Of the 17th century led to new limiting processes.į ′ ( x ) = 2 x įor this sum is F( b) − F( a). Presented by the analytical geometry and natural philosophy Rigorous limiting processes (see Function). The methods of Euclid and Archimedes were specimens of Polygon circumscribed to the same circle can be made less thanĪny assigned area by increasing the number of sides of the polygon. A simple example of itsĪpplication is the 6th proposition of Archimedes’ treatise On the Sphere and Cylinder, in which it is proved that the area containedīetween a regular polygon inscribed in a circle and a similar Of the magnitude to be evaluated between two others which canīe brought by a definite process to differ from each other by Of the proposed magnitudes.” The method adopted by Archimedes There will at length remain a magnitude less than the smaller Than its half, and from the remainder more than its half, and so on, “If from the greater of two magnitudes there be taken more Lemma prefixed to the 12th book of Euclid’s Elements as follows: The principle on which it is based was laid down in the One of these methods wasĪfterwards called the “method of exhaustions,” and Of tangents, but they devised methods for investigating the The Greek geometers made little progress with the problem Purpose for which the area of a curve was sought was often toįind something which is not an area-for instance, a length, or aģ. It may be seen that before the invention of the infinitesimalĬalculus the introduction of a curve into discussions of theĬourse of any phenomenon, and the problem of quadraturesįor that curve, were not exclusively of geometrical import the In that example the fixed ordinate vanishes. Galileo’s investigation may serve as an example. ![]() Two ordinates, of which one was regarded as fixed and the otherĪs variable. The area contained between the curve, the axis of abscissae and The “problem of quadratures.” It was sought to determine Was usually presented in a particular form in which it is called The problem of finding the area of a curve Problems of maxima and minima to problems of contact was The problem is to be solved touch each other. Roots, and, when this is the case, the curves by which Or minima arise when a certain equation has equal Of tangents was understood in the sense that maxima The problem of maxima and minima to the problem The problems of Maxima and Minima, Tangents, and Quadratures. The curve at an assigned point, and the problem ofĭetermining the area of the curve. Or a minimum, the problem of drawing a tangent to Problem of finding the points at which the ordinate is a maximum The most prominent problems in regard to a curve were the That the distance through which the body has fallen is represented The same method was used later by many writers,Īmong whom Johannes Kepler and Galileo Galilei may be mentioned. Joined the ends of all the lines of “latitude.” Oresme’s longitudeĪnd latitude were what we should now call the abscissa and Time was represented by the line, straight or curved, which He recognized that the variation of the temperature with the the temperature at the instant, by a length,Ĭalled the “latitude,” measured at right angles to this line. Since some epoch, by a length, called the “longitude,” measuredĪlong a particular line and he represented the other of the two Geometrical representation of Variable Quantities.Īnd afterwards taught at the Collège de Navarre in When it was employed by Nicole Oresme, who studied Variable quantities dates from the 14th century, ![]() To the mathematicians of the 17th century was that of the The guise in which variable quantities presented themselves
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