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II. History.

13. The new limiting processes which were introduced in the development of the higher analysis were in the first instance related to problems of the integral calculus. Johannes Kepler in his Astronomia nova . . . de motibus stellae Martis (1609) stated his laws of planetary motion, to Kepler’s methods
of Integration. the effect that the orbits of the planets are ellipses with the sun at a focus, and that the radii vectores drawn from the sun to the planets describe equal areas in equal times. From these statements it is to be concluded that Kepler could measure the areas of focal sectors of an ellipse. When he made out these laws there was no method of evaluating areas except the Greek methods. These methods would have sufficed for the purpose, but Kepler invented his own method. He regarded the area as measured by the “sum of the radii” drawn from the focus, and he verified his laws of planetary motion by actually measuring a large number of radii of the orbit, spaced according to a rule, and adding their lengths.

Fig. 5.

He had observed that the focal radius vector SP (fig. 5) is equal to the perpendicular SZ drawn from S to the tangent at p to the auxiliary circle, and he had further established the theorem which we should now express in the form—the differential element of the area ASp as Sp turns about S, is equal to the product of SZ and the differential adφ, where a is the radius of the auxiliary circle, and φ is the angle ACp, that is the eccentric angle of P on the ellipse. The area ASP bears to the area ASp the ratio of the minor to the major axis, a result known to Archimedes. Thus Kepler’s radii are spaced according to the rule that the eccentric angles of their ends are equidifferent, and his “sum of radii” is proportional to the expression which we should now write

$\int _{0}^{\phi }(a+ae\cos {\phi })\,d\phi ,$

where e is the eccentricity. Kepler evaluated the sum as proportional to φ + e sin φ.

Kepler soon afterwards occupied himself with the volumes of solids. The vintage of the year 1612 was extraordinarily abundant, and the question of the cubic content of wine casks was brought under his notice. This fact accounts for the title of his work, Nova stereometria doliorum; accessit stereometriae Archimedeae supplementum (1615). In this treatise he regarded solid bodies as being made up, as it were (veluti), of “infinitely” many “infinitely” small cones or “infinitely” thin disks, and he used the notion of summing the areas of the disks in the way he had previously used the notion of summing the focal radii of an ellipse.

14. In connexion with the early history of the calculus it must not be forgotten that the method by which logarithms were invented (1614) was effectively a method of infinitesimals. Natural logarithms were not invented as the indices of a certain base, and the notation e Logarithms. for the base was first introduced by Euler more than a century after the invention. Logarithms were introduced as numbers which increase in arithmetic progression when other related numbers increase in geometric progression. The two sets of numbers were supposed to increase together, one at a uniform rate, the other at a variable rate, and the increments were regarded for purposes of calculation as very small and as accruing discontinuously.

15. Kepler’s methods of integration, for such they must be called, were the origin of Bonaventura Cavalieri’s theory of the summation of indivisibles. The notion of a continuum, such as the area within a closed curve, as being made up of indivisible parts, “atoms” of Cavalieri’s Indivisibles. area, if the expression may be allowed, is traceable to the speculations of early Greek philosophers; and although the nature of continuity was better understood by Aristotle and many other ancient writers yet the unsound atomic conception was revived in the 13th century and has not yet been finally uprooted. It is possible to contend that Cavalieri did not himself hold the unsound doctrine, but his writing on this point is rather obscure. In his treatise Geometria indivisibilibus continuorum nova quadam ratione promota (1635) he regarded a plane figure as generated by a line moving so as to be always parallel to a fixed line, and a solid figure as generated by a plane moving so as to be always parallel to a fixed plane; and he compared the areas of two plane figures, or the volumes of two solids, by determining the ratios of the sums of all the indivisibles of which they are supposed to be made up, these indivisibles being segments of parallel lines equally spaced in the case of plane figures, and areas marked out upon parallel planes equally spaced in the case of solids. By this method Cavalieri was able to effect numerous integrations relating to the areas of portions of conic sections and the volumes generated by the revolution of these portions about various axes. At a later date, and partly in answer to an attack made upon him by Paul Guldin, Cavalieri published a treatise entitled Exercitationes geometricae sex (1647), in which he adapted his method to the determination of centres of gravity, in particular for solids of variable density.

Among the results which he obtained is that which we should now write

$\int _{0}^{x}x^{m}\,dx={\frac {x^{m+1}}{m+1}}$,(m integral).

He regarded the problem thus solved as that of determining the sum of the mth powers of all the lines drawn across a parallelogram parallel to one of its sides.

At this period scientific investigators communicated their results to one another through one or more intermediate persons. Such intermediaries were Pierre de Carcavy and Pater Marin Mersenne; and among the writers thus in communication were Bonaventura Cavalieri, Successors
of Cavalieri. Christiaan Huygens, Galileo Galilei, Giles Personnier de Roberval, Pierre de Fermat, Evangelista Torricelli, and a little later Blaise Pascal; but the letters of Carcavy or Mersenne would probably come into the hands of any man who was likely to be interested in the matters discussed. It often happened that, when some new method was invented, or some new result obtained, the method or result was quickly known to a wide circle, although it might not be printed until after the lapse of a long time. When Cavalieri was printing his two treatises there was much discussion of the problem of quadratures. Roberval (1634) regarded an area as made up of “infinitely” many “infinitely” narrow strips, each of which may be considered to be a rectangle, and he had similar ideas in regard to lengths and volumes. He knew how to approximate to the quantity which we express by $\int _{0}^{1}x^{m}dx$ by the process of forming the sum

${\frac {0^{m}+1^{m}+2^{m}+\ldots +(n-1)^{m}}{n^{m+1}}},$

and he claimed to be able to prove that this sum tends to 1/(m + 1), as n increases for all positive integral values of m. The method of integrating x^m by forming this sum was found also by Fermat (1636), who stated expressly that he arrived at it by generalizing a method employed by Fermat’s method
of Integration. Archimedes (for the cases m = 1 and m = 2) in his books on Conoids and Spheroids and on Spirals (see T. L. Heath, The Works of Archimedes, Cambridge, 1897). Fermat extended the result to the case where m is fractional (1644), and to the case where m is negative. This latter extension and the proofs were given in his memoir, Proportionis geometricae in quadrandis parabolis et hyperbolis usus, which appears to have received a final form before 1659, although not published until 1679. Fermat did not use fractional or negative indices, but he regarded his problems as the quadratures of parabolas and hyperbolas of various orders. His method was to divide the interval of integration into parts by means of intermediate points the abscissae of which are in geometric progression. In the process of § 5 above, the points M must be chosen according to this rule. This restrictive condition being understood, we may say that Fermat’s formulation of the problem of quadratures is the same as our definition of a definite integral.

The result that the problem of quadratures could be solved for any curve whose equation could be expressed in the form

$y=x^{m}\,$  $(m\neq -1),$

or in the form

$y=a_{1}x_{1}^{m}+a_{2}x_{2}^{m}+\ldots +a_{n}x_{n}^{m}\,$