INFINITESIMAL CALCULUS T HISTORICAL INTRODUCTION. HE mathematical and physical sciences owe their present great development to the introduction of the infinitesimal calculus. The power, for example, of that calculus as an instrument of analysis has vastly extended the science of geometry, so that the investigations of the ancient Greeks go but a short way into the field of know ledge which has been laid open by the modern method; the discoveries of Archimedes and Apollonius are now easy deductions from its more extended results. So long as the early geometers confined their speculations to the comparison of the areas of rectilinear figures they encountered little difficulty. They readily showed that the determination of the area of any such figure can be reduced to that of a rectangle, or of a square, and thus be completely effected. This process of finding areas was named the " method of quadratures." It failed, however, when they attempted to determine the areas bounded by curved lines, or the surfaces of the elementary solids such as the right cone and the sphere. In treating of these the ancients found it necessary to introduce new notions and modes of demonstration into geometry, and the diffi culty of comparing the areas of curvilinear with those of rectilinear figures gave rise to the " method of exhaus tions." The fundamental principle of this method con sists in conceiving the continual approach of two varying magnitudes to a fixed intermediate magnitude, with which they never become identical, though they may approach it to within less than any assignable difference. For example, a polygon may be inscribed in a circle, and another cir cumscribed to it, each differing from it by less than any assignable area ; hence the ancients may have concluded that areas of circles have to each other the same ratio as the similar polygons inscribed in or circumscribed to them, that is, the ratio of the squares of the radii. But, as this kind of proof was of a different nature from that by which the more elementary doctrines were established, the Greek geometers fortified it by a reductio ad absurdum, proving, in the above example, that the square of the radius of one circle is to that of another as the area of the former is to a space which is neither less nor greater than the latter, and therefore exactly equal to it. rchi- By the aid of this method Archimedes arrived at his edes. great geometrical discoveries. He determined that the ratio of the circumference to the diameter of a circle lies between 3| and 3|, by considering the regular polygons of ninety-six sides which may be inscribed in or circum scribed to the circle. He proved that the area of a segment of a parabola cut off by any chord equals two- thirds of a parallelogram included between the chord and the parallel tangent to the curve. He determined the f quadrature of the ellipse. In the curves named after him the " spirals of Archimedes," he showed how to draw a tangent at any point, and also determined the area of any portion. In space of three dimensions, Archimedes proved that the surface of a sphere equals four times that of one of its great circles, that the surface of a spherical cap is equal to the area of a circle the length of whose radius is the distance from the vertex of the cap to any point on its bounding circle; that a sphere has a volume which is two-thirds of that of a cylinder circumscribed to it, and that their surfaces are in the same ratio. Further, the same method of exhaustions furnished Archimedes with the cubature of conoids and spheroids, as he termed surfaces generated by the revolution of the parabola, the hyperbola, and the ellipse. During nearly two thousand years no new method Kepler, enabled mathematicians to rise to a higher generality than that attained in the works of the great Greek geometers. The celebrated Kepler was the first to extend the results of Archimedes. In his treatise entitled Nova Stereometria Dolionmi; accessit stereometric Archimedeae supplementum (1G15), 1 he discussed a number of solids of revolution, for example, those formed by the revolution of a conic section about any ordinate, or a tangent at the vertex, or any line within or without the curve. Thus ho con sidered some ninety new solids, and proposed problems concerning them; of these problems he resolved only a few of the most simple. In this treatise he introduced for the first time the name and notion of " infinity " into the language of geometry. Thus, ho considered a circle as composed of an "infinite" number of triangles, having their common vertex at the centre, and forming the circumference by their bases. In like manner lie regarded a cone as composed of an infinite number of pyramids, having their vertices at its vertex, and stand ing on an infinite number of triangular bases, bounded by the circular base of the cone. It may also be noted that Kepler was the first to observe that the increment of a variable the ordinate of a curve, for example is evan escent for values infinitely near a maximum or minimum value of the variable. This remark contains the germ of the rule for determining "maxima" and "minima," given by Format about twenty years subsequently. Several years after Kepler had given his method of Cava- determining volumes of revolution, another celebrated lieri - theory, of a similar kind, the " geometry of indivisibles " (1635) of Cavalieri, professor of geometry at Bologna, marks an epoch in the progress which science has made in modern times. In this work lines were considered as composed of an infinite number of points, surfaces of lines, and solids of surfaces. For example, if the perpendicular of a triangle be divided into an infinite number of equal parts, and through each point of division a line be drawn parallel to the base and terminated by the sides, then, according to Cavalieri, we may consider the area of the triangle as the sum of all these parallel lines, regarded as its elements. Again, as these parallels form a series in arithmetical progression, of which the first term is zero, this sum is represented by half the product of the last term and the number of terms. Now the base js the last term, and the altitude measures the number of terms ; thus he deduced the ordinary expression for the area of a triangle. Cavalieri applied his method to a number of problems, such as finding the volumes of pyramids, the areas of certain simple curves of the parabolic species, the determination of centres of gravity, &c. ; and it is remarkable that he was the first who gave an accurate demonstration of the well- known properties of the centre of gravity, originally announced by Pappus, but commonly called Guldin s theorems. It is accordingly to Cavalieri, and not to Guldin, that the credit is due of having made the first advance beyond Pappus. Cavalieri s method is analogous to that employed in the integral calculus, the " indivisible " being that which has since been styled the " differential element " of the integral. J This work is enlarged from his earlier Stereometria Dolionim Vinariorum (1605), which originated in a dispute with a eller ol wine as to the proper method of gauging the contents , accounts for its strange title.
