is
| a1n1 a2n2 ... | Γ (n1/α1) Γ (n2/α2) ... | . | |
| α1 α2 ... | Γ (1 + n1/α1 + n2/α2 + ... ) |
If, however, the object aimed at is an integration through the volume of an ellipsoid it is simpler to reduce the domain of integration to that within a sphere of radius unity by the transformation x = aξ, y = bη, z = cζ, and then to perform the integration through the sphere by transforming to polar coordinates as in (ix).
56. Methods of approximate integration began to be devised very early. Kepler’s practical measurement of the focal sectors Approximate and Mechanical Integration. of ellipses (1609) was an approximate integration, as also was the method for the quadrature of the hyperbola given by James Gregory in the appendix to his Exercitationes geometricae (1668). In Newton’s Methodus differentialis (1711) the subject was taken up systematically. Newton’s object was to effect the approximate quadrature of a given curve by making a curve of the type
y = a0 + a1 x + a2 x2 + ... + an xn
pass through the vertices of (n + 1) equidistant ordinates of the given curve, and by taking the area of the new curve so determined as an approximation to the area of the given curve. In 1743 Thomas Simpson in his Mathematical Dissertations published a very convenient rule, obtained by taking the vertices of three consecutive equidistant ordinates to be points on the same parabola. The distance between the extreme ordinates corresponding to the abscissae x = a and x = b is divided into 2n equal segments by ordinates y1, y2, ... y2n−1, and the extreme ordinates are denoted by y0, y2n. The vertices of the ordinates y0, y1, y2 lie on a parabola with its axis parallel to the axis of y, so do the vertices of the ordinates y2, y3, y4, and so on. The area is expressed approximately by the formula
{ (b − a)/6n } [y0 + y2n + 2 (y2 + y4 + ... + y2n−2) + 4 (y1 + y3 + ... + y2n−1) ],
which is known as Simpson’s rule. Since all simple integrals can be represented as areas such rules are applicable to approximate integration in general. For the recent developments reference may be made to the article by A. Voss in Ency. d. Math. Wiss., Bd. II., A. 2 (1899), and to a monograph by B. P. Moors, Valeur approximative d’une intégrale définie (Paris, 1905).
Many instruments have been devised for registering mechanically the areas of closed curves and the values of integrals. The best known are perhaps the “planimeter” of J. Amsler (1854) and the “integraph” of Abdank-Abakanowicz (1882).
Bibliography.—For historical questions relating to the subject the chief authority is M. Cantor, Geschichte d. Mathematik (3 Bde., Leipzig, 1894-1901). For particular matters, or special periods, the following may be mentioned: H. G. Zeuthen, Geschichte d. Math. im Altertum u. Mittelalter (Copenhagen, 1896) and Gesch. d. Math. im XVI. u. XVII. Jahrhundert (Leipzig, 1903); S. Horsley, Isaaci Newtoni opera quae exstant omnia (5 vols., London, 1779-1785); C. I. Gerhardt, Leibnizens math. Schriften (7 Bde., Leipzig, 1849-1863); Joh. Bernoulli, Opera omnia (4 Bde., Lausanne and Geneva, 1742). Other writings of importance in the history of the subject are cited in the course of the article. A list of some of the more important treatises on the differential and integral calculus is appended. The list has no pretensions to completeness; in particular, most of the recent books in which the subject is presented in an elementary way for beginners or engineers are omitted.—L. Euler, Institutiones calculi differentialis (Petrop., 1755) and Institutiones calculi integralis (3 Bde., Petrop., 1768-1770); J. L. Lagrange, Leçons sur le calcul des fonctions (Paris, 1806, Œuvres, t. x.), and Théorie des fonctions analytiques (Paris, 1797, 2nd ed., 1813, Œuvres, t. ix.); S. F. Lacroix, Traité de calcul diff. et de calcul int. (3 tt., Paris, 1808-1819). There have been numerous later editions; a translation by Herschel, Peacock and Babbage of an abbreviated edition of Lacroix’s treatise was published at Cambridge in 1816. G. Peacock, Examples of the Differential and Integral Calculus (Cambridge, 1820); A. L. Cauchy, Résumé des leçons ... sur le calcul infinitésimale (Paris, 1823), and Leçons sur le calcul différentiel (Paris, 1829; Œuvres, sér. 2, t. iv.); F. Minding, Handbuch d. Diff.-u. Int.-Rechnung (Berlin, 1836); F. Moigno, Leçons sur le calcul diff. (4 tt., Paris, 1840-1861); A. de Morgan, Diff. and Int. Calc. (London, 1842); D. Gregory, Examples on the Diff. and Int. Calc. (2 vols., Cambridge, 1841-1846); I. Todhunter, Treatise on the Diff. Calc. and Treatise on the Int. Calc. (London, 1852), numerous later editions; B. Price, Treatise on the Infinitesimal Calculus (2 vols., Oxford, 1854), numerous later editions; D. Bierens de Haan, Tables d’intégrales définies (Amsterdam, 1858); M. Stegemann, Grundriss d. Diff.- u. Int.-Rechnung (2 Bde., Hanover, 1862) numerous later editions; J. Bertrand, Traité de calc. diff. et int. (2 tt., Paris, 1864-1870); J. A. Serret, Cours de calc. diff. et int. (2 tt., Paris, 1868, 2nd ed., 1880, German edition by Harnack, Leipzig, 1884-1886, later German editions by Bohlmann, 1896, and Scheffers, 1906, incomplete); B. Williamson, Treatise on the Diff. Calc. (Dublin, 1872), and Treatise on the Int. Calc. (Dublin, 1874) numerous later editions of both; also the article “Infinitesimal Calculus” in the 9th ed. of the Ency. Brit.; C. Hermite, Cours d’analyse (Paris, 1873); O. Schlömilch, Compendium d. höheren Analysis (2 Bde., Leipzig, 1874) numerous later editions; J. Thomae, Einleitung in d. Theorie d. bestimmten Integrale (Halle, 1875); R. Lipschitz, Lehrbuch d. Analysis (2 Bde., Bonn, 1877, 1880); A. Harnack, Elemente d. Diff.- u. Int.-Rechnung (Leipzig, 1882, Eng. trans. by Cathcart, London, 1891); M. Pasch, Einleitung in d. Diff.- u. Int.-Rechnung (Leipzig, 1882); Genocchi and Peano, Calcolo differenziale (Turin, 1884, German edition by Bohlmann and Schepp, Leipzig, 1898, 1899); H. Laurent, Traité d’analyse (7 tt., Paris, 1885-1891); J. Edwards, Elementary Treatise on the Diff. Calc. (London, 1886), several later editions; A. G. Greenhill, Diff. and Int. Calc. (London, 1886, 2nd ed., 1891); É. Picard, Traité d’analyse (3 tt., Paris, 1891-1896); O. Stolz, Grundzüge d. Diff.- u. Int.-Rechnung (3 Bde., Leipzig, 1893-1899); C. Jordan, Cours d’analyse (3 tt., Paris, 1893-1896); L. Kronecker, Vorlesungen ü. d. Theorie d. einfachen u. vielfachen Integrale (Leipzig, 1894); J. Perry, The Calculus for Engineers (London, 1897); H. Lamb, An Elementary Course of Infinitesimal Calculus (Cambridge, 1897); G. A. Gibson, An Elementary Treatise on the Calculus (London, 1901); É. Goursat, Cours d’analyse mathématique (2 tt., Paris, 1902-1905); C.-J. de la Vallée Poussin, Cours d’analyse infinitésimale (2 tt., Louvain and Paris, 1903-1906); A. E. H. Love, Elements of the Diff. and Int. Calc. (Cambridge, 1909); W. H. Young, The Fundamental Theorems of the Diff. Calc. (Cambridge, 1910). A résumé of the infinitesimal calculus is given in the articles “Diff.- u. Int-Rechnung” by A. Voss, and “Bestimmte Integrale” by G. Brunel in Ency. d. math. Wiss. (Bde. ii. A. 2, and ii. A. 3, Leipzig, 1899, 1900). Many questions of principle are discussed exhaustively by E. W. Hobson, The Theory of Functions of a Real Variable (Cambridge, 1907).