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 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 “” 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).

INFINITIVE, a form of the verb, properly a noun with verbal functions, but usually taken as a mood (see ). The Latin grammarians gave it the name of infinitus or infinitivus modus, i.e. indefinite, unlimited mood, as not having definite persons or numbers.

 INFLEXION (from Lat. inflectere, to bend), the action of bending inwards, or turning towards oneself, or the condition of being bent or curved. In optics, the term “inflexion” was used by Newton for what is now known as “” (q.v.). For inflexion in geometry see. Inflexion when used of the voice, in speaking or singing, indicates a change in tone, pitch or expression. In (q.v.) inflexion indicates the changes which a word undergoes to bring it into correct relations with the other words with which it is used. In English grammar nouns, pronouns, adjectives (in their degrees of comparison), verbs and adverbs are inflected. Some grammarians, however, regard the inflexions of adverbs more as an actual change in word-formation.

 INFLUENCE (Late Lat. influentia, from influere, to flow in), a word whose principal modern meaning is that of power, control or action affecting others, exercised either covertly or without visible means or direct physical agency. It is one of those numerous terms of (q.v.) which have established themselves in current language. From the stars was supposed to flow an ethereal stream which affected the course of events on the earth and the fortunes and characters of men. For the law as to “undue influence” see.

 INFLUENZA (syn. “grip,” la grippe), a term applied to an infectious febrile disorder due to a specific bacillus, characterized specially by catarrh of the respiratory passages and alimentary canal, and occurring mostly as an epidemic. The Italians in the 17th century ascribed it to the influence of the stars, and hence the name “influenza.” The French name grippe came into use in 1743, and those of petite poste and petit courier in 1762, while général became another synonym in 1780. Apparently the scourge was common; in 1403 and 1557 the sittings of the Paris law courts had to be suspended through it, and in 1427 sermons had to be abandoned through the coughing and sneezing; in 1510 masses could not be sung. Epidemics occurred in 1580, 1676, 1703, 1732 and 1737, and their cessation was supposed to be connected with earthquakes and volcanic eruptions.

The disease is referred to in the works of the ancient physicians, and accurate descriptions of it have been given by medical writers during the last three centuries. These various accounts agree substantially in their narration of the phenomena and course of the disease, and influenza has in all times been regarded as fulfilling all the conditions of an epidemic in its sudden invasion, and rapid and extensive spread. Among the chief epidemics were those of 1762, 1782, 1787, 1803, 1833, 1837 and 1847. It appeared in fleets at sea away from all communication with land, and to such an extent as to disable them temporarily for service. This happened in 1782 in the case of the squadron of Admiral Richard Kempenfelt (1718–1782), which had to return to England from the coast of France in consequence of influenza attacking his crews.

Like cholera and plague, influenza reappeared in the last quarter of the 19th century, after an interval of many years, in epidemic or rather pandemic form. After the year 1848, in which 7963 deaths were directly attributed to influenza in England and Wales, the disease continued prevalent until 1860, with distinct but minor epidemic exacerbations in 1851, 1855 and 1858; during the next decade the mortality dropped rapidly though not steadily, and the diminution continued down to the year 1889, In which only 55 deaths were ascribed to this cause. It is not clear whether the disease ever disappears wholly, and the deaths registered in 1889 are the lowest recorded in any year since the registrar-general’s returns began. Occasionally local outbreaks of illness resembling epidemic influenza have been observed during the period of abeyance, as in Norfolk in 1878 and in Yorkshire in 1887; but whether such outbreaks and the so-called “sporadic” cases are nosologically identical with epidemic influenza is open to doubt. The relation seems rather to be similar to that between Asiatic cholera and “cholera nostras.” Individual cases may be indistinguishable, but as a factor in the public health the difference between sporadic and epidemic influenza is as great and unmistakable as that between the two forms of cholera. This fact, which had been forgotten by some since 1847 and never learnt by others, was brought home forcibly to all by the visitation of 1889.

According to the exhaustive report drawn up by Dr H. Franklin Parsons for the Local Government Board, the earliest appearances were observed in May 1889, and three localities are mentioned as affected at the same time, all widely separated from each other—namely, Bokhara in Central Asia, Athabasca in the north-west Territories of Canada and Greenland. About the middle of October it was reported at Tomsk in Siberia, and by the end of the month at St Petersburg. During November Russia became generally affected, and cases were noticed in Paris, Berlin, Vienna, London and Jamaica (?). In December epidemic influenza became established over the whole of Europe, along the Mediterranean, in Egypt and over a large area in the United States. It appeared in several towns in England, beginning with Portsmouth, but did not become generally