ew concepts, which did not fit into the old formal-logical categories of mathematics, were justified since they corresponded to the relationships of the real world. For example, the “reality” of the concept of the derivative stemmed from the “reality” of the concept of velocity in mechanics; therefore, the problem was not whether this concept could be justified logically but only how this was to be done.
The mathematical achievements of the 17th century began with the discovery of logarithms by J. Napier, who published his tables in 1614. In 1637, Descartes published Geometry, which contains the foundations of coordinate geometry and a classification of curves in which curves were subdivided into algebraic and transcendental. The real roots of an equation of any degree were investigated in algebra in close relation to the possibility of representing the roots of an equation P(x) = 0 by the points of intersection of the curve y = P(x) with the axis of the abscissas (Descartes, Newton, M. Rolle). P. Fermat’s investigations of maxima and minima and his search for tangents to curves in essence contained the methods of the differential calculus, but the methods themselves had not yet been isolated or developed. Another source of infinitesimal analysis was the development of the method of “indivisibles” by Kepler (1615) and B. Cavalieri (1635), both of whom applied this method to the determination of the volumes of solids of revolution and other problems. Thus, the foundations of the differential calculus were essentially created in geometric form.
A parallel development was the study of infinite series. The properties of very simple series, beginning with the geometric progression, were studied by J. Wallis (1685). N. Mercator (1668) expanded In (1 + x) in a power series. Newton obtained (1665-69) the binomial formula for any exponent and the power series expansions of the functions ex, sin x, and arc sin x. Nearly all 17th-century mathematicians, for example, Wallis, Huygens, Leibniz and Jakob Bernoulli, contributed to the subsequent development of the concept of infinite series.
With the creation of the coordinate method and the dissemination of the concept of directed mechanical quantities (velocity, acceleration), the concept of negative number became clear and acquired intuitive content. On the other hand, complex numbers remained a by-product of the algebraic apparatus, and for the most part continued to be merely an object of unproductive disputes.
The discovery of the differential and integral calculus in the proper sense of the terms dates to the last third of the 17th century. Leibniz, who gave an extensive exposition of the fundamental ideas of the new calculus in articles published between 1682 and 1686, holds priority with respect to this discovery insofar as publication is concerned. But insofar as the time of obtaining the actual results is concerned, there is every reason to believe that priority rests with Newton, who arrived at the fundamental ideas of the differential and integral calculus in 1665-66. In 1669, the manuscript of Newton’s On Analysis by Means of Equations was submitted to the British mathematicians I. Barrow and J. Collins and was acclaimed by British mathematicians. The Method of Fluxions, in which Newton gave a definitive systematic exposition of his theory, was written in 1670-71 but was published only in 1736. Leibniz, however, began his investigations of infinitesimal analysis only in 1673.
Newton and Leibniz were the first to consider in general form the operations of the differentiation and integration of functions, which were basic to the new calculus. They established the relation between these operations (the Newton-Leibniz formula) and worked out a general uniform algorithm for them. However, Newton and Leibniz differed in their approach. The concepts of “fluent” (a variable quantity) and its “fluxion” (rate of change) are the basic concepts for Newton. To the direct problem of finding fluxions and the relations between fluxions for given fluents (differentiation and construction o