Libro De La Historia De Mexico

few new branches of mathematics are the work of single individuals. the analytic geometry of descartes and fermat was certainly not the result of their investigation only, but was the outgrowth of several mathematical trends which converged in the sixteenth and seventeenth centuries. it was the result of the influence of apollonius, oresme viete and many others.
far less is the development ofthe calculus to be ascribed to one or two men. we have followed the long and uneven flow of thought which led from the philosophical speculations and mathematical demonstrations of the ancients to the remarkably successful heuristic methods of the seventeenth century. we have indicated that the procedures invented by fermat. for example, are almost identical with those found in the calculus, andthat the new propositions discovered by barrow include the geometrical equivalent of the basic theorem of the subject.
the time was indeed ripe, in the second half of the seventeenth century, for someone to organize the views, methods, and discoveries involved in the infinitesimal analysis in to a new subject characterized by a distinctive method of procedure. fermat had not done this, largelybecause of his failure to generalize his methods and to recogenize that the problems of tangents and quadratures were two aspects of a single mathematical analysis-that the one was the inverse of the other. barrow was unable to establish the new subject for, although the first to recognize clearly the unifying significante of this inverse property he failed to realize that his theorems were the basisfor a new subject. being unsympathetic with the cartesian mathematical analysis and the algebraic trend, he implied that his results were to be considered as rounding out the geometry of the ancients. the traditional view, therefore, ascribes the invention of the calculus to the more famous mathematicians, isaac newton and gottfriend Wilhelm von Leibniz. From the point of view of the developmentof the concepts involved, the aspect which concerns us chiefly here, it might be far better to speak of the evolution of the calculus. Nevertheless, inasmuch as newton and Leibniz, apparently independently, invented algorithmic procedures which were universally applicable and which were essentially the same as those employed at the present time in the calculus, and since such methods werenecessary for the later logical development of the conceptions of the derivative and the integral, there will be no inconsistency involved in thinking of these men as the inventors of the subject. In doing so, however, we are not to consider or to imply that they are responsible for the ideas and definitions underlying the subject at the present time; for these basic notions were to be rigorouslyelaborated only after two centuries of further effort in this direction. Furthermore, inasmuch as we are here more concerned with ideas than with rules of procedure, we shall not discuss the shamefully bitter controversy as to the priority and independence of the inventions by newton and Leibniz. Both men owed a very great deal to their immediate predecessors in the development of the new analysis, andthe resulting formulations of Newton and Leibniz were most probably the results of a common anterior, rather than a reciprocal coincident, influence.
Attempts have been made by historians of the calculus to trace two distinctly different threads of development: one, the kinematic, leading to newton through plato, Archimedes, Galileo, cavalieri, and barrow; and the other, the atomistic, tendingtoward Leibniz through Democritus, kepler, fermat ,pascal and Huygens. There is however a complete lack of recognition of such a cleavage by the mathematicians involved, nor can we now distinguish the views and methods of the one “group”, throughout the seventeenth century, from those of the other. Galileo, cavalieri, Torricelli, and barrow used both fluxionary and infinitesimal considerations, and...