![]() ![]() In that example the fixed ordinate vanishes. Galileo's investigation may serve as an example. The problem of finding the area of a curve was usually presented in a particular form in which it is called the " problem of quadratures." It was sought to determine the area contained between the curve, the axis of abscissae and two ordinates, of which one was regarded as fixed and the other as variable. The reduction of problems of maxima and minima to problems of contact was known to Pappus. The relation of the problem of maxima and minima to the problem of tangents was understood in the sense that maxima or minima arise when a certain equation has equal roots, and, when this is the case, the curves by which the problem is to be solved touch each other. The most prominent problems in regard to a curve were the problem of finding the points at which the ordinate is a maximum or a minimum, the problem of drawing a tangent to the curve at an assigned point, and the problem of determining the area of the curve. The velocity being proportional to the time, the " curve " obtained is a straight line OB, and Galileo showed that the distance through which the body has fallen is represented by the area of the triangle OAB. In Galileo's investigation of the motion of falling bodies (1638) the abscissa OA represents the time during which a body has been falling, and the ordinate AB represents the velocity acquired during that time (see fig. The same method was used later by many writers, among whom Johannes Kepler and Galileo Galilei may be mentioned. He recognized that the variation of the temperature with the time was represented by the line, straight or curved, which joined the ends of all the lines of " latitude." Oresme's Iongitude and latitude were what we should now call the abscissa and ordinate. the temperature at the instant, by a length, called the " latitude," measured at right angles to this line. Since some epoch, by a length, called the "longitude," measured along a particular line and he represented the other of the two quantities, e.g. He represented one of two Variable variable quantities, e.g. ![]() This method of representing variableuantities dates from the 14th century, quantities 4 ys metrical when it was employed by Nicole Oresme, who studied represent- and afterwards taught at the College de Navarre in ation of Paris from 1348 to 1361. The guise in which variable quantities presented themselves to the mathematicians of the 17th century was that of the lengths of variable lines. Hobson has well said, " pertinent criticism of fundamentals almost invariably gives rise to new construction." In the history of the infinitesimal calculus the 17th and 18th centuries were mainly a period of construction, the 19th century mainly a period of criticism. Critics of new theories are never lacking. Many, perhaps all, of the mathematical and physical theories which have survived have had a similar history - a history which may be divided roughly into two periods: a period of construction, in which results are obtained from partially formed notions, and a period of criticism, in which the fundamental notions become progressively more and more precise, and are shown to be adequate bases for the constructions previously built upon them. ![]() A similar statement might be made in regard to other theories included in mathematical analysis, such, for instance, as the theory of infinite series. There was no necessity for this confused conception, and it came to be understood that it can be dispensed with but the calculus was not developed by its first founders in accordance with logical principles from precisely defined notions, and it gained adherents rather through the impressiveness and variety of the results that could be obtained by using it than through the cogency of the arguments by which it was established. The name " infinitesimal " has been applied to the calculus because most of the leading results were first obtained by means of arguments about " infinitely small " quantities the " infinitely small " or " infinitesimal " quantities were vaguely conceived as being neither zero nor finite but in some intermediate, nascent or evanescent, state. The infinitesimal calculus is the body of rules and processes by means of which continuously varying magnitudes are dealt with in mathematical analysis.
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