Part 14

[5] The term _landsknecht_, it appears, was not confined to the right bank of the Rhine. The French "lansquenets" came largely from Alsace, according to General Hardy de Périni. In the Italian wars Francis I. had in his service a famous corps called the "black bands" which was recruited, in the lower Rhine countries.

[6] This practice of "maintenance" on a large scale continued to exist in France long afterwards. As late as the battle of Lens (1648) we find figuring in the king of France's army three "regiments of the House of Condé."

[7] Even as late as 1645 a battalion of infantry in England was called a "tercio" or "tertia" (see ARMY; _Spanish army_).

[8] In France it is recorded that the _Gardes françaises_, when warned for duty at the Louvre, used to stroll thither in twos and threes.

[9] About this time there was introduced, for resisting cavalry, the well-known hollow battalion square, which, replacing the former masses of pikes, represented up to the most modern times the defensive, as the line or column represented the offensive formation of infantry.

[10] The Prussian Grenadier battalions in the Silesian and Seven Years' Wars were more and more confined strictly to line-of-battle duties as the irregular light infantry developed in numbers.

[11] Even when the hostile artillery was still capable of fire these masses were used, for in no other formation could the heterogeneous and ill-trained infantry of Napoleon's vassal states (which constituted half of his army) be brought up at all.

[12] Rifles had, of course, been used by corps of light troops (both infantry and mounted) for many years. The British Rifle Brigade was formed in 1800, but even in the Seven Years' War there were rifle-corps or companies in the armies of Prussia and Austria. These older rifles could not compare in rapidity or volume of fire with the ordinary firelock.

[13] The Prussian company was about 250 strong (see below under "Organization"). This strength was adopted after 1870 by practically all nations which adopted universal service. The battalion had 4 companies.

[14] The 1902 edition of _Infantry Training_ indeed treated the new scouts as a thin advanced firing line, but in 1907, at which date important modifications began to be made in the "doctrine" of the British Army, the scouts were expressly restricted to the old-fashioned "skirmishing" duties.

[15] This is no new thing, but belongs, irrespective of armament, to the "War of masses." The king of Prussia's fighting instructions of the 10th of August 1813 lay down the principle as clearly as any modern work.

[16] In the British Service, men whose nerves betray them on the shooting range are ordered more gymnastics (_Musketry Regulations_, 1910).

[17] In 1870 the "preconceived idea" was practically confined to strategy, and the tactical improvisations of the Germans themselves deranged the execution of the plan quite as often as the act of the enemy. Of late years, therefore, the "preconceived idea" has been imposed on tactics also in that country. Special care and study is given to the once despised "early deployments" in cases where a fight is part of the "idea," and to the difficult problem of breaking off the action, when it takes a form that is incompatible with the development of the main scheme.

[18] In February 1910 a new _Infantry Training_ was said to be in preparation. The _I.T._ of 1905 is in some degree incompatible with the later and ruling doctrine of the _F.S. Regulations_, and in the winter of 1909 the Army Council issued a memorandum drawing attention to the different conceptions of the decisive attack as embodied in the latter and as revealed in manoeuvre procedure.

INFANT SCHOOLS. The provision in modern times of systematized training for children below the age when elementary education normally begins may be dated from the village school at Waldbach founded by Jean Frédéric Oberlin in 1774. Robert Owen started an infant school at New Lanark in 1800, and great interest in the question was taken in Great Britain during the early years of the 19th century, leading to the foundation in 1836 of the Home and Colonial School Society for the training of teachers in infant schools; this in turn reacted upon other countries, especially Germany. Further impetus and a new direction were given to the movement by Friedrich W. A. Froebel, and the methods of training adopted for children between the ages of three and six have in most countries been influenced by, if not based on, that system of directed

## activities which was the foundation of the type of "play-school" called

by him the _Kinder Garten_, or "children's garden." The growing tendency in England to lay stress on the mental training of very young children, and to use the "infant school" as preparatory to the elementary school, has led to a considerable reaction; medical officers of health have pointed out the dangers of infection to which children up to the age of five are specially liable when congregated together--also the physical effects of badly ventilated class-rooms, and there is a consensus of opinion that formal mental teaching is directly injurious before the age of six or even seven years. At the same time the increase in the industrial employment of married women, with the consequent difficulty of proper care of young children by the mother in the home, has somewhat shifted the ground from a purely educational to a social and physical aspect. While it is agreed that the ideal place for a young child is the home under the supervision of its mother, the present industrial conditions often compel a mother to go out to work, and leave her children either shut up alone, or free to play about the streets, or in the care of a neighbour or professional "minder." In each case the children must suffer. The provision by a public authority of opportunities for suitable training for such children seems therefore a necessity. The moral advantages gained by freeing the child from the streets, by the superintendence of a trained teacher over the games, by the early inculcation of habits of discipline and obedience; the physical advantages of cleanliness and tidiness, and the opportunity of disclosing incipient diseases and weaknesses, outweigh the disadvantages which the opponents of infant training adduce. It remains to give a brief account of what is done in Great Britain, the United States of America, and certain other countries. A valuable report was issued for the English Board of Education by a Consultative Committee upon the school attendance of children below the age of five (vol. 22 of the _Special Reports_, 1909), which also gives some account of the provision of day nurseries or _crèches_ for babies.

_United Kingdom._--Up to 1905 it was the general English practice since the Education Act of 1870 for educational authorities to provide facilities for the teaching of children between three and five years old whose parents desired it. In 1905, of an estimated 1,467,709 children between those ages, 583,268 were thus provided for in England and Wales. In 1905 the objections, medical and educational, already stated, coupled with the increasing financial strain on the local educational authorities, led to the insertion in the code of that year of Article 53, as follows: "Where the local education authority have so determined in the case of any school maintained by them, children who are under five years may be refused admission to that school." In consequence in 1907 the numbers were found to have fallen to 459,034 out of an estimated 1,480,550 children, from 39.74% in 1905 to 31%. In the older type of infant school stress was laid on the mental preparation of children for the elementary teaching which was to come later. This forcing on of young children was encouraged by the system under which the government grant was allotted; children in the infant division earned an annual grant of 17s. per head, on promotion to the upper school this would be increased to 22s. In 1909 the system was altered; a rate of 21s. 4d. was fixed as the grant for all children above five, and the grant for those below the age was reduced to 13s. 4d. Different methods of training the teachers in these schools as well as the children themselves have been now generally adopted. These methods are largely based on the Froebelian plan, and greater attention is being paid to physical development. In one respect England is perhaps behind the more progressive of other European countries, viz. in providing facilities for washing and attending to the personal needs of the younger children. There is no _femme de service_ as in Belgium on the staff of English schools. While in Ireland the children below the age of five attend the elementary schools in much the same proportion as in England and Wales, in Scotland it has never been the general custom for such children to attend school.

_United States of America._--In no country has the kindergarten system taken such firm root, and the provision made for children below the compulsory age is based upon it. In 1873 there were 42 kindergartens with 1252 pupils; in 1898 the numbers had risen to 2884 with 143,720 pupils; more than half these were private schools, managed by charitable institutions or by individuals for profit. In 1904-1905 there were 3176 public kindergartens with 205,118 pupils.

_Austria Hungary._--Provision in Austria is made for children under six by two types of institution, the Day Nursery (_Kinderbewahranstalten_) and the Kindergarten. In 1872 as the result of a State Commission the Kindergarten was established in the state system of education. Its aim is to "confirm and complete the home education of children under school age, so that through regulated exercise of body and mind they may be prepared for institution in the primary school." No regular teaching in ordinary school subjects is allowed; games, singing and handwork, and training of speech and observation by objects, tales and gardening are the means adopted. The training for teachers in these schools is regulated by law. No children are to be received in a kindergarten til! the beginning of the fourth and must leave at the end of the sixth year. In 1902-1903 there were 77,002 children in kindergartens and 74,110 in the day nurseries. In Hungary a law was passed in 1891 providing for the education and care of children between three and six, either by asyle or nurseries open all the year round in communes which contribute from £830 to £1250 in state taxation, or during the summer in those whose contribution is less. Communes above the higher sum must provide kindergartens. In 1904 there were over 233,000 children in such institutions.

_Belgium._--For children between three and six education and training are provided by _Écoles gardiennes_ or _Jardins d'enfants_. They are free but not compulsory, are provided and managed by the communes, receive a state grant, and are under government inspection. Schools provided by private individuals or institutions must conform to the conditions of the communal schools. There is a large amount of voluntary assistance especially in the provision of clothes and food for the poorer children. The state first recognized these schools in 1833. In 1881 there were 708 schools with accommodation for over 56,000 children; in 1907 there were 2837 and 264,845 children, approximately one-half of the total number of children in the country between the ages of three and six. In 1890 the minister of Public Instruction issued a code of rules on which is based the organization of the _Écoles gardiennes_ throughout Belgium, but some of the communes have regulations of their own. A special examination for teachers in the _Écoles gardiennes_ was started in 1898. All candidates must pass this examination before a _certificat de capacité_ is granted. The training includes a course in Froebelian methods. While Froebel's system underlies the training in these schools, the teaching is directed very much towards the practical education of the child, special stress being laid on manual dexterity. Reading, writing and arithmetic are also allowed in the classes for the older children. A marked feature of the Belgian schools is the close attention paid to health and personal cleanliness. In all schools there is a _femme de service_, not a teacher, but an attendant, whose duty it is to see to the tidiness and cleanliness of the children, and to their physical requirements.

_France._--The first regular infant school was established in Paris at the beginning of the 19th century and styled a _Salle d'essai_. In 1828 a model school, called a _Salle d'asile_, was started, followed shortly by similar institutions all over France. State recognition and inspection were granted, and by 1836 there were over 800 in Paris and the provinces. In 1848 they became establishments of public instruction, and the name _École maternelle_ which they have since borne was given them. Every commune with 2000 inhabitants must have one of these schools or a _Classe enfantine_. Admission is free, but not compulsory, for children between two and six. Food and clothes are provided in exceptional cases. Formal mental instruction is still given to a large extent, and the older children are taught reading, writing and arithmetic. Though the staffs of the school include _femmes de service_, not so much attention is paid to cleanliness as in Belgium, nor is so much stress laid on hygiene. In 1906-1907 there were 4111 public and private _Écoles maternelles_ in France, with over 650,000 pupils. The closing of the clerical schools has led to some diminution in the numbers.

_Germany.___--There are two classes of institution in Germany for children between the ages of 2½ or 3 and 6. These are the _Kleinkinderbewahranstalten_ and _Kindergarten_. The first are primarily social in purpose, and afford a place for the children of mothers who have to leave their homes for work. These institutions, principally conducted by religious or charitable societies, remain open all day and meals are provided. Many of them have a kindergarten attached, and others provide some training on Froebelian principles. The kindergartens proper are also principally in private hands, though most municipalities grant financial assistance. They are conducted on advanced Froebelian methods, and formal teaching in reading, writing and arithmetic is excluded. In Cologne, Düsseldorf, Frankfort and Munich there are municipal schools. The state gives no recognition to these institutions and they form no part of the public system of education.

_Switzerland._--In the German speaking cantons the smaller towns and villages provide for the younger children by _Bewahranstalten_, generally under private management with public financial help. The larger towns provide kindergartens where the training is free but not compulsory for children from four to six. These are generally conducted on Froebel's system and there is no formal instruction. In the French speaking cantons the _Écoles enfantines_ are recognized as the first stage of elementary education. They are free and not compulsory for children from three to six years of age. (C. We.)

INFINITE (from Lat. _in_, not, _finis_, end or limit; cf. _findere_, to cleave), a term applied in common usage to anything of vast size. Strictly, however, the epithet implies the absence of all limitation. As such it is used specially in (1) theology and metaphysics, (2) mathematics.

1. Tracing the history of the world to the earliest date for which there is any kind of evidence, we are faced with the problem that for everything there is a prior something: the mind is unable to conceive an absolute beginning ("ex nihilo nihil"). Mundane distances become trivial when compared with the distance from the earth of the sun and still more of other heavenly bodies: hence we infer infinite space. Similarly by continual subdivision we reach the idea of the infinitely small. For these inferences there is indeed no actual physical evidence: infinity is a mental concept. As such the term has played an important part in the philosophical and theological speculation. In early Greek philosophy the attempt to arrive at a physical explanation of existence led the Ionian thinkers to postulate various primal elements (e.g. water, fire, air) or simply the infinite [Greek: to ápeiron] (see IONIAN SCHOOL). Both Plato and Aristotle devoted much thought to the discussion as to which is most truly real, the finite objects of sense, or the universal idea of each thing laid up in the mind of God; what is the nature of that unity which lies behind the multiplicity and difference of perceived objects? The same problem, variously expressed, has engaged the attention of philosophers throughout the ages. In Christian theology God is conceived as infinite in power, knowledge and goodness, uncreated and immortal: in some Oriental systems the end of man is absorption into the infinite, his perfection the breaking down of his human limitations. The metaphysical and theological conception is open to the agnostic objection that the finite mind of man is by hypothesis unable to cognize or apprehend not only an infinite object, but even the very conception of infinity itself; from this standpoint the Infinite is regarded as merely a postulate, as it were an unknown quantity (cf. [root]-1 in mathematics). The same difficulty may be expressed in another way if we regard the infinite as unconditioned (cf. Sir William Hamilton's "philosophy of the unconditioned," and Herbert Spencer's doctrine of the infinite "unknowable"); if it is argued that knowledge of a thing arises only from the recognition of its differences from other things (i.e. from its limitations), it follows that knowledge of the infinite is impossible, for the infinite is by hypothesis unrelated.

With this conception of _the_ infinite as absolutely unconditioned should be compared what may be described roughly as lesser infinities which can be philosophically conceived and mathematically demonstrated. Thus a point, which is by definition infinitely small, is as compared with a line a unit: the line is infinite, made up of an infinite number of points, any pair of which have an infinite number of points between them. The line itself, again, in relation to the plane is a unit, while the plane is infinite, i.e. made up of an infinite number of lines; hence the plane is described as doubly infinite in relation to the point, and a solid as trebly infinite. This is Spinoza's theory of the "infinitely infinite," the limiting notion of infinity being of a numerical, quantitative series, each term of which is a qualitative determination itself quantitatively little, e.g. a line which is quantitatively unlimited (i.e. in length) is qualitatively limited when regarded as an infinitely small unit of a plane. A similar relation exists in thought between the various grades of species and genera; the highest genus is the "infinitely infinite," each subordinated genus being infinite in relation to the particulars which it denotes, and finite when regarded as a unit in a higher genus.

2. In mathematics, the term "infinite" denotes the result of increasing a variable without limit; similarly, the term "infinitesimal," meaning indefinitely small, denotes the result of diminishing the value of a variable without limit, with the reservation that it never becomes actually zero. The application of these conceptions distinguishes ancient from modern mathematics. Analytical investigations revealed the existence of series or sequences which had no limit to the number of terms, as for example the fraction 1/(1 - x) which on division gives the series. 1 + x + x²+ ...; the discussion of these so-called infinite sequences is given in the articles SERIES and FUNCTION. The doctrine of geometrical continuity (q.v.) and the application of algebra to geometry, developed in the 16th and 17th centuries mainly by Kepler and Descartes, led to the discovery of many properties which gave to the notion of infinity, as a localized space conception, a predominant importance. A line became continuous, returning into itself by way of infinity; two parallel lines intersect in a point at infinity; all circles pass through two fixed points at infinity (the circular points); two spheres intersect in a fixed circle at infinity; an asymptote became a tangent at infinity; the foci of a conic became the intersections of the tangents from the circular points at infinity; the centre of a conic the pole of the line at infinity, &c. In analytical geometry the line at infinity plays an important part in trilinear coordinates. These subjects are treated in GEOMETRY. A notion related to that of infinitesimals is presented in the Greek "method of exhaustion"; the more perfect conception, however, only dates from the 17th century, when it led to the infinitesimal calculus. A curve came to be treated as a sequence of infinitesimal straight lines; a tangent as the extension of an infinitesimal chord; a surface or area as a sequence of infinitesimally narrow strips, and a solid as a collection of infinitesimally small cubes (see INFINITESIMAL CALCULUS).

INFINITESIMAL CALCULUS. 1. The infinitesimal calculus is the body of rules and processes by means of which continuously varying magnitudes are dealt with in mathematical analysis. The name "infinitesimal" has been applied to the calculus because most of the leading results were first obtained by means of arguments about "infinitely small" quantities; the "infinitely small" or "infinitesimal" quantities were vaguely conceived as being neither zero nor finite but in some intermediate, nascent or evanescent, state. There was no necessity for this confused conception, and it came to be understood that it can be dispensed with; but the calculus was not developed by its first founders in accordance with logical principles from precisely defined notions, and it gained adherents rather through the impressiveness and variety of the results that could be obtained by using it than through the cogency of the arguments by which it was established. A similar statement might be made in regard to other theories included in mathematical analysis, such, for instance, as the theory of infinite series. Many, perhaps all, of the mathematical and physical theories which have survived have had a similar history--a history which may be divided roughly into two periods: a period of construction, in which results are obtained from

## partially formed notions, and a period of criticism, in which the

fundamental notions become progressively more and more precise, and are shown to be adequate bases for the constructions previously built upon them. These periods usually overlap. Critics of new theories are never lacking. On the other hand, as E. W. Hobson has well said, "pertinent criticism of fundamentals almost invariably gives rise to new construction." In the history of the infinitesimal calculus the 17th and 18th centuries were mainly a period of construction, the 19th century mainly a period of criticism.

I. _Nature of the Calculus._

Geometrical representation of Variable Quantities.