Part 22

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, _Oeuvres_, t. x.), and _Théorie des fonctions analytiques_ (Paris, 1797, 2nd ed., 1813, _Oeuvres_, 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; _Oeuvres_, 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). (A. E. H. L.)

INFINITIVE, a form of the verb, properly a noun with verbal functions, but usually taken as a mood (see GRAMMAR). 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 "diffraction of light" (q.v.). For inflexion in geometry see CURVE. Inflexion when used of the voice, in speaking or singing, indicates a change in tone, pitch or expression. In grammar (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 astrology (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 CONTRACT.

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 epidemic until the commencement of the new year. In London the full onset of unmistakable influenza dated from the 1st of January 1890. Everywhere it seems to have exhibited the same explosive character when once fully established. In St Petersburg, out of a government staff of 260 men, 220 were taken ill in one night, the 15th of November. During January 1890 the epidemic reached its height in London, and appeared in a large number of towns throughout the British Islands, though it was less prevalent in the north and north-west than in the south. January witnessed a great extension of the disease in Germany, Holland, Switzerland, Austria-Hungary, Italy, Spain and Portugal; but in Russia, Scandinavia and France it was already declining. The period of greatest

## activity in Europe was the latter half of December and the earlier half

of January, with the change of the year for a central point. Other parts of the world affected in January 1890 were Cape Town, Canada, the United States generally, Algiers, Tunis, Cairo, Corsica, Sardinia, Sicily, Honolulu, Mexico, the West Indies and Montevideo. In February the provincial towns of England were most severely affected, the death-rate rising to 27.4, but in London it fell from 28.1 to 21.2, and for Europe generally the back of the epidemic was broken. At the same time, however, it appeared in Ceylon, Penang, Japan, Hong Kong and India; also in West Africa, attacking Sierra Leone, and Gambia in the middle of the month; and finally in the west, where Newfoundland and Buenos Aires were invaded. In March influenza became widely epidemic in India,

## particularly in Bengal and Bombay, and made its appearance in Australia

and New Zealand. In April and May it was epidemic all over Australasia, in Central America, Brazil, Peru, Arabia and Burma. During the summer and autumn it reached a number of isolated islands, such as Iceland, St Helena, Mauritius and Réunion. Towards the close of the year it was reported from Yunnan in the interior of China, from the Shiré Highlands in Central Africa, Shoa in Abyssinia, and Gilgit in Kashmir. In the course of fifteen months, beginning with its undoubted appearance in Siberia in October 1889, it had traversed the entire globe.

The localities attacked by influenza in 1889-1890 appear in no case to have suffered severely for more than a month or six weeks. Thus in Europe and North America generally the visitation had come to an end in the first quarter of 1890. The earliest signs of an epidemic revival on a large scale occurred in March 1891, in the United States and the north of England. It was reported from Chicago and other large towns in the central states, whence it spread eastwards, reaching New York about the end of March. In England it began in the Yorkshire towns, particularly in Hull, and also independently in South Wales. In London influenza became epidemic for the second time about the end of April, and soon afterwards was widely distributed in England and Wales. The large towns in the north, together with London and Wales, suffered much more heavily in mortality than in the previous attack, but the south-west of England, Scotland and Ireland escaped with comparatively little sickness. The same may be said of the European continent generally, except parts of Russia, Scandinavia and perhaps the north of Germany. This second epidemic coincided with the spring and early summer; it had subsided in London by the end of June. The experience of Sheffield is interesting. In 1890 the attack, contrary to general experience, had been undecided, lingering and mild; in 1891 it was very sudden and extremely severe, the death-rate rising to 73.4 during the month of April, and subsiding with equal rapidity. During the third quarter of the year, while Europe was free, the antipodes had their second attack, which was more severe than the first. As in England, it reversed the previous order of things, beginning in the provinces and spreading thence to the capital towns. The last quarter of the year was signalized by another recrudescence in Europe, which reached its height during the winter. All parts, including Great Britain, were severely affected. In England those parts which had borne the brunt of the epidemic in the early part of the year escaped. In fact, these two revivals may be regarded as one, temporarily interrupted by the summer quarter.

The recrudescence at the end of 1891 lasted through mid-winter, and in many places, notably in London, it only reached its height in January 1892, subsiding slowly and irregularly in February and March. Brighton suffered with exceptional severity. The continent of Europe seems to have been similarly affected. In Italy the notifications of influenza were as follow: 1891--January to October, 0; November, 30; December, 6461; 1892--January, 84,543; February, 55,352; March, 28,046; April, 7962; May, 1468; June, 223. Other parts of the world affected were the West Indies, Tunis, Egypt, Sudan, Cape Town Teheran, Tongking and China. In August 1892 influenza was reported from Peru, and later in the year from various places in Europe.

A fourth recrudescence, but of a milder character, occurred in Great Britain in the spring of 1893, and a fifth in the following winter, but the year 1894 was freer from influenza than any since 1890. In 1895 another extensive epidemic took place. In 1896 influenza seemed to have spent its strength, but there was an increased prevalence of the disease in 1897, which was repeated on a larger scale in 1898, and again in 1899, when 12,417 deaths were recorded in England and Wales. This was the highest death-rate since 1892. After this the death-rate declined to half that amount and remained there with the slight upward variations until 1907, in which the total death-rate was 9257. The experience of other countries has been very similar; they have all been subjected to periodical revivals of epidemic influenza at irregular intervals and of varying intensity since its reappearance in 1889, but there has been a general though not a steady decline in its activity and potency. Its behaviour is, in short, quite in keeping with the experience of 1847-1860, though the later visitation appears to have been more violent and more fatal than the former. Its diffusion was also more rapid and probably more extensive.

The foregoing general summary may be supplemented by some further details of the incidence in Great Britain. The number of deaths directly attributed to influenza, and the death-rates per million in each year in England and Wales, are as follow:--

+------+--------+-------------+ | Year.| Deaths.| Death-rates | | | | per million.| +------+--------+-------------+ | 1890 | 4,523 | 157 | | 1891 | 16,686 | 574 | | 1892 | 15,737 | 534 | | 1893 | 9,669 | 325 | | 1894 | 6,625 | 220 | | 1895 | 12,880 | 424 | | 1896 | 3,753 | 122 | | 1897 | 6,088 | 196 | | 1898 | 10,405 | 331 | | 1899 | 12,417 | 389 | | 1900 | 16,245 | 504 | | 1901 | 5,666 | 174 | | 1902 | 7,366 | 223 | | 1903 | 6,322 | 189 | | 1904 | 5,694 | 168 | | 1905 | 6,953 | 204 | | 1906 | 6,310 | 183 | | 1907 | 9,257 | 265 | +------+--------+-------------+

It is interesting to compare these figures with the corresponding ones for the previous visitation:--

+------+--------+-------------+ | Year.| Deaths.| Death-rates | | | | per million.| +------+--------+-------------+ | 1847 | 4,881 | 285 | | 1848 | 7,963 | 460 | | 1849 | 1,611 | 92 | | 1850 | 1,380 | 78 | | 1851 | 2,152 | 120 | | 1852 | 1,359 | 76 | | 1853 | 1,789 | 99 | | 1854 | 1,061 | 58 | | 1855 | 3,568 | 193 | +------+--------+-------------+

The two sets of figures are not strictly comparable, because, during the first period, notification of the cause of death was not compulsory; but it seems clear that the later wave was much the more deadly. The average annual death-rate for the nine years is 320 in the one case against 162 in the other, or as nearly as possible double. In both epidemic periods the second year was far more fatal than the first, and in both a marked revival took place in the ninth year; in both also an intermediate recrudescence occurred, in the fifth year in one case, in the sixth in the other. The chief point of difference is the sudden and marked drop in 1849-1850, against a persistent high mortality in 1892-1893, especially in 1892, which was nearly as fatal as 1891.

To make the significance of these epidemic figures clear, it should be added that in the intervening period 1861-1889 the average annual death-rate from influenza was only fifteen, and in the ten years immediately preceding the 1890 outbreak it was only three. Moreover, in epidemic influenza, the mortality directly attributed to that disease is only a fraction of that actually caused by it. For instance, in January 1890 the deaths from influenza in London were 304, while the excess of deaths from respiratory diseases was 1454 and from all causes 1958 above the average.

We have seen above that the mortality was far greater in the second epidemic year than in the first, and this applies to all parts of England, and to rural as well as to urban communities, as the following table shows:--

_Deaths from Influenza._

+--------------------------------------+------+------+ | | 1890.| 1891.| +--------------------------------------+------+------+ | London | 624 | 2302 | | 24 Great Towns over 80,000 population| 439 | 2417 | | 35 Towns between 20,000 and 80,000 | 186 | 765 | | 21 Towns between 10,000 and 20,000 | 46 | 196 | | 60 Towns under 10,000 | 62 | 196 | | 85 Rural Sanitary Districts | 317 | 841 | +--------------------------------------+------+------+

In spite of these figures, it appears that the 1890 attack, which was in general much more sudden in its onset than that of 1891, also caused a great deal more sickness. More people were "down with influenza," though fewer died. For Instance, the number of persons treated at the Middlesex Hospital in the two months' winter epidemic of 1890 was 1279; in the far more fatal three months' spring epidemic of 1891 it was only 726. One explanation of this discrepancy between the incidence of sickness and mortality is that in the second attack, which was more protracted and more insidious, the stress of the disease fell more upon the lungs. Another is that its comparative mildness, combined with the time of year, in itself proved dangerous, because it tempted people to disregard the illness, whereas in the first epidemic they were too ill to resist. On the whole, rural districts showed a higher death-rate than towns, and small towns a higher one than large ones in both years. This is explained by the age distribution in such localities; influenza being

## particularly fatal to aged people, though no age is exempt. Certain

counties were much more severely affected than others. The eastern counties, namely, Essex, Suffolk and Norfolk, together with Hampshire and one or two others, escaped lightly in both years; the western counties, namely, North and South Wales, with the adjoining counties of Monmouth, Hereford and Shropshire, suffered heavily in both years.

It will be convenient to discuss _seriatim_ the various points of interest on which light has been thrown by the experience described above.