II and apparently with some real difficulties, was the IBM-machine adopted for military-scientific work connected with the atom bomb project at Los Alamos, and the pressure to improve the performance of computers carne directly from the needs to perform calculations related to the war effort.

Some problems are due to the inevitable limitations of numerical and symbolic algorithms; some are the result of poor programming or mathematical choices; others, because of design compromises. All present systems mishandle (complex) multivalued functions because there is no internal check on the validity of the domain of computations. Some mistakes might be rather devious since they are difficult to detect. Below are several examples from MAPLE V which show the kind problems presently encountered even by a very sophisticated CAS.

4Por a different opinion about the state of the Calculus, see Halmos (1990). 28 The recent trend to think of CAS as potential powerful tools for teaching and learning mathematics (at least at the undergraduate level) is, in a sense, an inevitable step in the sequence of changes brought about by a cognitive technology. g. language, print, pens and pencils, computers). This notion of CAS as teaching/learning tools is certainly reinforced and accelerated by several factors - societal pressure wherein computers have become the icon of modernity and progress, commercial interests of software and computer manufacturers, and the (recurrent) faith in a "technological flx" for pedagogical problems particularly in the current climate of general alarm about the health of mathematical instruction at the post-secondary level (at least in the United States).

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Analytical dynamics: being a synopsis of leading topics by Hathaway A.S.

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