Encyclopaedia Britannica, 11th Edition, "Indole" to "Insanity" / Volume 14, Slice 5

THE ENCYCLOPÆDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION

ELEVENTH EDITION

VOLUME XIV SLICE V
Indole to Insanity

Articles in This Slice

INDOLE, or Beńzopyrrol, C8H7N, a substance first prepared by A. Baeyer in 1868. It may be synthetically obtained by distilling oxindole (C8H8NO) with zinc dust; by heating ortho-nitrocinnamic acid with potash and iron filings; by the reduction of indigo blue; by the action of sodium ethylate on ortho-aminochlorstyrene; by boiling aniline with dichloracetaldehyde; by the dry distillation of ortho-tolyloxamic acid; by heating aniline with dichloracetal; by distilling a mixture of calcium formate and calcium anilidoacetate; and by heating pyruvic acid phenyl hydrazone with anhydrous zinc chloride. It is also formed in the pancreatic fermentation of albumen, and, in small quantities, by passing the vapours of mono- and dialkyl-anilines through a red-hot tube. It crystallizes in shining leaflets, which melt at 52° C. and boil at 245° C. (with decomposition), and is volatile in a current of steam. It is a feeble base, and gives a cherry-red coloration with a pine shaving. Many derivatives of indole are known. B-methylindol or skatole occurs in human faeces.

INDONESIAN, a term invented by James Richardson Logan to describe the light-coloured non-Malay inhabitants of the Eastern Archipelago. It now denotes all those peoples of Malaysia and Polynesia who are not to be classified as Malays or Papuans, but are of Caucasic type. Among these are the Battaks of north Sumatra; many of the Bornean Dyaks and Philippine Islanders, and the large brown race of east Polynesia which includes Samoans, Maoris, Tongans, Tahitians, Marquesas Islanders and the Hawaiians.

See J. Richardson Logan, The Languages and Ethnology of the Indian Archipelago (1857).

INDORE, a native state of India in the central India agency, comprising the dominions of the Maharaja Holkar. Its area, exclusive of guaranteed holdings on which it has claims, is 9500 sq. m. and the population in 1901 was 850,690, showing a decrease of 23% in the decade, owing to the results of famine. As in the case of most states in central India the territory is not homogeneous, but distributed over several political charges. It has portions in four out of the seven charges of central India, and in one small portion in the Rajputana agency. The Vindhya range traverses the S. division of the state in a direction from east to west, a small part of the territory lying to the north of the mountains, but by much the larger part to the south. The latter is a portion of the valley of the Nerbudda, and is bounded on the south by the Satpura hills. Basalt and other volcanic formations predominate in both ranges, although there is also much sandstone. The Nerbudda flows through the state; and the valley at Mandlesar, in the central part, is between 600 and 700 ft. above the sea. The revenue is estimated at £350,000. The metre gauge railway from Khandwa to Mhow and Indore city, continued to Neemuch and Ajmere, was constructed in 1876.

The state had its origin in an assignment of lands made early in the 18th century to Malhar Rao Holkar, who held a command in the army of the Mahratta Peshwa. Of the Dhangar or shepherd caste, he was born in 1694 at the village of Hol near Poona, and from this circumstance the family derives its surname of Holkar. Before his death in 1766 Malhar Rao had added to his assignment large territorial possessions acquired by his armed power during the confusion of the period. By the end of that century the rulership had passed to another leader of the same clan, Tukoji Holkar, whose son, Jaswant Rao, took an important part in the contest for predominance in the Mahratta confederation. He did not, however, join the combined army of Sindha and the raja of Berar in their war against the British in 1803, though after its termination he provoked hostilities which led to his complete discomfiture. At first he defeated a British force that had marched against him under Colonel Monson; but when he made an inroad into British territory he was completely defeated by Lord Lake, and compelled to sign a treaty which deprived him of a large portion of his possessions. After his death his favourite mistress, Tulsi Bai, assumed the regency, until in 1817 she was murdered by the military commanders of the Indore troops, who declared for the peshwa on his rupture with the British government. After their defeat at Mehidpur in 1818, the state submitted by treaty to the loss of more territory, transferred to the British government its suzerainty over a number of minor tributary states, and acknowledged the British protectorate. For many years afterwards the administration of the Holkar princes was troubled by intestine quarrels, misrule and dynastic contentions, necessitating the frequent interposition of British authority; and in 1857 the army, breaking away from the chief’s control, besieged the British residency, and took advantage of the mutiny of the Bengal sepoys to spread disorder over that part of central India. The country was pacified after some fighting. In 1899 a British resident was appointed to Indore, which had formerly been directly under the agent to the governor-general in central India. At the same time a change was made in the system of administration, which was from that date carried on by a council. In 1903 the Maharaja, Shivaji Rao Holkar, G.C.S.I., abdicated in favour of his son Tukoji Rao, a boy of twelve, and died in 1908.

The City of Indore is situated 1738 ft. above the sea, on the river Saraswati, near its junction with the Khan. Pop. (1901) 86,686. These figures do not include the tract assigned to the resident, known as “the camp” (pop. 11,118), which is under British administration. The city is one of the most important trading centres in central India.

Indore Residency, a political charge in central India, is not co-extensive with the state, though it includes all of it except some outlying tracts. Area, 8960 sq. m.; pop. (1901) 833,410.

(J. S. Co.)

INDORSEMENT, or Endorsement (from Med. Lat. indorsare, to write upon the dorsum, or back), anything written or printed upon the back of a document. In its technical sense, it is the writing upon a bill of exchange, cheque or other negotiable instrument, by one who has a right to the instrument and who thereby transmits the right and incurs certain liabilities. See [Bill of Exchange].

INDO-SCYTHIANS, a name commonly given to various tribes from central Asia, who invaded northern India and founded kingdoms there. They comprise the Sakas, the Yue-Chi or Kushans and the Ephthalites or Hūnas.

INDRA, in early Hindu mythology, god of the clear sky and greatest of the Vedic deities. The origin of the name is doubtful, but is by some connected with indu, drop. His importance is shown by the fact that about 250 hymns celebrate his greatness, nearly one-fourth of the total number in the Rig Veda. He is represented as specially lord of the elements, the thunder-god. But Indra was more than a great god in the ancient Vedic pantheon. He is the patron-deity of the invading Aryan race in India, the god of battle to whose help they look in their struggles with the dark aborigines. Indra is the child of Dyaus, the Heaven. In Indian art he is represented as a man with four arms and hands; in two he holds a lance and in the third a thunderbolt. He is often painted with eyes all over his body and then he is called Sahasraksha, “the thousand eyed.” He lost much of his supremacy when the triad Brahma, Siva and Vishnu became predominant. He gradually became identified merely with the headship of Swarga, a local vice-regent of the abode of the gods.

See A. A. Macdonell, Vedic Mythology (Strassburg, 1897).

INDRE, a department of central France, formed in 1790 from parts of the old provinces of Berry, Orléanais, Marche and Touraine. Pop. (1906) 290,216. Area 2666 sq. m. It is bounded N. by the department of Loir-et-Cher, E. by Cher, S. by Creuse and Haute-Vienne, S.W. by Vienne and N.W. by Indre-et-Loire. It takes its name from the river Indre, which flows through it. The surface forms a vast plateau divided into three districts, the Boischaut, Champagne and Brenne. The Boischaut is a large well-wooded plain comprising seven-tenths of the entire area and covering the south, east and centre of the department. The Champagne, a monotonous but fertile district in the north, produces abundant cereal crops, and affords excellent pasturage for large numbers of sheep, celebrated for the fineness of their wool. The Brenne, which occupies the west of the department, was formerly marshy and unhealthy, but draining and afforestation have brought about considerable improvement.

The department is divided into the arrondissements of Châteauroux, Le Blanc, La Châtre and Issoudun, with 23 cantons and 245 communes. At Neuvy-St-Sépulchre there is a circular church of the 11th century, to which a nave was added in the 12th century, and at Mézières-en-Brenne there is an interesting church of the 14th century. At Levroux there is a fine church of the 13th century and the remains of a feudal fortress, and there is a magnificent château in the Renaissance style at Valençay.

INDRE-ET-LOIRE, a department of central France, consisting of nearly the whole of the old province of Touraine and of small portions of Orléanais, Anjou and Poitou. Pop. (1906) 337,916. Area 2377 sq. m. It is bounded N. by the departments of Sarthe and Loir-et-Cher, E. by Loir-et-Cher and Indre, S. and S.W. by Vienne and W. by Maine-et-Loire. It takes its name from the Loire and its tributary the Indre, which enter it on its eastern border and unite not far from its western border. The other chief affluents of the Loire in the department are the Cher, which joins it below Tours, and the Vienne, which waters the department’s southern region. Indre-et-Loire is generally level and comprises the following districts: the Gâtine, a pebbly and sterile region to the north of the Loire, largely consisting of forests and heaths with numerous small lakes; the fertile Varenne or valley of the Loire; the Champeigne, a chain of vine-clad slopes, separating the valleys of the Cher and Indre; the Véron, a region of vines and orchards, in the angle formed by the Loire and Vienne; the plateau of Sainte-Maure, a hilly and unproductive district in the centre of which are found extensive deposits of shell-marl; and in the south the Brenne, traversed by the Claise and the Creuse and forming part of the marshy territory which extends under the same name into Indre.

Indre-et-Loire is divided into the arrondissements of Tours, Loches and Chinon, with 24 cantons and 282 communes. The chief town is Tours, which is the seat of an archbishopric; and Chinon, Loches, Amboise, Chenonceaux, Langeais and Azay-le-Rideau are also important places with châteaus. The Renaissance château of Ussé, and those of Luynes (15th and 16th centuries) and Pressigny-le-Grand (17th century) are also of note. Montbazon possesses the imposing ruins of a square donjon of the 11th and 12th centuries. Preuilly has the most beautiful Romanesque church in Touraine. The Sainte Chapelle (16th century) at Champigny is a survival of a château of the dukes of Bourbon-Montpensier. The church of Montrésor (1532) with its mausoleum of the family of Montrésor; that of St Denis-Hors (12th and 16th century) close to Amboise, with the curious mausoleum of Philibert Babou, minister of finance under Francis I. and Henry II.; and that of Ste Catherine de Fierbois, of the 15th century, are of architectural interest. The town of Richelieu, founded 1631 by the famous minister of Louis XIII., preserves the enceinte and many of the buildings of the 17th century. Megalithic monuments are numerous in the department.

INDRI, a Malagasy word believed to mean “there it goes,” but now accepted as the designation of the largest of the existing Malagasy (and indeed of all) lemurs. Belonging to the family Lemuridae (see [Primates]) it typifies the subfamily Indrisinae, which includes the avahi and the sifakas (q.v.). From both the latter it is distinguished by its rudimentary tail, measuring only a couple of inches in length, whence its name of Indris brevicaudatus. Measuring about 24 in. in length, exclusive of the tail, the indri varies considerably in colour, but is usually black, with a variable number of whitish patches, chiefly about the loins and on the fore-limbs. The forests of a comparatively small tract on the east coast of Madagascar form its home. Shoots, flowers and berries form the food of the indri, which was first discovered by the French traveller and naturalist Pierre Sonnerat in 1780.

(R. L.*)

INDUCTION (from Lat. inducere, to lead into; cf. Gr. ἐπαγωγή), in logic, the term applied to the process of discovering principles by the observation and combination of particular instances. Aristotle, who did so much to establish the laws of deductive reasoning, neglected induction, which he identified with a complete enumeration of facts; and the schoolmen were wholly concerned with syllogistic logic. A new era opens with Bacon, whose writings all preach the principle of investigating the laws of nature with the purpose of improving the conditions of human life. Unluckily his mind was still enslaved by the formulae of the quasi-mechanical scholastic logic. He supposed that natural laws would disclose themselves by the accumulation and due arrangement of instances without any need for original speculation on the part of the investigator. In his Novum Organum there are directions for drawing up the various kinds of lists of instances. For two hundred years after Bacon’s death little was done towards the theory of induction; the reason being, probably, that the practical scientists knew no logic, while the university logicians, with their conservative devotion to the syllogism, knew no science. Whewell’s Philosophy of the Inductive Sciences (1840), the work of a thoroughly equipped scientist, if not of a great philosopher, shows due appreciation of the cardinal point neglected by Bacon, the function of theorizing in inductive research. He saw that science advances only in so far as the mind of the inquirer is able to suggest organizing ideas whereby our observations and experiments are colligated into intelligible system. In this respect J. S. Mill is inferior to Whewell: throughout his System of Logic (1843) he ignores the constitutive work of the mind, and regards knowledge as the merely passive reception of sensuous impressions. His work was intended mainly to reduce the procedure of induction to a regular demonstrative system like that of the syllogism; and it was for this purpose that he formulated his famous Four Methods of Experimental Inquiry. His work has contributed greatly to the systematic treatment of induction. But it must be remarked that his Four Methods are not methods of formal proof, as their author supposed, but methods whereby hypotheses are suggested or tested. The actual proof of an hypothesis is never formal, but always lies in the tests of experiment or observation to which it is subjected.

The current theory of induction as set forth in the standard works is so far satisfactory that it combines the merit of Whewell’s treatment with that of Mill’s; and yet it is plain that there is much for the logician of the future to accomplish. The most important faculty in scientific inquiry is the faculty of suggesting new and valuable hypotheses. But no one has ever given any explanation how the hypotheses arise in the mind: we attribute it to “genius,” which, of course, is no explanation at all. The logic of discovery, in the higher sense of the term, simply has no existence. Another important but neglected province of the subject is the relation of scientific induction to the inductions of everyday life. There are some who think that a study of this relation would quite transform the accepted view of induction. Consider such a piece of reasoning as may be heard any day in a court of justice, a detective who explains how in his opinion a certain burglary was effected. If all reasoning is either deductive or inductive, this must be induction. And yet it does not answer to the accepted definition of induction, “the process of discovering a general principle by observation of particular instances”: what the detective does is to reconstruct a particular crime; he evolves no general principle. Such reasoning is used by every man in every hour of his life: by it we understand what people are doing around us, and what is the meaning of the sense-impressions which we receive. In the logic of the future it will probably be recognized that scientific induction is only one form of this universal constructive or reconstructive faculty. Another most important question closely akin to that just mentioned is the true relation between these reasoning processes and our general life as active intelligent beings. How is it that the detective is able to understand the burglar’s plan of action?—the military commander to forecast the enemy’s plan of campaign? Primarily, because he himself is capable of making such plans. Men as active creatures co-operating with their fellow-men are incessantly engaged in forming plans and in apprehending the plans of those around them. Every plan may be viewed as a form of induction; it is a scheme invented to meet a given situation, an hypothesis which is put to the test of events, and is verified or refuted by practical success or failure. Such considerations widen still farther our view of scientific induction and help us to understand its relation to ordinary human thought and activity. The scientific investigator in his inductive stage is endeavouring to make out the plan on which his material is constructed. The phenomena serve as indications to help him in framing his hypothesis, generally a guess at first, which he proceeds to verify by experiment and the collection of additional facts. In the deductive stage he assumes that he has made out the plan and can apply it to the discovery of further detail. He has the capacity of detecting plans in nature because he is wont to form plans for practical purposes.

There are good recent accounts of induction in Welton’s Manual of Logic, ii., in H. W. B. Joseph’s Introduction to Logic, and in W. R. Boyce Gibson’s Problem of Logic; see also [Logic].

(H. St.)

INDUCTION COIL, an electrical instrument consisting of two coils of wire wound one over the other upon a core consisting of a bundle of iron wires. One of these circuits is called the primary circuit and the other the secondary circuit. If an alternating or intermittent continuous current is passed through the primary circuit, it creates an alternating or intermittent magnetization in the iron core, and this in turn creates in the secondary circuit a secondary current which is called the induced current. For most purposes an induction coil is required which is capable of giving in the secondary circuit intermittent currents of very high electromotive force, and to attain this result the secondary circuit must as a rule consist of a very large number of turns of wire. Induction coils are employed for physiological purposes and also in connexion with telephones, but their great use at the present time is in connexion with the production of high frequency electric currents, for Röntgen ray work and wireless telegraphy.

The instrument began to be developed soon after Faraday’s discovery of induced currents in 1831, and the subsequent researches of Joseph Henry, C. G. Page and W. Sturgeon on the induction of a current. N. J. Callan Early history. described in 1836 the construction of an electromagnet with two separate insulated wires, one thick and the other thin, wound on an iron core together. He provided the primary circuit of this instrument with an interrupter, and found that when the primary current was rapidly intermitted, a series of secondary currents was induced in the fine wire, of high electromotive force and considerable strength. Sturgeon in 1837 constructed a similar coil, and provided the primary circuit with a mercury interrupter operated by hand. Various other experimentalists took up the construction of the induction coil, and to G. H. Bachhoffner is due the suggestion of employing an iron core made of a bundle of fine iron wires. At a somewhat later date Callan constructed a very large induction coil containing a secondary circuit of very great length of wire. C. G. Page and J. H. Abbot in the United States, between 1838 and 1840, also constructed some large induction coils.[1] In all these cases the primary circuit was interrupted by a mechanically worked interrupter. On the continent of Europe the invention of the automatic primary circuit interrupter is generally attributed to C. E. Neeff and to J. P. Wagner, but it is probable that J. W. M’Gauley, of Dublin, independently invented the form of hammer break now employed. In this break the magnetization of the iron core by the primary current is made to attract an iron block fixed to the end of a spring, in such a way that two platinum points are separated and the primary circuit thus interrupted. It was not until 1853 that H. L. Fizeau added to the break the condenser which greatly improved the operation of the coil. It 1851 H. D. Rühmkorff (1803-1877), an instrument-maker in Paris, profiting by all previous experience, addressed himself to the problem of increasing the electromotive force in the secondary circuit, and induction coils with a secondary circuit of long fine wire have generally, but unnecessarily, been called Rühmkorff coils. Rühmkorff, however, greatly lengthened the secondary circuit, employing in some coils 5 or 6 m. of wire. The secondary wire was insulated with silk and shellac varnish, and each layer of wire was separated from the next by means of varnished silk or shellac paper; the secondary circuit was also carefully insulated from the primary circuit by a glass tube. Rühmkorff, by providing with his coil an automatic break of the hammer type, and equipping it with a condenser as suggested by Fizeau, arrived at the modern form of induction coil. J. N. Hearder in England and E. S. Ritchie in the United States began the construction of large coils, the last named constructing a specially large one to the order of J. P. Gassiot in 1858. In the following decade A. Apps devoted great attention to the production of large induction coils, constructing some of the most powerful coils in existence, and introduced the important improvement of making the secondary circuit of numerous flat coils of wire insulated by varnished or paraffined paper. In 1869 he built for the old Polytechnic Institution in London a coil having a secondary circuit 150 m. in length. The diameter of the wire was 0.014 in., and the secondary bobbin when complete had an external diameter of 2 ft. and a length of 4 ft. 10 ins. The primary bobbin weighed 145 ℔, and consisted of 6000 turns of copper wire 3770 yds. in length, the wire being .095 of an inch in diameter. Excited by the current from 40 large Bunsen cells, this coil could give secondary sparks 30 in. in length. Subsequently, in 1876, Apps constructed a still larger coil for William Spottiswoode, which is now in the possession of the Royal Institution. The secondary circuit consisted of 280 m. of copper wire about 0.01 of an inch in diameter, forming a cylinder 37 in. long and 20 in. in external diameter; it was wound in flat disks in a large number of separate sections, the total number of turns being 341,850. Various primary circuits were employed with this coil, which when at its best could give a spark of 42 in. in length.

A general description of the mode of constructing a modern induction coil, such as is used for wireless telegraphy or Röntgen ray apparatus, is as follows: The iron core consists of a bundle of soft iron wires inserted in the Construction. interior of an ebonite tube. On the outside of this tube is wound the primary circuit, which generally consists of several distinct wires capable of being joined either in series or parallel as required. Over the primary circuit is placed another thick ebonite tube, the thickness of the walls of which is proportional to the spark-producing power of the secondary circuit. The primary coil must be wholly enclosed in ebonite, and the tube containing it is generally longer than the secondary bobbin. The second circuit consists of a number of flat coils wound up between paraffined or shellaced paper, much as a sailor coils a rope. It is essential that no joints in this wire shall occur in inaccessible places in the interior. A machine has been devised by Leslie Miller for winding secondary circuits in flat sections without any joints in the wire at all (British Patent, No. 5811, 1903). A coil intended to give a 10 or 12 in. spark is generally wound in this fashion in several hundred sections, the object of this mode of division being to prevent any two parts of the secondary circuit which are at great differences of potential from being near to one another, unless effectively insulated by a sufficient thickness of shellaced or paraffined paper. A 10-in. coil, a size very commonly used for Röntgen ray work or wireless telegraphy, has an iron core made of a bundle of soft iron wires No. 22 S.W.G., 2 in. in diameter and 18 in. in length. The primary coil wound over this core consists of No. 14 S.W.G. copper wire, insulated with white silk laid on in three layers and having a resistance of about half an ohm. The insulating ebonite tube for such a coil should not be less than ¼ in. in thickness, and should have two ebonite cheeks on it placed 14 in. apart. This tube is supported on two hollow pedestals down which the ends of the primary wire are brought. The secondary coil consists of No. 36 or No. 32 silk-covered copper wire, and each of the sections is prepared by winding, in a suitable winding machine, a flat coiled wire in such a way that the two ends of the coil are on the outside. The coil should not be wound in less than a hundred sections, and a larger number would be still better. The adjacent ends of consecutive sections are soldered together and insulated, and the whole secondary coil should be immersed in paraffin wax. The completed coil (fig. 1) is covered with a sheet of ebonite and mounted on a base board which, in some cases, contains the primary condenser within it and carries on its upper surface a hammer break. For many purposes, however, it is better to separate the condenser and the break from the coil. Assuming that a hammer break is employed, it is generally of the Apps form. The interruption of the primary circuit is made between two contact studs which ought to be of massive platinum, and across the break points is joined the primary condenser. This consists of a number of sheets of paraffined paper interposed between sheets of tin foil, alternate sheets of the tin foil being joined together (see Leyden Jar). This condenser serves to quench the break spark. If the primary condenser is not inserted, the arc or spark which takes place at the contact points prolongs the fall of magnetism in the core, and since the secondary electromotive force is proportional to the rate at which this magnetism changes, the secondary electromotive force is greatly reduced by the presence of an arc-spark at the contact points. The primary condenser therefore serves to increase the suddenness with which the primary current is interrupted, and so greatly increases the electromotive force in the secondary circuit. Lord Rayleigh showed (Phil. Mag., 1901, 581) that if the primary circuit is interrupted with sufficient suddenness, as for instance if it is severed by a bullet from a gun, then no condenser is needed. No current flows in the secondary circuit so long as a steady direct current is passing through the primary, but at the moments that the primary circuit is closed and opened two electromotive forces are set up in the secondary; these are opposite in direction, the one induced by the breaking of the primary circuit being by far the stronger. Hence the necessity for some form of circuit breaker, by the continuous action of which there results a series of discharges from one secondary terminal to the other in the form of sparks.

The hammer break is somewhat irregular in action and gives a good deal of trouble in prolonged use; hence many other forms of primary circuit interrupters have been devised. These may be classified as (1) hand- or motor-worked Interrupters or Breaks. dipping interrupters employing mercury or platinum contacts; (2) turbine mercury interrupters; (3) electrolytic interrupters. In the first class a steel or platinum point, operated by hand or by a motor, is periodically immersed in mercury and so serves to close the primary circuit. To prevent oxidation of the mercury by the spark and break it must be covered with oil or alcohol. In some cases the interruption is caused by the continuous rotation of a motor either working an eccentric which operates the plunger, or, as in the Mackenzie-Davidson break, rotating a slate disk having a metal stud on its surface, which is thus periodically immersed in mercury in a vessel. A better class of interrupter is the mercury turbine interrupter. In this some form of rotating turbine pump pumps mercury from a vessel and squirts it in a jet against a copper plate. Either the copper plate or the jet is made to revolve rapidly by a motor, so that the jet by turns impinges against the plate and escapes it; the mercury and plate are both covered with a deep layer of alcohol or paraffin oil, so that the jet is immersed in an insulating fluid. In a recent form the chamber in which the jet works is filled with coal gas. The current supplied to the primary circuit of the coil travels from the mercury in the vessel through the jet to the copper plate, and hence is periodically interrupted when the jet does not impinge against the plate. Mercury turbine breaks are much employed in connexion with large induction coils used for wireless telegraphy on account of their regular action and the fact that the number of interruptions per second can be controlled easily by regulating the speed of the motor which rotates the jet. But all mercury breaks employing paraffin or alcohol as an insulating medium are somewhat troublesome to use because of the necessity of periodically cleaning the mercury. Electrolytic interrupters were first brought to notice by Dr A. R. B. Wehnelt in 1898 (Elektrotechnische Zeitschrift, January 20th, 1899). He showed that if a large lead plate was placed in dilute sulphuric acid as a cathode, and a thick platinum wire protruding for a distance of about one millimetre beyond a glass or porcelain tube into which it tightly fitted was used as an anode, such an arrangement when inserted in the circuit of a primary coil gave rise to a rapid intermittency in the primary current. It is essential that the platinum wire should be the anode or positive pole. The frequency of the Wehnelt break can be adjusted by regulating the extent to which the platinum wire protrudes through the porcelain tube, and in modern electrolytic breaks several platinum anodes are employed. This break can be employed with any voltage between 30 and 250. The Caldwell interrupter, a modification of the Wehnelt break, consists of two electrodes immersed in dilute sulphuric acid, one of them being enclosed by a glass vessel which has a small hole in it capable of being more or less closed by a tapered glass plug. It differs from the Wehnelt break in that there is no platinum to wear away and it requires less current; hence finer regulation of the coil to the current can be obtained. It will also work with either direct or alternating currents. The hammer and mercury turbine breaks can be arranged to give interruptions from about 10 per second up to about 50 or 60. The electrolytic breaks are capable of working at a higher speed, and under some conditions will give interruptions up to a thousand per second. If the secondary terminals of the induction coils are connected to spark balls placed a short distance apart, then with an electrolytic break the discharge has a flame-like character resembling an alternating current arc. This type of break is therefore preferred for Röntgen ray work since it makes less flickering upon the screen, but its advantages in the case of wireless telegraphy are not so marked. In the Grisson interrupter the primary circuit of the induction coil is divided into two parts by a middle terminal, so that a current flowing in at this point and dividing equally between the two halves does not magnetize the iron. This terminal is connected to one pole of the battery, the other two terminals being connected alternately to the opposite pole by means of a revolving commutator which (1) passes a current through one half of the primary, thus magnetizing the core; (2) passes a current through both halves in opposite directions, thus annulling the magnetization; (3) passes a current through the second half of the primary, thus reversing the magnetization of the core; and (4) passes a current in both halves through opposite directions, thus again annulling the magnetization. As this series of operations can be performed without interrupting a large current through the inductive circuit there is not much spark at the commutator, and the speed of commutation can be regulated so as to obtain the best results due to a resonance between the primary and secondary circuits. Another device due to Grisson is the electrolytic condenser interrupter. If a plate of aluminium and one of carbon or iron is placed in an electrolyte yielding oxygen, this aluminium-carbon or aluminium-iron cell can pass current in one direction but not in the other. Much greater resistance is experienced by a current flowing from the aluminium to the iron than in the opposite direction, owing to the formation of a film of aluminic hydroxide on the aluminium. If then a cell consisting of a number of aluminium plates alternating with iron plates or carbon in alkaline solution is inserted in the primary circuit of an induction coil, the application of an electromotive force in the right direction will cause a transitory current to flow through the coil until the electrolytic condenser is charged. By the use of a proper commutator the position of the electrolytic cell in the circuit can be reversed and another transitory primary current created. This interrupted flow of electricity through the primary circuit provides the intermittent magnetization of the core necessary to produce the secondary electromotive force. This operation of commutation can be conducted without much spark at the commutator because the circuit is interrupted at the time when there is no current in it. In the case of the electrolytic condenser no supplementary paraffined paper condenser is necessary as in the case of the hammer or mercury interrupters.

Fig. 2.—Arrangements for producing High Frequency Currents.
T, Transformer or induction coil. Q, Q, Choking coils. D, Spark balls. C, Condenser.	L, Inductance. P, Primary circuit of high frequency coil. S, Secondary circuit.

An induction coil for the transformation of alternating current is called a transformer (q.v.). One type of high frequency current transformer is called an oscillation transformer or sometimes a Tesla coil. The construction of such High Frequency Coils. a coil is based on different principles from that of the coil just described. If the secondary terminals of an ordinary induction coil or transformer are connected to a pair of spark balls (fig. 2), and if these are also connected to a glass plate condenser or Leyden jar of ordinary type joined in series with a coil of wire of low resistance and few turns, then at each break of the primary circuit of the ordinary induction coil a secondary electromotive force is set up which charges the Leyden jar, and if the spark balls are set at the proper distance, this charge is succeeded by a discharge consisting of a movement of electricity backwards and forwards across the spark gap, constituting an oscillatory electric discharge (see [Electrokinetics]). Each charge of the jar may produce from a dozen to a hundred electric oscillations which are in fact brief electric currents of gradually decreasing strength. If the circuit of few turns and low resistance through which this discharge takes place is overlaid with another circuit well insulated from it consisting of a large number of turns of finer wire, the inductive action between the two circuits creates in the secondary a smaller series of electric oscillations of higher potential. Between the terminals of this last-named coil we can then produce a series of discharges each of which consists in an extremely rapid motion of electricity to and fro, the groups of oscillations being separated by intervals of time corresponding to the frequency of the break in the primary circuit of the ordinary induction coil charging the Leyden jar or condenser. These high frequency discharges differ altogether in character from the secondary discharges of the ordinary induction coil. Theory shows that to produce the best results the primary circuit of the oscillation transformer should consist of only one thick turn of wire or, at most, but of a few turns. It is also necessary that the two circuits, primary and secondary, should be well insulated from one another, and for this purpose the oscillation transformer is immersed in a box or vessel full of highly insulating oil. For full details N. Tesla’s original Papers must be consulted (see Journ. Inst. Elect. Eng. 21, 62).

In some cases the two circuits of the Tesla coil, the primary and secondary, are sections of one single coil. In this form the arrangement is called a resonator or auto transformer, and is much used for producing high frequency discharges for medical purposes. The construction of a resonator is as follows: A bare copper wire is wound upon an ebonite or wooden cylinder or frame, and one end of it is connected to the outside of a Leyden jar or battery of Leyden jars, the inner coating of which is connected to one spark ball of the ordinary induction coil. The other spark ball is connected to a point on the above-named copper wire not very far from the lower end. By adjusting this contact, which is movable, the electric oscillations created in the short section of the resonator coil produce by resonance oscillations in the longer free section, and a powerful high frequency electric brush or discharge is produced at the free end of the resonator spiral. An electrode or wire connected with this free end therefore furnishes a high frequency glow discharge which has been found to have valuable therapeutic powers.

Fig. 3.
C1, Condenser in primary circuit. C2, Condenser in secondary circuit.	L1, Inductance in primary circuit. L2, Inductance in secondary circuit.

The general theory of an oscillation transformer containing capacity and inductance in each circuit has been given by Oberbeck, Bjerknes and Drude.[2] Suppose there are two circuits, each consisting of a coil of wire, the two being superimposed Theory of Oscillation Transformers. or adjacent, and let each circuit contain a condenser or Leyden jar in series with the circuit, and let one of these circuits contain a spark gap, the other being closed (fig. 3). If to the spark balls the secondary terminals of an ordinary induction coil are connected, and these spark balls are adjusted near one another, then when the ordinary coil is set in operation, sparks pass between the balls and oscillatory discharges take place in the circuit containing the spark gap. These oscillations induce other oscillations in the second circuit. The two circuits have a certain mutual inductance M, and each circuit has self inductance L1 and L2. If then the capacities in the two circuits are denoted by C1 and C2 the following simultaneous equations express the relation of the currents, i1 and i2, and potentials, v1, and v2, in the primary and secondary circuits respectively at any instant:—

R1 and R2 being the resistances of the two circuits. If for the moment we neglect the resistances of the two circuits, and consider that the oscillations in each circuit follow a simple harmonic law i = I sin pt we can transform the above equations into a biquadratic

The capacity and inductance in each circuit can be so adjusted that their products are the same number, that is C1L1 = C2L2 = CL. The two circuits are then said to be in resonance or to be tuned together. In this particular and unique case the above biquadratic reduces to

where k is written for M √ (L1L2) and is called the coefficient of coupling. In this case of resonant circuits it can also be shown that the maximum potential differences at the primary and secondary condenser terminals are determined by the rule V1/V2 = 2√C2/√C1. Hence the transformation ratio is not determined by the relative number of turns on the primary and secondary circuits, as in the case of an ordinary alternating current transformer (see [Transformers]), but by the ratio of the capacity in the two oscillation circuits. For full proofs of the above the reader is referred to the original papers.

Each of the two circuits constituting the oscillation transformer taken separately has a natural time period of oscillation; that is to say, if the electric charge in it is disturbed, it oscillates to and fro in a certain constant period like a pendulum and therefore with a certain frequency. If the circuits have the same frequency when separated they are said to be isochronous. If n stands for the natural frequency of each circuit, where n = p/2π the above equations show that when the two circuits are coupled together, oscillations set up in one circuit create oscillations of two frequencies in the secondary circuit. A mechanical analogue to the above electrical effect can be obtained as follows: Let a string be strung loosely between two fixed points, and from it let two other strings of equal length hang down at a certain distance apart, each of them having a weight at the bottom and forming a simple pendulum. If one pendulum is set in oscillation it will gradually impart this motion to the second, but in so doing it will bring itself to rest; in like manner the second pendulum being set in oscillation gives back its motion to the first. The graphic representation, therefore, of the motion of each pendulum would be a line as in fig. 4. Such a curve represents the effect in music known as beats, and can easily be shown to be due to the combined effect of two simple harmonic motions or simple periodic curves of different frequency superimposed. Accordingly, the effect of inductively coupling together two electrical circuits, each having capacity and inductance, is that if oscillations are started in one circuit, oscillations of two frequencies are found in the secondary circuit, the frequencies differing from one another and differing from the natural frequency of each circuit taken alone. This matter is of importance in connexion with wireless telegraphy (see [Telegraph]), as in apparatus for conducting it, oscillation transformers as above described, having two circuits in resonance with one another, are employed.

References.—J. A. Fleming, The Alternate Current Transformer (2 vols., London, 1900), containing a full history of the induction coil; id., Electric Wave Telegraphy (London, 1906), dealing in chap. i., with the construction of the induction coil and various forms of interrupter as well as with the theory of oscillation transformers; A. T. Hare, The Construction of Large Induction Coils (London, 1900); J. Trowbridge, “On the Induction Coil,” Phil. Mag. (1902), 3, p. 393; Lord Rayleigh, “On the Induction Coil,” Phil. Mag. (1901), 2, p. 581; J. E. Ives, “Contributions to the Study of the Induction Coil,” Physical Review (1902), vols. 14 and 15.

(J. A. F.)

[1] For a full history of the early development of the induction coil see J. A. Fleming, The Alternate Current Transformer, vol. ii., chap. i.

[2] See A. Oberbeck, Wied. Ann. (1895), 55, p. 623; V. F. R. Bjerknes, d. (1895), 55, p. 121, and (1891), 44, p. 74; and P. K. L. Drude, Ann. Phys. (1904), 13, p. 512.

INDULGENCE (Lat. indulgentia, indulgere, to grant, concede), in theology, a term defined by the official catechism of the Roman Catholic Church in England as “the remission of the temporal punishment which often remains due to sin after its guilt has been forgiven.” This remission may be either total (plenary) or partial, according to the terms of the Indulgence. Such remission was popularly called a pardon in the middle ages—a term which still survives, e.g. in Brittany.

The theory of Indulgences is based by theologians on the following texts: 2 Samuel (Vulgate, 2 Kings) xii. 14; Matt. xvi. 19 and xviii. 17, 18; 1 Cor. v. 4, 5; 2 Cor. ii. 6-11; but the practice itself is confessedly of later growth. As Bishop Fisher says in his Confutation of Luther, “in the early church, faith in Purgatory and in Indulgences was less necessary than now.... But in our days a great part of the people would rather cast off Christianity than submit to the rigour of the [ancient] canons: wherefore it is a most wholesome dispensation of the Holy Ghost that, after so great a lapse of time, the belief in purgatory and the practice of Indulgences have become generally received among the orthodox” (Confutatio, cap. xviii.; cf. Cardinal Caietan, Tract. XV. de Indulg. cap. i.). The nearest equivalent in the ancient Church was the local and temporary African practice of restoring lapsed Christians to communion at the intercession of confessors and prospective martyrs in prison. But such reconciliations differed from later Indulgences in at least one essential particular, since they brought no remission of ecclesiastical penance save in very exceptional cases. However, as the primitive practice of public penance for sins died out in the Church, there grew up a system of equivalent, or nominally equivalent, private penances. Just as many of the punishments enjoined by the Roman criminal code were gradually commuted by medieval legislators for pecuniary fines, so the years or months of fasting enjoined by the earlier ecclesiastical codes were commuted for proportionate fines, the recitation of a certain number of psalms, and the like. “Historically speaking, it is indisputable that the practice of Indulgences in the medieval church arose out of the authoritative remission, in exceptional cases, of a certain proportion of this canonical penalty.” At the same time, according to Catholic teaching, such Indulgence was not a mere permission to omit or postpone payment, but was in fact a discharge from the debt of temporal punishment which the sinner owed. The authority to grant such discharge was conceived to be included in the power of binding and loosing committed by Christ to His Church; and when in the course of time the vaguer theological conceptions of the first ages of Christianity assumed scientific form and shape at the hands of the Schoolmen, the doctrine came to prevail that this discharge of the sinner’s debt was made through an application to the offender of what was called the “Treasure of the Church” (Thurston, p. 315). “What, then, is meant by the ‘Treasure of the Church’?... It consists primarily and completely of the merit and satisfaction of Christ our Saviour. It includes also the superfluous merit and satisfaction of the Blessed Virgin and the Saints. What do we mean by the word ‘superfluous’? In one way, as I need not say, a saint has no superfluous merit. Whatever he has, he wants it all for himself, because, the more he merits on earth (by Christ’s grace) the greater is his glory in heaven. But, speaking of mere satisfaction for punishment due, there cannot be a doubt that some of the Saints have done more than was needed in justice to expiate the punishment due to their own sins.... It is this ‘superfluous’ expiation that accumulates in the Treasure of the Church” (Bp. of Newport, p. 166). It must be noted that this theory of the “Treasure” was not formulated until some time after Indulgences in the modern sense had become established in practice. The doctrine first appeared with Alexander of Hales (c. 1230) and was at once adopted by the leading schoolmen. Clement VI. formally confirmed it in 1350, and Pius VI. still more definitely in 1794.

The first definite instance of a plenary Indulgence is that of Urban II. for the First Crusade (1095). A little earlier had begun the practice of partial Indulgences, which are always expressed in terms of days or years. However definite may have been the ideas originally conveyed by these notes of time, their first meaning has long since been lost. Eusebius Amort, in 1735, admits the gravest differences of opinion; and the Bishop of Newport writes (p. 163) “to receive an Indulgence of a year, for example, is to have remitted to one so much temporal punishment as was represented by a year’s canonical penance. If you ask me to define the amount more accurately, I say that it cannot be done. No one knows how severe or how long a Purgatory was, or is, implied in a hundred days of canonical penance.” The rapid extension of these time-Indulgences is one of the most remarkable facts in the history of the subject. Innocent II., dedicating the great church of Cluny in 1132, granted as a great favour a forty days’ Indulgence for the anniversary. A hundred years later, all churches of any importance had similar indulgences; yet Englishmen were glad even then to earn a pardon of forty days by the laborious journey to the nearest cathedral, and by making an offering there on one of a few privileged feast-days. A century later again, Wycliffe complains of Indulgences of two thousand years for a single prayer (ed. Arnold, i. 137). In 1456, the recitation of a few prayers before a church crucifix earned a Pardon of 20,000 years for every such repetition (Glassberger in Analecta Franciscana, ii. 368): “and at last Indulgences were so freely given that there is now scarcely a devotion or good work of any kind for which they cannot be obtained” (Arnold & Addis, Catholic Dictionary, s.v.). To quote again from Father Thurston (p. 318): “In imitation of the prodigality of her Divine Master, the Church has deliberately faced the risk of depreciation to which her treasure was exposed.... The growing effeminacy and corruption of mankind has found her censures unendurable ... and the Church, going out into the highways and the hedges, has tried to entice men with the offer of generous Indulgence.” But it must be noted that, according to the orthodox doctrine, not only can an Indulgence not remit future sins, but even for the past it cannot take full effect unless the subject be truly contrite and have confessed (or intend shortly to confess) his sins.

This salutary doctrine, however, has undoubtedly been obscured to some extent by the phrase a poena et a culpa, which, from the 13th century to the Reformation, was applied to Plenary Indulgences. The prima-facie meaning of the phrase is that the Indulgence itself frees the sinner not only from the temporal penalty (poena) but also from the guilt (culpa) of all his sins: and the fact that a phrase so misleading remained so long current shows the truth of Father Thurston’s remark: “The laity cared little about the analysis of it, but they knew that the a culpa et poena was the name for the biggest thing in the nature of an Indulgence which it was possible to get” (Dublin Review, Jan. 1900). The phrase, however, was far from being confined to the unlearned. Abbot Gilles li Muisis, for instance, records how, at the Jubilee of 1300, all the Papal Penitentiaries were in doubt about it, and appealed to the Pope. Boniface VIII. did indeed take the occasion of repeating (in the words of his Bull) that confession and contrition were necessary preliminaries; but he neither repudiated the misleading words nor vouchsafed any clear explanation of them. (Chron. Aegidii li Muisis ed. de Smet, p. 189.) His predecessor, Celestine V., had actually used them in a Bull.

The phrase exercised the minds of learned canonists all through the middle ages, but still held its ground. The most accepted modern theory is that it is merely a catchword surviving from a longer phrase which proclaimed how, during such Indulgences, ordinary confessors might absolve from sins usually “reserved” to the Bishop or the Pope. Nobody, however, has ventured exactly to reconstitute this hypothetical phrase; nor is the theory easy to reconcile with (i.) the uncertainty of canonists at the time when the locution was quite recent, (ii.) the fact that Clement V. and Cardinal Cusanus speak of absolution a poena et a culpa as a separate thing from (a) plenary absolution and (b) absolution from “reserved” sins (Clem. lib. v. tit. ix. c. 2, and Johann Busch (d. c. 1480) Chron. Windeshemense, cap. xxxvi.). But, however it originated, the phrase undoubtedly contributed to foster popular misconceptions as to the intrinsic value of Indulgences, apart from repentance and confession; though Dr Lea seems to press this point unduly (p. 54 ff.), and should be read in conjunction with Thurston (p. 324 ff.).

These misconceptions were certainly widespread from the 13th to the 16th century, and were often fostered by the “pardoners,” or professional collectors of contributions for Indulgences. This can best be shown by a few quotations from eminent and orthodox churchmen during those centuries. Berthold of Regensburg (c. 1270) says, “Fie, penny-preacher! ... thou dost promise so much remission of sins for a mere halfpenny or penny, that thousands now trust thereto, and fondly dream to have atoned for all their sins with the halfpenny or penny, and thus go to hell” (ed. Pfeiffer, i. 393).[1] A century later, the author of Piers Plowman speaks of pardoners who “give pardon for pence poundmeal about” (i.e. wholesale; B. ii. 222); and his contemporary, Pope Boniface IX., complained of their absolving even impenitent sinners for ridiculously small sums (pro qualibet parva pecuniarum summula, Raynaldus, Ann. Ecc. 1390). In 1450 Thomas Gascoigne, the great Oxford Chancellor, wrote: “Sinners say nowadays ‘I care not how many or how great sins I commit before God, for I shall easily and quickly get plenary remission of any guilt and penalty whatsoever (cujusdam culpae et poenae) by absolution and indulgence granted to me from the Pope, whose writing and grant I have bought for 4d. or 6d. or for a game of tennis’”—or sometimes, he adds, by a still more disgraceful bargain (pro actu meretricio, Lib. Ver. p. 123, cf. 126). In 1523 the princes of Germany protested to the Pope in language almost equally strong (Browne, Fasciculus, i. 354). In 1562 the Council of Trent abolished the office of “pardoner.”

The greatest of all Plenary Indulgences is of course the Roman Jubilee. This was instituted in 1300 by Boniface VIII., who pleaded a popular tradition for its celebration every hundredth year, though no written evidence could be found. Clement VI. shortened the period to 50 years (1350): it was then further reduced to 33, and again in 1475 to 25 years.

See also the article on [Luther]. The latest and fullest authority on this subject is Dr H. C. Lea, Hist, of Auricular Confession and Indulgences in the Latin Church (Philadelphia, 1896); his standpoint is frankly non-Catholic, but he gives ample materials for judgment. The greatest orthodox authority is Eusebius Amort, De Origine, &c., indulgentiarum (1735). More popular and more easily accessible are Father Thurston’s The Holy Year of Jubilee (1900), and an article by the Bishop of Newport in the Nineteenth Century for January 1901, with a reply by Mr Herbert Paul in the next number.

(G. G. Co.)

[1] Equally strong assertions were made by the provincial council of Mainz in 1261; and Lea (p. 287) quotes the complaints of 36 similar church councils before 1538.

INDULINES, a series of dyestuffs of blue, bluish-red or black shades, formed by the interaction of para-amino azo compounds with primary monamines in the presence of a small quantity of a mineral acid. They were first discovered in 1863 (English patent 3307) by J. Dale and H. Caro, and since then have been examined by many chemists (see O. N. Witt, Ber., 1884, 17, p. 74; O. Fischer and E. Hepp, Ann., 1890, 256, pp. 233 et seq.; F. Kehrmann, Ber., 1891, 24, pp. 584, 2167 et seq.). They are derivatives of the eurhodines (aminophenazines, aminonaphthophenazines), and by means of their diazo derivatives can be de-amidated, yielding in this way azonium salts; consequently they may be considered as amidated azonium salts. The first reaction giving a clue to their constitution was the isolation of the intermediate azophenin by O. Witt (Jour. Chem. Soc., 1883, 43, p. 115), which was proved by Fischer and Hepp to be dianilidoquinone dianil, a similar intermediate compound being found shortly afterwards in the naphthalene series. Azophenin, C30H24N4, is prepared by warming quinone dianil with aniline; by melting together quinone, aniline and aniline hydrochloride; or by the action of aniline on para-nitrosophenol or para-nitrosodiphenylamine. The indulines are prepared as mentioned above from aminoazo compounds:

or by condensing oxy- and amido-quinones with phenylated ortho-diamines (F. Kehrmann, Ber., 1895, 28, p. 1714):

The indulines may be subdivided into the following groups:— (1) benzindulines, derivatives of phenazine; (2) isorosindulines; and (3) rosindulines, both derived from naphthophenazine; and (4) naphthindulines, derived from naphthazine.

The rosindulines and naphthindulines have a strongly basic character, and their salts possess a marked red colour and fluorescence. Benzinduline (aposafranine), C18H13N3, is a strong base, but cannot be diazotized, unless it be dissolved in concentrated mineral acids. When warmed with aniline it yields anilido-aposafranine, which may also be obtained by the direct oxidation of ortho-aminodiphenylamine. Isorosinduline is obtained from quinone dichlorimide and phenyl-β-naphthylamine; rosinduline from benzene-azo-α-naphthylamine and aniline and naphthinduline from benzene-azo-α-naphthylamine and naphthylamine.

INDULT (Lat. indultum, from indulgere, grant, concede, allow), a, papal licence which authorizes the doing of something not sanctioned by the common law of the church; thus by an indult the pope authorizes a bishop to grant certain relaxations during the Lenten fast according to the necessities of the situation, climate, &c., of his diocese.

INDUNA, a Zulu-Bantu word for an officer or head of a regiment among the Kaffir (Zulu-Xosa) tribes of South Africa. It is formed from the inflexional prefix in and duna, a lord or master. Indunas originally obtained and retained their rank and authority by personal bravery and skill in war, and often proved a menace to their nominal lord. Where, under British influence, the purely military system of government among the Kaffir tribes has broken down or been modified, indunas are now administrators rather than warriors. They sit in a consultative gathering known as an indaba, and discuss the civil and military affairs of their tribe.

INDUS, one of the three greatest rivers of northern India.

A considerable accession of exact geographical knowledge has been gained of the upper reaches of the river Indus and its tributaries during those military and political movements which have been so constant on the northern In the Himalaya. frontiers of India of recent years. The sources of the Indus are to be traced to the glaciers of the great Kailas group of peaks in 32° 20′ N. and 81° E., which overlook the Mansarowar lake and the sources of the Brahmaputra, the Sutlej and the Gogra to the south-east. Three great affluents, flowing north-west, unite in about 80° E. to form the main stream, all of them, so far as we know at present, derived from the Kailas glaciers. Of these the northern tributary points the road from Ladakh to the Jhalung goldfields, and the southern, or Gar, forms a link in the great Janglam—the Tibetan trade route—which connects Ladakh with Lhasa and Lhasa with China. Gartok (about 50 m. from the source of this southern head of the Indus) is an important point on this trade route, and is now made accessible to Indian traders by treaty with Tibet and China. At Leh, the Ladakh capital, the river has already pursued an almost even north-westerly course for 300 m., except for a remarkable divergence to the south-west which carries it across, or through, the Ladakh range to follow the same course on the southern side that had been maintained on the north. This very remarkable instance of transverse drainage across a main mountain axis occurs in 79° E., about 100 m. above Leh. For another 230 m., in a north-westerly direction, the Indus pursues a comparatively gentle and placid course over its sandy bed between the giant chains of Ladakh to the north and Zaskar (the main “snowy range” of the Himalaya) to the south, amidst an array of mountain scenery which, for the majesty of sheer altitude, is unmatched by any in the world. Then the river takes up the waters of the Shyok from the north (a tributary nearly as great as itself), having already captured the Zasvar from the south, together with innumerable minor glacier-fed streams. The Shyok is an important feature in The Shyok affluent. Trans-Himalayan hydrography. Rising near the southern foot of the well-known Karakoram pass on the high road between Ladakh and Kashgar, it first drains the southern slopes of the Karakoram range, and then breaks across the axis of the Muztagh chain (of which the Karakoram is now recognized as a subsidiary extension northwards) ere bending north-westwards to run a parallel course to the Indus for 150 m. before its junction with that river. The combined streams still hold on their north-westerly trend for another 100 m., deep hidden under the shadow of a vast array of snow-crowned summits, until they arrive within sight of the Rakapushi peak which pierces the north-western sky midway between Gilgit and Hunza. Here the great change of direction to the south-west occurs, which is thereafter maintained till the Indus reaches the ocean. At this point it receives the Gilgit river from the north-west, having dropped The Gilgit affluent. from 15,000 to 4000 ft. (at the junction of the rivers) after about 500 m. of mountain descent through the independent provinces of northern Kashmir. (See [Gilgit].) A few miles below the junction it passes Bunji, and from that point to a point beyond Chilas (50 m. below Bunji) it runs within the sphere of British interests. Then once again it resumes its “independent” course through the wild mountains of Kohistan and Hazara, receiving tribute from both sides (the Buner contribution being the most noteworthy) till it emerges into the plains of the Punjab below Darband, in 34° 10′ N. All this part of the river has been mapped in more or less detail of late years. The hidden strongholds of those Hindostani fanatics who had found a refuge on its banks since Mutiny days have been swept clean, and many ancient mysteries have been solved in the course of its surveying.

From its entrance into the plains of India to its disappearance in the Indian Ocean, the Indus of to-day is the Indus of the ’fifties—modified only in some interesting particulars. It has been bridged at several important points. There Indus of the plains. are bridges even in its upper mountain courses. There is a wooden pier bridge at Leh of two spans, and there are native suspension bridges of cane or twig-made rope swaying uneasily across the stream at many points intervening between Leh and Bunji; but the first English-made iron suspension bridge is a little above Bunji, linking up the highroad between Kashmir and Gilgit. Next occurs the iron girder railway bridge at Attock, connecting Rawalpindi with Peshawar, at which point the river narrows almost to a gorge, only 900 ft. above sea-level. Twenty miles below Attock the river has carved out a central trough which is believed to be 180 ft. deep. Forty miles below Attock another great bridge has been constructed at Kushalgarh, which carries the railway to Kohat and the Kurram valley. At Mari, beyond the series of gorges which continue from Kushalgarh to the borders of the Kohat district, on the Sind-Sagar line, a boat-bridge leads to Kalabagh (the Salt city) and northwards to Kohat. Another boat-bridge opposite Dera Ismail Khan connects that place with the railway; but there is nothing new in these southern sections of the Indus valley railway system except the extraordinary development of cultivation in their immediate neighbourhood. The Lansdowne bridge at Sukkur, whose huge cantilevers stand up as a monument of British enterprise visible over the flat plains for many miles around, is one of the greatest triumphs of Indian bridge-making. Kotri has recently been connected with Hyderabad in Sind, and the Indus is now one of the best-bridged rivers in India. The intermittent navigation which was maintained by the survivals of the Indus flotilla as far north as Dera Ismail Khan long after the establishment of the railway system has ceased to exist with the dissolution of the fleet, and the high-sterned flat Indus boats once again have the channels and sandbanks of the river all to themselves.

Within the limits of Sind the vagaries of the Indus channels have necessitated a fresh survey of the entire riverain. The results, however, indicate not so much a marked departure in the general course of the river as a great Lower Indus and delta. variation in the channel beds within what may be termed its outside banks. Collaterally much new information has been obtained about the ancient beds of the river, the sites of ancient cities and the extraordinary developments of the Indus delta. The changing channels of the main stream since those prehistoric days when a branch of it found its way to the Runn of Cutch, through successive stages of its gradual shift westwards—a process of displacement which marked the disappearance of many populous places which were more or less dependent on the river for their water supply—to the last and greatest change of all, when the stream burst its way through the limestone ridges of Sukkur and assumed a course which has been fairly constant for 150 years, have all been traced out with systematic care by modern surveyors till the medieval history of the great river has been fully gathered from the characters written on the delta surface. That such changes of river bed and channel should have occurred within a comparatively limited period of time is the less astonishing if we remember that the Indus, like many of the greatest rivers of the world, carries down sufficient detritus to raise its own bed above the general level of the surrounding plains in an appreciable and measurable degree. At the present time the bed of the Indus is stated to be 70 ft. above the plains of the Sind frontier, some 50 m. to the west of it.

The total length of the Indus, measured directly, is about 1500 m. With its many curves and windings it stretches to about 2000 m., the area of its basin being computed at 372,000 sq. m. Even at its lowest in winter it is 500 ft. wide at Iskardo (near Statistics. the Gilgit junction) and 9 or 10 ft. deep. The temperature of the surface water during the cold season in the plains is found to be 5° below that of the air (64° and 69° F.). At the beginning of the hot season, when the river is bringing down snow water, the difference is 14° (87° and 101° June). At greater depths the difference is still greater. At Attock, where the river narrows between rocky banks, a height of 50 ft. in the flood season above lowest level is common, with a velocity of 13 m. per hour. The record rise (since British occupation of the Punjab) is 80 ft. At its junction with the Panjnad (the combined rivers of the Punjab east of the Indus) the Panjnad is twice the width of the Indus, but its mean depth is less, and its velocity little more than one-third. This discharge of the Panjnad at low season is 69,000 cubic ft. per second, that of the Indus 92,000. Below the junction the united discharge in flood season is 380,000 cubic ft., rising to 460,000 (the record in August). The Indus after receiving the other rivers carries down into Sind, in the high flood season, turbid water containing silt to the amount of 1⁄229 part by weight, or 1⁄410 by volume—equal to 6480 millions of cubic ft. in the three months of flood. This is rather less than the Ganges carries. The silt is very fine sand and clay. Unusual floods, owing to landslips or other exceptional causes, are not infrequent. The most disastrous flood of this nature occurred in 1858. It was then that the river rose 80 ft. at Attock. The most striking result of the rise was the reversal of the current of the Kabul river, which flowed backwards at the rate of 10 m. per hour, flooding Nowshera and causing immense damage to property. The prosperity of the province of Sind depends almost entirely on the waters of the Indus, as its various systems of canals command over nine million acres out of a cultivable area of twelve and a half million acres.

See Maclagan, Proceedings R.G.S., vol. iii.; Haig, The Indus Delta Country (London, 1894); Godwin-Austen, Proceedings R.G.S. vol. vi.

(T. H. H.*)

INDUSTRIA (mod. Monteù da Po), an ancient town of Liguria, 20 m. N.E. of Augusta Taurinorum. Its original name was Bodincomagus, from the Ligurian name of the Padus (mod. Po), Bodincus, i.e. bottomless (Plin. Hist. Nat. iii. 122), and this still appears on inscriptions of the early imperial period. It stood on the right bank of the river, which has now changed its course over 1 m. to the north. It was a flourishing town, with municipal rights, as excavations (which have brought to light the forum, theatre, baths, &c.) have shown, but appears to have been deserted in the 4th century A.D.

See A. Fabietti in Atti della Società di Archeologia di Torino, iii, 17 seq.; Th. Mommsen in Corp. Inscrip. Lat. v. (Berlin, 1877), p. 845; E. Ferrero in Notizie degli Scavi (1903), p. 43.

INDUSTRIAL SCHOOL, in England a school, generally established by voluntary contributions, for the industrial training of children, in which children are lodged, clothed and fed, as well as taught. Industrial schools are chiefly for vagrant and neglected children and children not convicted of theft. Such schools are for children up to the age of fourteen, and the limit of detention is sixteen. They are regulated by the Children Act 1908, which repealed the Industrial Schools Act 1866, as amended by Acts of 1872, 1891 and 1901, and parallel legislation in the various Elementary Education Acts, besides some few local acts. The home secretary exercises powers of supervision, &c. See [Juvenile Offenders].

INDUSTRY (Lat. industria, from indu-, a form of the preposition in, and either stare, to stand, or struere, to pile up), the quality of steady application to work, diligence; hence employment in some particular form of productive work, especially of manufacture; or a particular class of productive work itself, a trade or manufacture. See [Labour Legislation], &c.

INE, king of the West Saxons, succeeded Ceadwalla in 688, his title to the crown being derived from Ceawlin. In the earlier part of his reign he was at war with Kent, but peace was made in 694, when the men of Kent gave compensation for the death of Mul, brother of Ceadwalla, whom they had burned in 687. In 710 Ine was fighting in alliance with his kinsman Nun, probably king of Sussex, against Gerent of West Wales and, according to Florence of Worcester, he was victorious. In 715 he fought a battle with Ceolred, king of Mercia, at Woodborough in Wiltshire, but the result is not recorded. Shortly after this time a quarrel seems to have arisen in the royal family. In 721 Ine slew Cynewulf, and in 722 his queen Aethelburg destroyed Taunton, which her husband had built earlier in his reign. In 722 the South Saxons, previously subject to Ine, rose against him under the exile Aldbryht, who may have been a member of the West Saxon royal house. In 725 Ine fought with the South Saxons and slew Aldbryht. In 726 he resigned the crown and went to Rome, being succeeded by Aethelheard in Wessex. Ine is said to have built the minster at Glastonbury. The date of his death is not recorded. He issued a written code of laws for Wessex, which is still preserved.

See Bede, Hist. Eccl. (Plummer), iv. 15, v. 7; Saxon Chronicle (Earle and Plummer), s.a. 688e, 694, 710, 715, 721, 722, 725, 728; Thorpe, Ancient Laws, i. 2-25; Sehmid, Gesetze der Angelsachsen (Leipzig, 1858); Liebermann, Gesetzeder Angelsachsen (Halle, 1898-99).

INEBOLI, a town on the north coast of Asia Minor, 70 m. W. of Sinūb (Sinope). It is the first place of importance touched at by mercantile vessels plying eastwards from Constantinople, being the port for the districts of Changra and Kastamuni, and connected with the latter town by a carriage road (see [Kastamuni]). The roadstead is exposed, having no protection for shipping except a jetty 300 ft. long, so that in rough weather landing is impracticable. The exports (chiefly wool and mohair) are about £248,000 annually and the imports £200,000. The population is about 9000 (Moslems 7000, Christians 2000). Ineboli represents the ancient Abonou-teichos, famous as the birthplace of the false prophet Alexander, who established there (2nd century A.D.) an oracle of the snake-God Glycon-Asclepius. This impostor, immortalized by Lucian, obtained leave from the emperor Marcus Aurelius to change the name of the town to Ionopolis, whence the modern name is derived (see [Alexander the Paphlagonian]).

INEBRIETY, LAW OF. The legal relations to which inebriety (Lat. in, intensive, and ebrietas, drunkenness) gives rise are partly civil and partly criminal.

I. Civil Capacity.—The law of England as to the civil capacity of the drunkard is practically identified with, and has passed through substantially the same stages of development as the law in regard to the civil capacity of a person suffering from mental disease (see [Insanity]). Unless (see III. inf.) a modification is effected in his condition by the fact that he has been brought under some form of legal control, a man may, in spite of intoxication, enter into a valid marriage or make a valid will, or bind himself by a contract, if he is sober enough to know what he is doing, and no improper advantage of his condition is taken (cf. Matthews v. Baxter, 1873, L.R. 8 Ex. 132; Imperial Loan Co. v. Stone, 1892, 1 Q.B. 599). The law is the same in Scotland and in Ireland; and the Sale of Goods Act 1893 (which applies to the whole United Kingdom) provides that where necessaries are sold and delivered to a person who by reason of drunkenness is incompetent to contract, he must pay a reasonable price for them; “necessaries” for the purposes of this provision mean goods suitable to the condition in life of such person and to his actual requirements at the time of the sale and delivery.

Under the Roman law, and under the Roman Dutch law as applied in South Africa, drunkenness, like insanity, appears to vitiate absolutely a contract made by a person under its influence (Molyneux v. Natal Land and Colonization Co., 1905, A.C. 555).

In the United States, as in England, intoxication does not vitiate contractual capacity unless it is of such a degree as to prevent the person labouring under it from understanding the nature of the transaction into which he is entering (Bouvier, Law Dict., s.v. “Drunkenness”; and cf. Waldron v. Angleman, 1004, 58 Atl. 568; Fowler v. Meadow Brook Water Co., 1904, 57 Atl. 959; 208 Penn., 473). The same rule is by implication adopted in the Indian Contract Act (Act ix. of 1872), which provides (s. 12) that “a person is ... of sound mind for the purpose of making a contract if, at the time when he makes it, he is capable of understanding it and of forming a rational judgment as to its effect upon his interests.” In some legal systems, however, habitual drunkenness is a ground for divorce or judicial separation (Sweden, Law of the 27th of April 1810; France, Code Civil, Art. 231, Hirt v. Hirt, Dalloz, 1898, pt. ii., p. 4, and n. 4).

II. Criminal Responsibility.—In English law, drunkenness, unlike insanity, was at one time regarded as in no way an excuse for crime. According to Coke (Co. Litt., 247) a drunkard, although he suffers from acquired insanity, dementia affectata, is voluntarius daemon, and therefore has no privilege in consequence of his state; “but what hurt or ill soever he doth, his drunkenness doth aggravate it.” Sir Matthew Hale (P.C. 32) took a more moderate view, viz. that a person under the influence of this voluntarily contracted madness “shall have the same judgment as if he were in his right senses”; and admitted the existence of two “allays” or qualifying circumstances: (1) temporary frenzy induced by the unskilfulness of physicians or by drugging; and (2) habitual or fixed frenzy. Those early authorities have, however, undergone considerable development and modification.

Although the general principle that drunkenness is not an excuse for crime is still steadily maintained (see Russell, Crimes, 6th ed., i. 144; Archbold, Cr. Pl., 23rd ed., p. 29), it is settled law that where a particular intent is one of the constituent elements of an offence, the fact that a prisoner was intoxicated at the time of its commission is relevant evidence to show that he had not the capacity to form that intent. Drunkenness is also a circumstance of which a jury may take account in considering whether an act was premeditated, or whether a prisoner acted in self-defence or under provocation, when the question is whether the danger apprehended or the provocation was sufficient to justify his conduct or to alter its legal character. Moreover, delirium tremens, if it produce such a degree of madness as to render a person incapable of distinguishing right from wrong, relieves him from criminal responsibility for any act committed by him while under its influence; and in one case at nisi prius (R. v. Baines, The Times, 25th Jan. 1886) this doctrine was extended by Mr Justice Day to temporary derangement occasioned by drink. The law of Scotland accepts, if it does not go somewhat beyond, the later developments of that of England in regard to criminal responsibility in drunkenness. Indian law on the point is similar to the English (Indian Penal Code, Act. xlv. of 1860, ss. 85, 86; Mayne, Crim. Law of India, ed. 1896, p. 391). In the United States the same view is the prevalent legal doctrine (see Bishop, Crim. Law, 8th ed., i, ss. 397-416). The Criminal Code of Queensland (No. 9 of 1899, Art. 28) provides that a person who becomes intoxicated intentionally is responsible for any crime that he commits while so intoxicated, whether his voluntary intoxication was induced so as to afford an excuse for the commission of an offence or not. As in England, however, when an intention to cause a specific result is an element of an offence, intoxication, whether complete or partial, and whether intentional or unintentional, may be regarded for the purpose of ascertaining whether such intention existed or not. There is a similar provision in the Penal Code of Ceylon (No. 2 of 1883, Art. 79). The Criminal Codes of Canada (1892, c. 29, ss. 7 et seq.) and New Zealand (No. 56 of 1893, ss. 21 et seq.) are silent on the subject of intoxication as an excuse for crime. The Criminal Code of Grenada (No. 2 of 1897, Art. 51) provides that “a person shall not, on the ground of intoxication, be deemed to have done any act involuntarily, or be exempt from any liability to punishment for any act: and a person who does an act while in a state of intoxication shall be deemed to have intended the natural and probable consequences of his act.” There is a similar provision in the Criminal Code of the Gold Coast Colony (No. 12 of 1892, s. 54). Under the French Penal Code (Art. 64), “il n’y a ni crime, ni délit, lorsque le prévenu était en état de démence au temps de l’action ou lorsqu’il aura été contraint par une force à laquelle il n’ a pu résister.” According to the balance of authority (Dalloz, Rép. tit., Peine, ss. 402 et seq.) intoxication is not assimilated to insanity, within the meaning of this article, but it may be and is taken account of by juries as an extenuating circumstance (Ortolan, Droit Pénal i. s. 323: Chauveau et Hélie i. s. 360). A provision in the German Penal Code (Art. 51) that an act is not punishable if its author, at the time of committing it, was in a condition of unconsciousness, or morbid disturbance of the activity of his mind which prevented the free exercise of his will, has been held not to extend to intoxication (Clunet, 1883, p. 311). But in Germany as in France, intoxication may apparently be an extenuating circumstance. Under the Italian Penal Code (Arts. 46-49) intoxication—unless voluntarily induced so as to afford an excuse for crime—may exclude or modify responsibility.

So far only the question whether drunkenness is an excuse for offences committed under its influence has been dealt with. There remains the question how far drunkenness itself is a crime. Mere private intoxication is not, either in England or in the United States (Bishop, Crim. Law, 8th ed., i. s. 399) indictable as an offence at common law; but in all civilized countries public drunkenness is punishable when it amounts to a breach of the peace (see [Liquor Laws]) or contravention of public order; and modern legislation in many countries provides for deprivation of personal liberty for long periods in case of a frequent repetition of the offence. Reference may be made in this connexion to the Inebriates Acts 1898, 1899 and 1900 (see iii. inf.), and also to similar legislation in the British colonies and in foreign legal systems (e.g. Cape of Good Hope, No. 32 of 1896; Ceylon, Licensing Ordinance 1891, ss. 23, 24, 29; New South Wales, Vagrants Punishment Act 1866; Massachusetts, Acts of 1891, c. 427, 1893, cc. 414, 44; France, Law of 23rd of Jan. 1873, Art. 6).

III. State Action in Regard to Inebriety.—This assumes a variety of forms. (a) Measures regulating the punishment of occasional or habitual drunkenness by fines or short terms of imprisonment. (b) Control in penal establishments for lengthened periods. (c) Laws prohibiting the sale of liquor to persons who are known inebriates: e.g. in England (Licensing Act 1902); Ontario (Rev. Stats. 1897, c. 245, ss. 124, 125); New South Wales (Liquor Act 1898, ss. 52, 53); Cape of Good Hope (No. 28 of 1883, s. 89); New York (Rev. Stats. 1889-1892, c. 20, Title iv.); California (Act to prevent sale of liquor to drunkards, 1889); Massachusetts (Pub. Stats., ed. 1902, c. 100, s. 9). (d) Laws regulating the appointment of some person or persons to act as guardian or guardians, or who may be endowed with legal powers over the person and estate of an inebriate. Thus in France (Code Civil, Arts. 489 et seq.), Germany (Civil Code, Art. 6 (39)) and Austria-Hungary (Bürgerliches Gesetz-Buch, ss. 21, 269, 270, 273), an inebriate may be judicially interdicted if he is squandering his property and thereby exposing his family to future destitution. Provision is also made for the interdiction of inebriates by the laws of Nova Scotia (Rev. Stats. 1900, c. 126, s. 2), Manitoba (Rev. Stat. 1902, c. 103, ss. 30 et seq.), British Columbia (Rev. Stat. 1897, c. 66), New South Wales (Inebriates Act 1900, s. 5), Tasmania (Inebriates Act 1885, No. 17, s. 23); Canton of Bâle (Trustee Law of the 23rd of Feb. 1880, s. 11), Orange River Colony (Code Laws, c. 108, s. 30), Maryland (Code General Laws, c. 474, s. 47). (e) Control for the purpose of reformation. Legislation of this character provides reformatory treatment: (1) for the inebriate who makes a voluntary application for admission; (2) by compulsory seclusion for the inebriate who refuses consent to treatment and yet manages to keep out of the reach of the law; (3) for the inebriate who is a police-court recidivist, or who has committed crime, caused or contributed to by drink. The legislation of the Cape of Good Hope (Inebriates Act 1896) and of North Dakota (Habitual Drunkards Act 1895) provides for the first of these methods of treatment alone. Compulsory detention for ordinary inebriates only is provided for by the laws of Delaware (Act of 1898), Massachusetts (Rev. Laws, c. 87), and of the Cantons of Berne (Law of the 24th of Nov. 1883) and Bâle (Law of the 21st of Feb. 1901). All three methods of treatment are in force in New South Wales (Inebriates Act 1900), Queensland (Inebriates Institutions Act 1896) and South Australia (Inebriates Act 1881). Provision is made only for voluntary application and compulsory detention of ordinary inebriates in Victoria (Inebriates Act 1890), Tasmania (Inebriates Act 1885; Inebriates Hospitals Act 1892) and New Zealand (Inebriates Institutions Act 1898). The legislation of the United Kingdom (Inebriates Acts 1879-1900) deals both with voluntary application and with the committal of criminal inebriates or of police-court recidivists. A brief sketch of the English system must suffice.

The Inebriates Acts of 1870-1900 deal in the first place with non-criminal, and in the second place with criminal, habitual drunkards.

For the purposes of the acts the term “habitual drunkard” means “a person who, not being amenable to any jurisdiction in lunacy, is notwithstanding, by reason of habitual intemperate drinking of intoxicating liquor, at times dangerous to himself or herself, or incapable of managing himself or herself and his or her affairs.” A person would become amenable to the lunacy jurisdiction not only where habitual drunkenness made him a “lunatic” in the legal sense of the term, but where it created, such a state of disease and consequential “mental infirmity” as to bring his case within section 116 of the Lunacy Act 1890, the effect of which is explained in the article Insanity. Any “habitual drunkard” within the above definition may obtain admission to a “licensed retreat” on a written application to the licensee, stating the time (the maximum period is two years) that he undertakes to remain in the retreat. The application must be accompanied by the statutory declaration of two persons that the applicant is an habitual drunkard, and its signature must be attested by a justice of the peace who has satisfied himself as to the fact, and who is required to state that the applicant understood the nature and effect of his application. Licences (each of which is subject to a duty and is impressed with a stamp of £5, and 10s. for every patient above ten in number) are granted for retreats by the borough council and the town clerk in boroughs, and elsewhere by the county council and the clerk of the county council. The maximum period for which a licence may be granted is two years, but licences may be renewed by the licensing authority on payment of a stamp duty of the same amount as on the original grant. When an habitual drunkard has once been committed to a retreat, he must remain in the retreat for the time that he has fixed in his application, subject to certain statutory provisions similar to those prescribed by the Lunacy Acts for asylums as to leave of absence and discharge; and he may be retaken and brought back to the retreat under a justice’s warrant. The term of detention may be extended on its expiry, or an inebriate may be readmitted, on a fresh application, without any statutory declaration, and without the attesting justice being required to satisfy himself that the applicant is an habitual drunkard. Licensed retreats are subject to inspection by an Inspector of Retreats appointed by the Home Secretary, to whom he makes an annual report. The Home Secretary is empowered to make rules and regulations for the management of retreats, and “regulations and orders,” not inconsistent with such rules, are to be prepared by the licensee within a month after the granting of his licence, and submitted to the inspector for approval. The rules now in force are dated as regards (a) England, 28th Feb. 1902; (b) Scotland, 14th April 1902; (c) Ireland, 3rd Feb. 1903. There are also statutory provisions, similar to those of the Lunacy Acts, as to offences—(i.) by licensees failing to comply with the requirements of the acts; (ii) by persons ill-treating patients, or helping them to escape, or unlawfully supplying them with intoxicating liquor; (iii.) by patients refusing to comply with the rules. The Home Secretary may (i.) authorize the establishment of “State Inebriate Reformatories,” to be paid for out of moneys provided by parliament; and (ii.) sanction “Certified Inebriates’ Reformatories” on the application of any borough or county council, or any person whatever, if satisfied concerning the reformatory and the persons proposing to maintain it. An Inspector of Certified Inebriate Reformatories has been appointed. Regulations for State Inebriate Reformatories and for Certified Inebriate Reformatories have been made, dated as follows: State Inebriate Reformatories:—England, 21st of June 1901, 29th of Dec. 1903, 29th of April 1904; Scotland, 9th of March 1900; Ireland, 16th of March 1899, 16th of April 1901, 10th of Feb. 1904. Certified Inebriate Reformatories:—England, Model Regulations, 17th of Dec. 1898; Scotland, Regulations, 14th of Feb. 1899; Ireland, Model Regulations, 29th of April 1899.

Any person convicted on indictment of an offence punishable with imprisonment or penal servitude (i.e. of any non-capital felony and of most misdemeanours), if the court is satisfied from the evidence that the offence was committed under the influence of drink, or that drink was a contributing cause of the offence, may, if he admits that he is, or is found by the jury to be, an habitual drunkard, in addition to or in substitution for any other sentence, be ordered to be detained in a state or certified inebriate reformatory, the managers of which are willing to receive him. Again, any habitual drunkard who is found drunk in any public place, or who commits any other of a series of similar offences under various statutes, after having within twelve months been convicted at least three times of a similar offence, may, on conviction on indictment, or, if he consent, on summary conviction, be sent for detention in any certified inebriate reformatory. The expenses of prosecuting habitual drunkards under the above provisions are payable out of the local rates upon an order to that effect by the judge of assize or chairman of quarter-sessions if the prosecution be on indictment, or by a court of summary jurisdiction if the offence is dealt with summarily.

Authorities.—As to the history of legislation on the subject see Parl. Paper No. 242 of 1872; 1893 C. 7008. See also Wyatt Paine, Inebriate Reformatories and Retreats (London, 1899); Blackwell, Inebriates Acts, 1879-1898 (London, 1899); Wood Renton, Lunacy (London and Edinburgh, 1896); Kerr, Inebriety (3rd ed., London, 1894). An excellent account of the systems in force in other countries for the treatment of inebriates will be found in Parl. Pap. (1902), cd. 1474.

(A. W. R.)

INFALLIBILITY (Fr. infaillibilité and infallibilité, the latter now obsolete, Med. Lat. infallibilitas, infallibilis, formed from fallor, to make a mistake), the fact or quality of not being liable to err or fail. The word has thus the general sense of “certainty”; we may, e.g., speak of a drug as an infallible specific, or of a man’s judgment as infallible. In these cases, however, the “infallibility” connotes certainty only in so far as anything human can be certain. In the language of the Christian Church the word “infallibility” is used in a more absolute sense, as the freedom from ail possibility of error guaranteed by the direct action of the Spirit of God. This belief in the infallibility of revelation is involved in the very belief in revelation itself, and is common to all sections of Christians, who differ mainly as to the kind and measure of infallibility residing in the human instruments by which this revelation is interpreted to the world. Some see the guarantee, or at least the indication, of infallibility in the consensus of the Church (quod semper, ubique, et ab omnibus) expressed from time to time in general councils; others see it in the special grace conferred upon St Peter and his successors, the bishops of Rome, as heads of the Church; others again see it in the inspired Scriptures, God’s Word. This last was the belief of the Protestant Reformers, for whom the Bible was in matters of doctrine the ultimate court of appeal. To the translation and interpretation of the Scriptures men might bring a fallible judgment, but this would be assisted by the direct action of the Spirit of God in proportion to their faith. As for infallibility, this was a direct grace of God, given only to the few. “What ever was perfect under the sun,” ask the translators of the Authorized Version (1611) in their preface, “where apostles and apostolick men, that is, men endued with an extraordinary measure of God’s Spirit, and privileged with the privilege of infallibility, had not their hand?” In modern Protestantism, on the other hand, the idea of an infallible authority whether in the Church or the Bible has tended to disappear, religious truths being conceived as valuable only as they are apprehended and made real to the individual mind and soul by the grace of God, not by reason of any submission to an external authority. (See also [Inspiration].)

At the present time, then, the idea of infallibility in religious matters is most commonly associated with the claim of the Roman Catholic Church, and more especially of the pope personally as head of that Church, to possess the privilege of infallibility, and it is with the meaning and limits of this claim that the present article deals.

The substance of the claim to infallibility made by the Roman Catholic Church is that the Church and the pope cannot err when solemnly enunciating, as binding on all the faithful, a decision on a question of faith or morals. The infallibility of the Church, thus limited, is a necessary outcome of the fundamental conception of the Catholic Church and its mission. Every society of men must have a supreme authority, whether individual or collective, empowered to give a final decision in the controversies which concern it. A community whose mission it is to teach religious truth, which involves on the part of its members the obligation of belief in this truth, must, if it is not to fail of its object, possess an authority capable of maintaining the faith in its purity, and consequently capable of keeping it free from and condemning errors. To perform this function without fear of error, this authority must be infallible in its own sphere. The Christian Church has expressly claimed this infallibility for its formal dogmatic teaching. In the very earliest centuries we find the episcopate, united in council, drawing up symbols of faith, which every believer was bound to accept under pain of exclusion, condemning heresies, and casting out heretics. From Nicaea and Chalcedon to Florence and Trent, and to the present day, the Church has excluded from her communion all those who do not profess her own faith, i.e. all the religious truths which she represents and imposes as obligatory. This is infallibility put into practice by definite acts.

The infallibility of the pope was not defined until 1870 at the Vatican Council; this definition does not constitute, strictly speaking, a dogmatic innovation, as if the pope had not hitherto enjoyed this privilege, or as if the Church, as a whole, had admitted the contrary; it is the newly formulated definition of a dogma which, like all those defined by the Councils, continued to grow into an ever more definite form, ripening, as it were, in the always living community of the Church. The exact formula for the papal infallibility is given by the Vatican Council in the following terms (Constit. Pastor aeternus, cap. iv.); “we teach and define as a divinely revealed dogma, that the Roman Pontiff, when he speaks ex cathedra—i.e. when, in his character as Pastor and Doctor of all Christians, and in virtue of his supreme apostolic authority, he lays down that a certain doctrine concerning faith or morals is binding upon the universal Church,—possesses, by the Divine assistance which was promised to him in the person of the blessed Saint Peter, that same infallibility with which the Divine Redeemer thought fit to endow His Church, to define its doctrine with regard to faith and morals; and, consequently, that these definitions of the Roman Pontiff are irreformable in themselves, and not in consequence of the consent of the Church.” A few notes will suffice to elucidate this pronouncement.

(a) As the Council expressly says, the infallibility of the pope is not other than that of the Church; this is a point which is too often forgotten or misunderstood. The pope enjoys it in person, but solely qua head of the Church, and as the authorized organ of the ecclesiastical body. For this exercise of the primacy as for the others, we must conceive of the pope and the episcopate united to him as a continuation of the Apostolic College and its head Peter. The head of the College possesses and exercises by himself alone the same powers as the College which is united with him; not by delegation from his colleagues, but because he is their established chief. The pope when teaching ex cathedra acts as head of the whole episcopal body and of the whole Church.

(b) If the Divine constitution of the Church has not changed in its essential points since our Lord, the mode of exercise of the various powers of its head has varied; and that of the supreme teaching power as of the others. This explains the late date at which the dogma was defined, and the assertion that the dogma was already contained in that of the papal primacy established by our Lord himself in the person of St Peter. A certain dogmatic development is not denied, nor an evolution in the direction of a centralization in the hands of the pope of the exercise of his powers as primate; it is merely required that this evolution should be well understood and considered as legitimate.

(c) As a matter of fact the infallibility of the pope, when giving decisions in his character as head of the Church, was generally admitted before the Vatican Council. The only reservation which the most advanced Gallicans dared to formulate, in the terms of the celebrated declaration of the clergy of France (1682), had as its object the irreformable character of the pontifical definitions, which, it was claimed, could only have been acquired by them through the assent of the Church. This doctrine, rather political than theological, was a survival of the errors which had come into being after the Great Schism, and especially at the council of Constance; its object was to put the Church above its head, as the council of Constance had put the ecumenical council above the pope, as though the council could be ecumenical without its head. In reality it was Gallicanism alone which was condemned at the Vatican Council, and it is Gallicanism which is aimed at in the last phrase of the definition we have quoted.

(d) Infallibility is the guarantee against error, not in all matters, but only in the matter of dogma and morality; everything else is beyond its power, not only truths of another order, but even discipline and the ecclesiastical laws, government and administration, &c.

(e) Again, not all dogmatic teachings of the pope are under the guarantee of infallibility; neither his opinions as private instructor, nor his official allocutions, however authoritative they may be, are infallible; it is only his ex cathedra instruction which is guaranteed; this is admitted by everybody.

But when does the pope speak ex cathedra, and how is it to be distinguished when he is exercising his infallibility? As to this point there are two schools, or rather two tendencies, among Catholics: some extend the privilege of infallibility to all official exercise of the supreme magisterium, and declare infallible, e.g. the papal encyclicals.[1] Others, while recognizing the supreme authority of the papal magisterium in matters of doctrine, confine the infallibility to those cases alone in which the pope chooses to make use of it, and declares positively that he is imposing on all the faithful the obligation of belief in a certain definite proposition, under pain of heresy and exclusion from the Church; they do not insist on any special form, but only require that the pope should clearly manifest his will to the Church. This second point of view, as clearly expounded by Mgr Joseph Fessler (1813-1872), bishop of St Pölten, who was secretary to the Vatican Council, in his work Die wahre und die falsche Unfehlbarkeit der Päpste (French trans. La vraie et la fausse infaillibilité, Paris, 1873), and by Cardinal Newman in his “Letter to the Duke of Norfolk,” is the correct one, and this is clear from the fact that it has never been blamed by the ecclesiastical authority. Those who hold the latter opinion have been able to assert that since the Vatican Council no infallible definition had yet been formulated by the popes, while recognizing the supreme authority of the encyclicals of Leo XIII.

It is remarkable that the definition of the infallibility of the pope did not appear among the projects (schemata) prepared for the deliberations of the Vatican Council (1869). It doubtless arose from the proposed forms for the definitions of the primacy and the pontifical magisterium. The chapter on the infallibility was only added at the request of the bishops and after long hesitation on the part of the cardinal presidents. The proposed form, first elaborated in the conciliary commission de fide, was the object of long public discussions from the 50th general congregation (May 13th, 1870) to the 85th (July 13th); the constitution as a whole was adopted at a public session, on the 18th, of the 535 bishops present, two only replied “Non placet”; but about 50 had preferred not to be present. The controversies occasioned by this question had started from the very beginning of the Council, and were carried on with great bitterness on both sides. The minority, among whom were prominent Cardinals Rauscher and Schwarzenberg, Hefele, bishop of Rotterdam (the historian of the councils) Cardinal Mathieu, Mgr Dupanloup, Mgr Maret, &c., &c., did not pretend to deny the papal infallibility; they pleaded the inopportuneness of the definition and brought forward difficulties mainly of an historical order, in particular the famous condemnation of Pope Honorius by the 6th ecumenical council of Constantinople in 680. The majority, in which Cardinal Manning played a very active part, took their stand on theological reasons of the strongest kind; they invoked the promises of Our Lord to St Peter: “Thou art Peter, and upon this rock will I build my Church, and the gates of hell shall not prevail against her”; and again, “I have prayed for thee, Peter, that thy faith fail not; and do thou in thy turn confirm thy brethren”; they showed the popes, in the course of the ages, acting as the guardians and judges of the faith, arousing or welcoming dogmatic controversies and authoritatively settling them, exercising the supreme direction in the councils and sanctioning their decisions; they explained that the few historical difficulties did not involve any dogmatic defect in the teaching of the popes; they insisted upon the necessity of a supreme tribunal giving judgment in the name of the whole of the scattered Church; and finally, they considered that the definition had become opportune for the very reason that under the pretext of its inopportuneness the doctrine itself was being attacked.

The definition once proclaimed, controversies rapidly ceased; the bishops who were among the minority one after the other formulated their loyal adhesion to the Catholic dogma. The last to do so in Germany was Hefele, who published the decrees of the 10th of April 1871, thus breaking a long friendship with Döllinger; in Austria, where the government had thought good to revive for the occasion the royal placet, Mgr Haynald and Mgr Strossmayer delayed the publication, the former till the 15th of September 1871, the latter till the 26th of December 1872. In France the adhesion was rapid, and the publication was only delayed by some bishops in consequence of the disastrous war with Prussia. Though no bishops abandoned it, a few priests, such as Father Hyacinthe Loyson, and a few scholars at the German universities refused their adhesion. The most distinguished among the latter was Döllinger, who resisted all the advances of Mgr Scherr, archbishop of Munich, was excommunicated on the 17th of April 1871, and died unreconciled, though without joining any separate group. After him must be mentioned Friedrich of Munich, several professors of Bonn, and Reinkens of Breslau, who was the first bishop of the “Old Catholics.” These professors formed the “Committee of Bonn,” which organized the new Church. It was recognized and protected first in Bavaria, thanks to the minister Freiherr Johann von Lutz, then in Saxony, Baden, Württemberg, Prussia, where it was the pretext for, if not the cause of, the Kulturkampf, and finally in Switzerland, especially at Geneva.

For the theological aspects of the dogma of infallibility, see, among many others, L. Billot, S.J., De Ecclesia Christi (3 vols., Rome, 1898-1900); or G. Wilmers, S.J., De Christi Ecclesia (Regensburg, 1897). The most accessible popular work is that of Mgr Fessler already mentioned. For the history of the definition see [Vatican Council]; also [Papacy], [Gallicanism], [Febronianism], [Old Catholics], &c.

(A. Bo.*)

[1] It was in this sense that it was understood by Döllinger, who pointed out that the definition of the dogma would commit the Church to all past official utterances of the popes, e.g. the Syllabus of 1864, and therefore to a war à outrance against modern civilization. This view was embodied in the circular note to the Powers, drawn up by Döllinger and issued by the Bavarian prime minister Prince Hohenlohe-Schillingsfürst on April 9, 1869. It was also the view universally taken by the German governments which supported the Kulturkampf in a greater or less degree.—Ed.

INFAMY (Lat. infamia), public disgrace or loss of character. Infamy (infamia) occupied a prominent place in Roman law, and took the form of a censure on individuals pronounced by a competent authority in the state, which censure was the result either of certain actions which they had committed or of certain modes of life which they had pursued. Such a censure involved disqualification for certain rights both in public and in private law (see A. H. J. Greenidge, Infamia, its Place in Roman Public and Private Law, 1894). In English law infamy attached to a person in consequence of conviction of some crime. The effect of infamy was to render a person incompetent to give evidence in any legal proceeding. Infamy as a cause of incompetency was abolished by an act of 1843 (6 & 7 Vict. c. 85).

The word “infamous” is used in a particular sense in the English Medical Act of 1858, which provides that if any registered medical practitioner is judged by the General Medical Council, after due inquiry, to have been guilty of infamous conduct in any professional respect, his name may be erased from the Medical Register. The General Medical Council are the sole judges of whether a practitioner has been guilty of conduct infamous in a professional respect, and they act in a judicial capacity, but an accused person is generally allowed to appear by counsel. Any action which is regarded as disgraceful or dishonourable by a man’s professional brethren—such, for example, as issuing advertisements in order to induce people to consult him in preference to other practitioners—may be found infamous.

INFANCY, in medical practice, the nursing age, or the period during which the child is at the breast. As a matter of convenience it is usual to include in it children up to the age of one year. The care of an infant begins with the preparations necessary for its birth and the endeavour to ensure that taking place under the best possible sanitary conditions. On being born the normal infant cries lustily, drawing air into its lungs. As soon as the umbilical cord which unites the child to the mother has ceased to pulsate, it is tied about 2 in. from the child’s navel and is divided above the ligature. The cord is wrapped in a sterilized gauze pad and the dressing is not removed until the seventh to the tenth day, when the umbilicus is healed.

The baby is now a separate entity, and the first event in its life is the first bath. The room ready to receive a new-born infant should be kept at a temperature of 70° F. The temperature of the first bath should be 100° F. The child should be well supported in the bath by the left hand of the nurse, and care should be taken to avoid wetting the gauze pad covering the cord. In some cases infants are covered with a white substance termed “vernix caseosa,” which may be carefully removed by a little olive oil. Sponges should never be used, as they tend to harbour bacteria. A soft pad of muslin or gauze which can be boiled should take its place. After the first ten days 94° F. is the most suitable temperature for a bath. When the baby has been well dried the skin may be dusted with pure starch powder to which a small quantity of boric acid has been added. The most important part of the toilet of a new-born infant is the care of the eyes, which should be carefully cleansed with gauze dipped in warm water and one drop of a 2% solution of nitrate of silver dropped into each eye. The clothes of a newly born child should consist exclusively of woollen undergarments, a soft flannel binder, which should be tied on, being placed next the skin, with a long-sleeved woven wool vest and over this a loose garment of flannel coming below the feet and long enough to tuck up. Diapers should be made of soft absorbent material such as well-washed linen and should be about two yards square and folded in a three-cornered shape. An infant should always sleep in a bed or cot by itself. In 1907, of 749 deaths from violence in England and Wales of children under one month, 445 were due to suffocation in bed with adults. A healthy infant should spend most of its time asleep and should be laid into its cot immediately after feeding.

The normal infant at birth weighs about 7 ℔. During the two or three days following birth a slight decrease in weight occurs, usually 5 to 6 oz. When nursing begins the child increases in weight up to the seventh day, when the infant will have regained its weight at birth. From the second to the fourth week after birth (according to Camerer) an infant should gain 1 oz. daily or 1½ to 2 ℔ monthly, from the fourth to the sixth month ½ to 2⁄3 of an oz. daily or 1 ℔ monthly, from the sixth to the twelfth month ½ oz. daily or less than 1 ℔ monthly. At the sixth month it should be twice the weight at birth. The average weight at the twelfth month is 20 to 21 ℔. The increase of weight in artificially fed is less regular than in breast-fed babies.

Food.—There is but one proper food for an infant, and that is its mother’s milk, unless when in exceptional circumstances the mother is not allowed to nurse her child. Artificially fed children are much more liable to epidemic diseases. The child should be applied to the breast the first day to induce the flow of milk. The first week the child should be fed at intervals of two hours, the second week eight to nine times, and the fourth week eight times at intervals of two and a half hours. At two months the child is being suckled six times daily at intervals of three hours, the last feed being at 11 P.M. Where a mother cannot nurse a child the child must be artificially fed. Cow’s milk must be largely diluted to suit the new-born infant. Armstrong gives the following table of dilution:—

Koplik has drawn out a table of the amounts to be given as follows:—

In cities it is advisable that milk should be either sterilized by boiling or pasteurized, i.e. subjected to a form of heating which, while destroying pathogenic bacteria, does not alter the taste. The milk in a suitable apparatus is subjected to a temperature of 65° C. (149° F.) for half an hour and is then rapidly cooled to 20° C. (68° F.). Children fed on pasteurized milk should be given a teaspoonful of fresh orange juice daily to supply the missing acid and salts.

Artificial feeding is given by means of a bottle. In France all bottles with rubber tubes have been made illegal. They are a fruitful source of infection, as it is impossible to keep them clean. The best bottle is the boat-shaped one, with a wide mouth at one end, to which is attached a rubber teat, while the other end has a screw stopper. This is readily cleansed and a stream of water can be made to flow through it. All bottle teats should be boiled at least once a day for ten minutes with soda and kept in a glass-covered jar until required. A feed should be given at the temperature of 100° F.

At the ninth month a cereal may be added to the food. Before that the infant is unable to digest starchy foods. Much starch tends to constipation, and it is rarely wise to give starchy preparations in a proportion of more than 3% to children under a year old. A child who is carefully fed in a cleanly manner should not have diarrhoea, and its appearance indicates carelessness somewhere. The English registrar-general’s returns for 1906 show that in the seventy-six largest towns in England and Wales 14,306 deaths of infants under one year from diarrhoea took place in July, August and September alone. These deaths are largely preventable; when Dr Budin of Paris established his “Consultations de Nourissons” the infant mortality of Paris amounted to 178 per 1000, but at the consultation the rate was 46 per 1000. At Varengeville-sur-mer a consultation for nurslings was instituted under Dr Poupalt of Dieppe in 1904. During the seven previous years the infant mortality had averaged 145 per 1000. In 1904-1905 not one infant at the consultation died, though it was a summer of extreme heat, and in 1898 when similar heat had prevailed the infant mortality was 285 per 1000. The deaths of infants under one year in England and Wales, taken from the registrar-general’s returns for 1907, amounted to 117.62 per 1000 births, an alarming sacrifice of life. France has been turning her attention to the establishment of infant consultations on the lines of Dr Budin’s, and similar dispensaries under the designation “Gouttes de lait” have been widely established in that country; gratifying results in the fall in infant mortality have followed. At the Fécamp dispensary the mortality from diarrhoea has fallen to 2.8, while that in neighbouring towns is from 50 to 76 per 1000 (Sir A. Simpson). It has been left to private enterprise in England to deal with this problem. The St Pancras “School for Mothers” was established in 1907 in north-west London. Though started by private persons it was in 1909 worked in connexion with the Health Department of the Borough Council, but was supported by charitable subscriptions and by a small contribution from the student mothers. There are classes for mothers on the care of their health during pregnancy, infant feeding, home nursing, cooking and needlework. Poor mothers unable to contribute get free dinners for three months previous to the birth of their child and for nine months after if the child is breast-fed. Two doctors are in attendance, and mothers are encouraged to bring their children fortnightly to be weighed, and receive advice. The average attendance is ninety. A baby is said to have “graduated” when it is a year old. An interesting development in connexion with the scheme is a class for fathers at which the medical officer of health for the district lectures on the duties of fatherhood. Similar schools for mothers are now established in Fulham and Stepney. Weighing centres have been established at Dundee, Sheffield, Nottingham, Birmingham, Aberdeen, Bolton, Belfast, and Newcastle-on-Tyne. An infants’ milk depôt has been established at Finsbury, and effort is being made to establish milk laboratories where separate nursing portions of sterile milk could be supplied to poor mothers. The Walker-Gordon milk laboratories in the United States are a step in this direction.

The average length of a child at birth is 19½ in. and during the first year the average increase is 77⁄8 in. A new-born infant is deaf (Koplik). This is supposed to be due to the blocking of the eustachian tubes with mucus. On the fourth day there is some evidence of hearing, and at the fifth week noises in the room disturb it. A healthy infant may be taken out of doors when a fortnight old in summer, after which it should have a daily outing, the eyes being protected from the direct rays of the sun. On the second day the eyes are sensitive to light, in the second month the infant notices colours, at the sixth month it knows its parents, and should be able to hold its head up. At the sixth month the baby begins to cut its temporary teeth. After their appearance they should be cleaned once a day by a piece of gauze moistened in boric acid solution. Attempts to stand are made about the tenth month, and walking begins about the fourteenth month. By this time the intelligence should be developed and memory is observed. A child a year old should be able to articulate a few small words. With the advent of walking and speech the period of infancy may be said to end.

See Pierre Budin, The Nursling (1907); Henry Koplik, Disease of Infancy and Childhood (1906); Eric Pritchard, The Physiological Feeding of Infants (1904); Eric Pritchard, Infant Education (1907); John Grimshaw, Your Child’s Health (1908).

(H. L. H.)

INFANT (in early forms enfaunt, enfant, through the Fr. enfant, from Lat. infans, in, not, and fans, the present participle of fari, to speak), a child; in non-legal use, a very young child, a baby, or one of an age suitable to be taught in an “infant school”; in law, a person under full age, and therefore subject to disabilities not affecting persons who have attained full age.

This article deals with “infants” in the last sense; for the more general sense see [Infancy] and [Child]. The period of full age varies widely in different systems, as do also the disabilities attaching to nonage (non-age). In Roman law, the age of puberty, fixed at fourteen for males and twelve for females, was recognized as a dividing line. Under that age a child was under the guardianship of a tutor, but several degrees of infancy were recognized. The first was absolute infancy; after that, until the age of seven, a child was infantiae proximus; and from the eighth year to puberty he was pubertati proximus. An infant in the last stage could, with the assent of his tutor, act so as to bind himself by stipulations; in the earlier stages he could not, although binding stipulations could be made to him in the second stage. After puberty, until the age of twenty-five years, a modified infancy was recognized, during which the minor’s acts were not void altogether, but voidable, and a curator was appointed to manage his affairs. The difference between the tutor and the curator in Roman law was marked by the saying that the former was appointed for the care of the person, the latter for the estate of the pupil. These principles apply only to children who are sui juris. The patria potestas, so long as it lasts, gives to the father the complete control of the son’s actions. The right of the father to appoint tutors to his children by will (testamentarii) was recognized by the Twelve Tables, as was also the tutorship of the agnati (or legal as distinct from natural relations) in default of such an appointment. Tutors who held office in virtue of a general law were called legitimi. Besides and in default of these, tutors dativi were appointed by the magistrates. These terms are still used in much the same sense in modern systems founded on the Roman law, as may be seen in the case of Scotland, noticed below.

By the law of England full age is twenty-one, and all minors alike are subject to incapacities. The period of twenty-one years is regarded as complete at the beginning of the day before the birthday: for example, an infant born on the first day of January attains his majority at the first moment of the 31st of December. The incapacity of an infant is designed for his own protection, and its general effect is to prevent him from binding himself absolutely by obligations. Of the contracts of an infant which are binding ab initio, the most important are those relating to “necessaries.” By the Sale of Goods Act 1893, an infant liable on a contract for necessaries can be sued only for a reasonable price, not necessarily the price he agreed to pay. The same statute declares “necessaries” to mean “goods suitable to the condition in life of the infant, and to his actual requirements at the time of the sale and delivery.” In the case of goods having a market price, the market price is reasonable. In all other cases the question is one of fact for the jury. The protection of infants extends sometimes to transactions completed after full age; the relief of heirs who have been induced to barter away their expectations is an example. “Catching bargains,” as they are called, throw on the persons claiming the benefit of them the burden of proving their substantial righteousness.

At common law a bargain made by an infant might be ratified by him after full age, and would then become binding. Lord Tenterden’s act required the ratification to be in writing. But now, by the Infants’ Relief Act 1874, “all contracts entered into by infants for the repayment of money lent or to be lent, or for goods supplied or to be supplied (other than contracts for necessaries), and all accounts stated, shall be absolutely void,” and “no action shall be brought whereby to charge any person upon any promise made after full age to pay any debt contracted during infancy, or upon any ratification made after full age of any promise or contract made during infancy, whether there shall or shall not be any new consideration for such promise or ratification after full age.” For some years after the passage of this statute highly conflicting views were held as to the meaning of the part of section 2 whereby it was enacted that “no action shall be brought whereby to charge any person ... upon any ratification made after full age of any promise or contract made during infancy.” Some authorities were of opinion that the section only applied to the three classes of contract made void by the previous section, viz. for goods supplied, money lent and on account stated. Others thought the effect to be that no contract, except for necessaries, made during infancy could be enforced after the infant came to full age. After several conflicting decisions it has been settled that both these views were wrong. Of the infant’s contracts voidable at common law there were two kinds. The first kind became void at full age, unless expressly ratified. The second kind were valid, unless repudiated within a reasonable time after full age was attained by the infant. The Infants’ Relief Act (section 2) strikes only at the first class and leaves the second untouched. Thus a promise of marriage made during infancy cannot be ratified so as to become actionable: but an infant’s marriage settlement, being of the second class, is valid, unless it is repudiated within a reasonable time after the infant attains full age. What is a reasonable time depends on all the circumstances of the case. In a case decided in 1893 a settlement made by a female infant was allowed to be repudiated thirty years after she attained full age, but the circumstances were exceptional. A contract of marriage may be lawfully made by persons under age. Marriageable age is fourteen in males and twelve in females. So, generally, an infant may bind himself by contract of apprenticeship or service. Since the passing of the Wills Act, an infant, except he be a soldier in actual military service or a seaman at sea, is unable to make a will. Infancy is in general a disqualification for public offices and professions, e.g. to be a member of parliament or an elector, a mayor or burgess, a priest or deacon, a barrister or solicitor, &c.

Before 1886 the custody of an infant belonged in the first place, and against all other persons, to the father, who was said to be “the guardian of his children by nature and nurture”; and the father might by deed or will dispose of the custody or tuition of his children until the age of twenty-one.

The Guardianship of Infants Act 1886 placed the mother almost on the same footing as the father as to guardianship of infants. On the death of the father the mother becomes guardian under the statute, either alone when no guardian has been appointed by the father, or jointly with any guardian appointed by him under 12 Chas. II. c. 24. A change of the law even more important is that whereby the mother may by deed or will appoint a guardian or guardians of her infant children to act after her death. If the father survives the mother, the mother’s guardian can only act if it be shown to the satisfaction of the court that the father is unfitted to be the sole guardian. On the death of the father, the guardian so appointed by the mother acts jointly with any guardian appointed by the father. The Guardianship of Infants Act 1886 also gives power to the high court and to county courts to make orders, upon the application of the mother, regarding the custody of an infant, and the right of access thereto of either parent. The court must take into consideration “the welfare of the infant, and ... the conduct of the parents, and ... the wishes as well of the mother as of the father.” The same statute also empowers the high court of justice, “on being satisfied that it is for the welfare of the infant,” to “remove from his office any testamentary guardian or any guardian appointed or acting by virtue of this act,” and also to appoint another in place of the guardian so removed.

The same statute gives power to a court sitting in divorce practically to take away from a parent guilty of a matrimonial offence all rights of guardianship. When a decree for judicial separation or divorce is pronounced, the court pronouncing it may at the same time declare the parent found guilty of misconduct to be unfit to have the custody of the children of the marriage. “In such case the parent so declared to be unfit shall not, upon the death of the other parent, be entitled as of right to the custody or guardianship of such children.” The court exercises this power very sparingly. When the declaration of unfitness is made, the practical effect is to give to the innocent parent the sole guardianship, as well as power to appoint a testamentary guardian to the exclusion of the guilty parent.

Another radical change has been made in the rights of parents as to guardianship of their children. In consequence of several cases where, after children had been rescued by philanthropic persons from squalid homes and improper surroundings, the courts had felt bound by law to redeliver them to their parents, the Custody of Children Act 1891 was passed. It provides that when the parent of a child applies to the court for a writ or order for the production of the child, and the court is of opinion that the parent has abandoned or deserted the child, or that he has otherwise so conducted himself that the court should refuse to enforce his right to the custody of the child, the court may, in its discretion, decline to issue the writ or make the order. If the child, in respect of whom the application is made, is being brought up by another person (“person” includes “school or institution”), or is being boarded out by poor-law guardians, the court may, if it orders the child to be given up to the parent, further order the parent to pay all or part of the cost incurred by such person or guardians in bringing up the child.

A parent who has abandoned or deserted his child is, prima facie, unfit to have the custody of the child. And before the court can make an order giving him the custody, the onus lies on him to prove that he is fit. The same rule applies where the child has been allowed by the parent “to be brought up by another person at that person’s expense, or by the guardians of a poor-law union, for such a length of time and under such circumstances as to satisfy the court that the parent was unmindful of his parental duties.”

The 4th section of the Custody of Children Act 1891 preserves the right of the parent to control the religious training of the infant. The father, however unfit he may be to have the custody of his child, has the legal right to require the child to be brought up in his own religion. If the father is dead, and has left no directions on the point, the mother may assert a similar right. But the court may consult the wishes of the child; and when an infant has been allowed by the father to grow up in a faith different from his own, the court will not, as a rule, order any change in the character of religious instruction. This is especially the case where the infant appears to be settled in his convictions.

In the same direction as the Custody of Children Act 1891 is the Children Act 1908, whereby considerable powers have been conferred on courts of summary jurisdiction (see [Children, Law Relating to]).

There is not at common law any corresponding obligation on the part of either parent to maintain or educate the children. The legal duties of parents in this respect are only those created by the poor laws, the Education Acts and the Children Act 1908.

An infant is liable to a civil action for torts and wrongful acts committed by him. But, as it is possible so to shape the pleadings as to make what is in substance a right arising out of contract take the form of a right arising from civil injury, care is taken that an infant in such a case shall not be held liable. With respect to crime, mere infancy is not a defence, but a child under seven years of age is presumed to be incapable of committing a crime, and between seven and fourteen his capacity requires to be affirmatively proved. After fourteen an infant is doli capax.

The law of Scotland follows the leading principles of the Roman law. The period of minority (which ends at twenty-one) is divided into two stages, that of absolute incapacity (until the age of fourteen in males, and twelve in females), during which the minor is in pupilarity, and that of partial incapacity (between fourteen and twenty-one), during which he is under curators. The guardians (or tutors) of the pupil are either tutors-nominate (appointed by the father in his will); tutors-at-law (being the next male agnate of twenty-five years of age), in default of tutors-nominate; or tutors-dative, appointed by royal warrant in default of the other two. No act done by the pupil, or action raised in his name, has any effect without the interposition of a guardian. After fourteen, all acts done by a minor having curators are void without their concurrence. Every deed in nonage, whether during pupilarity or minority, and whether authorized or not by tutors or curators, is liable to reduction on proof of “lesion,” i.e. of material injury, due to the fact of nonage, either through the weakness of the minor himself or the imprudence or negligence of his curators. Damage in fact arising on a contract in itself just and reasonable would not be lesion entitling to restitution. Deeds in nonage, other than those which are absolutely null ab initio, must be challenged within the quadriennium utile, or four years after majority.

The Guardianship of Infants Act 1886, the Custody of Children Act 1891 and the Children Act 1908, mentioned above, all apply to Scotland.

In the United States, the principles of the English common law as to infancy prevail, generally the most conspicuous variations being those affecting the age at which women attain majority. In many states this is fixed at eighteen. There is some diversity of practice as to the age at which a person can make a will of real or personal estate.

INFANTE (Spanish and Portuguese form of Lat. infans, young child), a title of the sons of the sovereign of Spain and Portugal, the corresponding infanta being given to the daughters. The title is not borne by the eldest son of the king of Spain, who is prince of Asturias, Il principe de Asturias. Until the severance of Brazil from the Portuguese monarchy, the eldest son was prince of Brazil. While a son or daughter of the sovereign of Spain is by right infante or infanta of Spain, the title, alone, is granted to other members of the blood royal by the sovereign.

INFANTICIDE, the killing of a newly-born child or of the matured foetus. When practised by civilized peoples the subject of infanticide concerns the criminologist and the jurist; but its importance in anthropology, as it involves a widespread practice among primitive or savage nations, requires more detailed attention. J. F. McLennan (Studies in Ancient History, pp. 75 et seq.) suggests that the practice of female infanticide was once universal, and that in it is to be found the origin of exogamy. Much evidence, however, has been adduced against this hypothesis by Herbert Spencer and Edward Westermarck. Infanticide, both of males and females, is far less widespread among savage races than McLennan supposed. It certainly is common in many lands, and more females are killed than males; but among many fierce and savage peoples it is almost unknown. Thus among the Tuski, Ahts, Western Eskimo and the Botocudos new-born children are killed now and then, if they are weak and deformed, or for some other reason (such as the superstition attaching to birth of twins) but without distinction of sex. Among the Dakota Indians and Crees female infanticide is rare. The Blackfoot Indians believe that a woman guilty of such an act will never reach “the Happy Mountain” after death, but will hover round the scene of her misdeed with branches of trees tied to her legs. The Aleutians hold that child-murder brings misfortune on the whole village. Among the Abipones it is common, but the boys are usually the victims, because it is customary to buy a wife for a son, whereas a grown daughter will always command a price. In Africa, where a warm climate and abundance of food simplify the problem of existence, the crime is not common. Herr Valdau relates that a Bakundu woman, accused of it, was condemned to death. In Samoa, in the Mitchell and Hervey Islands, and in parts of New Guinea, it was unheard of; while among the cannibals, the Solomon Islanders, it occurred rarely. A theory has been advanced by L. Fison (Kamilaroi and Kurnai, 1880) that female infanticide is far less common among the lower savages than among the more advanced tribes. Among some of the most degraded of human beings, such as the Yahgans of Tierra del Fuego, the crime was unknown, except when committed by the mother “from jealousy or hatred of her husband or because of desertion and wretchedness.” It is said that certain Californian Indians were never guilty of child-murder before the arrival of the whites; while Wm. Ellis (Polynesian Researches, i. 249) thinks it most probable that the custom was less prevalent in earlier than later Polynesian history. The weight of evidence tends to support Darwin’s theory that during the earliest period of human development man did not lose that strong instinct, the love of his young, and consequently did not practice infanticide; that, in short, the crime is not characteristic of primitive races.

Infanticide may be said to arise from four reasons. It may be (1) an act of callous brutality or to satisfy cannibalistic cravings. A Fuegian, Darwin relates, dashed his child’s brains out for upsetting a basket of fish. An Australian, seeing his infant son ill, killed, roasted and ate him. In some parts of Africa the negroes bait lion-traps with their own children. Some South American Indians, such as the Moxos, abandon or kill them without reason; while African and Polynesian cannibals eat them without the excuse of the periodic famines which made the Tasmanians regard the birth of a child as a piece of good fortune.

2. Or infanticide may be the result of the struggle for existence. Thus in Polynesia, while the climate ensures food in plenty, the relative smallness of the islands imposed the custom on all families without distinction. In the Hawaiian Islands all children, after the third or fourth, were strangled or buried alive. At Tahiti fathers had the right (and used it) of killing their newly-born children by suffocation. The chiefs were obliged by custom to kill all their daughters. The society of the Areois, famous in the Society Islands, imposed infanticide upon the women members by oath. In other islands all girl-children were spared, but only two boys in each family were reared. The difficulties of suckling partly explain the custom of killing twins. For the same reason the Eskimo and Red Indians used to bury the infant with the mother who died in childbirth. Among warrior and hunter tribes, where women could not act as beasts of burden as in agricultural communities, and where a large number of girls were likely to attract the hostile attentions of neighbouring tribesmen, girl-babies were murdered. Arabs, in ancient times, buried alive the majority of female children. In many lands infanticide was regarded as a meritorious act on the part of a parent, done, as a precaution against famine, in the interests of the tribe. In other parts of the world, infanticide results from customs which impose heavy burdens on child-rearing. Of these artificial hardships the best example is afforded by India. There the practice, though forbidden by both the Vedas and the Koran, prevailed among the Rajputs and certain aboriginal tribes. Among the aristocratic Rajputs, it was thought dishonourable that a girl should remain unmarried. Moreover, a girl may not marry below her caste; she ought to marry her superior, or at least her equal. This reasoning was most powerful with the highest castes, in which the disproportion of the sexes was painfully apparent. But, assuming marriage to be possible, it was ruinously expensive to the bride’s father, the cost in the case of some rajahs having been known to exceed £100,000. To avoid all this, the Rajput killed a proportion of his daughters—sometimes in a very singular way. A pill of tobacco and bhang might be given to the new-born child; or it was drowned in milk;[1] or the mother’s breast was smeared with opium or the juice of the poisonous datura. A common method was to cover the child’s mouth with a plaster of cow-dung, before it drew breath. Infanticide was also practised to a small extent by some sects of the aboriginal Khonds and by the poorer hill-tribes of the Himalayas. Where infanticide occurs in India, though it really rests on the economic facts stated, there is usually some poetical tradition of its origin. Infanticide from motives of prudence was common among some American Indian tribes of the north-west, with whom the “potlatch” was an essential part of their daughter’s marriage ceremonies.

3. Or infanticide may be in the nature of a religious observance. The gods must be appeased with blood, and it is believed that no sacrifice can be so pleasing to them as the child of the worshipper. Such were the motives impelling parents to the burning of children in the worship of Moloch. In India children were thrown into the sacred river Ganges, and adoration paid to the alligators who fed on them. Where the custom prevails as a sacrifice the male child is usually the victim.

4. Or, finally, infanticide may have a social or political reason. Thus at Sparta (and in other places in early Greek and Roman history) weakly or deformed children were killed by order of the state, a custom approved in the ideal systems of Aristotle and Plato, and still observed among the Eskimo and the Kamchadales.

Authorities.—Herbert Spencer, Principles of Sociology, i. 614-619; McLennan, Studies in Ancient History, pp. 75 et seq.; McLennan, “Exogamy and Endogamy” in the Fortnightly Review, xxi. 884 et seq.; Darwin, Descent of Man, ii. 400 et seq.; L. Fison, and A. W. Howitt, Kamilaroi and Kurnai (1880); Westermarck, History of Human Marriage (1894); Browne, Infanticide: Its Origin, Progress and Suppression (London, 1857); Lord Avebury, Prehistoric Times (1900), and Origin of Civilization (1902).

Law.—The crime of infanticide among civilized nations is still frequent. It is however due in most cases to abnormal causes, such as a sudden access of insanity, privation, unreasoning dislike to the child, &c. It is most closely connected with illegitimacy in the class of farm and domestic servants, the more common motive being the terror of the mother of incurring the disgrace with which society visits the more venial offence. Often, however, it is inspired by no better motive than the wish to escape the burden of the child’s support. The granting of affiliation orders thus tends to save the lives of many children, though it provides a motive for the paramour sometimes to share in the crime. The laws of the European states differ widely on this subject—some of them treating infanticide as a special crime, others regarding it merely as a case of murder of unusually difficult proof. In the law of England infanticide is murder or manslaughter according to the presence or absence of deliberation. The infant must be a human being in the legal sense; and “a child becomes a human being when it has completely proceeded in a living state from the body of its mother, whether it has breathed or not, and whether it has an independent circulation or not, and whether the navel-string is severed or not; and the killing of such a child is homicide when it dies after birth in consequence of injuries received before, during or after birth.” À child in the womb or in the act of birth, though it may have breathed, is therefore not a human being, the killing of which amounts to homicide. The older law of child murder under a statute of James I. consisted of cruel presumptions against the mother, and it was not till 1803 that trials for that offence were placed under the ordinary rules of evidence. The crown now takes upon itself the onus of proving in every case that the child has been alive. This is often a matter of difficulty, and hence a frequent alternative charge is that of concealment of birth (see [Birth]), or concealment of pregnancy in Scotland. It is the opinion of the most eminent of British medical jurists that this presumption has tended to increase infanticide. Apart from this, the technical definition of human life has excited a good deal of comment and some indignation. The definition allows many wicked acts to go unpunished. The experience of assizes in England shows that many children are killed when it is impossible to prove that they were wholly born. The distinction taken by the law was probably comprehended by the minds of the class to which most of the unhappy mothers belong. Partly to meet this complaint it was suggested to the Royal Commission of 1866 that killing during birth, or within seven days thereafter, should be an offence punishable with penal servitude. The second complaint is of an opposite character—partly that infanticide by mothers is not a fit subject for capital punishment, and partly that, whatever be the intrinsic character of the act, juries will not convict or the executive will not carry out the sentence. Earl Russell gave expression to this feeling when he proposed that no capital sentence should be pronounced upon mothers for the killing of children within six months after birth. When there has been a verdict of murder, sentence of death must be passed, but the practice of the Home Office, as laid down in 1908, is invariably to commute the death sentence to penal servitude for life. The circumstances of the case and the disposition and general progress of the prisoners under discipline in a convict prison are then determining factors in the length of subsequent detention, which rarely exceeds three years. After release, the prisoner’s further progress is carefully watched, and if it is seen to be to her advantage the conditions of her release are cancelled and she is restored to complete freedom.

In India measures against the practice were begun towards the end of the 18th century by Jonathan Duncan and Major Walker. They were continued by a series of able and earnest officers during the 19th century. One of its chief events, representing many minor occurrences, was the Amritsar durbar of 1853, which was arranged by Lord Lawrence. At that meeting the chiefs residing in the Punjab and the trans-Sutlej states signed an agreement engaging to expel from caste every one who committed infanticide, to adopt fixed and moderate rates of marriage expenses, and to exclude from these ceremonies the minstrels and beggars who had so greatly swollen the expense. According to the present law, if the female children fall below a certain percentage in any tract or among any tribe in northern India where infanticide formerly prevailed, the suspected village is placed under police supervision, the cost being charged to the locality. By these measures, together with a strictly enforced system of reporting births and deaths, infanticide has been almost trampled out; although some of the Rājput clans keep their female offspring suspiciously close to the lowest average which secures them from surveillance.

It is difficult to say to what extent infanticide prevails in the United Kingdom. At one time a large number of children were murdered in England for the purpose of obtaining the burial money from a benefit club,[2] but protection against this risk has been provided for by the Friendly Societies Act 1896, and the Collecting Societies Act 1896. The neglect or killing of nurse-children is treated under [Baby-farming], and [Children, Law Relating to].

In the United States, the elements of this offence are practically the same as in England. The wilful killing of an unborn child is not manslaughter unless made so by statute. To constitute manslaughter under Laws N.Y. 1869, ch. 631, by attempts to produce miscarriage, the “quickening” of the child must be averred and proved (Evans v. People, 49 New York Rep. 86; see also Wallace v. State, 7 Texas app. 570).

[1] In Baluchistan, where children are often drowned in milk, there is a euphemistic proverb: “The lady’s daughter died drinking milk.”

[2] See Report on the Sanitary Condition of the Labouring Classes, “Supplementary Report on Interment in Towns,” by Edwin Chadwick (Parl. Papers, 1843, xii. 395); and The Social Condition and Education of the People, by Joseph Kay (1850).

INFANTRY, the collective name of soldiers who march and fight on foot and are armed with hand-weapons. The word is derived ultimately from Lat. infans, infant, but it is not clear how the word came to be used to mean soldiers. The suggestion that it comes from a guard or regiment of a Spanish infanta about the end of the 15th century cannot be maintained in view of the fact that Spanish foot-soldiers of the time were called soldados and contrasted with French fantassins and Italian fanteria. The New English Dictionary suggests that a foot-soldier, being in feudal and early modern times the varlet or follower of a mounted noble, was called a boy (cf. Knabe, garçon, footman, &c., and see [Valet]).

Historical Sketch

The importance of the infantry arm, both in history and at the present time, cannot be summed up better and more concisely than in the phrase used by a brilliant general of the Napoleonic era, General Morand—“L’infanterie, c’est l’armée.”

It may be confidently asserted that the original fighting man was a foot-soldier. But infantry was differentiated as an “arm” considerably later than cavalry; for when a new means of fighting (a chariot or a horse) presented itself, it was assimilated by relatively picked men, chiefs and noted warriors, who ipso facto separated themselves from the mass or reservoir of men. How this mass itself ceased to be a mere residue and developed special characteristics; how, instead of the cavalry being recruited from the best infantry, cavalry and infantry came to form two distinct services; and how the arm thus constituted organized itself, technically and tactically, for its own work—these are the main questions that constitute the historical side of the subject. It is obvious that as the “residue” was far the greatest part of the army, the history of the foot-soldier is practically identical with the history of soldiering.

It was only when a group of human beings became too large to be surprised and assassinated by a few lurking enemies, that proper fighting became the normal method of settling a quarrel or a rivalry. Two groups, neither of which had been able to surprise the other, had to meet face to face, and the instinct of self-preservation had to be reconciled with the necessity of victory. From this it was an easy step to the differentiation of the champion, the proved excellent fighting man, and to providing this man, on whom everything depended, with all assistance that better arms, armour, horse or chariot could give him. But suppose our champion slain, how are we to make head against the opposing champion? For long ages, we may suppose, the latter, as in the Iliad, slaughtered the sheep who had lost their shepherd, but in the end the “residue” began to organize itself, and to oppose a united front to the enemy’s champions—in which term we include all selected men, whether horsemen, charioteers or merely specially powerful axemen and swordsmen. But once the individual had lost his commanding position, the problem presented itself in a new form—how to ensure that every member of the group did his duty by the others—and the solution of this problem for the conditions of the ancient hand-to-hand struggle marks the historical beginning of infantry tactics.

Gallic warriors bound themselves together with chains. The Greeks organized the city state, which gave each small army solidarity and the sense of duty to an ideal, and the phalanx, in which the file-leaders were in a sense champions yet were The phalanx and the legion. made so chiefly by the unity of the mass. But the Romans went farther. Besides developing solidarity and a sense of duty, they improved on this conception of the battle to such a degree that as a nation they may be called the best tacticians who ever existed. Giving up the attempt to make all men fight equally well, they dislocated the mass of combatants into three bodies, of which the first, formed of the youngest and most impressionable men, was engaged at the outset, the rest, more experienced men, being kept out of the turmoil. This is the very opposite of the “champion” system. Those who would have fled after the fall of the champions are engaged and “fought out” before the champions enter the area of the contest, while the champions, who possess in themselves the greatest power of resisting and mastering the instinct of self-preservation, are kept back for the moment when ordinary men would lose heart.

It might be said with perfect justice that without infantry there would never have been discipline, for cavalry began and continued as a crowd of champions. Discipline, which created and maintained the intrinsic superiority of the Roman legion, depended first on the ideal of patriotism. This was ingrained into every man from his earliest years and expressed in a system of rewards and punishments which took effect from the same ideal, in that rewards were in the main honorary in character (mural crowns, &c.), while no physical punishment was too severe for the man who betrayed, by default or selfishness, the cause of Rome. Secondly, though every man knew his duty, not every man was equal to doing it, and in recognition of this fact the Romans evolved the system of three-line tactics in which the strong parts of the machine neutralized the weak. The first of these principles, being psychological in character, rose, flourished and decayed with the moral of the nation. The second, deduced from the first, varied with it, but as it was objectively expressed in a system of tactics, which had to be modified to suit each case, it varied also in proportion as the combat took more or less abnormal forms. So closely knit were the parts of the system that not only did the decadence of patriotism sap the legionary organization, but also the unsuitability of that organization to new conditions of warfare reacted unfavourably, even disastrously, on the moral of the nation. Between them, the Roman infantry fell from its proud place, and whereas in the Republic it was familiarly called the “strength” (robur), by the 4th century A.D. it had become merely the background for a variety of other arms and corps. Luxury produced “egoists,” to whom the rewards meant nothing and the punishments were torture for the sake of torture. When therefore the Roman imperium extended far enough to bring in silks from China and ivory from the forests of central Africa, the citizen-army ceased to exist, and the mere necessity for garrisoning distant savage lands threw the burden of service upon the professional soldier.

The natural consequence of this last was the uniform training of every man. There were no longer any primary differences between one cohort and another, and though the value of the three-line system in itself ensured its continuance, The Roman Imperial Army. any cohort, however constituted, might find itself serving in any one of the three lines, i.e. the moral of the last line was no better than that of the first. The best guarantee of success became uniform regimental excellence, and whereas Camillus or Scipio found useful employment in battle for every citizen, Caesar complained that a legion which had been sent him was too raw, though it had been embodied for nine years. The conditions which were so admirably met by the old system never reappeared; for before armies resumed a “citizen” character the invention of firearms had subjected all ranks and lines alike to the same ordeal of facing unseen death, and the old soldiers were better employed in standing shoulder to shoulder with the young. In brief, the old Roman organization was based on patriotism and experience, and when patriotism gave place to “egoism,” and the experience of the citizen who spent every other summer in the field of war gave place to the formal training of the paid recruit, it died, unregretted either by the citizen or by the military chieftain. The latter knew how to make the army his devoted servant, while the former disliked military service and failed to prepare himself for the day when the military chief and the mercenary overrode his rights and set up a tyranny, and ultimately the inner provinces of the empire came to be called inermes—unarmed, defenceless—in contrast to the borderland where the all-powerful professional legions lay in garrison.

In these same frontier provinces the tactical disintegration of the legion slowly accomplished itself. Originally designed for the exigencies of the normal pitched battle on firm open fields, and even after its professionalization retaining its character as a large battle unit, it was soon fragmented through the exigencies of border warfare into numerous detachments of greater or less size, and when the military frontier of the empire was established, the legion became an almost sedentary corps, finding the garrisons for the blockhouses on its own section of the line of defence. Further, the old heavy arms and armour which had given it the advantage in wars of conquest—in which the barbarians, gathering to defend their homes, offered a target for the blow of an army—were a great disadvantage when it became necessary to police the conquered territory, to pounce upon swiftly moving bodies of raiders before they could do any great harm. Thus gradually cavalry became more numerous, and light infantry of all sorts more useful, than the old-fashioned linesman. To these corps went the best recruits and the smartest officers, the opportunities for good service and the rewards for it. The legion became once more the “residue.” Thus when the “champion” reappeared on the battlefield the solidarity that neutralized his power had ceased to exist.

The battle of Adrianople, the “last fight of the legion,” illustrates this. The frontal battle was engaged in the ordinary way, and the cohorts of the first line of the imperial army were fighting man to man with the front ranks of the Gothic infantry (which had indeed a solidarity of its own, unlike the barbarians of the early empire, and was further guaranteed against moral over-pressure by a wagon laager), when suddenly the armoured heavy cavalry of the Goths burst upon their flank and rear. There were no longer Principes and Triarii of the old Republican calibre, but only average troops, in the second and third lines, and they were broken at once. The first line felt the battle in rear as well as in front and gave way. Thereafter the victors, horse and foot, slaughtered unresisting herds of men, not desperate soldiers, and on this day the infantry arm, as an arm, ceased to exist.

Of course, not every soldier became a horseman, and still fewer could provide themselves with armour. Regular infantry, too, was still maintained for siege, mountain and forest warfare. But the robur, the kernel of the line The Dark Ages. of battle, was gone, and though a few of the peoples that fought their way into the area of civilization in the dark ages brought with them the natural and primitive method of fighting on foot, it was practically always a combination of mighty champions and “residue,” even though the latter bound themselves together by locked shields, as the Gauls had bound themselves long before with chains, to prevent “skulking.” These infantry nations, without any infantry system comparable to that of the Greeks and Romans, succumbed in turn to the crowd of mounted warriors—not like the Greeks and Romans for want of good military qualities, but for want of an organization which would have distributed their fighting powers to the best advantage. One has only to study the battle of Hastings to realize how completely the infantry masses of the English slipped from the control of their leaders directly the front ranks became seriously engaged. For many generations after Hastings there was no attempt to use infantry as the kernel of armies, still less to organize it as such beforehand. Indeed, except in the Crusades, where men of high and of low degree alike fought for their common faith, and in sieges, where cavalry was powerless and the services of archers and labourers were at a premium, it became quite unusual for infantry to appear on the field at all.

The tactics of feudal infantry at its best were conspicuously illustrated in the battle of Bouvines, where besides the barons, knights and sergeants, the Brabançon mercenaries (heavy foot) and the French communal militia opposed one another. Bouvines. On the French right wing, the opportune arrival of a well-closed mass of cavalry and infantry in the flank of a loose crowd of men-at-arms which had already been thoroughly engaged, decided the fight. In the centre, the respective infantries were in first line, the nobles and knights, with their sovereigns, in second, yet it was a mixed mass of both that, after a period of confused fighting, focussed the battle in the persons of the emperor and the king of France, and if the personal encounters of the two bodies of knights gave the crowded German infantry a momentary chance to strike down the king, the latter was soon rescued by a half-dozen of heavy cavalrymen. On the left wing, the count of Boulogne made a living castle of his Brabançon pikes, whence with his men-at-arms he sallied forth from time to time and played the champion. Lastly, the Constable Montmorency brought over what was still manageable of the corps that had defeated the cavalry on the right (nearly all mounted men) and gave the final push to the allied centre and right in succession. Then the imperial army fled and was slaughtered without offering much resistance. Of infantry in this battle there was enough and to spare, but its only opportunities for decisive action were those afforded by the exhaustion of the armoured men or by the latter becoming absorbed in their own single combats to the exclusion of their proper work in the line of battle. As usual the infantry suffered nine-tenths of the casualties. For all their numbers and apparent tactical distribution on this field, they were “residue,” destitute of special organization, training or utility; and the only suggestion of “combined tactics” is the expedient adopted by the count of Boulogne, rings of spearmen to serve as pavilions served in the tournament—to secure a decorous setting for a display of knightly prowess.

In those days in truth the infantry was no more the army than to-day the shareholders of a limited company are the board of directors. They were deeply, sometimes vitally, interested in the result, but they contributed little or nothing to bringing it about, except when the opposing cavalries were in a state of moral equilibrium, and in these cases anything suffices—the appearance of camp followers on a “Gillies Hill,” as at Bannockburn or the sound of half-a-dozen trumpets—to turn the scale. Once it turned, the infantry of the beaten side was cut down unresistingly, while the more valuable prisoners were admitted to ransom. Thereafter, feudal tactics were based principally on the ideas of personal glory—won in single combat, champion against champion, and of personal profit—won by the knight in holding a wealthy and well-armed baron to ransom and by the foot-soldier in plundering while his masters were fighting. In the French army, the term bidaux, applied in the days of Bouvines to all the infantry other than archers and arblasters, came by a quite natural process to mean the laggards, malingerers and skulkers of the army.

But even this infantry contained within itself two half-smothered sparks of regeneration, the idea of archery and the idea of communal militia. Archery, in whatever form practised, was the one special form of military Revival of infantry. activity with which the heavy gendarme (whether he fought on horseback or dismounted) had no concern. Here therefore infantry had a special function, and in so far ceased to be “residue.” The communal militia was an early and inadequate expression of the town-spirit that was soon to produce the solid burgher-militia of Flanders and Germany and after that the trained bands of the English cities and towns. It therefore represented the principles of solidarity, of combination, of duty to one’s comrade and to the common cause—principles which had disappeared from feudal warfare.[1] It was under the influence of these two ideas or forces that infantry as an arm began once again, though slowly and painfully, to differentiate itself from the mass of bidaux until in the end the latter practically contained only the worthless elements.

The first true infantry battle since Hastings was fought at Courtrai in 1302, between the burghers of Bruges and a feudal army under Count Robert of Artois. The citizens, arrayed in heavy masses, and still armed with miscellaneous weapons, were Courtrai. careful to place themselves on ground difficult of access—dikes, pools and marshes—and to fasten themselves together, like the Gauls of old. Their van was driven back by the French communal infantry and professional crossbowmen, whereupon Robert of Artois, true feudal leader as he was, ordered his infantry to clear the way for the cavalry and without even giving them time to do so pushed through their ranks with a formless mass of gendarmerie. This, in attempting to close with the enemy, plunged into the canals and swamped lands, and was soon immovably fastened in the mud. The citizens swarmed all round it and with spear, cleaver and flail destroyed it. Robert himself with a party of his gendarmerie strove to break through the solid wall of spears, but in vain. He was killed and his army perished with him, for the citizens did not regard war as a game and ransom as the loser’s forfeit. As for the communal infantry which had won the first success, it had long since disappeared from the field, for when count Robert ordered his heavy cavalry forward, they had thought themselves attacked in rear by a rush of hostile cavalry—as indeed they were, for the gendarmerie rode them down—and melted away.

Crécy (q.v.) was fought forty-four years after Courtrai. Here the knights had open ground to fight on, and many boasted that they would revenge themselves. But they encountered not merely infantry, but infantry tactics, and were for the second, and not the last, time destroyed. The English army included a large feudal element, but the spirit of indiscipline had been crushed by a series of iron-handed kings, and for more than a century the nobles, in so far as they had been bad subjects, had been good Englishmen. The English yeomen had reached a level of self-discipline and self-respect which few even of the great continental cities had attained. They had, lastly, made the powerful long-bow (see [Archery]) their own, and Edward I. had combined the shock of the heavy cavalry with the slow searching preparatory rain of arrows (see [Falkirk]). That is, infantry tactics and cavalry tactics were co-ordinated by a general, and the special point of this for the present purpose is that instead of being, as in France, the unstable base of the so-called “feudal pyramid,” infantry has become an arm, capable of offence and defence and having its own special organization, function in the line of battle and tactical method. This last, indeed, like every other tactical method, rested ultimately on the moral of the men who had to put it into execution. Archer tactics did not serve against the disciplined rush of Joan of Arc’s gendarmerie, for the solidarity of the archer companies that tried to stop it had long been undermined.

Yet we cannot overrate the importance of the archer in this period of military history. In the city militias solidarity had been obtained through the close personal relationship of the trade gilds and by the elimination of the champion. The English archer. Therefore, as every offensive in war rests upon boldness, these militias were essentially defensive, for they could only hope to ward off the feudal champion, not to outfight him (Battle of Legnano, 1176. See Oman, p. 442). England, however, had evolved a weapon which no armour could resist, and a race of men as fully trained to use it as the gendarme was to use the lance.[2] This weapon gave them the power of killing without being killed, which the citizens’ spears and maces and voulges did not. But like all missiles, arrows were a poor stand-by in the last resort if determined cavalry crossed the “beaten zone” and closed in, and besides pavises and pointed stakes the English archers were given the support of the knights, nobles and sergeants—the armoured champions—whose steady lances guaranteed their safety. Here was the real forward stride in infantry tactics. Archery had existed from time immemorial, and a mere technical improvement in its weapon could hardly account for its suddenly becoming the queen of the battlefield. The defensive power of the “dark impenetrable wood” of spears had been demonstrated again and again, but when the cavalry had few or no preliminary difficulties to face, the chances of the infantry mass resisting long-continued pressure was small. It was the combination of the two elements that made possible a Crécy and a Poitiers, and this combination was the result of the English social system which produced the camaraderie of knight and yeoman, champion and plain soldier. Fortified by the knight’s unshakeable steadiness, the yeoman handled his bow and arrows with cool certainty and rapidity, and shot down every rush of the opposing champions. This was camaraderie de combat indeed, and in such conditions the offensive was possible and even easy. The English conquered whole countries while the Flemish and German spearmen and vougiers merely held their own. For them, decisive victories were only possible when the enemy played into their hands, but for the English the guarantee of such victories was the specific character of their army itself and the tactical methods resulting from and expressing that character.

But the war of conquest embodied in these decisive victories dwindled in its later stages to a war of raids. The feudal lord, like the feudal vassal, returned home and gave place to the professional man-at-arms and the professional The Hundred Years’ War. captain. Ransom became again the chief object, and except where a great leader, such as Bertrand Du Guesclin, compelled the mercenaries to follow him to death or victory, a battle usually became a mêlée of irregular duels between men-at-arms, with all the selfishness and little of the chivalry of the purely feudal encounter. The war went on and on, the gendarmes thickened their armour, and the archers found more difficulty in penetrating it. Moreover, in raids for devastation and booty, the slow-moving infantryman was often a source of danger to his comrades. In this guerrilla the archer, though he kept his place, soon ceased to be the mainstay of battle. It had become customary since Crécy (where the English knights and sergeants were dismounted to protect the archers) for all mounted men to send away their horses before engaging. Here and there cavalry masses were used by such energetic leaders as the Black Prince and Du Guesclin, and more often a few men remained mounted for work requiring exceptional speed and courage,[3] but as a general rule the man-at-arms was practically a mounted infantryman, and when he dismounted he stood still. Thus two masses of dismounted lances, mixed with archers, would meet and engage, but the archers, the offensive element, were now far too few in proportion to the lances, the purely defensive element, and battles became indecisive skirmishes instead of overwhelming victories.

Cavalry therefore became, in a very loose sense of the word, infantry. But we are tracing the history not of all troops that stood on their feet to fight, but of infantry and the special tactics of infantry, and the period before and after 1370, when the moral foundations of the new English tactics had disappeared, and the personality of Du Guesclin gave even the bandits of the “free companies” an intrinsic, if slight, superiority over the invaders, is a period of deadlock. Solidarity, such as it was, had gone over to the side of the heavy cavalry. But the latter had deliberately forfeited their power of forcing the decision by fighting on foot, and the English archer, the cadre of the English tactical system, though diminished in numbers, prestige and importance, held to existence and survived the deadlock. Infantry of that type indeed could never return to the “residue” state, and it only needed a fresh moral impetus, a Henry V., to set the old machinery to work again for a third great triumph. But again, after Agincourt, the long war lapsed into the hands of the soldiers of fortune, the basis of Edward’s and Henry’s tactics crumbled, and, led by a greater than Du Guesclin, the knights and the nobles of France, and the mercenary captains and men-at-arms as well, rode down the stationary masses of the English, lances and bowmen alike.

The net result of the Hundred Years’ War therefore was to re-establish the two arms, cavalry and infantry, side by side, the one acting by shock, and the other by fire. The lesson of Crécy was “prepare your charge before delivering it,” and for that purpose great bodies of infantry armed with bows, arblasts and handguns were brought into existence in France. When the French king in 1448 put into force the “lessons of the war” and organized a permanent army, it consisted in the main of heavy cavalry (knights and squires in the “ordonnance” companies, soldiers of fortune in the paid companies) and archers and arblasters (francs-archers recruited nationally, arblasters as a rule mercenaries, though largely recruited in Gascony). To these armes de jet were added, in ever-increasing numbers, hand firearms. Thus the “fire” principle of attack was established, and the defensive principle of “mass” relegated to the background. In such circumstances cavalry was of course the decisive arm, and the reputation of the French gendarmerie was such as to justify this bold elimination of the means of passive defence.[4]

The foot-soldier of Germany and the Low Countries had followed a very different line of development. Here the rich commercial cities scarcely concerned themselves with the quarrels or revolts of neighbouring nobles, Burgher militias. but they resolutely defended their own rights against feudal interference, and enforced them by an organized militia, opposing the strict solidarity of their own institutions to the prowess of the champion who threatened them. The struggle was between “you shall” on the part of the baron and “we will not” on the part of the citizens, the offensive versus the defensive in the simplest and plainest form. The latter was a policy of unbreakable squares, and wherever possible, strong positions as well. Sometimes the citizens, sometimes the nobles gained the day, but the general result was that steady infantry in proper formation could not be ridden down, and as yeomen-archers of the English type to “prepare” the charge were not obtainable from amongst the serf populations of the countryside, the problem of the attack was, for Central Europe, insoluble.

The unbreakable square took two forms, the wagenburg with artillery, and the infantry mass with pikes. The first was no more, in the beginning, than an expedient for the safe and rapid crossing of wider stretches of open country The Wagenburg. than would have been possible for dismounted men, whom the cavalry headed off as soon as they ventured far enough from the shelter of walls. The men rode not on horses but on carriages, and the carriages moved over the plains in laager formation, the infantrymen standing ready with halbert and voulge or short stabbing spear, and the gunners crouching around the long barrelled two-pounders and the “ribaudequins”—the early machine guns—which were mounted on the wagons. These wagenburgen combined in themselves the due proportions of mobility and passive defence, and in the skilled hands of Ziska they were capable of the boldest offensive. But such a tactical system depended first of all on drill, for the armoured cavalry would have crowded through the least gap in the wagon line, and the necessary degree of drill in those days could only be attained by an army which had both a permanent existence and some bond of solidarity more powerful than the incentive to plunder—that is, in practice, it was only attained in full by the Hussite insurgents. The cavalry, too, learned its lesson, and pitted mobile three-pounders against the foot-soldiers’ one- and two-pounders, and the wagenburg became no more than a helpless target. Thus when, not many years after the end of the Hussite wars, the Wars of the Roses eliminated the English model and the English tactics from the military world of Europe, the French system of fire tactics—masses of archers, arblasters and handgun-men, with some spearmen and halberdiers to stiffen them—was left face to face with that of the Swiss and Landsknechts, the system of the “long pike.”

A series of victories ranging from Morgarten (1315) to Nancy (1477) had made the Swiss the most renowned infantry in Europe. Originally their struggles with would-be oppressors had taken the form, often seen elsewhere, of arraying solid The Swiss. masses of men, united in purpose and fidelity to one another rather than by any material or tactical cohesion. Like the men of Bruges at Courtrai, the Swiss had the advantage of broken ground, and the still greater advantage of being opposed by reckless feudal cavalry. Their armament at this stage was not peculiar—voulges, gisarmes, halberts and spears—though they were specially adept in the use of the two-handed sword. But as time went on the long pike (said to have originated in Savoy or the Milanese about 1330) became more and more popular until at last on the verge of their brief ascendancy (about 1475-1515) the Swiss armed as much as one quarter of their troops with it. The use of firearms made little or no progress amongst them, and the Swiss mercenaries of 1480, like their forerunners of Morgarten and Sempach, fought with the arme blanche alone. But in a very few years after the Swiss nation had become soldiers of fortune en masse, the more open lands of Swabia entered into serious and bitter competition with them. From these lands came the Landsknechts, whose order was as strong as, and far less unwieldy than, that of the Swiss, whose armament included a far greater proportion of firearms, and who established a regimental system that left a permanent mark on army organization. The Landsknecht was the prototype of the infantryman of the 16th and 17th centuries, but his right to indicate the line of evolution had to be wrung from many rivals.

The year 1480 indeed was a turning-point in military history. Within the three years preceding it the battles of Nancy and Guinegate had destroyed both the old feudalism of Charles the Bold and the new cavalry tactics of the The long pike. French gendarmerie. The former was an anachronism, while the latter, when the great wars came to an end and there was no longer either a national impulse or a national leader, had lapsed into the old vices of ransom and plunder. With these, on the same fields, the franc-archer system of infantry tactics perished ignominiously. It rested, as we know, on the principle that the fire of the infantry was to be combined with and completed by the shock of the gendarmerie, and when the latter were found wanting as at Guinegate, the masses of archers and arblasters, which were only feebly supported by a few handfuls of pikemen and halberdiers, were swept away by the charge of some heavy battalions of Swabian and Flemish pikes. Guinegate was the début of the Landsknecht infantry as Nancy was that of the Swiss, and the lesson could not be misread. Louis XI. indeed hanged some of his franc-archers and dismissed the rest, and in their place raised “bands” of regular infantry, one of which bore for the first time the historic name of Picardie. But these “bands” were not self-contained. Armed for the most part with armes de jet they centred on the 6000 Swiss pikemen whom Louis XI., in 1480, took into his service, and for nearly fifty years thereafter the French foot armies are always composed of two elements, the huge battalions of Swiss or Landsknechts,[5] armed exclusively with the long pike (except for an ever-decreasing proportion of halberts, and a few arquebuses), and for their support and assistance, French and mercenary “bands.”

The Italian wars of 1494-1544, in which the principles of fire and shock were readjusted to meet the conditions created by firearms, were the nursery of modern infantry. The combinations of Swiss, Landsknechts, Spanish “tercios” and French “bands” that figured on the battlefields of the early 16th century were infinitely various. But it is not difficult to find a thread that runs through the whole.

The essence of the Swiss system was solidity. They arrayed themselves in huge oblongs of 5000 men and more, at the corners of which, like the tower bastions of a 16th-century fortress, stood small groups of arquebusiers. The The Italian Wars, 1494-1525. Landsknechts and the Romagnols of Italy, imitated and rivalled them, though as a rule developing more front and less depth. At this stage solidity was everything and fire-power nothing. At Fornuovo (1495) the mass of arquebusiers and arblasters in the French army did little or nothing; it was the Swiss who were l’espérance de l’ost. At Agnadello or Vailá in 1509 the ground and the “encounter-battle” character of the engagement gave special chances of effective employment to the arquebusiers on either side. Along the front the Venetian marksmen, secure behind a bank, picked off the leaders of the enemy as they came near. On the outer flank of the battle the bands of Gascon arquebusiers, which would otherwise have been relegated to an unimportant place in the general line of battle, lapped round the enemy’s flank in broken ground and produced great and almost decisive effect. But this was only an afterthought of the king of France and Bayard. In the rest of the battle the huge masses of Swiss pikes were thrown upon the enemy much as the old feudal cavalry had been, regardless of ditches, orchards and vineyards.

Then for a moment the problem was solved, or partially solved, by the artillery. From Germany the material, though not—at least to the same extent—the principle, of the wagenburg penetrated, in the first years of the 16th century, to Italy and thence to France. Thus by degrees a very numerous and exceedingly handy light artillery—“carts with gonnes,” as they were called in England—came into play on the Italian battlefields, and took over from the dying franc-archer system the work of preparing the assault by fire. For mere skirmishing the Swiss and Landsknechts had arquebusiers enough, without needing to call on the masses of Gascons, &c., and pari passu with the development of this artillery, the “bands,” other than Swiss and Landsknechts, began to improve themselves into pikemen and halberdiers. At Ravenna (1512) the bands of Gascony and Picardy, as well as the French aventuriers (the “bands of Piedmont,” afterwards the second senior regiment of the French line) fought in the line of battle shoulder to shoulder with the Landsknechts. On this day the fire action of the new artillery was extraordinarily murderous, ploughing lanes in the immobile masses of infantry. At Marignan the French gendarmerie and artillery, closely and skilfully combined, practically destroyed the huge masses of the Swiss, and so completely had “infantry” and “fire” become separate ideas that on the third day of this tremendous battle we find even the “bands of Piedmont” cutting their way into the Swiss masses.

But from this point the lead fell into the hands of the Spaniards. These were originally swift and handy light infantry, capable—like the Scottish Highlanders at Prestonpans and Falkirk long afterwards—of sliding The Spanish infantry and the arquebus. under the forest of pikes and breaking into the close-locked ranks with buckler and stabbing sword. For troops of this sort the arquebus was an ideal weapon, and the problem of self-contained infantry was solved by Gonsalvo de Cordoba, Pescara and the great Spanish captains of the day by intercalating small closed bodies of arquebusiers with rather larger, but not inordinately large, bodies of pikes. These arquebusiers formed separate, fully organized sections of the infantry regiment. In close defence they fought on the front and flanks of the pikes, but more usually they were pushed well to the front independently, their speed and excellent fire discipline enabling them to do what was wholly beyond the power of the older type of firing infantry—to take advantage of ground, to run out and reopen fire during a momentary pause in the battle of lance and pike, and to run back to the shelter of their own closed masses when threatened by an oncoming charge. When this system of tactics was consecrated by the glorious success of Pavia (1525), the “cart with gonnes” vanished and the system of fighting everywhere and always “at push of pike” fell into the background.

The lessons of Pavia can be read in Francis I.’s instructions to his newly formed Provincial (militia) Legions in 1534 and in the battle of Cerisoles ten years later. The “legion” was ordered to be composed of six “bands”—battalions we should 16th Century-tactics. call them now, but in those days the term “battalion” was consecrated to a gigantic square of the Swiss type—each of 800 pikes (including a few halberts) and 200 arquebusiers. The pikes, 4800 strong, of each legion were grouped in one large battalion, and covered on the front and flanks by the 1200 arquebuses, the latter working in small and handy squads. These “legions” did not of course count as good troops, but their organization and equipment, designed deliberately in peace time, and not affected by the coming and going of soldiers of fortune, represent therefore the theoretically perfect type for the 16th century. Cerisoles represents the system in practice, with veteran regular troops. On the side of the French most of the arquebuses were grouped on the right wing, in a long irregular line of companies or strong squads, supported at a moderate distance by companies or small battalions of “corselets” (pikes of the French bands of Picardy and Piedmont); the rest of the line of battle was composed of Landsknechts, &c., similarly arrayed, except that the arquebusiers were on the flanks and immediate front of the “corselets” and behind the arquebuses and corselets of the right wing came a Swiss monster of the old type. On the imperial side of the Landsknechts, Spanish and Italian infantry were drawn up in seven or eight battalions, each with its due proportion of pikes and “shot.” The course of the battle demonstrated both the active tactical power of the new form of fire-action and the solidity of the pike nucleus, the former in the attack and defence of hills, woods and localities, the latter in an episode in which a Spanish battalion, after being ridden through from corner to corner by the French gendarmes, continued on its way almost unchecked and quite unbroken. This combination of arquebusiers supported by corselets in first line and corselets with a few arquebusiers in second, reappeared at Renty (1554), and St Quentin (1557), and was in fact the typical disposition of infantry from about 1530 to 1600.

By 1550, then, infantry had entirely ceased to be an auxiliary arm. It contained within itself, and (what is more important) within its regimental units, the power of fighting effectively and decisively both at close quarters and at a distance—the principal characteristic of the arm to-day. It had, further, developed a permanent regimental existence, both in Spain and in France, and in the former country it had progressed so far from the “residue” state that young nobles preferred to trail a pike in the ranks of the foot to service in the gendarmerie or light horse. The service battalions were kept up to war strength by the establishment of depots and the preliminary training there of recruits. In France, apart from Picardie and the other old regiments, every temporary regiment, on disbandment, threw off a depot company of the best soldiers, on which nucleus the regiment was reconstituted for the next campaign. Moreover, the permanent establishment was augmented from time to time by the colonel-general of the foot “giving his white flag” to temporary regiments.

The organization of the French infantry in 1570 presents some points of interest. The former broad classification of au delà and en deça des monts or “Picardie” and “Piedmont,” representing the home and Italian armies, had disappeared, and The French infantry in 1570. instead the whole of the infantry, under one colonel-general, was divided into the regiments of Picardie, Piedmont and French Guards, each of which had its own colonel and its own colours. Besides these, three newer corps were entretenus par le Roy—“Champagne,” practically belonging to the Guise[6] family, and two others formed out of the once enormous regiment of Marshal de Cossé-Brissac. At the end of a campaign all temporary regiments were disbanded, but in imitation of the Spanish depot system, each, on disbandment, threw off a depot company of picked men who formed the nucleus for the next year’s augmentation. The regiment consisted of 10-16 “ensigns” or companies, each of about 150 pikemen and 50 arquebusiers. Each company had a proprietary captain, the owners of the first two companies being the colonel-general and the colonel (mestre de camp). The senior captain was called the sergeant-major, and performed the duties of a second in command and an adjutant or brigade-major. Unlike the regimental commander, the sergeant-major was always mounted, and it is recorded that one officer newly appointed to the post incurred the ridicule of the army by dismounting to speak to the king. “Some veteran officers,” wrote a contemporary, “are inclined to think that the regimental commander should be mounted as well as the sergeant-major.” The regiment was as a rule formed for parade and battle either in line 10 deep or in “battalion” (i.e. mass), Swiss fashion. The captain occupied the front, the ensigns with the company colours the centre, and the lieutenants the rear place in the file. The sergeants, armed with the halbert, marched on each side of the battalion or company. Though the musket was gradually being introduced, and had powerful advocates in Marshal Strozzi and the duke of Guise, the bulk of the “shot” still carried the arquebus, the calibre of which had been, thanks to Strozzi’s efforts, standardized (see [Caliver]) so that all the arms took the same sizes of ball. The pikeman had half-armour and a 14-ft. pike, the arquebusier beside the firearm a sword which he was trained to use in the manner of the former Spanish light infantry. The arquebusiers were arrayed in 3 ranks in front of the pikes or in 10 deep files on either flank.

The wars in which this system was evolved were wars for prestige and aggrandizement. They were waged, therefore, by mercenary soldiers, whose main object was to live, and who were officered either by men of their own stamp, or by nobles eager to win military glory. But the Wars of Religion raised questions of life and death for the Frenchmen of either faith, and such public opinion as there was influenced the method of operations so far that a decision and not a prolongation of the struggle began to be the desired end of operations. Hence in those wars the relatively immobile “battalion” of pikes diminishes in importance and the arquebusiers and musketeers grow more and more efficient. Armies, too, became smaller, and marched more rapidly. Encounter-battles became more frequent than “pitched” battles, and in these the musketeer was at a great advantage. Thus by 1600 the proportions between pikes and musketeers in the French army had come to be 6 pikes to 4 muskets or arquebuses, and the bataillon de combat or brigade was normally no more than 1200 strong. In the Netherlands, however, the war of consciences was fought out between the best regular army in the world and burgher militias. Even the French fantassins were second in importance to the Spanish soldados. The latter continued to hold the pre-eminent position they had gained at Pavia.[7] They improved the arquebus into the musket, a heavier and much more powerful weapon (fired from a rest) which could disable a horse at 500 paces.

At this moment the professional soldier was at the high-water mark of his supremacy. The musket was too complicated to be rapidly and efficiently used by any but a highly trained man; the pike, probably because it had now Alva. to protect two or three ranks of “shot” in front of the leading rank of pikemen, as well as the pikemen themselves, had grown longer (up to 18 ft.); and drill and manœuvre had become more important than ever, for in the meantime cavalry had mostly abandoned the massive armour and the long lance in favour of half-armour and the pistol, and their new tactics made them both swifter to charge groups of musketeers and more deadly to the solid masses of pikemen. This superiority of the regular over the irregular was most conspicuously shown in Alva’s war against the Netherlands patriots. Desperately as the latter fought, Spanish captains did not hesitate to attack patriot armies ten times their own strength. If once or twice this contempt led them to disaster, as at Heiligerlee in 1568 (though here, after all, Louis of Nassau’s army was chiefly composed of trained mercenaries), the normal battle was of the Jemmingen type—seven soldados dead and seven thousand rebels.

As regards battles in the open field, such results as these naturally confirmed the “Spanish system” of tactics. The Dutch themselves, when they evolved reliable field armies, copied it with few modifications, and by degrees it was spread over Europe by the professional soldiers on both sides. There was plenty of discussion and readjustment of details. For example, the French, with their smaller battalions and more rapid movements, were inclined to disparage both the cuirass and the pike, and only unwillingly hampered themselves with the long heavy Spanish musket, which had to be fired from a rest. In 1600, nearly fifty years after the introduction of the musket, this most progressive army still deliberately preferred the old light arquebus, and only armed a few selected men with the larger weapons. On the other hand, the Spaniards, though supreme in the open, had for the most part to deal with desperate men behind fortifications. Fighting, therefore, chiefly at close quarters with a fierce enemy, and not disposing either of the space or of the opportunity for “manœuvre-battles,” they sacrificed all their former lightness and speed, and clung to armour, the long pike and the heavy 2½ oz. bullet. But the principles first put into practice by Gonsalvo de Cordoba, the combination, in the proportions required in each case, of fire and shock elements in every body of organized infantry however small, were maintained in full vigour, and by now the superiority of the infantry arm in method, discipline and technique, which had long before made the Spanish nobles proud to trail a pike in the ranks, began to impress itself on other nations. The relative value of horse and foot became a subject for expert discussion instead of an axiom of class pride. The question of cavalry versus infantry, hotly disputed in all ages, is a matter affecting general tactics, and does not come within the scope of the present article (see further [Cavalry]). Expert opinion indeed was still on the side of the horsemen. It was on their cavalry, with its speed, its swords and its pistols that the armies of the 16th century relied in the main to produce the decision Infantry in 1600. in battle. Sir Francis Vane, speaking of the battle of Nieupoort in 1600, says, “Whereas most commonly in battles the success of the foot dependeth on that of the horse, here it was clean contrary, for so long as the foot held good the horse could not be beaten out of the field.” The “success” of the foot in Vane’s eyes is clearly resistance to disintegration rather than ability to strike a decisive blow.

Plate I.

Plate II.

It must be remembered, however, that Vane is speaking of the Low Countries, and that in France at any rate the solidity which saved the day at Nieupoort was less appreciated than the élan which had won so many smart engagements in the Wars of Religion. Moreover, it was the offensive, the decision-compelling faculty of the foot that steadily developed during the 17th century. To this, little by little, the powers of passive resistance to which Vane did homage, valuable as they were, were sacrificed, until at last the long pike disappeared altogether and the firearm, provided with a bayonet, was the uniform weapon of the foot-soldier. This stage of infantry history covers almost exactly a century. As far as France was concerned, it was a natural evolution. But the acceptance of the principle by the rest of the military world, imposed by the genius of Gustavus Adolphus, was rather revolution than evolution.

In the army which Louis XIII. led against his revolted barons of Anjou in 1620, the old regiments (les vieux—Picardie, Piedmont, &c.) seem to have marched in an open chequer-wise formation of companies which is interesting not only Gustavus Adolphus. as a deliberate imitation of the Roman legion (all soldiers of that time, in the prevailing confusion of tactical ideas, sought guidance in the works of Xenophon, Aelian and Vegetius), but as showing that flexibility and handiness was not the monopoly of the Swedish system that was soon to captivate military Europe. The formations themselves are indeed found in the Spanish and Dutch armies, but the equipment of the men, and the general character of the operations in which they were engaged, probably failed to show off the advantages of this articulation, for the generals of the Thirty Years’ War, trained in this school, formed their infantry into large battalions (generally a single line of masses). Experience certainly gave the troops that used these unwieldy formations a relatively high manœuvring capacity, for Tilly’s army at Breitenfeld (1631) “changed front half-left” in the course of the battle itself. But the manœuvring power of the Swedes was higher still. Each party represented one side of the classical revival, the Swedes the Roman three-line manipular tactics, the Imperialists and Leaguers those of the Greek line of phalanxes. The former, depending as it did on high moral in the individual foot-soldier, was hardly suitable to such a congeries of mercenaries as those that Wallenstein commanded, and later in the Thirty Years’ War, when the old native Swedish and Scottish brigades had been annihilated, the Swedish infantry was little if at all better than the rest.

But its tactical system, sanctified by victory, was eagerly caught up by military Europe. The musket, though it had finally driven out the arquebus, had been lightened by Gustavus Adolphus so far that it could be fired without a rest. Rapidity in loading had so far improved that a company could safely be formed six deep instead of ten, as in the Spanish and Dutch systems. Its fire power was further augmented by the addition of two very light field-guns to each battalion; these could inflict loss at twice the effective range of the shortened musket. Above all, Gustavus introduced into the military systems of Europe a new discipline based on the idea of exact performance of duty, which made itself felt in every part of the service, and was a welcome substitute for the former easy-going methods of regimental existence.[8] The adoption of Swedish methods indeed was facilitated by the disrepute into which the older systems had fallen. Men were beginning to see that armies raised by contract for a few months’ work possessed inherent vices that made it impossible to rely upon them in small things. Courage the mercenary certainly possessed, but his individual sense of honour, code of soldierly morals, and sometimes devotion to a particular leader did not compensate for the absence of a strong motive for victory and for his general refractoriness in matters of detail, such as march-discipline and punctuality, which had become essential since the great Swedish king had reintroduced order, method and definiteness of purpose into the conduct of military operations. In the old-fashioned masses, moreover, individual weaknesses, both moral and physical, counted for little or were suppressed in the general soldierly feeling of the whole body. But the six-deep line used by Gustavus demanded more devotion and exact obedience in the individual and a more uniform method of drill and handling arms. So shallow an order was not strong enough, under any other conditions, to resist the shock of cavalry or even of pikemen. Indeed, had not the cavalry (who, after Gustavus’s death, were uninspired mercenaries like the rest) ceased to charge home in the fashion that Gustavus exacted of them, it is possible that the new-fashioned line would not have stood the test, and that infantry would have reverted to the early 16th-century type.

The problem of combining the maximum of fire power with the maximum of control over the individual firer was not fully solved until 1740, but the necessity of attempting the problem was realised from the first. In the Swedish The Great Rebellion. army, before it was corrupted by the atmosphere of the Thirty Years’ War, duty to God and to country were the springs of the punctual discipline, in small things and in great, which made it the most formidable army, unit for unit, in the world. In the English Civil War (in which the adherents of the “Swedish system” from the first ousted those of the “Dutch”) the difficulty was more acute, for although the mainsprings of action were similar, the technical side of the soldier’s business—the regimental organization, drill and handling of arms—had all to be improvised. Now in the beginning the Royalist cavalry was recruited from “gentlemen that have honour and courage and resolution”; later, Cromwell raised a cavalry force that was even more thoroughly imbued with the spirit of duty, “men who made some conscience of what they did,” and throughout the Civil War, consequently, the mounted arm was the queen of the battlefield.

The Parliamentary foot too “made some conscience of what it did,” more especially in the first years of the war. But its best elements—the drilled townsmen—were rather of a defensive than of an offensive character, and towards the close of the struggle, when the foot on both sides came to be formed of professional soldiers, the defensive element decreased, as it had decreased in France and elsewhere. The war was like Gustavus’s German campaign, one of rapid and far-ranging marches, and the armoured pikeman had either to shorten his pike and to cast off his armour or to be left at home with the heavy artillery (see Firth’s Cromwell’s Army, ch. iv.). Fights “at push of pike” were rare enough to be specially mentioned in reports of battles. Sir James Turner says that in 1657, when he was commissioned with others to raise regiments for the king of Denmark, “those of the Privy Council would not suffer one word to be mentioned of a pike in our Commissions.” It was the same with armour. In 1658 Lockhart, the commander of the English contingent in France, specially asked for a supply of cuirasses and headpieces for his pikemen in order to impress his allies. In 1671 Sir James Turner says, “When we see battalions of pikes, we see them everywhere naked unless it be in the Netherlands.” But a small proportion of pikes was still held to be necessary by experienced soldiers, for as yet the socket bayonet had not been invented, and there was still cavalry in Europe that could be trusted to ride home.

While such cavalry existed, the development of fire power was everywhere hindered by the necessity of self-defence. On the other hand the hitherto accepted defensive means militated against efficiency in many ways, and about 1670, when Louis Disuse of the pike. XIV. and Louvois were fashioning the new standing army that was for fifty years the model for Europe, the problem was how to improve the drill and efficiency of the musketeers so far that the pikes could be reduced to a minimum. In 1680 the firelock was issued instead of the matchlock to all grenadiers and to the four best shots in each French Company. The bayonet—in its primitive form merely a dagger that was fixed into the muzzle of the musket—was also introduced, and the pike was shortened. The proportion of pikes to muskets in Henry IV.’s day, 2 to 1 or 3 to 2, and in Gustavus’s 2 to 3, had now fallen to 1 to 3.

The day of great causes that could inspire the average man with the resolution to conquer or die was, however, past, and the “shallow order” (l’ordre mince), with all its demands on the individual’s sense of duty, had become an integral part of the military system. How then was the sense of duty to be created? Louis and Louvois and their contemporaries sought to create it by taking raw recruits in batches, giving them a consistent training, quartering them in barracks and uniforming them. Henceforward the soldier was not a unit, self-taught and free to enter the service of any master. He had no existence as a soldier apart from his regiment, and within it he was taught that the regiment was everything and the individual nothing. Thus by degrees the idea of implicit obedience to orders and of esprit de corps was absorbed. But the self-respecting Englishman or the quick ardent Frenchman was not the best raw material for quasi-automatic regiments, and it was not until an infinitely more rigorous system of discipline was applied to an unimaginative army that the full possibilities of this enforced sense of duty were realized.

The method of delivering fire originally used by the Spaniards, in which each man in succession fired and fell back to the rear of the file to reload, required for its continued and exact performance a degree of coolness and individual smartness Methods of fire before 1740. which was probably rarely attained in practice. This was not of serious moment when the “shot” were simple auxiliaries, but when under Gustavus the offensive idea came to the front, and the bullets of the infantry were expected to do something more than merely annoy the hostile pikemen, a more effective method had to be devised. First, the handiness of the musket was so far improved that one man could reload while five, instead of as formerly ten, fired. Then, as the enhanced rate of fire made the file-firing still more disorderly than before, two ranks and three were set to fire “volews” or “salvees” together, and before 1640 it had become the general custom for the musketeers to fire one or two volleys and then, along with the pikemen, to “fall on.” It was of course no mean task to charge even a disordered mass of pikes with a short sword or a clubbed musket, and usually after a few minutes the combatants would drift apart and the musketeers on either side would keep up an irregular fire until the officers urged the whole forward for a second attempt.

With the general disuse of the lance, the disappearance of the personal motives that formerly made the cavalryman charge home, the adoption of the flintlock musket and the invention of the socket bayonet (the fixing of which did not prevent fire The bayonet. being delivered), all reason for retaining the pike vanished, and from about 1700 to the present day, therefore, the invariable armament of infantry has been the musket (or rifle) and bayonet. The manner of employing the weapons, however, changed but slowly. In the French army in 1688, for instance (15 years before the abolition of the pike), the old file-fire was still officially recognized, though rarely employed, the more usual method being for the musketeers in groups of 12 to 30 men to advance to the front and deliver their volleys in turn, these groups corresponding in size to one of the musketeer wings (manches) of a company or double company. But the fire and shock action of infantry were still distinct, the idea of “push of pike” remained, the bayonet (as at Marsaglia) taking the place of the pike, and musketry methods were still and throughout the War of the Spanish Succession somewhat half-hearted and tentative. Two generals so entirely different in genius and temperament as Saxe and Catinat could agree on this point, that attacking infantry ought to close with the enemy, bayonets fixed, without firing a shot. Catinat’s orders to his army in 1690, indeed, seem rather to indicate that he expected his troops to endure the enemy’s first fire without replying in order that their own volley, when it was at last delivered at a few paces distance, should be as murderous as possible, while Saxe, who was a dreamer as well as a practical commander of troops, advocated the pure bayonet charge. But the fact that is common to both is the relative ineffectiveness of musketry before the Prussian era, whether this musketry was delivered by groups of men running forward and returning in line or even by companies in a long line of battle.

This ineffectiveness was due chiefly to the fact that fire and movement were separate matters. The enemy’s volley, that Catinat and others ordered their troops to endure without flinching, was sometimes (as at Fontenoy) absolutely crushing. But as a rule it inflicted an amount of loss that was not sufficient to put the advancing troops out of action, and experienced officers were aware that to halt to reply gave the enemy time to reload, and that once the fight became an interchange of partial and occasional volleys or a general tiraillerie, there was an end to the attack.

Meanwhile, the tactics of armies had been steadily crystallizing into the so-called “linear” form, which, as far as concerns the infantry, is simply two long lines of battalions (three, four or five deep) and gave the utmost possible development Linear tactics. to fire-power. The object of the “line” was to break or beat down the opposing line in the shortest possible time, whether by fire action or shock action, but fire action was only decisive at so short a range that the principal volley could be followed immediately by a charge over a few score paces at most and the crossing of bayonets. Fire was, however, effective at ranges outside charging distance, especially from the battalion guns, and however the decision was achieved in the end, it was necessary to cross the zone between about 300 yds. and 50 yds. range as quickly as possible. It was therefore the business of the regimental officer to force his men across this zone before fire was opened. If, as Catinat recommended, decisive range was reached with every musket loaded and the troops well in hand, their fire when finally it was delivered might well be decisive. But in practice this rarely happened, and though here and there such expedients as a skirmishing line were employed to assist the advance by disturbing the enemy’s fire the most that was hoped by the average colonel or captain was that in the advance fire should be opened as late as possible and that the officers should strive to keep in their hands the power of breaking off the fire-fight and pushing the troops forward again. Theorists were already proposing column formations for shock action, and initiating the long controversy between l’ordre mince and l’ordre profonde, but this was for the time being pure speculation. The linear system rested on the principle that the maximum weight of controlled fire at short range was decisive, and the practical problem of infantry tactics was how to obtain this. The question of fire versus shock had been answered in favour of the former, and henceforward for many years the question of fire versus movement held the first place. The purpose was settled, and it remained to discover the means.

This means was Prussian fire-discipline, which was elaborated by Leopold of Dessau and Frederick William I., and practically applied by Frederick the Great. It consisted first in the combination, instead of the alternation, of fire and movement, and secondly in the thorough efficiency of the fire in itself. But both these demanded a more stringent and technically more perfect drill than had ever before been imagined, or, for that matter, has ever since been attained. A hundred years before the steady drill of the Spanish veterans at Rocroi, who at the word of command opened their ranks to let the cannon fire from the rear and again closed them, impressed every soldier in Europe. But such drill as this was child’s play compared with the Old Dessauer’s.

On approaching the enemy the marching columns of the Prussians, which were generally open columns of companies 4 deep, wheeled, in succession to the right or left (almost always to the right) and thus passed along the front of the enemy at a distance Prussian fire discipline, 1740. of 800-1200 yds. until the rear company had wheeled. Then the whole together (or in the case of a deployment to the left, in succession) wheeled into line facing the enemy. These movements, if intervals and distances were preserved with proper precision, brought the infantry into two long well-closed lines, and parade-ground precision was actually attained, thanks to remorseless drilling and to the reintroduction of the march in step to music. Of course such movements were best executed on a firm plain, and as far as possible the attack and defence of woods and villages was left to light infantry and grenadiers. But even in marshes and scrub, the line managed to manœuvre with some approach to the precision of the barrack square.[9] Now, this precision allowed Frederick to take risks that no former commander would have dared to take. At Hohenfriedberg the infantry columns crossed a marshy stream almost within cannon shot of the enemy; at Kolin (though there this insolence was punished) the army filed past the Imperialist skirmishers within less than musket shot, and the climax of this daring was the “oblique order” attack of Leuthen. With this was bound up a fire discipline that was more extraordinary than any perfection of manœuvre. Before Hohenfriedberg the king gave orders that “pelotonfeuer” was to be opened at 200 paces from the enemy and continued up to 30 paces, when the line was to fall on with the bayonet. The possibility of this combination of fire and movement was the work of Leopold, who gave the Prussian infantry iron ramrods, and by sheer drill made the soldier a machine capable of delivering (with the flintlock muzzle-loading muskets, be it observed) five volleys a minute. This pelotonfeuer or company volleys replaced the old fire by ranks practised in other armies. Fire began from the flanks of the battalion, which consisted of eight companies (for firing, 3 deep). When the right company commander gave “fire,” the commander of No. 2 gave “ready,” followed in turn by other companies up to the centre. The same process having been gone through on the left flank, by the time the two centre companies had fired the two flank companies were ready to recommence, and thus a continuous series of rolling volleys was delivered, at one or two seconds’ interval only between companies. In attack this fire was combined with movement, each company in turn advancing a few paces after “making ready.” In square, old-fashioned methods of fire were employed. Square was an indecisive and defensive formation, rarely used, and in the advance of the deployed line, the offensive and decision-seeking formation par excellence, the special Prussian fire-discipline gave Frederick an advantage of five shots to two against all opponents. The bayonet-attack, if the rolling volleys had done their work, was merely “presenting the cheque for payment” as a modern German writer puts it. The cheque had been drawn, the decision given, in the fire-fight.

For some years this method of infantry training gave the Prussians a decisive superiority in whatever order they fought. But their enemies improved and also grew in numbers, while the Prussian army’s resources were strictly Leuthen. limited. Thus in the Seven Years’ War, after the two costly battles of Prague and Kolin (1757) especially, it became necessary to manœuvre with the object of bringing the Prussian infantry into contact with an equal or if possible smaller portion of the enemy’s line. If this could be achieved, victory was as certain as ever, but the difficulties of bringing about a successful manœuvre were such that the classical “oblique order” attack was only once completely executed. This was at Leuthen, December 5th, 1757, perhaps the greatest day in the history of the Prussian army. Here, in a rolling plain country occasionally broken by marshes and villages, the “oblique order” was executed at high speed and with clockwork precision. Frederick’s object was to destroy the left of the Austrian army (which far outnumbered his own) before the rest of their deployed line of battle could change front to intervene. His method was to place his own line, by a concealed flank march, opposite the point where he desired to strike, and then to advance, not in two long lines but in échelon of battalions from the right (see [Leuthen]). The échelon was not so deep but that each battalion was properly supported by the following one on its left (100 paces distance), and each, as it came within 200 yds. of the Austrian battalion facing it, opened its “rolling volleys” while continuing to advance; thus long before the left and most backward battalions were committed to the fight, the right battalions were crumbling the Austrian infantry units one by one from left to right. It was the same, without parade manœuvres, when at last the Austrians managed to organize a line of defence about Leuthen village. Unable to make an elaborate change of front with the whole centre and right wing for want of time, they could do no more than crowd troops about Leuthen, on a short fighting front, and this crumbled in turn before the Prussian volleys.

One lesson of Leuthen that contemporary soldiers took to heart was that even a two-to-one superiority in numbers could not remedy want of manoeuvring capacity. It might be hoped that with training and drill an Austrian battalion could be made equal to a Prussian one in the front-to-front fight, and in fact, as losses told more and more heavily on Frederick’s army as years went on, the specific superiority of his infantry disappeared. From 1758 therefore, to the end of the war, there were no more Rossbachs and Leuthens. Superiority in efficiency through previous training having exhausted its influence, superiority in force through manœuvre began to be the general’s ideal, and as it was a more familiar notion to the average Prussian general, trained to manœuvre, than to his opponent, whose idea of “manœuvre” was to sidle carefully from one position to another, Prussian generalship maintained its superiority, in spite of many reverses, to the end. The last campaigns were indeed a war of positions, because Frederick had no longer the men available for forcing the Austrians out of them, and on many occasions he was so weak that the most passive defensive and the most elaborate entrenchments barely sufficed to save him. But whenever opportunity offered itself, the king sought a decisive success by bringing the whole of his infantry against part of the enemy’s—the principle of Leuthen put in practice over a wider area and with more elastic manœuvre methods. The long échelon of battalions directed against a part of the hostile line developed quite naturally into an irregular échelon of brigade columns directed against a part of the enemy’s position. But the history of the “cordon system” which followed this development belongs rather to the subject of tactics in general than to that of infantry fighting methods. Within the unit the tactical method scarcely varied. In a battle each battalion or brigade fought as a unit in line, using company volleys and seeking the decision by fire.

In this, and in even the most minute details of drill and uniform, military Europe slavishly copied Prussia for twenty years after the Seven Years’ War. The services of ex-Prussian officers were at a premium just as those of Controversies and developments, 1760-1790. Gustavus’s officers had been 150 years before. Military missions from all countries went to Potsdam or to the “Reviews” to study Prussian methods, with as simple a faith in their adequacy as that shown to-day by small states and half-civilized kingdoms who send military representatives to serve in the great European armies. And withal, the period 1763-1792 is full of tactical and strategical controversies. The principal of these, as regards infantry, was that between “fire” and “shock” revived about 1710 by Folard, and about 1780 the American War of Independence complicated it by introducing a fresh controversy between skirmishing and close order. As to the first, in Folard’s day as in Frederick’s, fire action at close range was the deciding factor in battle, but in Frederick’s later campaigns, wherein he no longer disposed of the old Prussian infantry and its swift mechanical fire-discipline, there sprang up a tendency to trust to the bayonet for the decision. If the (so-called) Prussian infantry of 1762 could be in any way brought to close with the enemy, it had a fair chance of victory owing to its leaders’ previous dispositions, and then the advocates of “shock,” who had temporarily been silenced by Mollwitz and Hohenfriedberg, again took courage. The ordinary line was primarily a formation for fire, and only secondarily or by the accident of circumstances for shock, and, chiefly perhaps under Saxe’s influence, the French army had for many years been accustomed to differentiate between “linear” formations for fire and “columnar” for attack—thus reverting to 16th-century practice. While, therefore, the theoreticians pleaded for battalion columns and the bayonet or for line and the bullet, the practical soldier used both. Many forms of combined line and column were tried, but in France, where the question was most assiduously studied, no agreement had been arrived at when the advent of the skirmisher further complicated the issues.

In the early Silesian wars, when armies fought in open country in linear order, the outpost service scarcely concerned the line troops sufficiently to cause them to get under arms at the sound of firing on the sentry line. It was performed by irregular light troops, recruited from wild characters of all nations, who were also charged with the preliminary skirmishing necessary to clear up the situation before the deployment of the battle-army, but once the line opened fire their work was done and they cleared away to the flanks (generally in search of plunder). Later, however, as the preliminary manœuvring before the battle grew in importance and the ground taken into the manœuvring zone was more varied and extended than formerly, light infantry was more and more in demand—in a “cordon” defensive for patrolling the intervals between the various detachments of line troops, in an attack for clearing the way for the deployment of each column. Yet in all this there was no suggestion that light troops or skirmishers were capable of bringing about the decision in an armed conflict. When Frederick gained a durable peace in 1763 he dismissed his “free battalions” without mercy, and by 1764 not more than one Prussian soldier in eleven was an “irregular,” either of horse or foot.[10]

But in the American War of Independence the line was pitted against light infantry in difficult country, and the British and French officers who served in it returned to Europe full of enthusiasm for the latter. Nevertheless, their light infantry Light Infantry. was, unlike Frederick’s, selected line infantry. The light infantry duties—skirmishing, reconnaissance, outposts—were grafted on to a thorough close-order training. At first these duties fell to the grenadiers and light companies of each battalion, but during the struggle in the colonies, the light companies of a brigade were so frequently massed in one battalion that in the end whole regiments were converted into light infantry. This combination of “line” steadiness and “skirmisher” freedom was the keynote of Sir John Moore’s training system fifteen years later, and Moore’s regiments, above all the 52nd, 43rd (now combined as the Oxfordshire Light Infantry) and 95th Rifles (Rifle Brigade), were the backbone of the British Army throughout the Peninsular War. At Waterloo the 52nd, changing front in line at the double, flung itself on the head and flank of the Old Guard infantry, and with the “rolling volleys” inherited from the Seven Years’ War, shattered it in a few minutes. Such an exploit would have been absolutely inconceivable in the case of one of the old “free battalions.” But the light infantry had not merely been levelled up to the line, it had surpassed it, and in 1815 there were no troops in Europe, whether trained to fight in line or column or skirmishers, who could rival the three regiments named, the “Light Division” of Peninsular annals. For meantime the infantry organization and tactics of the old régime, elsewhere than in England, had been disintegrated by the flames of the French Revolution, and from their ashes a new system had arisen, which forms the real starting-point of the infantry tactics of to-day.

The controversialists of Louis XVI.’s time, foremost of whom were Guibert, Joly de Maizeroy and Menil Durand (see Max Jähns, Gesch. d. Kriegswissenschaften, vol. iii.), were agreed that shock action should be the work of troops The French Revolution. formed in column, but as to the results to be expected from shock action, the extent to which it should be facilitated by a previous fire preparation, and the formations In which fire should be delivered (line, line with skirmishers or “swarms”) discussion was so warm that it sometimes led to wrangles in ladies’ drawing-rooms and meetings in the duelling field. The drill-book for the French infantry issued shortly before the Revolution was a common-sense compromise, which in the main adhered to the Frederician system as modified by Guibert, but gave an important place in infantry tactics to the battalion “columns of attack,” that had hitherto appeared only spasmodically on the battlefields of the French army and never elsewhere. This, however, and the quick march (100 paces to the minute instead of the Frederician 75) were the only prescriptions in the drill-book that survived the test of a “national” war, to which within a few years it was subjected (see [French Revolutionary Wars]). The rest, like the “linear system” of organization and manœuvre to which it belonged (see [Army], §§ 30-33; [Conscription], &c.) was ignored, and circumstances and the practical troop-leaders evolved by circumstances fashioned the combination of close-order columns and loose-order skirmishers which constituted essentially the new tactics of the Revolutionary and Napoleonic infantry.

The process of evolution cannot be stated in exact terms, more especially as the officers, as they grew in wisdom through experience, learned to apply each form in accordance with ground and circumstances, and to reject, when unsuitable, not only the Tactical evolution in France 1792-1807. forms of the drill-book, but the forms proposed by themselves to replace those of the drill-book. But certain tendencies are easily discernible. The first tendency was towards the dissolution of all tactical links. The earlier battles were fought partly in line for fire action, partly in columns for the bayonet attack. Now the linear tactics depended on exact preservation of dressing, intervals and distances, and what required in the case of the Prussians years of steady drill at 76 paces to the minute was hardly attainable with the newly levied ardent Frenchmen marching at 100 to 120. Once, therefore, the line moved, it broke up into an irregular swarm of excited firers, and experience soon proved that only the troops kept out of the turmoil, whether in line or in column, were susceptible of manœuvre and united action. Thus from about 1795 onwards the forms of the old régime, with half the troops in front in line of battle (practically in dense hordes of firers) and the other half in rear in line or line of columns, give way to new ones in which the skirmishers are fewer and the closed troops more numerous, and the decision rests no longer with the fire of the leading units (which of course could not compare in effectiveness with the rolling volleys of the drilled line) but with the bayonets of the second and third lines—the latter being sometimes in line but more often, owing to the want of preliminary drill, in columns. The skirmishers tended again to become pure light infantry, whose rôle was to prepare, not to give, the decision, and who fought in a thin line, taking every advantage of cover and marksmanship. In the Consulate and early Empire, indeed, we commonly find, in the closed troops destined for the attack, mixed line and column formations combining in themselves shock and controlled close-order fire—absolutely regardless of the skirmishers in front.

In sum, then, from 1792 to 1795 the fighting methods of the French infantry, of which so much has been written and said, are, as they have aptly been called, “horde-tactics.” From 1796 onwards to the first campaigns of the Empire, on the other hand, there is an ever-growing tendency to combine skirmishers, properly so called, with controlled and well-closed bodies in rear, the first to prepare the attack to the best of their ability by individual courage and skill at arms, the second to deliver it at the right moment (thanks to their retention of manœuvre formations), and with all possible energy (thanks to the cohesion, moral and material, which carried forward even the laggards). Even when in the long wars of the Empire the quality of the troops progressively deteriorated, infantry tactics within the regiment or brigade underwent no radical alteration. The actual formations were most varied, but they always contained two of the three elements, column, line and skirmishers. Column (generally two lines of battalions in columns of double-companies) was for shock or attack, line for fire-effect, and skirmishers to screen the advance, to scout the ground and to disturb the enemy’s aim. Of these, except on the defensive (which was rare in a Napoleonic battle), the “column” of attack was by far the most important. The line formations for fire, with which it was often combined, rarely accounted for more than one-quarter of the brigade or division, while the skirmishers were still less numerous. Withal, these formations in themselves were merely fresh shapes for old ideas. The armament of Napoleon’s troops was almost identical with that of Frederick’s or Saxe’s. Line, column and combinations of the two were as old as Fontenoy and were, moreover, destined to live for many years after Napoleon had fallen. “Horde-tactics” did not survive the earlier Revolutionary campaigns. Wherein then lies the change which makes 1792 rather than 1740 the starting-point of modern tactics?

The answer, in so far as so comprehensive a question can be answered from a purely infantry standpoint, is that whereas Frederick, disposing of a small and highly finished instrument, used its manœuvre power and regimental Napoleon’s infantry and artillery tactics, 1807-1815. efficiency to destroy one part of his enemy so swiftly that the other had no time to intervene, Napoleon, who had numbers rather than training on his side, only delivered his decisive blow after he had “fixed” all bodies of the enemy which would interfere with his preparations—i.e. had set up a physical barrier against the threatened intervention. This new idea manifested itself in various forms. In strategy (q.v.) and combined tactics it is generally for convenience called “economy of force.” In the domain of artillery (see [Artillery]) it marked a distinction, that has revived in the last twenty years, between slow disintegrating fire and sudden and overpowering “fire-preparation.” As regards infantry the effect of it was revolutionary. Regiments and brigades were launched to the attack to compel the enemy to defend himself, and fought until completely dissolved to force him to use up his reserves. “On s’engage partout et puis l’on voit” is Napoleon’s own description of his holding attack, which in no way resembled the “feints” of previous generations. The self-sacrifice of the men thus engaged enabled their commander to “see,” and to mass his reserves opposite a selected point, while little by little the enemy was hypnotized by the fighting. Lastly, when “the battle was ripe” a hundred and more guns galloped into close range and practically annihilated a part of the defender’s line. They were followed up by masses of reserve infantry, often more solidly formed at the outset than the old Swiss masses of the 16th century.[11] If the moment was rightly chosen these masses, dissolved though they soon were into dense formless crowds, penetrated the gap made by the guns (with their arms at the slope) and were quickly followed by cavalry divisions to complete the enemy’s defeat. Here, too, it is to be observed there is no true shock. The infantry masses merely “present the cheque for payment,” and apart from surprises, ambushes and fights in woods and villages there are few recorded cases of bayonets being crossed in these wars. Napoleon himself said “Le feu est tout, le reste peu de chose,” and though a mere plan of his dispositions suggests that he was the disciple of Folard and Menil Durand, in reality he simply applied “fire-power” in the new and grander form which his own genius imagined.

The problem, then, was not what it had been one hundred and fifty years before. The business of the attack was not to break down the passive resistance of the defence, but to destroy or to evade its fire-power. No attack with the bayonet could succeed if this remained effective and unbroken, and no resistance (in the open field at least) availed when it had been mastered or evaded. In Napoleon’s army, the circumstance that the infantry was (after 1807) incapable of carrying out its own fire-preparation forced the task into the hands of the field artillery. In other armies the 18th-century system had been discredited by repeated disasters, and the infantry, as it became “nationalized,” was passing slowly through the successive phases of irregular lines, “swarms,” skirmishers and line-and-column formations that the French Revolutionary armies had traversed before them—none of them methods that in themselves had given decisive results.

In all Europe the only infantry that represented the Frederician tradition and prepared its own charge by its own fire was the British. Eye-witnesses who served in the ranks of the French have described the sensation of powerlessness The British Peninsular infantry. that they felt as their attacking column approached the line and watched it load and come to the present. The column stopped short, a few men cheered, others opened a ragged individual fire, and then came the volleys and the counter-attack that swept away the column. Sometimes this counterstroke was made, as in the famous case of Busaco, from an apparently unoccupied ridge, for the British line, under Moore’s guidance, had shaken off the Prussian stiffness, fought 2 deep instead of 3 and was able to take advantage of cover. The “blankness of the battlefield” noted by so many observers to-day in the South African and Manchurian Wars was fully as characteristic of Wellington’s battles from Vimeiro to Waterloo, in spite of close order and red uniforms. But these battles were of the offensive-defensive type in the main, and for various reasons this type could not be accepted as normal by the rest of Europe. Nonchalance was not characteristic of the eager national levies of 1813 and 1814, and the Wellington method of infantry tactics, though it had brought about the failure of Napoleon’s last effort, was still generally regarded as an illustration of the already recognized fact that on the defensive the fire-power of the line, unless partly or wholly evaded by rapidity in the advance and manœuvring power or mastered and extinguished by the fire-power of the attack, made the front of the defence impregnable. There was indeed nothing in the English tactics at Waterloo that, standing out from the incidents of the battle, offered a new principle of winning battles.

Nor indeed did Europe at large desire a fresh era of warfare. Only the French, and a few unofficial students of war elsewhere, realized the significance of the rejuvenated “line.” For every one else, the later Napoleonic battle was the model, and as the great wars had ended before the “national” spirit had been exhausted or misused in wars of aggrandizement, infantry tactics retained, in Germany, Austria and Russia, the characteristic Napoleonic formations, lines of battalion or regimental columns, sometimes combined with linear formations for fire, and always covered by skirmishers. That these columns must in action dissolve sooner or later into dense irregular swarms was of course foreseen, but Napoleon had accustomed the world to long and costly fire-fighting as the preliminary to the attack of the massed reserves, and for the short remainder of the period of smooth-bore muskets, troops were always launched to the attack in columns covered by a thin line of picked shots as skirmishers. The moral power of the offensive “will to conquer” and the rapidity of the attack itself were relied upon to evade and disconcert the fire-power of the defence. If the attack failed to do so, the ranges at which infantry fire was really destructive were so small that it was easy for the columns to deploy or disperse and open a fire-fight to prepare the way for the next line of columns. And after a careful study of the battle of the Alma, in which the British line won its last great victory in the open field, Moltke himself only proposed such modifications in the accepted tactical system as would admit of the troops being deployed for defence instead of meeting attack, as the Russians met it, in solid and almost stationary columns. Fire in the attack, in fact, had come to be considered as chiefly the work of artillery, and as artillery, being an expensive arm, had been reduced during the period of military stagnation following Waterloo, and was no longer capable of Napoleonic feats, the attack was generally a bayonet attack pure and simple. Infantry methods, 1815-1870. Waterloo and the Alma were credited, not to fire-power, but to English solidity, and as Ardant du Picq observes, “All the peoples of Europe say ‘no one can resist our bayonet attack if it is made resolutely’—and all are right.... Bayonet fixed or in the scabbard, it is all the same.” Since the disappearance of the “dark impenetrable wood” of spears, the question has always turned on the word “resolute.” If the defence cannot by any means succeed in mastering the resolution of the assailant, it is doomed. But the means (moral and material) at the disposal of the defence for the purpose of mastering this resolution were, within a few years of the Crimean War, revolutionized by the general adoption of the rifle, the introduction of the breech-loader and the revival of the “nation in arms.”

Thirty years before the Crimea the flint-lock had given way to the percussion lock (see [Gun]), which was more certain in its action and could be used in all weathers. But fitting a copper cap on the nipple was not so simple a matter for nervous fingers as priming with a pinch of powder, and the usual rate of fire had fallen from the five rounds a minute of Frederick’s day to two or three at the most. “Fire-power” therefore was at a low level until the general introduction[12] of the rifled barrel, which while further diminishing the rate of fire, at any rate greatly increased the range at which volleys were thoroughly effective. Artillery (see [Artillery], § 13), the fire-weapon of the attack, made no corresponding progress, and even as early as the Alma and Inkerman (where the British troops used the Minie rifle) the dense columns had suffered heavily without being able to retaliate by “crossing bayonets.” Fire power, therefore, though still the special prerogative of the defence, began to reassert its influence, and for a brief period the defensive was regarded as the best form of tactics. But the low rate of fire was still a serious objection. Many incidents in the American Civil War showed this, notably Fredericksburg, where the key of the Confederate position was held—against a simple frontal attack unsupported by effective artillery fire—by three brigades in line one behind the other, i.e. by a six-deep firing line. No less force could guarantee the “inviolability of the front,” and even when, in this unnatural and uneconomical fashion, the rate of fire was augmented as well as the effective range, a properly massed and well-led attack in column (or in a rapid succession of deployed lines) generally reached the defender’s position, though often in such disorder that a resolute counterstroke drove it back again. The American fought over more difficult country and with less previous drill-training than the armies of the Old World. The fire-power of the defence, therefore, that even in America did not always prevail over the resolution of the attack, entirely failed in the Italian war of 1859 to stop the swiftly moving, well-drilled columns of the French professional army, in which the national élan had not as yet been suppressed, as it was a few years later, by the doctrine that “the new arms found their greatest scope in the defence.” The Austrians, who had pinned their faith to this doctrine, deserted their false gods, forbade any mention of the defensive in their drill-books, and brought back into honour the bayonet tactics of the old wars.

The need of artillery support for the attack was indeed felt (though the gunners had not as yet evolved any substitute for the case-shot preparation of Napoleon’s time), but men remembered that artillery was used by the great captain, not so much to enable good troops to close with the enemy, as to win battles with masses of troops of an inferior stamp, and contemporary experience seemed to show that (if losses were accepted as inevitable) good and resolute troops could overpower the defence, even in face of the rifle and without the aid of case shot. But a revolution was at hand.

In 1861 Moltke, discussing the war in Italy, wrote, “General Niel attributes his victory (at Solferino) to the bayonet. But that does not imply that the attack was often followed by a hand-to-hand fight. In principle, when one The breech-loading rifle. makes a bayonet charge, it is because one supposes that the enemy will not await it.... To approach the enemy closely, pouring an efficacious fire into him—as Frederick the Great’s infantry did—is also a method of the offensive.” This method was applicable at that time for the Prussians alone, for they alone possessed a breech-loading firearm. The needle-gun was a rudimentary weapon in many respects, but it allowed of maintaining more than twice the rate of fire that the muzzle-loader could give, and, moreover, it permitted the full use of cover, because the firer could lie down to fire without having to rise between every round to load. Further, he could load while actually running forward, whereas with the old arms loading not only required complete exposure but also checked movement. The advantages of the Prussian weapon were further enhanced, in the war against Austria, by the revulsion of feeling in the Imperial army in favour of the pure bayonet charge in masses that had followed upon Magenta and Solferino.

With the stiffly drilled professional soldier of England, Austria and Russia the handiness of the new weapon could hardly have been exploited, for (in Russia at any rate) even skirmishers had to march in step. The Prussians were drilled nominally in accordance with regulations dating from 1812, and therefore suitable, if not to the new weapon, at least to the “swarm” fighting of an enthusiastic national army, but upon these regulations a mass of peace-time amendments had been superposed, and in theory their drill was as stiff as that of the Russians. But, as in France in 1793-1796, the composition of their army—a true “nation in arms”—and the character of the officers evolved by the universal service system saved them from their regulations. The offensive spirit was inculcated as thoroughly as elsewhere, and in a much more practical form. Dietrich von Bülow’s predictions of the future battle of “skirmishers” (meaning thereby a dense but irregular firing line) had captivated the younger school of officers, while King William and the veterans of Napoleon’s wars were careful to maintain small columns (sometimes company[13] columns of 240 rifles, but quite as often half-battalion and battalion columns) as a solid background to the firing line. Thus in 1866 (see [Seven Weeks’ War]), as Moltke had foreseen, the attacking infantry fought its way to close quarters by means of its own fire, and the bayonet charge again became, in his own words, “not the first, but the last, phase of the combat,” immediately succeeding a last burst of rapid fire at short range and carried out by the company and battalion reserves in close order. Against the Austrians, whose tactics alternated between unprepared bayonet rushes by whole brigades and a passive slow-firing defensive, victory was easily achieved.

But immediately after Königgrätz the French army was served out with a breech-loading rifle greatly superior in every respect to the needle-gun, and after four years’ tension France pitted breech-loader against breech-loader. Infantry in the war of 1870. In the first battles (see [Wörth], and [Metz]: Battles) the decision-seeking spirit of the “armed nation,” the inferior range of the needle-gun as compared with that of the chassepot, and the recollections of easy triumphs in 1864 and 1866, all combined to drive the German infantry forward to within easy range before they began to make use of their weapons. Their powerful artillery would have sufficed of itself to enable them to do this (see [Sedan]), had they but waited for its fire to take effect. But they did not, and they suffered accordingly, for, owing to the ineffectiveness of their rifle between 1000 and 400 yds. range, they had to advance, as the Austrians and Russians had done in previous wars, without firing a shot. In these circumstances their formations, whether line or column, broke up, and the whole attacking force dissolved into long irregular swarms. These swarms were practically composed only of the brave men, while the rest huddled together in woods and valleys. When, therefore, at last the firing line came within 400 or 500 yds. of the French, it was both severely tried and numerically weak, but the fact that it was composed of the best men only enabled it to open and to maintain an effective fire. Even then the French, highly disciplined professional soldiers that they were, repeatedly swept them back by counterstrokes, but these counterstrokes were subjected to the fire of the German guns and were never more than locally and momentarily effective. More and more German infantry was pushed forward to support the firing line, and, like its predecessors, each reinforcement, losing most of its unwilling men as it advanced over the shot-swept ground, consisted on arrival of really determined men, and closing on the firing line pushed it forward, sometimes 20 yds., sometimes 100, until at last rapid fire at the closest ranges dislodged the stubborn defenders. Bayonets (as usual) were never actually used, save in sudden encounters in woods and villages. The decisive factors were, first the superiority of the Prussian guns, secondly, heavy and effective fire delivered at short range, and above all the high moral of a proportion of resolute soldiers who, after being subjected for hours to the most demoralizing influences, had still courage left for the final dash. These three factors, in spite of changes in armament, rule the infantry attack of to-day.

Infantry Tactics Since 1870

The net result of the Franco-German War on infantry tactics, as far as it can be summed up in a single phrase, was to transfer the fire-fight to the line of skirmishers. Henceforward the old and correct sense of the word “skirmishers” is lost. They have nothing to do with a “skirmish,” but are the actual organ of battle, and their old duties of feeling the way for the battle-formations have been taken over by “scouts.” The last-named were not, however, fully recognized in Great Britain[14] till long after the war—not in fact until the war in South Africa had shown that the “skirmisher” or firing line was too powerful an engine to be employed in mere “feeling.” In most European armies “combat patrols,” which work more freely, are preferred to scouts, but the idea is the same.

The fire-fight on the line of skirmishers, now styled the firing line, is the centre of gravity of the modern battle. In 1870, owing to the peculiar circumstances of unequal armament, the “fire-fight” was insufficiently developed Lessons of 1870. and uneconomically used, and after the war tacticians turned their attention to the evolution of better methods than those of Wörth and Gravelotte, Europe in general following the lead of Prussia. Controversy, in the early stages, took the form of a contest between “drill” and “individualism,” irrespective of formations and technical details, for until about 1890 the material efficiency of the gun and the rifle remained very much what it had been in 1870, and the only new factor bearing on infantry tactics was the general adoption of a “national army” system similar to Prussia’s and of rifles equal, and in some ways superior, to the chassepot. All European armies, therefore, had to consider equality in artillery power, equality in the ballistics of rifles, and equal intensity of fighting spirit as the normal conditions of the next battle of nations. Here, in fact, was an equilibrium, and in such conditions how was the attacking infantry to force its way forward, whether by fire or movement or by both? France sought the answer in the domain of artillery. Under the guidance of General Langlois, she re-created the Napoleonic hurricane of case-shot (represented in modern conditions by time shrapnel), while from the doctrine formed by Generals Maillard and Bonnal there came a system of infantry tactics derived fundamentally from the tactics of the Napoleonic era. This, however, came later; for the moment (viz. from 1871 to about 1890) the lead in infantry training was admittedly in the hands of the Prussians.

German officers who had fought through the war had seen the operations, generally speaking, either from the staff officer’s or from the regimental officer’s point of view. To the former and to many of the latter the most indelible impression of the battlefield was what they called Massen-Drückebergertum or “wholesale skulking.” The rest, who had perhaps in most cases led the brave remnant of their companies in the final assaults, believed that battles were won by the individual soldier and his rifle. The difference between the two may be said to lie in this, that the first sought a remedy, the second a method. The remedy was drill, the method extended order.

The extreme statement of the case in favour of drill pure and simple is to be found in the famous anonymous pamphlet A Summer Night’s Dream, in which a return to the “old Prussian fire-discipline” of Frederick’s day was offered as the solution of the problem, how to give “fire” its maximum efficacity. Volleys and absolutely mechanical obedience to word of command represent, of course, the most complete application of fire-power that can be conceived. But the proposals of the extreme close-order school were nevertheless merely pious aspirations, not so much because of the introduction of the breech-loader as because the short-service “national” army can never be “drilled” in the Frederician sense. The proposals of the other school were, however, even more impracticable, in that they rested on the hypothesis that all men were brave, and that, consequently, all that was necessary was to teach the recruit how to shoot and to work with other individuals in the squad or company. Disorder of the firing line was accepted, not as an unavoidable evil, but as a condition in which individuality had full play, and as dense swarm formations were quite as vulnerable as an ordinary line, it was an easy step from a thick line of “individuals” to a thin one. The step was, in fact, made in the middle of the war of 1870, though it was hardly noticed that extension only became practicable in proportion as the quality of the enemy decreased and the Germans became acclimatized to fire.

Between these extremes, a moderate school, with the emperor William (who had more experience of the human being in battle than any of his officers) at its head, spent a few years in groping for close-order formations which admitted of control without vulnerability, then laid down the principle and studied the method of developing the greatest fire-power of which short-service infantry was supposed capable, ultimately combined the “drill” and teaching ideas in the German infantry regulations of 1888, which at last abolished those of 1812 with their multitudinous amendments.

The necessity for “teaching” arose partly out of the new conditions of service and the relative rarity of wars. The soldier could no longer learn the ordinary rules of safety in action and comfort in bivouac by experience, Conditions of the modern battle. and had to be taught. But it was still more the new conditions of fighting that demanded careful individual training. Of old, the professional soldier (other than the man belonging to light troops or the ground scout) was, roughly speaking, either so far out of immediate danger as to preserve his reasoning faculties, or so deep in battle that he became the unconscious agent of his inborn or acquired instincts. But the increased range of modern arms prolonged the time of danger, and although (judged by casualty returns) the losses to-day are far less than those which any regiment of Frederick’s day was expected to face without flinching, and actual fighting is apparently spasmodic, the period in which the individual soldier is subjected to the fear of bullets is greatly increased. Zorndorf, the most severe of Frederick’s battles, lasted seven hours, Vionville twelve and Wörth eleven. The battle of the future in Europe, without being as prolonged as Liao-Yang, Shaho and Mukden, will still be undecided twenty-four hours after the advanced guards have taken contact. Now, for a great part of this time, the “old Prussian fire-discipline,” which above all aims at a rapid decision, will be not only unnecessary, but actually hurtful to the progress of the battle as a whole. As in Napoleon’s day (for reasons presently to be mentioned) the battle must resolve itself into a preparative and a decisive phase.[15] In the last no commander could desire a better instrument (if such were attainable with the armies of to-day) than Frederick’s forged steel machine, in which every company was human mitrailleuse. But the preparatory combat not only will be long, but also must be graduated in intensity at different times and places in accordance with the commander’s will, and the Frederician battalion only attained its mechanical perfection by the absolute and permanent submergence of the individual qualities of each soldier, with the result that, although it furnished the maximum effort in the minimum time, it was useless once it fell apart into ragged groups. The individual spirit of earnestness and intelligence in the use of ground by small fractions, which in Napoleon’s day made the combat d’usure possible, was necessarily unknown in Frederick’s. On the other hand, graduation implies control on the part of the leaders, and this the method of irregular swarms of individual fighters imagined by the German progressives merely abdicates. At most such swarms—however close or extended—can only be tolerated as an evil that no human power can avert when the battle has reached a certain stage of intensity. Even the latest German Infantry Training (1906) is explicit on this point. “It must never be forgotten that the obligation of abandoning close order is an evil which can often be avoided when” &c. &c. (par. 342). The consequences of this evil, further, are actually less serious in proportion as the troops are well drilled—not to an unnecessary and unattainable ideal of mechanical perfection, but to a state of instinctive self-control in danger. Drill, therefore, carried to such a point that it has eliminated the bad habits of the recruit without detriment to his good habits, is still the true basis of all military training, whether training be required for the swift controlled movements of bodies of infantry in close order, for the cool and steady fire of scattered groups of skirmishers, or for the final act of the resolute will embodied in the “decisive attack.” Unfortunately for the solution of infantry problems “drill” and “close order” are often confused, owing chiefly to the fact that in the 1870 battles the dissolution of close order formations practically meant the end of control as control was then understood. Both the material and objective, and the inward and spiritual significances of “drill” are, however, independent of “close order.” In fact, in modern history, when a resolute general has made a true decisive attack with half-drilled troops, he has generally arrayed them in the closest possible formations.

Drill is the military form of education by repetition and association (see G. le Bon, Psychologie de l’éducation). Materially it consists in exercises frequently repeated by bodies of soldiers with a view to ensuring the harmonious action of each individual Drill. in the work to be performed by the mass—in a word, rehearsals. Physical “drill” is based on physiology and gymnastics, and aims at the development of the physique and the individual will power.[16] But the psychological or moral is incomparably the most important side of drill. It is the method or art of discipline. Neither self-control nor devotion in the face of imminent danger can as a rule come from individual reasoning. A commander-in-chief keeps himself free from the contact with the turmoil of battle so long as he has to calculate, to study reports or to manœuvre, and commanders of lower grades, in proportion as their duty brings them into the midst of danger, are subjected to greater or less disturbing influences. The man in the fighting line where the danger is greatest is altogether the slave of the unconscious. Overtaxed infantry, whether defeated or successful, have been observed to present an appearance of absolute insanity. It is true that in the special case of great war experience reason resumes part of its dominion in proportion as the fight becomes the soldier’s habitual milieu. Thus towards the end of a long war men become skilful and cunning individual fighters; sometimes, too, feelings of respect for the enemy arise and lead to interchange of courtesies at the outposts, and it has also been noticed that in the last stage of a long war men are less inclined to sacrifice themselves. All this is “reason” as against inborn or inbred “instinct.” But in the modern world, which is normally at peace, some method must be found of ensuring that the peace-trained soldier will carry out his duties when his reason is submerged. Now we know that the constant repetition of a certain act, whether on a given impulse or of the individual’s own volition, will eventually make the performance of that act a reflex action. For this reason peace-drilled troops have often defeated a war-trained enemy, even when the motives for fighting were equally powerful on each side. The mechanical performance of movements, and loading and firing at the enemy, under the most disturbing conditions can be ensured by bringing the required self-control from the domain of reason into that of instinct. “L’éducation,” says le Bon, “est l’art de faire passer le conscient dans l’inconscient.” Lastly, the instincts of the recruit being those special to his race or nation, which are the more powerful because they are operative through many generations, it is the drill sergeant’s business to bring about, by disuse, atrophy of the instincts which militate against soldierly efficiency, and to develop, by constant repetition and special preparation, other useful instincts which the Englishman or Frenchman or German does not as such possess. In short, as regards infantry training, there is no real distinction between drill and education, save in so far as the latter term covers instruction in small details of field service which demand alertness, shrewdness and technical knowledge (as distinct from technical training). As understood by the controversialists of the last generation, drill was the antithesis of education. To-day, however, the principle of education having prevailed against the old-fashioned notion of drill, it has been discovered that after all drill is merely an intensive form of education. This discovery (or rather definition and justification of an existing empirical rule) is attributable chiefly to a certain school of French officers, who seized more rapidly than civilians the significance of modern psycho-physiology. In their eyes, a military body possesses in a more marked degree than another, the primary requisite of the “psychological crowd,” studied by Gustave le Bon, viz. the orientation of the wills of each and all members of the crowd in a determined direction. Such a crowd generates a collective will that dominates the wills of the individuals composing it. It coheres and acts on the common property of all the instincts and habits in which each shares. Further it tends to extremes of baseness and heroism—this being particularly marked in the military crowd—and lastly it reacts to a stimulus. The last is the keynote of the whole subject of infantry training as also, to a lesser degree, of that of the other arms. The officer can be regarded practically as a hypnotist playing upon the unconscious activities of his subject. In the lower grades, it is immaterial whether reason, caprice or a fresh set of instincts stimulated by an outside authority, set in motion the “suggestion.” The true leader, whatever the provenance of his “suggestion,” makes it effective by dominating the “psychological crowd” that he leads. On the other hand, if he fails to do so, he is himself dominated by the uncontrolled will of the crowd, and although leaderless mobs have at times shown extreme heroism, it is far more usual to find them reverting to the primitive instinct of brutality or panic fear. A mob, therefore, or a raw regiment, requires greater powers of suggestion in its leader, whereas a thorough course of drill tunes the “crowd” to respond to the stimulus that average officers can apply.

So far from diminishing, drill has increased in importance under modern conditions of recruiting. It has merely changed in form, and instead of being repressive it has become educative. The force of modern short-service troops, as troops, is far sooner spent than that of the old-fashioned automatic regiments, while the reserve force of its component parts, remaining after the dissolution, is far higher than of old. But this uncontrolled, force is liable to panic as well as amenable to an impulse of self-sacrifice. In so far, then, it is necessary to adopt the catchword of the Bülow school and to “organize disorder,” and the only known method of doing so is drill. “Individualism” pure and simple had certainly a brief reign during and after the South African War, especially in Great Britain, and both France and Germany coquetted with “Boer tactics,” until the Russo-Japanese war brought military Europe back to the old principles.

But the South African War came precisely at the point of time when the controversies of 1870 had crystallized into a form of tactics that was not suitable to the conditions of that war, while about the same time the relations of infantry The South African War. and artillery underwent a profound change. As regards the South African War, the clear atmosphere, the trained sight of the Boers, and the alternation of level plain and high concave kopjes which constituted the usual battlefield, made the front to front infantry attacks not merely difficult but almost impossible. For years, indeed ever since the Peninsular War, the tendency of the British army to deploy early had afforded a handle to European critics of its tactical methods. It was a tendency that survived with the rest of the “linear” tradition. But in South Africa, owing to the special advantages of the defenders, which denied to the assailant all reliable indications of the enemy’s strength and positions, this early deployment had to take a non-committal form—viz. many successive lines of skirmishers. The application of this form was, indeed, made easy by the openness of the ground, but like all “schematic” formations, open or close, it could not be maintained under fire, with the special disadvantage that the extensions were so wide as to make any manœuvring after the fight had cleared up a situation a practical impossibility. Hence some preconceived idea of an objective was an essential preliminary, and as the Boer mounted infantry hardly ever stood to defend any particular position to the last (as they could always renew the fight at some other point in their vast territory), the preconceived idea was always, after the early battles, an envelopment in which the troops told off to the frontal holding attack were required, not to force their advance to its logical conclusion, but to keep the fight alive until the flank attack made itself felt. The principal tendency of British infantry tactics after the Boer War was therefore quite naturally, under European as well as colonial conditions, to deploy at the outset in great depth, i.e. in many lines of skirmishers, each line, when within about 1400 yds. of the enemy’s position, extending to intervals of 10 to 20 paces between individuals. The reserves were strong and their importance was well marked in the 1902 training manual, but their functions were rather to extend or feed the firing line, to serve as a rallying point in case of defeat and to take up the pursuit (par. 220, Infantry Training, 1902), than to form the engine of a decisive attack framed by the commander-in-chief after “engaging everywhere and then seeing” as Napoleon did. The 1905 regulations adhered to this theory of the attack in the main, only modifying a number of tactical prescriptions which Formulation of the British “Doctrine.” had not proved satisfactory after their transplantation from South Africa to Europe, but after the Russo-Japanese War a series of important amendments was issued which gave greater force and still greater elasticity to the attack procedure, and in 1909 the tactical “doctrine” of the British army was definitively formulated in Field Service Regulations, paragraph 102, of which after enumerating the advantages and disadvantages of the “preconceived idea” system, laid it down, as the normal procedure of the British Army, that the general should “obtain the decision by manœuvre on the battlefield with a large general reserve maintained in his own hand” and “strike with his reserve at the right place and time.”

The rehabilitation of the Napoleonic attack idea thus frankly accepted in Great Britain had taken place in France several years before the South African War, and neither this war nor that in Manchuria effectively shook the faith of the French army in the principle, while on the other hand Germany remains faithful to the “preconceived idea,” both in strategy and tactics.[17] This essential difference in the two rival “doctrines” is intimately connected with the revival of the Napoleonic artillery attack, in the form of concentrated time shrapnel.

The Napoleonic artillery preparation, it will be remembered, was a fire of overwhelming intensity delivered against the selected point of the enemy’s position, at the moment of the massed and decisive assault of the reserves. In Napoleon’s time the artillery went in to within 300 or 400 yds. range for this act, i.e. in front of the infantry, whereas now the guns fire over the heads of the infantry and concentrate shells instead of guns on the vital point. The principle is, however, the same. A model infantry attack in the Napoleonic manner was that of Okasaki’s brigade on the Terayama hill at the battle of Shaho, described by Sir Ian Hamilton in his Staff Officer’s Scrap-Book. The Japanese, methodical and cautious as they were, only sanctioned a pure open force assault as a last resort. Then the brigadier Okasaki, a peculiarly resolute leader, arrayed his brigade in a “schematic” attack formation of four lines, the first two in single rank, the third in line and the fourth in company columns. Covered by a powerful converging shrapnel fire, the brigade covered the first 900 yds. of open plain without firing a shot. Then, however, it disappeared from sight amongst the houses of a village, and the spectators watched the thousands of flashes fringing the further edge that indicated a fire-fight at decisive range (the Terayama was about 600 yds. beyond the houses). Forty minutes passed, and the army commander Kuroki said, “He cannot go forward. We are in check to-day all along the line.” But at that moment Okasaki’s men, no longer in a “schematic” formation but in many irregularly disposed groups—some of a dozen men and some of seventy, some widely extended and some practically in close order—rushed forward at full speed over 600 yds. of open ground, and stormed the Terayama with the bayonet.

Such an attack as that at the battle of Shaho is rare, but so it has always been with masterpieces of the art of war. We have only to multiply the front of attack by two and the forces engaged by five—and to find the resolute The decisive attack. general to lead them—to obtain the ideal decisive attack of a future European war. Instead of the bare open plain over which the advance to decisive range was made, a European general would in most cases dispose of an area of spinneys, farm-houses and undulating fields. The schematic approach-march would be replaced in France and England by a forward movement of bodies in close order, handy enough to utilize the smallest covered ways. Then the fire of both infantry and artillery would be augmented to its maximum intensity, overpowering that of the defence, and the whole of the troops opposite the point to be stormed would be thrown forward for the bayonet charge. The formation for this scarcely matters. What is important is speed and the will to conquer, and for this purpose small bodies (sections, half-companies or companies), not in the close order of the drill book but grouped closely about the leader who inspires and controls them, are as potent an instrument as a Frederician line or a Napoleonic column.

Controversy, in fact, does not turn altogether on the method of the assault, or even on the method of obtaining the fire-superiority of guns and rifles that justifies it. Although one nation may rely on its guns more than on the rifles, or vice versa, all are agreed that at decisive range the firing line should contain as many men as can use their rifles effectually. Perhaps the most disputed point is the form of the “approach-march,” viz. the dispositions and movements of the attacking infantry between about 1400 and about 600 yds. from the position of the enemy.

The condition of the assailant’s infantry when it reaches decisive ranges is largely governed by the efforts it has expended and the losses it has suffered in its progress. Sometimes even after a firing line of some strength has been The approach-march. established at decisive range, it may prove too difficult or too costly for the supports (sent up from the rear to replace casualties and to augment fire-power) to make their way to the front. Often, again, it may be within the commander’s intentions that his troops at some particular point in the line should not be committed to decisive action before a given time—perhaps not at all. It is obvious then that no “normal” attack procedure which can be laid down in a drill book (though from time to time the attempt has been made, as in the French regulations of 1875) can meet all cases. But here again, though all armies formally and explicitly condemn the normal attack, each has its own well-marked tendencies.

The German regulations of 1906 define the offensive as “transporting fire towards the enemy, if necessary to his immediate proximity”; the bayonet attack “confirms” the victory. Every attack begins with deployment Current views on the infantry attack. into extended order, and the leading line advances as close to the enemy as possible before opening fire. In ground offering cover, the firing line has practically its maximum density at the outset. In open ground, however, half-sections, groups and individuals, widely spaced out, advance stealthily one after the other till all are in position. It is on this position, called the “first fire position” and usually about 1000 yds. from the enemy, that the full force of the attack is deployed, and from this position, as simultaneously as possible, it opens the fight for fire-superiority. Then, each unit covering the advance of its neighbours, the whole line fights its way by open force to within charging distance. If at any point a decision is not desired, it is deliberately made impossible by employing there such small forces as possess no offensive power. Where the attack is intended to be pushed home, the infantry units employed act as far as possible simultaneously, resolutely and in great force (see the German Infantry Regulations, 1906, §§ 324 et seq.).

While in Germany movement “transports the fire,” in France fire is regarded as the way to make movement possible. It is considered (see Grandmaison, Dressage de l’infanterie) that a premature and excessive deployment enervates the attack, that the ground (i.e. covered ways of approach for small columns, not for troops showing a fire front) should be used as long as possible to march “en troupe” and that a firing line should only be formed when it is impossible to progress without acting upon the enemy’s means of resistance. Thereafter each unit, in such order as its chief can keep, should fight its way forward, and help others to do so—like Okasaki’s brigade in the last stage of its attack—utilizing bursts of fire or patches of wood or depressions in the ground, as each is profitable or available to assist the advance. “From the moment when a fighting unit is ‘uncoupled,’ its action must be ruled by two conditions, and by those only: the one material, an object to be reached; the other moral, the will to reach the object.”

The British Field Service Regulations of 1909 are in spirit more closely allied to the French than to the German. “The climax of the infantry attack is the assault, which is made possible by superiority of fire” is the principle (emphasized in the book itself by the use of conspicuous type), and a “gradual building up of the firing line within close range of the position,” coupled with the closest artillery support, and the final blow of the reserves delivered “unexpectedly and in the greatest possible strength” are indicated as the means.[18]

The defence, as it used to be understood, needs no description. To-day in all armies the defence is looked upon not as a means of winning a battle, but as a means of temporizing and avoiding the decision until the commander of Defence. the defending party is enabled, by the general military situation or by the course and results of the defensive battle itself, to take the offensive. In the British Field Service Regulations it is laid down that when an army acts on the defensive no less than half of it should if possible be earmarked, suitably posted and placed under a single commander, for the purpose of delivering a decisive counter-attack. The object of the purely defensive portion, too, is not merely to hold the enemy’s firing line in check, but to drive it back so that the enemy may be forced to use up his local reserve resources to keep the fight alive. A firing line covered and steadied by entrenchments, and restless local reserves ever on the look-out for opportunities of partial counterstrokes, are the instruments of this policy.

A word must be added on the use of entrenchments by infantry, a subject the technical aspect of which is fully dealt with and illustrated in [Fortification and Siegecraft]: Field Defences. Entrenchments of greater or less strength by themselves Entrenchments. have always been used by infantry on the defensive, especially in the wars of position of the 17th and 18th centuries. In the Napoleonic and modern “wars of movement,” they are regarded, not as a passive defence—they have long ceased to present a physical barrier to assault—but as fire positions so prepared as to be defensible by relatively few men. Their purpose is, by economizing force elsewhere, to give the maximum strength to the troops told off for the counter-offensive. In the later stages of the American Civil War, and also in the Russo-Japanese War of 1904-1905—each in its way an example of a “war of positions”—the assailant has also made use of the methods of fortification to secure every successive step of progress in the attack. The usefulness and limitations of this procedure are defined in generally similar terms in the most recent training manuals of nearly every European army. Section 136, § 7 of the British Infantry Training (1905, amended 1907) says: “During the process of establishing a superiority of fire, successive fire positions will be occupied by the firing line. As a rule those affording natural cover will be chosen, but if none exist and the intensity of the hostile fire preclude any immediate further advance, it may be expedient for the firing line to create some. This hastily constructed protection will enable the attack to cope with the defender’s fire and thus prepare the way for a farther advance. The construction of cover during an attack, however, will entail delay and a temporary loss of fire effect and should therefore be resorted to only when absolutely necessary.... As soon as possible the advance should be resumed, &c.” The German regulations are as follows (Infantry Training, 1906, § 313): “In the offensive the entrenching tool may be used where it is desired, for the moment, to content one’s self with maintaining the ground gained.... The entrenching tool is only to be used with the greatest circumspection, because of the great difficulty of getting an extended line to go forward under fire when it has expended much effort in digging cover for itself. The construction of trenches must never paralyze the desire for the irresistible advance, and above all must not kill the spirit of the offensive.”

Organization and Equipment

The organization of infantry varies rather more than that of other arms in different countries. Taking the British system first, the battalion (and not as elsewhere the regiment of two, three or more battalions) is the administrative and manœuvre unit. It is about 1000 strong, and is commanded by a lieutenant-colonel, who has a major and an adjutant (captain or lieutenant) to assist him, and an officer of lieutenant’s or captain’s rank (almost invariably promoted from the ranks), styled the quartermaster, to deal with supplies, clothing, &c. There are eight companies of a nominal strength of about 120 each. These are commanded by captains (or by junior majors), and each captain has or should have two lieutenants or second lieutenants to assist him. Machine guns are in Great Britain distributed to the battalions and not massed in permanent batteries. In addition there are various regimental details, such as orderly-room staff, cooks, cyclists, signallers, band and ambulance men. The company is divided into four sections of thirty men each and commanded by sergeants. A half-company of two sections is under the control of a subaltern officer. A minor subdivision of the section into two “squads” is made unless the numbers are insufficient to warrant it. In administrative duties the captain’s principal assistant is the colour-sergeant or pay-sergeant, who is not assigned to a section command. The lieutenant-colonel, the senior major and the adjutant are mounted. The commanding officer is assisted by a battalion staff, at the head of which is the adjutant. The sergeant-major holds a “warrant” from the secretary of state for war, as does the bandmaster. Other members of the battalion staff are non-commissioned officers, appointed by the commanding officer. The most important of these is the quartermaster-sergeant, who is the assistant of the quartermaster. The two colours (“king’s” and “regimental”) are in Great Britain carried by subalterns and escorted by colour-sergeants (see [Colours]).

The “tactical” unit of infantry is now the company, which varies very greatly in strength in the different armies. Elsewhere the company of 250 rifles is almost universal, but in Great Britain the company has about 110 men in the ranks, forming four sections. These sections, each of about 28 rifles, are the normal “fire-units,” that is to say, the unit which delivers its fire at the orders of and with the elevation and direction given by its commander. This, it will be observed, gives little actual executive work for the junior officers. But a more serious objection than this (which is modified in practice by arrangement and circumstances) is the fact that a small unit is more affected by detachments than a large one. In the home battalions of the Regular Army such detachments are very large, what with finding drafts for the foreign service battalions and for instructional courses, while in the Territorial Force, where it is so rarely possible to assemble all the men at once, the company as organized is often too small to drill as such. On the other hand, the full war-strength company is an admirable unit for control and manœuvre in the field, owing to its rapidity of movement, handiness in using accidents of ground and cover, and susceptibility to the word of command of one man. But as soon as its strength falls below about 80 the advantages cease to counterbalance the defects. The sections become too small as fire-units to effect really useful results, and the battalion commander has to coordinate and to direct 8 comparatively ineffective units instead of 4 powerful ones. The British regular army, therefore, has since the South African War, adopted the double company as the unit of training. This gives at all times a substantial unit for fire and manœuvre training, but the disadvantage of having a good many officers only half employed is accentuated. As to the tactical value of the large or double company, opinions differ. Some hold that as the small company is a survival from the days when the battalion was the tactical unit and the company was the unit of volley-fire, it is unsuited to the modern exigencies that have broken up the old rigid line into several independent and co-operating fractions. Others reply that the strong continental company of 250 rifles came into existence in Prussia in the years after Waterloo, not from tactical reasons, but because the state was too poor to maintain a large establishment of officers, and that in 1870, at any rate, there were many instances of its tactical unwieldiness. The point that is common to both organizations is the fact that there is theoretically one subaltern to every 50 or 60 rifles, and this reveals an essential difference between the British and the Continental systems, irrespective of the sizes or groupings of companies. The French or German subaltern effectively commands his 50 men as a unit, whereas the British subaltern supervises two groups of 25 to 30 men under responsible non-commissioned officers. That is to say, a British sergeant may find himself in such a position that he has to be as expert in controlling and obtaining good results from collective fire as a German lieutenant. For reasons mentioned in [Army], § 40, non-commissioned officers, of the type called by Kipling the “backbone of the army,” are almost unobtainable with the universal service system, and the lowest unit that possesses any independence is the lowest unit commanded by an officer. But apart from the rank of the fire-unit commander, it is questionable whether the section, as understood in England, is not too small a fire-unit, for European warfare at any rate. The regulations of the various European armies, framed for these conditions, practically agree that the fire-unit should be commanded by an officer and should be large enough to ensure good results from collective fire. The number of rifles meeting this second condition is 50 to 80 and their organization a “section” (corresponding to the British half-company) under a subaltern officer. The British army has, of course, to be organized and trained for an infinitely wider range of activity, and no one would suggest the abolition of the small section as a fire-unit. But in a great European battle it would be almost certainly better to group the two sections into a real unit for fire effect. (For questions of infantry fire tactics see [Rifle]: § Musketry.)

On the continent of Europe the “regiment,” which is a unit, acting in peace and war as such, consists normally of three battalions, and each battalion of four companies or 1000 rifles. The company of 250 rifles is commanded by a captain, who is mounted. In France the company has four sections, commanded in war by the three subalterns and the “adjudant” (company sergeant-major); the sections are further grouped in pairs to constitute pelotons (platoons) or half-companies under the senior of the two section leaders. In peace there are two subalterns only, and the peloton is the normal junior officer’s command. The battalion is commanded by a major (commandant or strictly chef de bataillon), the regiment (three or four battalions) by a colonel with a lieutenant-colonel as second. An organization of 3-battalion regiments and 3-company battalions was proposed in 1910.

In Germany, where what we have called the continental company originated, the regiment is of three battalions under majors, and the battalion of four companies commanded by captains. The company is divided into three Züge (sections), each under a subaltern, who has as his second a sergeant-major, a “vice-sergeant-major” or a “sword-knot ensign” (aspirant officer). In war there is one additional officer for company. The Zug at war-strength has therefore about 80 rifles in the ranks, as compared with the French “section” of 50, and the British section of 30.

The system prevailing in the United States since the reorganization of 1901 is somewhat remarkable. The regiment, which is a tactical as well as an administrative unit, consists of three battalions. Each battalion has four companies of (at war-strength) 3 officers and 150 rifles each. The regiment in war therefore consists of about 1800 rifles in three small and handy battalions of 600 each. The circumstances in which this army serves, and in particular the maintenance of small frontier posts, have always imposed upon subalterns the responsibilities of small independent commands, and it is fair to assume that the 75 rifles at a subaltern’s disposal are regarded as a tactical unit.

In sum, then, the infantry battalion is in almost every country about 1000 rifles strong in four companies. In the United States it is 600 strong in four companies, and in Great Britain it is 1000 strong in eight. The captain’s command is usually 200 to 250 men, in the United States 150, and in Great Britain 120. The lieutenant or second lieutenant commands in Germany 80 rifles, in France 50, in the United States 75, as a unit of fire and manœuvre. In Great Britain he commands, with relatively restricted powers, 60.

A short account of the infantry equipments—knapsack or valise, belt, haversack, &c.—in use in various countries will be found in [Uniforms, Naval and Military]. The armament of infantry is, in all countries, the magazine rifle (see [Rifle]) and bayonet (q.v.), for officers and for certain under-officers sword (q.v.) and pistol (q.v.). Ammunition (q.v.) in the British service is carried (a) by the individual soldier, (b) by the reserves (mules and carts) in regimental charge, some of which in action are assembled from the battalions of a brigade to form a brigade reserve, and (c) by the ammunition columns.

Bibliography.—The following works are selected to show (1) the historical development of the arm, and (2) the different “doctrines” of to-day as to its training and functions:—Ardant du Picq, Études sur le combat; C. W. C. Oman, The Art of War: Middle Ages; Biottot, Les Grands Inspirés—Jeanne d’Arc; Hardy de Périni, Batailles françaises; C. H. Firth, Cromwell’s Army; German official history of Frederick the Great’s wars, especially Erster Schlesische Krieg, vol. i.; Susane, Histoire de l’infanterie française; French General Staff, La Tactique au XVIIIme—l’infanterie and La Tactique et la discipline dans les armées de la Révolution—Général Schauenbourg; J. W. Fortescue, History of the British Army; Moorsom, History of the 52nd Regiment; de Grandmaison, Dressage de l’infanterie (Paris, 1908); works of W. v. Scherff; F. N. Maude, Evolution of Infantry Tactics and Attack and Defence; [Meckel] Ein Sommernachtstraum (Eng. trans, in United Service Magazine, 1890); J. Meckel, Taktik; Malachowski, Scharfe- und Revuetaktik; H. Langlois, Enseignements de deux guerres; F. Hoenig, Tactics of the Future and Twenty-four Hours of Moltke’s Strategy (Eng. trans.); works of A. von Boguslowski; British Officers’ Reports on the Russo-Japanese War; H. W. L. Hime, Stray Military Papers; Grange, “Les Réalités du champ de bataille—Woerth” (Rev. d’infanterie, 1908-1909); V. Lindenau, “The Boer War and Infantry Attack” (Journal R. United Service Institution, 1902-1903); Janin, “Aperçus sur la tactique—Mandchourie” (Rev. d’infanterie, 1909); Soloviev, “Infantry Combat in the Russo-Jap. War” (Eng. trans. Journal R.U.S.I., 1908); British Official Field Service Regulations, part i. (1909), and Infantry Training (1905); German drill regulations of 1906 (Fr. trans.); French drill regulations of 1904; Japanese regulations 1907 (Eng. trans.). The most important journals devoted to the infantry arm are the French official Revue d’infanterie (Paris and Limoges), and the Journal of the United Stales Infantry Association (Washington, D. C).

(C. F. A.)

[1] At Bouvines, it is recorded with special emphasis that Guillaume des Barres, when in the act of felling the emperor, heard the call to rescue King Philip Augustus and, forfeiting his rich prize, made his way back to help his own sovereign.

[2] Crossbows indeed were powerful, and also handled by professional soldiers (e.g. the Genoese at Crécy), but they were slow in action, six times as slow as the long bow, and the impatient gendarmerie generally became tired of the delay and crowded out or rode over the crossbowmen.

[3] As for instance when thirty men-at-arms “cut out” the Captal de Buch from the midst of his army at Cocherel.

[4] This tendency of the French military temperament reappears at almost every stage in the history of armies.

[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 manœuvre 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 τὸ ἄπειρον (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. √−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.

2. The guise in which variable quantities presented themselves to the mathematicians of the 17th century was that of the lengths of variable lines. This method of representing variable quantities dates from the 14th century, Geometrical representation of Variable Quantities. when it was employed by Nicole Oresme, who studied and afterwards taught at the Collège de Navarre in Paris from 1348 to 1361. He represented one of two variable quantities, e.g. the time that has elapsed since some epoch, by a length, called the “longitude,” measured along a particular line; and he represented the other of the two quantities, e.g. the temperature at the instant, by a length, called the “latitude,” measured at right angles to this line. He recognized that the variation of the temperature with the time was represented by the line, straight or curved, which joined the ends of all the lines of “latitude.” Oresme’s longitude and latitude were what we should now call the abscissa and ordinate. The same method was used later by many writers, among whom Johannes Kepler and Galileo Galilei may be mentioned. In Galileo’s investigation of the motion of falling bodies (1638) the abscissa OA represents the time during which a body has been falling, and the ordinate AB represents the velocity acquired during that time (see fig. 1). The velocity being proportional to the time, the “curve” obtained is a straight line OB, and Galileo showed that the distance through which the body has fallen is represented by the area of the triangle OAB.

The most prominent problems in regard to a curve were the problem of finding the points at which the ordinate is a maximum or a minimum, the problem of drawing a tangent to the curve at an assigned point, and the problem of The problems of Maxima and Minima, Tangents, and Quadratures. determining the area of the curve. The relation of the problem of maxima and minima to the problem of tangents was understood in the sense that maxima or minima arise when a certain equation has equal roots, and, when this is the case, the curves by which the problem is to be solved touch each other. The reduction of problems of maxima and minima to problems of contact was known to Pappus. The problem of finding the area of a curve was usually presented in a particular form in which it is called the “problem of quadratures.” It was sought to determine the area contained between the curve, the axis of abscissae and two ordinates, of which one was regarded as fixed and the other as variable. Galileo’s investigation may serve as an example. In that example the fixed ordinate vanishes. From this investigation it may be seen that before the invention of the infinitesimal calculus the introduction of a curve into discussions of the course of any phenomenon, and the problem of quadratures for that curve, were not exclusively of geometrical import; the purpose for which the area of a curve was sought was often to find something which is not an area—for instance, a length, or a volume or a centre of gravity.

3. The Greek geometers made little progress with the problem of tangents, but they devised methods for investigating the problem of quadratures. One of these methods was afterwards called the “method of exhaustions,” and Greek methods. the principle on which it is based was laid down in the lemma prefixed to the 12th book of Euclid’s Elements as follows: “If from the greater of two magnitudes there be taken more than its half, and from the remainder more than its half, and so on, there will at length remain a magnitude less than the smaller of the proposed magnitudes.” The method adopted by Archimedes was more general. It may be described as the enclosure of the magnitude to be evaluated between two others which can be brought by a definite process to differ from each other by less than any assigned magnitude. A simple example of its application is the 6th proposition of Archimedes’ treatise On the Sphere and Cylinder, in which it is proved that the area contained between a regular polygon inscribed in a circle and a similar polygon circumscribed to the same circle can be made less than any assigned area by increasing the number of sides of the polygon. The methods of Euclid and Archimedes were specimens of rigorous limiting processes (see [Function]). The new problems presented by the analytical geometry and natural philosophy of the 17th century led to new limiting processes.

4. In the problem of tangents the new process may be described as follows. Let P, P′ be two points of a curve (see fig. 2). Let x, y be the coordinates of P, and x + Δx, y + Δy those of P′. The symbol Δx means “the difference of two Differentiation. x’s” and there is a like meaning for the symbol Δy. The fraction Δy/Δx is the trigonometrical tangent of the angle which the secant PP′ makes with the axis of x. Now let Δx be continually diminished towards zero, so that P′ continually approaches P. If the curve has a tangent at P the secant PP′ approaches a limiting position (see § 33 below). When this is the case the fraction Δy/Δx tends to a limit, and this limit is the trigonometrical tangent of the angle which the tangent at P to the curve makes with the axis of x. The limit is denoted by

If the equation of the curve is of the form y = ƒ(x) where ƒ is a functional symbol (see [Function]), then

and

The limit expressed by the right-hand member of this defining equation is often written

ƒ′(x),

and is called the “derived function” of ƒ(x), sometimes the “derivative” or “derivate” of ƒ(x). When the function ƒ(x) is a rational integral function, the division by Δx can be performed, and the limit is found by substituting zero for Δx in the quotient. For example, if ƒ(x) = x2, we have

and

ƒ′(x) = 2x.

The process of forming the derived function of a given function is called differentiation. The fraction Δy/Δx is called the “quotient of differences,” and its limit dy/dx is called the “differential coefficient of y with respect to x.” The rules for forming differential coefficients constitute the differential calculus.

The problem of tangents is solved at one stroke by the formation of the differential coefficient; and the problem of maxima and minima is solved, apart from the discrimination of maxima from minima and some further refinements, by equating the differential coefficient to zero (see [Maxima and Minima]).

5. The problem of quadratures leads to a type of limiting process which may be described as follows: Let y = ƒ(x) be the equation of a curve, and let AC and BD be the ordinates of the points C and D (see fig. 3). Let a, b be the abscissae of these Integration. points. Let the segment AB be divided into a number of segments by means of intermediate points such as M, and let MN be one such segment. Let PM and QN be those ordinates of the curve which have M and N as their feet. On MN as base describe two rectangles, of which the heights are the greatest and least values of y which correspond to points on the arc PQ of the curve. In fig. 3 these are the rectangles RM, SN. Let the sum of the areas of such rectangles as RM be formed, and likewise the sum of the areas of such rectangles as SN. When the number of the points such as M is increased without limit, and the lengths of all the segments such as MN are diminished without limit, these two sums of areas tend to limits. When they tend to the same limit the curvilinear figure ACDB has an area, and the limit is the measure of this area (see § 33 below). The limit in question is the same whatever law may be adopted for inserting the points such as M between A and B, and for diminishing the lengths of the segments such as MN. Further, if P′ is any point on the arc PQ, and P′M′ is the ordinate of P′, we may construct a rectangle of which the height is P′M′ and the base is MN, and the limit of the sum of the areas of all such rectangles is the area of the figure as before. If x is the abscissa of P, x + Δx that of Q, x′ that of P′, the limit in question might be written

lim. Σba ƒ(x′) Δx,

where the letters a, b written below and above the sign of summation Σ indicate the extreme values of x. This limit is called “the definite integral of ƒ(x) between the limits a and b,” and the notation for it is

∫ba ƒ(x) dx.

The germs of this method of formulating the problem of quadratures are found in the writings of Archimedes. The method leads to a definition of a definite integral, but the direct application of it to the evaluation of integrals is in general difficult. Any process for evaluating a definite integral is a process of integration, and the rules for evaluating integrals constitute the integral calculus.

6. The chief of these rules is obtained by regarding the extreme ordinate BD as variable. Let ξ now denote the abscissa of B. The area A of the figure ACDB is represented by the Theorem of Inversion. integral ∫ξa ƒ(x)dx, and it is a function of ξ. Let BD be displaced to B′D′ so that ξ becomes ξ + δξ (see fig. 4). The area of the figure ACD′B′ is represented by the integral ∫ξ+Δξa ƒ(x)dx, and the increment ΔA of the area is given by the formula

ΔA = ∫ξ+Δξξ ƒ(x) dx,

which represents the area BDD′B′. This area is intermediate between those of two rectangles, having as a common base the segment BB′, and as heights the greatest and least ordinates of points on the arc DD′ of the curve. Let these heights be H and h. Then ΔA is intermediate between HΔξ and hΔξ, and the quotient of differences ΔA/Δξ is intermediate between H and h. If the function ƒ(x) is continuous at B (see Function), then, as Δξ is diminished without limit, H and h tend to BD, or ƒ(ξ), as a limit, and we have

The introduction of the process of differentiation, together with the theorem here proved, placed the solution of the problem of quadratures on a new basis. It appears that we can always find the area A if we know a function F(x) which has ƒ(x) as its differential coefficient. If ƒ(x) is continuous between a and b, we can prove that

A = ∫ba ƒ(x) dx = F(b) − F(a).

When we recognize a function F(x) which has the property expressed by the equation

we are said to integrate the function ƒ(x), and F(x) is called the indefinite integral of ƒ(x) with respect to x, and is written

∫ ƒ(x) dx.

7. In the process of § 4 the increment Δy is not in general equal to the product of the increment Δx and the derived Differentials. function ƒ′(x). In general we can write down an equation of the form

Δy = ƒ′(x) Δx + R,

in which R is different from zero when Δx is different from zero; and then we have not only

lim.Δx=0 R = 0,

but also

We may separate Δy into two parts: the part ƒ′(x)Δx and the part R. The part ƒ′(x)Δx alone is useful for forming the differential coefficient, and it is convenient to give it a name. It is called the differential of ƒ(x), and is written dƒ(x), or dy when y is written for ƒ(x). When this notation is adopted dx is written instead of Δx, and is called the “differential of x,” so that we have

dƒ(x) = ƒ′(x) dx.

Thus the differential of an independent variable such as x is a finite difference; in other words it is any number we please. The differential of a dependent variable such as y, or of a function of the independent variable x, is the product of the differential of x and the differential coefficient or derived function. It is important to observe that the differential coefficient is not to be defined as the ratio of differentials, but the ratio of differentials is to be defined as the previously introduced differential coefficient. The differentials are either finite differences, or are so much of certain finite differences as are useful for forming differential coefficients.

Again let F(x) be the indefinite integral of a continuous function ƒ(x), so that we have

When the points M of the process explained in § 5 are inserted between the points whose abscissae are a and b, we may take them to be n − 1 in number, so that the segment AB is divided into n segments. Let x1, x2, ... xn−1 be the abscissae of the points in order. The integral is the limit of the sum

ƒ(a) (x1 − a) + ƒ(x1) (x2 − x1) + ... + ƒ(xr) (xr+1 − xr) + ... + ƒ(xn−1) (b − xn−1),

every term of which is a differential of the form ƒ(x)dx. Further the integral is equal to the sum of differences

{F(x1) − F(a)} + {F(x2) − F(x1)} + ... + {F(xr+1) − F(xr)} + ... + {F(b) − F(xn−1)},

for this sum is F(b) − F(a). Now the difference F(xr+1) − F(xr) is not equal to the differential ƒ(xr) (xr+1 − xr), but the sum of the differences is equal to the limit of the sum of these differentials. The differential may be regarded as so much of the difference as is required to form the integral. From this point of view a differential is called a differential element of an integral, and the integral is the limit of the sum of differential elements. In like manner the differential element ydx of the area of a curve (§ 5) is not the area of the portion contained between two ordinates, however near together, but is so much of this area as need be retained for the purpose of finding the area of the curve by the limiting process described.

8. The notation of the infinitesimal calculus is intimately bound up with the notions of differentials and sums of elements. The letter Notation.
Fundamental Artifice. “d” is the initial letter of the word differentia (difference) and the symbol ∫ is a conventionally written “S,” the initial letter of the word summa (sum or whole). The notation was introduced by Leibnitz (see §§ 25-27, below).

9. The fundamental artifice of the calculus is the artifice of forming differentials without first forming differential coefficients. From an equation containing x and y we can deduce a new equation, containing also Δx and Δy, by substituting x + Δx for x and y + Δy for y. If there is a differential coefficient of y with respect to x, then Δy can be expressed in the form φ.Δx + R, where lim.Δx=0 (R/Δx) = 0, as in § 7 above. The artifice consists in rejecting ab initio all terms of the equation which belong to R. We do not form R at all, but only φ·Δx, or φ.dx, which is the differential dy. In the same way, in all applications of the integral calculus to geometry or mechanics we form the element of an integral in the same way as the element of area y·dx is formed. In fig. 3 of § 5 the element of area y·dx is the area of the rectangle RM. The actual area of the curvilinear figure PQNM is greater than the area of this rectangle by the area of the curvilinear figure PQR; but the excess is less than the area of the rectangle PRQS, which is measured by the product of the numerical measures of MN and QR, and we have

Thus the artifice by which differential elements of integrals are formed is in principle the same as that by which differentials are formed without first forming differential coefficients.

10. This principle is usually expressed by introducing the notion of orders of small quantities. If x, y are two variable numbers which are Orders of small quantities. connected together by any relation, and if when x tends to zero y also tends to zero, the fraction y/x may tend to a finite limit. In this case x and y are said to be “of the same order.” When this is not the case we may have either

or

In the former case y is said to be “of a lower order” than x; in the latter case y is said to be “of a higher order” than x. In accordance with this notion we may say that the fundamental artifice of the infinitesimal calculus consists in the rejection of small quantities of an unnecessarily high order. This artifice is now merely an incident in the conduct of a limiting process, but in the 17th century, when limiting processes other than the Greek methods for quadratures were new, the introduction of the artifice was a great advance.

11. By the aid of this artifice, or directly by carrying out the appropriate limiting processes, we may obtain the Rules of Differentiation. rules by which differential coefficients are formed. These rules may be classified as “formal rules” and “particular results.” The formal rules may be stated as follows:—

(i.) The differential coefficient of a constant is zero.

(ii.) For a sum u + v + ... + z, where u, v, ... are functions of x,

(iii.) For a product uv

(iv.) For a quotient u/v

(v.) For a function of a function, that is to say, for a function y expressed in terms of a variable z, which is itself expressed as a function of x,

In addition to these formal rules we have particular results as to the differentiation of simple functions. The most important results are written down in the following table:—

Each of the formal rules, and each of the particular results in the table, is a theorem of the differential calculus. All functions (or rather expressions) which can be made up from those in the table by a finite number of operations of addition, subtraction, multiplication or division can be differentiated by the formal rules. All such functions are called explicit functions. In addition to these we have implicit functions, or such as are determined by an equation containing two variables when the equation cannot be solved so as to exhibit the one variable expressed in terms of the other. We have also functions of several variables. Further, since the derived function of a given function is itself a function, we may seek to differentiate it, and thus there arise the second and higher differential coefficients. We postpone for the present the problems of differential calculus which arise from these considerations. Again, we may have explicit functions which are expressed as the results of limiting operations, or by the limits of the results obtained by performing an infinite number of algebraic operations upon the simple functions. For the problem of differentiating such functions reference may be made to [Function].

12. The processes of the integral calculus consist largely in transformations Indefinite Integrals. of the functions to be integrated into such forms that they can be recognized as differential coefficients of functions which have previously been differentiated. Corresponding to the results in the table of § 11 we have those in the following table:—

The formal rules of § 11 give us means for the transformation of integrals into recognizable forms. For example, the rule (ii.) for a sum leads to the result that the integral of a sum of a finite number of terms is the sum of the integrals of the several terms. The rule (iii.) for a product leads to the method of integration by parts. The rule (v.) for a function of a function leads to the method of substitution (see § 48 below.)

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 Kepler’s methods of Integration. (1609) stated his laws of planetary motion, to 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.

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

∫φ0 (a + ae cos φ) dφ,

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 Logarithms. as the indices of a certain base, and the notation e 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, Cavalieri’s Indivisibles. as being made up of indivisible parts, “atoms” of 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

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 Successors of Cavalieri. in communication were Bonaventura 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 ∫10 xmdx by the process of forming the sum

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 xm by forming this sum was found also by Fermat (1636), who stated expressly that he Fermat’s method of Integration. arrived at it by generalizing a method employed by 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 = xm (m ≠ −1),

or in the form

y = a1 xm1 + a2 xm2 + ... + an xmn,

where none of the indices is equal to −1, was used by John Various Integrations. Wallis in his Arithmetica infinitorum (1655) as well as by Fermat (1659). The case in which m = −1 was that of the ordinary rectangular hyperbola; and Gregory of St Vincent in his Opus geometricum quadraturae circuli et sectionum coni (1647) had proved by the method of exhaustions that the area contained between the curve, one asymptote, and two ordinates parallel to the other asymptote, increases in arithmetic progression as the distance between the ordinates (the one nearer to the centre being kept fixed) increases in geometric progression. Fermat described his method of integration as a logarithmic method, and thus it is clear that the relation between the quadrature of the hyperbola and logarithms was understood although it was not expressed analytically. It was not very long before the relation was used for the calculation of logarithms by Nicolaus Mercator in his Logarithmotechnia (1668). He began by writing the equation of the curve in the form y = 1/(1 + x), expanded this expression in powers of x by the method of division, and integrated it term by term in accordance with the well-understood rule for finding the quadrature of a curve given by such an equation as that written at the foot of p. 325.

By the middle of the 17th century many mathematicians could perform integrations. Very many particular results had been obtained, and applications of them had been Integration before the Integral Calculus. made to the quadrature of the circle and other conic sections, and to various problems concerning the lengths of curves, the areas they enclose, the volumes and superficial areas of solids, and centres of gravity. A systematic account of the methods then in use was given, along with much that was original on his part, by Blaise Pascal in his Lettres de Amos Dettonville sur quelques-unes de ses inventions en géométrie (1659).

16. The problem of maxima and minima and the problem of tangents had also by the same time been effectively solved. Oresme in the 14th century knew that at a point where the ordinate of a curve is a maximum or a minimum Fermat’s methods of Differentiation. its variation from point to point of the curve is slowest; and Kepler in the Stereometria doliorum remarked that at the places where the ordinate passes from a smaller value to the greatest value and then again to a smaller value, its variation becomes insensible. Fermat in 1629 was in possession of a method which he then communicated to one Despagnet of Bordeaux, and which he referred to in a letter to Roberval of 1636. He communicated it to René Descartes early in 1638 on receiving a copy of Descartes’s Géométrie (1637), and with it he sent to Descartes an account of his methods for solving the problem of tangents and for determining centres of gravity.

Fermat’s method for maxima and minima is essentially our method. Expressed in a more modern notation, what he did was to begin by connecting the ordinate y and the abscissa x of a point of a curve by an equation which holds at all points of the curve, then to subtract the value of y in terms of x from the value obtained by substituting x + E for x, then to divide the difference by E, to put E = 0 in the quotient, and to equate the quotient to zero. Thus he differentiated with respect to x and equated the differential coefficient to zero.

Fermat’s method for solving the problem of tangents may be explained as follows:—Let (x, y) be the coordinates of a point P of a curve, (x′, y′), those of a neighbouring point P′ on the tangent at P, and let MM′ = E (fig. 6).

From the similarity of the triangles P′TM′, PTM we have

y′ : A − E = y : A,

where A denotes the subtangent TM. The point P′ being near the curve, we may substitute in the equation of the curve x − E for x and (yA − yE)/A for y. The equation of the curve is approximately satisfied. If it is taken to be satisfied exactly, the result is an equation of the form φ(x, y, A, E) = 0, the left-hand member of which is divisible by E. Omitting the factor E, and putting E = 0 in the remaining factor, we have an equation which gives A. In this problem of tangents also Fermat found the required result by a process equivalent to differentiation.

Fermat gave several examples of the application of his method; among them was one in which he showed that he could differentiate very complicated irrational functions. For such functions his method was to begin by obtaining a rational equation. In rationalizing equations Fermat, in other writings, used the device of introducing new variables, but he did not use this device to simplify the process of differentiation. Some of his results were published by Pierre Hérigone in his Supplementum cursus mathematici (1642). His communication to Descartes was not published in full until after his death (Fermat, Opera varia, 1679). Methods similar to Fermat’s were devised by René de Sluse (1652) for tangents, and by Johannes Hudde (1658) for maxima and minima. Other methods for the solution of the problem of tangents were devised by Roberval and Torricelli, and published almost simultaneously in 1644. These methods were founded upon the composition of motions, the theory of which had been taught by Galileo (1638), and, less completely, by Roberval (1636). Roberval and Torricelli could construct the tangents of many curves, but they did not arrive at Fermat’s artifice. This artifice is that which we have noted in § 10 as the fundamental artifice of the infinitesimal calculus.

17. Among the comparatively few mathematicians who before 1665 could perform differentiations was Isaac Barrow. In his book entitled Lectiones opticae et geometricae, written apparently in 1663, 1664, and published in Barrow’s Differential Triangle. 1669, 1670, he gave a method of tangents like that of Roberval and Torricelli, compounding two velocities in the directions of the axes of x and y to obtain a resultant along the tangent to a curve. In an appendix to this book he gave another method which differs from Fermat’s in the introduction of a differential equivalent to our dy as well as dx. Two neighbouring ordinates PM and QN of a curve (fig. 7) are regarded as containing an indefinitely small (indefinite parvum) arc, and PR is drawn parallel to the axis of x. The tangent PT at P is regarded as identical with the secant PQ, and the position of the tangent is determined by the similarity of the triangles PTM, PQR. The increments QR, PR of the ordinate and abscissa are denoted by a and e; and the ratio of a to e is determined by substituting x + e for x and y + a for y in the equation of the curve, rejecting all terms which are of order higher than the first in a and e, and omitting the terms which do not contain a or e. This process is equivalent to differentiation. Barrow appears to have invented it himself, but to have put it into his book at Newton’s request. The triangle PQR is sometimes called “Barrow’s differential triangle.”

The reciprocal relation between differentiation and integration (§ 6) was first observed explicitly by Barrow in the book cited above. If the quadrature of a curve y = ƒ(x) is known, so that the area up to the ordinate x is given by F(x), the curve Barrow’s Inversion-theorem. y = F(x) can be drawn, and Barrow showed that the subtangent of this curve is measured by the ratio of its ordinate to the ordinate of the original curve. The curve y = F(x) is often called the “quadratrix” of the original curve; and the result has been called “Barrow’s inversion-theorem.” He did not use it as we do for the determination of quadratures, or indefinite integrals, but for the solution of problems of the kind which were then called “inverse problems of tangents.” In these problems it was sought to determine a curve from some property of its tangent, e.g. the property that the subtangent is proportional to the square of the abscissa. Such problems are now classed under “differential equations.” When Barrow wrote, quadratures were familiar and differentiation unfamiliar, just as hyperbolas were trusted while logarithms were strange. The functional notation was not invented till long afterwards (see [Function]), and the want of it is felt in reading all the mathematics of the 17th century.

18. The great secret which afterwards came to be called the “infinitesimal calculus” was almost discovered by Fermat, and still more nearly by Barrow. Barrow went farther than Fermat in the theory of differentiation, though not in the Nature of the discovery called the Infinitesimal Calculus. practice, for he compared two increments; he went farther in the theory of integration, for he obtained the inversion-theorem. The great discovery seems to consist partly in the recognition of the fact that differentiation, known to be a useful process, could always be performed, at least for the functions then known, and partly in the recognition of the fact that the inversion-theorem could be applied to problems of quadrature. By these steps the problem of tangents could be solved once for all, and the operation of integration, as we call it, could be rendered systematic. A further step was necessary in order that the discovery, once made, should become accessible to mathematicians in general; and this step was the introduction of a suitable notation. The definite abandonment of the old tentative methods of integration in favour of the method in which this operation is regarded as the inverse of differentiation was especially the work of Isaac Newton; the precise formulation of simple rules for the process of differentiation in each special case, and the introduction of the notation which has proved to be the best, were especially the work of Gottfried Wilhelm Leibnitz. This statement remains true although Newton invented a systematic notation, and practised differentiation by rules equivalent to those of Leibnitz, before Leibnitz had begun to work upon the subject, and Leibnitz effected integrations by the method of recognizing differential coefficients before he had had any opportunity of becoming acquainted with Newton’s methods.

19. Newton was Barrow’s pupil, and he knew to start with in 1664 all that Barrow knew, and that was practically all that was known about the subject at that time. His original thinking on the subject dates from the year Newton’s investigations. of the great plague (1665-1666), and it issued in the invention of the “Calculus of Fluxions,” the principles and methods of which were developed by him in three tracts entitled De analysi per aequationes numero terminorum infinitas, Methodus fluxionum et serierum infinitarum, and De quadratura curvarum. None of these was published until long after they were written. The Analysis per aequationes was composed in 1666, but not printed until 1711, when it was published by William Jones. The Methodus fluxionum was composed in 1671 but not printed till 1736, nine years after Newton’s death, when an English translation was published by John Colson. In Horsley’s edition of Newton’s works it bears the title Geometria analytica. The Quadratura appears to have been composed in 1676, but was first printed in 1704 as an appendix to Newton’s Opticks.

20. The tract De Analysi per aequationes ... was sent by Newton to Barrow, who sent it to John Collins with a request that it might be made known. One way of making it known would have been to print it in the Philosophical Transactions Newton’s method of Series. of the Royal Society, but this course was not adopted. Collins made a copy of the tract and sent it to Lord Brouncker, but neither of them brought it before the Royal Society. The tract contains a general proof of Barrow’s inversion-theorem which is the same in principle as that in § 6 above. In this proof and elsewhere in the tract a notation is introduced for the momentary increment (momentum) of the abscissa or area of a curve; this “moment” is evidently meant to represent a moment of time, the abscissa representing time, and it is effectively the same as our differential element—the thing that Fermat had denoted by E, and Barrow by e, in the case of the abscissa. Newton denoted the moment of the abscissa by o, that of the area z by ov. He used the letter v for the ordinate y, thus suggesting that his curve is a velocity-time graph such as Galileo had used. Newton gave the formula for the area of a curve v = xm(m ± −1) in the form z = xm+1/(m + 1). In the proof he transformed this formula to the form zn = cn xp, where n and p are positive integers, substituted x + o for x and z + ov for z, and expanded by the binomial theorem for a positive integral exponent, thus obtaining the relation

zn + nzn−1 ov + ... = cn (xp + pxp−1 o + ...),

from which he deduced the relation

nzn−1 v = cn pxp−1

by omitting the equal terms zn and cnxp and dividing the remaining terms by o, tacitly putting o = 0 after division. This relation is the same as v = xm. Newton pointed out that, conversely, from the relation v = xm the relation z = xm+1 / (m + 1) follows. He applied his formula to the quadrature of curves whose ordinates can be expressed as the sum of a finite number of terms of the form axm; and gave examples of its application to curves in which the ordinate is expressed by an infinite series, using for this purpose the binomial theorem for negative and fractional exponents, that is to say, the expansion of (1 + x)n in an infinite series of powers of x. This theorem he had discovered; but he did not in this tract state it in a general form or give any proof of it. He pointed out, however, how it may be used for the solution of equations by means of infinite series. He observed also that all questions concerning lengths of curves, volumes enclosed by surfaces, and centres of gravity, can be formulated as problems of quadratures, and can thus be solved either in finite terms or by means of infinite series. In the Quadratura (1676) the method of integration which is founded upon the inversion-theorem was carried out systematically. Among other results there given is the quadrature of curves expressed by equations of the form y = xn (a + bxm)p; this has passed into text-books under the title “integration of binomial differentials” (see § 49). Newton announced the result in letters to Collins and Oldenburg of 1676.

21. In the Methodus fluxionum (1671) Newton introduced his characteristic notation. He regarded variable quantities as generated by the motion of a point, or line, or plane, and called the generated quantity a “fluent” and its rate of generation Newton’s method of Fluxions. a “fluxion.” The fluxion of a fluent x is represented by x, and its moment, or “infinitely” small increment accruing in an “infinitely” short time, is represented by ẋo. The problems of the calculus are stated to be (i.) to find the velocity at any time when the distance traversed is given; (ii.) to find the distance traversed when the velocity is given. The first of these leads to differentiation. In any rational equation containing x and y the expressions x + ẋo and y +ẏo are to be substituted for x and y, the resulting equation is to be divided by o, and afterwards o is to be omitted. In the case of irrational functions, or rational functions which are not integral, new variables are introduced in such a way as to make the equations contain rational integral terms only. Thus Newton’s rules of differentiation would be in our notation the rules (i.), (ii.), (v.) of § 11, together with the particular result which we write

a result which Newton obtained by expanding (x = ẋo)m by the binomial theorem. The second problem is the problem of integration, and Newton’s method for solving it was the method of series founded upon the particular result which we write

Newton added applications of his methods to maxima and minima, tangents and curvature. In a letter to Collins of date 1672 Newton stated that he had certain methods, and he described certain results which he had found by using them. These methods and results are those which are to be found in the Methodus fluxionum; but the letter makes no mention of fluxions and fluents or of the characteristic notation. The rule for tangents is said in the letter to be analogous to de Sluse’s, but to be applicable to equations that contain irrational terms.

22. Newton gave the fluxional notation also in the tract De Quadratura curvarum (1676), and he there added to it notation for the higher differential coefficients and for indefinite integrals, as we call them. Just as x, y, z, ... are fluents Publication of the Fluxional Notation. of which ẋ, ẏ, ̇z, ... are the fluxions, so ẋ, ẏ, ̇z, ... can be treated as fluents of which the fluxions may be denoted by ẍ, ̈y, ̈z,... In like manner the fluxions of these may be denoted by ẍ, ̈y, ̈z, ... and so on. Again x, y, z, ... may be regarded as fluxions of which the fluents may be denoted by ́x, ́y, ́z, ... and these again as fluxions of other quantities denoted by ̋x, ̋y, ̋z, ... and so on. No use was made of the notation ́x, ̋x, ... in the course of the tract. The first publication of the fluxional notation was made by Wallis in the second edition of his Algebra (1693) in the form of extracts from communications made to him by Newton in 1692. In this account of the method the symbols 0, ẋ, ẍ, ... occur, but not the symbols ́x, ̋x, .... Wallis’s treatise also contains Newton’s formulation of the problems of the calculus in the words Data aequatione fluentes quotcumque quantitates involvente fluxiones invenire et vice versa (“an equation containing any number of fluent quantities being given, to find their fluxions and vice versa”). In the Philosophiae naturalis principia mathematica (1687), commonly called the “Principia,” the words “fluxion” and “moment” occur in a lemma in the second book; but the notation which is characteristic of the calculus of fluxions is nowhere used.

23. It is difficult to account for the fragmentary manner of publication of the Fluxional Calculus and for the long delays which took place. At the time (1671) when Newton composed the Methodus fluxionum he contemplated Retarded Publication of the method of Fluxions. bringing out an edition of Gerhard Kinckhuysen’s treatise on algebra and prefixing his tract to this treatise. In the same year his “Theory of Light and Colours” was published in the Philosophical Transactions, and the opposition which it excited led to the abandonment of the project with regard to fluxions. In 1680 Collins sought the assistance of the Royal Society for the publication of the tract, and this was granted in 1682. Yet it remained unpublished. The reason is unknown; but it is known that about 1679, 1680, Newton took up again the studies in natural philosophy which he had intermitted for several years, and that in 1684 he wrote the tract De motu which was in some sense a first draft of the Principia, and it may be conjectured that the fluxions were held over until the Principia should be finished. There is also reason to think that Newton had become dissatisfied with the arguments about infinitesimals on which his calculus was based. In the preface to the De quadratura curvarum (1704), in which he describes this tract as something which he once wrote (“olim scripsi”) he says that there is no necessity to introduce into the method of fluxions any argument about infinitely small quantities; and in the Principia (1687) he adopted instead of the method of fluxions a new method, that of “Prime and Ultimate Ratios.” By the aid of this method it is possible, as Newton knew, and as was afterwards seen by others, to found the calculus of fluxions on an irreproachable method of limits. For the purpose of explaining his discoveries in dynamics and astronomy Newton used the method of limits only, without the notation of fluxions, and he presented all his results and demonstrations in a geometrical form. There is no doubt that he arrived at most of his theorems in the first instance by using the method of fluxions. Further evidence of Newton’s dissatisfaction with arguments about infinitely small quantities is furnished by his tract Methodus diferentialis, published in 1711 by William Jones, in which he laid the foundations of the “Calculus of Finite Differences.”

24. Leibnitz, unlike Newton, was practically a self-taught mathematician. He seems to have been first attracted to mathematics as a means of symbolical expression, and on the occasion of his first visit to London, early in Leibnitz’s course of discovery. 1673, he learnt about the doctrine of infinite series which James Gregory, Nicolaus Mercator, Lord Brouncker and others, besides Newton, had used in their investigations. It appears that he did not on this occasion become acquainted with Collins, or see Newton’s Analysis per aequationes, but he purchased Barrow’s Lectiones. On returning to Paris he made the acquaintance of Huygens, who recommended him to read Descartes’ Géométrie. He also read Pascal’s Lettres de Dettonville, Gregory of St Vincent’s Opus geometricum, Cavalieri’s Indivisibles and the Synopsis geometrica of Honoré Fabri, a book which is practically a commentary on Cavalieri; it would never have had any importance but for the influence which it had on Leibnitz’s thinking at this critical period. In August of this year (1673) he was at work upon the problem of tangents, and he appears to have made out the nature of the solution—the method involved in Barrow’s differential triangle—for himself by the aid of a diagram drawn by Pascal in a demonstration of the formula for the area of a spherical surface. He saw that the problem of the relation between the differences of neighbouring ordinates and the ordinates themselves was the important problem, and then that the solution of this problem was to be effected by quadratures. Unlike Newton, who arrived at differentiation and tangents through integration and areas, Leibnitz proceeded from tangents to quadratures. When he turned his attention to quadratures and indivisibles, and realized the nature of the process of finding areas by summing “infinitesimal” rectangles, he proposed to replace the rectangles by triangles having a common vertex, and obtained by this method the result which we write

1⁄4π = 1 − 1⁄3 + 1⁄5 − 1⁄7 + ...

In 1674 he sent an account of his method, called “transmutation,” along with this result to Huygens, and early in 1675 he sent it to Henry Oldenburg, secretary of the Royal Society, with inquiries as to Newton’s discoveries in regard to quadratures. In October of 1675 he had begun to devise a symbolical notation for quadratures, starting from Cavalieri’s indivisibles. At first he proposed to use the word omnia as an abbreviation for Cavalieri’s “sum of all the lines,” thus writing omnia y for that which we write “∫ ydx,” but within a day or two he wrote “∫ y”. He regarded the symbol “∫” as representing an operation which raises the dimensions of the subject of operation—a line becoming an area by the operation—and he devised his symbol “d” to represent the inverse operation, by which the dimensions are diminished. He observed that, whereas “∫” represents “sum,” “d” represents “difference.” His notation appears to have been practically settled before the end of 1675, for in November he wrote ∫ ydy = ½ y2, just as we do now.

25. In July of 1676 Leibnitz received an answer to his inquiry in regard to Newton’s methods in a letter written by Newton to Oldenburg. In this letter Newton gave a general statement of the binomial theorem and many results Correspondence of Newton and Leibnitz. relating to series. He stated that by means of such series he could find areas and lengths of curves, centres of gravity and volumes and surfaces of solids, but, as this would take too long to describe, he would illustrate it by examples. He gave no proofs. Leibnitz replied in August, stating some results which he had obtained, and which, as it seemed, could not be obtained easily by the method of series, and he asked for further information. Newton replied in a long letter to Oldenburg of the 24th of October 1676. In this letter he gave a much fuller account of his binomial theorem and indicated a method of proof. Further he gave a number of results relating to quadratures; they were afterwards printed in the tract De quadratura curvarum. He gave many other results relating to the computation of natural logarithms and other calculations in which series could be used. He gave a general statement, similar to that in the letter to Collins, as to the kind of problems relating to tangents, maxima and minima, &c., which he could solve by his method, but he concealed his formulation of the calculus in an anagram of transposed letters. The solution of the anagram was given eleven years later in the Principia in the words we have quoted from Wallis’s Algebra. In neither of the letters to Oldenburg does the characteristic notation of the fluxional calculus occur, and the words “fluxion” and “fluent” occur only in anagrams of transposed letters. The letter of October 1676 was not despatched until May 1677, and Leibnitz answered it in June of that year. In October 1676 Leibnitz was in London, where he made the acquaintance of Collins and read the Analysis per aequationes, and it seems to have been supposed afterwards that he then read Newton’s letter of October 1676, but he left London before Oldenburg received this letter. In his answer of June 1677 Leibnitz gave Newton a candid account of his differential calculus, nearly in the form in which he afterwards published it, and explained how he used it for quadratures and inverse problems of tangents. Newton never replied.

26. In the Acta eruditorum of 1684 Leibnitz published a short memoir entitled Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus. Leibnitz’s Differential Calculus. In this memoir the differential dx of a variable x, considered as the abscissa of a point of a curve, is said to be an arbitrary quantity, and the differential dy of a related variable y, considered as the ordinate of the point, is defined as a quantity which has to dx the ratio of the ordinate to the subtangent, and rules are given for operating with differentials. These are the rules for forming the differential of a constant, a sum (or difference), a product, a quotient, a power (or root). They are equivalent to our rules (i.)-(iv.) of § 11 and the particular result

d(xm) = mxm−1 dx.

The rule for a function of a function is not stated explicitly but is illustrated by examples in which new variables are introduced, in much the same way as in Newton’s Methodus fluxionum. In connexion with the problem of maxima and minima, it is noted that the differential of y is positive or negative according as y increases or decreases when x increases, and the discrimination of maxima from minima depends upon the sign of ddy, the differential of dy. In connexion with the problem of tangents the differentials are said to be proportional to the momentary increments of the abscissa and ordinate. A tangent is defined as a line joining two “infinitely” near points of a curve, and the “infinitely” small distances (e.g., the distance between the feet of the ordinates of such points) are said to be expressible by means of the differentials (e.g., dx). The method is illustrated by a few examples, and one example is given of its application to “inverse problems of tangents.” Barrow’s inversion-theorem and its application to quadratures are not mentioned. No proofs are given, but it is stated that they can be obtained easily by any one versed in such matters. The new methods in regard to differentiation which were contained in this memoir were the use of the second differential for the discrimination of maxima and minima, and the introduction of new variables for the purpose of differentiating complicated expressions. A greater novelty was the use of a letter (d), not as a symbol for a number or magnitude, but as a symbol of operation. None of these novelties account for the far-reaching effect which this memoir has had upon the development of mathematical analysis. This effect was a consequence of the simplicity and directness with which the rules of differentiation were stated. Whatever indistinctness might be felt to attach to the symbols, the processes for solving problems of tangents and of maxima and minima were reduced once for all to a definite routine.

27. This memoir was followed in 1686 by a second, entitled De Geometria recondita et analysi indivisibilium atque infinitorum, in which Leibnitz described the method of using his new differential calculus for the problem of quadratures. Development of the Calculus. This was the first publication of the notation ∫ ydx. The new method was called calculus summatorius. The brothers Jacob (James) and Johann (John) Bernoulli were able by 1690 to begin to make substantial contributions to the development of the new calculus, and Leibnitz adopted their word “integral” in 1695, they at the same time adopting his symbol “∫.” In 1696 the marquis de l’Hospital published the first treatise on the differential calculus with the title Analyse des infiniment petits pour l’intelligence des lignes courbes. The few references to fluxions in Newton’s Principia (1687) must have been quite unintelligible to the mathematicians of the time, and the publication of the fluxional notation and calculus by Wallis in 1693 was too late to be effective. Fluxions had been supplanted before they were introduced.

The differential calculus and the integral calculus were rapidly developed in the writings of Leibnitz and the Bernoullis. Leibnitz (1695) was the first to differentiate a logarithm and an exponential, and John Bernoulli was the first to recognize the property possessed by an exponential (ax) of becoming infinitely great in comparison with any power (xn) when x is increased indefinitely. Roger Cotes (1722) was the first to differentiate a trigonometrical function. A great development of infinitesimal methods took place through the founding in 1696-1697 of the “Calculus of Variations” by the brothers Bernoulli.

28. The famous dispute as to the priority of Newton and Leibnitz in the invention of the calculus began in 1699 through the publication by Nicolas Fatio de Duillier of a tract in which he stated that Newton was not only the Dispute concerning Priority. first, but by many years the first inventor, and insinuated that Leibnitz had stolen it. Leibnitz in his reply (Acta Eruditorum, 1700) cited Newton’s letters and the testimony which Newton had rendered to him in the Principia as proofs of his independent authorship of the method. Leibnitz was especially hurt at what he understood to be an endorsement of Duillier’s attack by the Royal Society, but it was explained to him that the apparent approval was an accident. The dispute was ended for a time. On the publication of Newton’s tract De quadratura curvarum, an anonymous review of it, written, as has since been proved, by Leibnitz, appeared in the Acta Eruditorum, 1705. The anonymous reviewer said: “Instead of the Leibnitzian differences Newton uses and always has used fluxions ... just as Honoré Fabri in his Synopsis Geometrica substituted steps of movements for the method of Cavalieri.” This passage, when it became known in England, was understood not merely as belittling Newton by comparing him with the obscure Fabri, but also as implying that he had stolen his calculus of fluxions from Leibnitz. Great indignation was aroused; and John Keill took occasion, in a memoir on central forces which was printed in the Philosophical Transactions for 1708, to affirm that Newton was without doubt the first inventor of the calculus, and that Leibnitz had merely changed the name and mode of notation. The memoir was published in 1710. Leibnitz wrote in 1711 to the secretary of the Royal Society (Hans Sloane) requiring Keill to retract his accusation. Leibnitz’s letter was read at a meeting of the Royal Society, of which Newton was then president, and Newton made to the society a statement of the course of his invention of the fluxional calculus with the dates of particular discoveries. Keill was requested by the society “to draw up an account of the matter under dispute and set it in a just light.” In his report Keill referred to Newton’s letters of 1676, and said that Newton had there given so many indications of his method that it could have been understood by a person of ordinary intelligence. Leibnitz wrote to Sloane asking the society to stop these unjust attacks of Keill, asserting that in the review in the Acta Eruditorum no one had been injured but each had received his due, submitting the matter to the equity of the Royal Society, and stating that he was persuaded that Newton himself would do him justice. A committee was appointed by the society to examine the documents and furnish a report. Their report, presented in April 1712, concluded as follows:

“The differential method is one and the same with the method of fluxions, excepting the name and mode of notation; Mr Leibnitz calling those quantities differences which Mr Newton calls moments or fluxions, and marking them with the letter d, a mark not used by Mr Newton. And therefore we take the proper question to be, not who invented this or that method, but who was the first inventor of the method; and we believe that those who have reputed Mr Leibnitz the first inventor, knew little or nothing of his correspondence with Mr Collins and Mr Oldenburg long before; nor of Mr Newton’s having that method above fifteen years before Mr. Leibnitz began to publish it in the Acta Eruditorum of Leipzig. For which reasons we reckon Mr Newton the first inventor, and are of opinion that Mr Keill, in asserting the same, has been no ways injurious to Mr Leibnitz.”

The report with the letters and other documents was printed (1712) under the title Commercium Epistolicum D. Johannis Collins et aliorum de analysi promota, jussu Societatis Regiae in lucem editum, not at first for publication. An account of the contents of the Commercium Epistolicum was printed in the Philosophical Transactions for 1715. A second edition of the Commercium Epistolicum was published in 1722. The dispute was continued for many years after the death of Leibnitz in 1716. To translate the words of Moritz Cantor, it “redounded to the discredit of all concerned.”

29. One lamentable consequence of the dispute was a severance of British methods from continental ones. In Great Britain it became a point of honour to use fluxions and other Newtonian methods, while on the continent the British and Continental Schools of Mathematics. notation of Leibnitz was universally adopted. This severance did not at first prevent a great advance in mathematics in Great Britain. So long as attention was directed to problems in which there is but one independent variable (the time, or the abscissa of a point of a curve), and all the other variables depend upon this one, the fluxional notation could be used as well as the differential and integral notation, though perhaps not quite so easily. Up to about the middle of the 18th century important discoveries continued to be made by the use of the method of fluxions. It was the introduction of partial differentiation by Leonhard Euler (1734) and Alexis Claude Clairaut (1739), and the developments which followed upon the systematic use of partial differential coefficients, which led to Great Britain being left behind; and it was not until after the reintroduction of continental methods into England by Sir John Herschel, George Peacock and Charles Babbage in 1815 that British mathematics began to flourish again. The exclusion of continental mathematics from Great Britain was not accompanied by any exclusion of British mathematics from the continent. The discoveries of Brook Taylor and Colin Maclaurin were absorbed into the rapidly growing continental analysis, and the more precise conceptions reached through a critical scrutiny of the true nature of Newton’s fluxions and moments stimulated a like scrutiny of the basis of the method of differentials.

30. This method had met with opposition from the first. Christiaan Huygens, whose opinion carried more weight than that of any other scientific man of the day, declared that the employment of differentials was unnecessary, Oppositions to the calculus. and that Leibnitz’s second differential was meaningless (1691). A Dutch physician named Bernhard Nieuwentijt attacked the method on account of the use of quantities which are at one stage of the process treated as somethings and at a later stage as nothings, and he was especially severe in commenting upon the second and higher differentials (1694, 1695). Other attacks were made by Michel Rolle (1701), but they were directed rather against matters of detail than against the general principles. The fact is that, although Leibnitz in his answers to Nieuwentijt (1695), and to Rolle (1702), indicated that the processes of the calculus could be justified by the methods of the ancient geometry, he never expressed himself very clearly on the subject of differentials, and he conveyed, probably without intending it, the impression that the calculus leads to correct results by compensation of errors. In England the method of fluxions had to face similar attacks. George Berkeley, bishop and philosopher, wrote in 1734 a tract entitled The Analyst; or a Discourse addressed to an Infidel Mathematician, in which he proposed to destroy the presumption that the The “Analyst” controversy. opinions of mathematicians in matters of faith are likely to be more trustworthy than those of divines, by contending that in the much vaunted fluxional calculus there are mysteries which are accepted unquestioningly by the mathematicians, but are incapable of logical demonstration. Berkeley’s criticism was levelled against all infinitesimals, that is to say, all quantities vaguely conceived as in some intermediate state between nullity and finiteness, as he took Newton’s moments to be conceived. The tract occasioned a controversy which had the important consequence of making it plain that all arguments about infinitesimals must be given up, and the calculus must be founded on the method of limits. During the controversy Benjamin Robins gave an exceedingly clear explanation of Newton’s theories of fluxions and of prime and ultimate ratios regarded as theories of limits. In this explanation he pointed out that Newton’s moment (Leibnitz’s “differential”) is to be regarded as so much of the actual difference between two neighbouring values of a variable as is needful for the formation of the fluxion (or differential coefficient) (see G. A. Gibson, “The Analyst Controversy,” Proc. Math. Soc., Edinburgh, xvii., 1899). Colin Maclaurin published in 1742 a Treatise of Fluxions, in which he reduced the whole theory to a theory of limits, and demonstrated it by the method of Archimedes. This notion was gradually transferred to the continental mathematicians. Leonhard Euler in his Institutiones Calculi differentialis (1755) was reduced to the position of one who asserts that all differentials are zero, but, as the product of zero and any finite quantity is zero, the ratio of two zeros can be a finite quantity which it is the business of the calculus to determine. Jean le Rond d’Alembert in the Encyclopédie méthodique (1755, 2nd ed. 1784) declared that differentials were unnecessary, and that Leibnitz’s calculus was a calculus of mutually compensating errors, while Newton’s method was entirely rigorous. D’Alembert’s opinion of Leibnitz’s calculus was expressed also by Lazare N. M. Carnot in his Réflexions sur la métaphysique du calcul infinitésimal (1799) and by Joseph Louis de la Grange (generally called Lagrange) in writings from 1760 onwards. Lagrange proposed in his Théorie des fonctions analytiques (1797) to found the whole of the calculus on the theory of series. It was not until 1823 that a treatise on the differential calculus founded upon the method of limits was published. The treatise was the Résumé des leçons ... sur Cauchy’s method of limits. le calcul infinitésimal of Augustin Louis Cauchy. Since that time it has been understood that the use of the phrase “infinitely small” in any mathematical argument is a figurative mode of expression pointing to a limiting process. In the opinion of many eminent mathematicians such modes of expression are confusing to students, but in treatises on the calculus the traditional modes of expression are still largely adopted.

31. Defective modes of expression did not hinder constructive work. It was the great merit of Leibnitz’s symbolism that a mathematician who used it knew what was to be done in order to formulate any problem analytically, Arithmetical basis of modern analysis. even though he might not be absolutely clear as to the proper interpretation of the symbols, or able to render a satisfactory account of them. While new and varied results were promptly obtained by using them, a long time elapsed before the theory of them was placed on a sound basis. Even after Cauchy had formulated his theory much remained to be done, both in the rapidly growing department of complex variables, and in the regions opened up by the theory of expansions in trigonometric series. In both directions it was seen that rigorous demonstration demanded greater precision in regard to fundamental notions, and the requirement of precision led to a gradual shifting of the basis of analysis from geometrical intuition to arithmetical law. A sketch of the outcome of this movement—the “arithmetization of analysis,” as it has been called—will be found in [Function]. Its general tendency has been to show that many theories and processes, at first accepted as of general validity, are liable to exceptions, and much of the work of the analysts of the latter half of the 19th century was directed to discovering the most general conditions in which particular processes, frequently but not universally applicable, can be used without scruple.

III. Outlines of the Infinitesimal Calculus.

32. The general notions of functionality, limits and continuity are explained in the article [Function]. Illustrations of the more immediate ways in which these notions present themselves in the development of the differential and integral calculus will be useful in what follows.

33. Let y be given as a function of x, or, more generally, let x and y be given as functions of a variable t. The first of these cases is included in the second by putting x = t. If certain conditions are satisfied the aggregate of the points determined Geometrical limits. by the functional relations form a curve. The first condition is that the aggregate of the values of t to which values of x and y correspond must be continuous, or, in other words, that these values must consist of all real numbers, or of all those real numbers which lie between assigned extreme numbers. When this condition is satisfied the points are “ordered,” and their order is determined by the order of the numbers t, supposed to be arranged in order of increasing or decreasing magnitude; also there are two senses of description of the curve, according as t is taken to increase or to diminish. The second condition is that the aggregate of the points which are determined by the functional relations must be “continuous.” This condition means that, if any point P determined by a value of t is taken, and any distance δ, however small, is chosen, it is possible to find two points Q, Q′ of the aggregate which are such that (i.) P is between Q and Q′, (ii.) if R, R′ are any points between Q and Q′ the distance RR′ is less than δ. The meaning of the word “between” in this statement is fixed by the ordering of the points. Sometimes additional conditions are imposed upon the functional relations before they are regarded as defining a curve. An aggregate of points which satisfies the two conditions stated above is sometimes called a “Jordan curve.” It by no means follows that every curve of this kind has a tangent. In order that the curve Tangents. may have a tangent at P it is necessary that, if any angle α, however small, is specified, a distance δ can be found such that when P is between Q and Q′, and PQ and PQ′ are less than δ, the angle RPR′ is less than α for all pairs of points R, R′ which are between P and Q, or between P and Q′ (fig. 8). When this condition is satisfied y is a function of x which has a differential coefficient. The only way of finding out whether this condition is satisfied or not is to attempt to form the differential coefficient. If the quotient of differences Δy/Δx has a limit when Δx tends to zero, y is a differentiable function of x, and the limit in question is the differential coefficient. The derived function, or differential coefficient, of a function ƒ(x) is always defined by the formula

Rules for the formation of differential coefficients in particular cases have been given in § 11 above. The definition of a differential coefficient, and the rules of differentiation are quite independent of any geometrical interpretation, such as that concerning tangents to a curve, and the tangent to a curve is properly defined by means of the differential coefficient of a function, not the differential coefficient by means of the tangent.

It may happen that the limit employed in defining the differential coefficient has one value when h approaches zero through positive values, and a different value when h approaches zero Progressive and Regressive Differential Coefficients. through negative values. The two limits are then called the “progressive” and “regressive” differential coefficients. In applications to dynamics, when x denotes a coordinate and t the time, dx/dt denotes a velocity. If the velocity is changed suddenly the progressive differential coefficient measures the velocity just after the change, and the regressive differential coefficient measures the velocity just before the change. Variable velocities are properly defined by means of differential coefficients.

All geometrical limits may be specified in terms similar to those employed in specifying the tangent to a curve; in difficult cases they must be so specified. Geometrical intuition may fail to answer the question of the existence or non-existence Areas. of the appropriate limits. In the last resort the definitions of many quantities of geometrical import must be analytical, not geometrical. As illustrations of this statement we may take the definitions of the areas and lengths of curves. We may not assume that every curve has an area or a length. To find out whether a curve has an area or not, we must ascertain whether the limit expressed by ƒydx exists. When the limit exists the curve has an area. The definition of the integral is quite independent of any geometrical interpretation. The length of a curve again is defined by means of a limiting process. Let P, Q be two points of a curve, and R1, R2, ... Rn−1 a set of intermediate points of the curve, supposed to be described in the sense in which Q comes after P. The points R are supposed to be reached successively in the order of the suffixes when the curve is described in this sense. We form a sum of lengths of chords

PR1 + R1R2 + ... + Rn−1Q.

If this sum has a limit when the number of the points R is increased indefinitely and the lengths of all the chords are diminished indefinitely, this limit is the length of the arc PQ. The limit Lengths of Curves. is the same whatever law may be adopted for inserting the intermediate points R and diminishing the lengths of the chords. It appears from this statement that the differential element of the arc of a curve is the length of the chord joining two neighbouring points. In accordance with the fundamental artifice for forming differentials (§§ 9, 10), the differential element of arc ds may be expressed by the formula

ds = √ { (dx)2 + (dy)2 },

of which the right-hand member is really the measure of the distance between two neighbouring points on the tangent. The square root must be taken to be positive. We may describe this differential element as being so much of the actual arc between two neighbouring points as need be retained for the purpose of forming the integral expression for an arc. This is a description, not a definition, because the length of the short arc itself is only definable by means of the integral expression. Similar considerations to those used in defining the areas of plane figures and the lengths of plane curves are applicable to the formation of expressions for differential elements of volume or of the areas of curved surfaces.

34. In regard to differential coefficients it is an important theorem Constants of Integration. that, if the derived function ƒ′(x) vanishes at all points of an interval, the function ƒ(x) is constant in the interval. It follows that, if two functions have the same derived function they can only differ by a constant. Conversely, indefinite integrals are indeterminate to the extent of an additive constant.

35. The differential coefficient dy/dx, or the derived function ƒ′(x), is itself a function of x, and its differential coefficient is denoted Higher Differential Coefficients. by ƒ″(x) or d2y/dx2. In the second of these notations d/dx is regarded as the symbol of an operation, that of differentiation with respect to x, and the index 2 means that the operation is repeated. In like manner we may express the results of n successive differentiations by ƒ(n)(x) or by dny/dxn. When the second differential coefficient exists, or the first is differentiable, we have the relation

(i.)

The limit expressed by the right-hand member of this equation may exist in cases in which ƒ′(x) does not exist or is not differentiable. The result that, when the limit here expressed can be shown to vanish at all points of an interval, then ƒ(x) must be a linear function of x in the interval, is important.

The relation (i.) is a particular case of the more general relation

ƒ(n)(x) = lim.h=0 h−n [ ƒ(x + nh) − nf {(x + (n − 1) h }

(ii.)

As in the case of relation (i.) the limit expressed by the right-hand member may exist although some or all of the derived functions ƒ′(x), ƒ″(x), ... ƒ(n−1)(x) do not exist.

Corresponding to the rule iii. of § 11 we have the rule for forming the nth differential coefficient of a product in the form

where the coefficients are those of the expansion of (1 + x)n in powers of x (n being a positive integer). The rule is due to Leibnitz, (1695).

Differentials of higher orders may be introduced in the same way as the differential of the first order. In general when y = ƒ(x), the nth differential dny is defined by the equation

dny = ƒ(n) (x) (dx)n,

in which dx is the (arbitrary) differential of x.

When d/dx is regarded as a single symbol of operation the symbol ƒ ... dx represents the inverse operation. If the former is denoted by D, the latter may be denoted by D−1. Dn means that Symbols of operation. the operation D is to be performed n times in succession; D−n that the operation of forming the indefinite integral is to be performed n times in succession. Leibnitz’s course of thought (§ 24) naturally led him to inquire after an interpretation of Dn. where n is not an integer. For an account of the researches to which this inquiry gave rise, reference may be made to the article by A. Voss in Ency. d. math. Wiss. Bd. ii. A, 2 (Leipzig, 1889). The matter is referred to as “fractional” or “generalized” differentiation.

36. After the formation of differential coefficients the most important theorem of the differential calculus is the theorem of intermediate value Theorem of Intermediate Value. (“theorem of mean value,” “theorem of finite increments,” “Rolle’s theorem,” are other names for it). This theorem may be explained as follows: Let A, B be two points of a curve y = ƒ(x) (fig. 9). Then there is a point P between A and B at which the tangent is parallel to the secant AB. This theorem is expressed analytically in the statement that if ƒ′(x) is continuous between a and b, there is a value x1 of x between a and b which has the property expressed by the equation

(i.)

The value x1 can be expressed in the form a + θ(b − a) where θ is a number between 0 and 1.

A slightly more general theorem was given by Cauchy (1823) to the effect that, if ƒ′(x) and F′(x) are continuous between x = a and x = b, then there is a number θ between 0 and 1 which has the property expressed by the equation

The theorem expressed by the relation (i.) was first noted by Rolle (1690) for the case where ƒ(x) is a rational integral function which vanishes when x = a and also when x = b. The general theorem was given by Lagrange (1797). Its fundamental importance was first recognized by Cauchy (1823). It may be observed here that the theorem of integral calculus expressed by the equation

F(b) − F(a) = ∫ba F′(x) dx

follows at once from the definition of an integral and the theorem of intermediate value.

The theorem of intermediate value may be generalized in the statement that, if ƒ(x) and all its differential coefficients up to the nth inclusive are continuous in the interval between x = a and x = b, then there is a number θ between 0 and 1 which has the property expressed by the equation

(ii.)

37. This theorem provides a means for computing the values of a function at points near to an assigned point when the value of the function and its differential coefficients at the assigned point are known. The function is expressed by a terminated Taylor’s Theorem. series, and, when the remainder tends to zero as n increases, it may be transformed into an infinite series. The theorem was first given by Brook Taylor in his Methodus Incrementorum (1717) as a corollary to a theorem concerning finite differences. Taylor gave the expression for ƒ(x + z) in terms of ƒ(x), ƒ′(x), ... as an infinite series proceeding by powers of z. His notation was that appropriate to the method of fluxions which he used. This rule for expressing a function as an infinite series is known as Taylor’s theorem. The relation (i.), in which the remainder after n terms is put in evidence, was first obtained by Lagrange (1797). Another form of the remainder was given by Cauchy (1823) viz.,

The conditions of validity of Taylor’s expansion in an infinite series have been investigated very completely by A. Pringsheim (Math. Ann. Bd. xliv., 1894). It is not sufficient that the function and all its differential coefficients should be finite at x = a; there must be a neighbourhood of a within which Cauchy’s form of the remainder tends to zero as n increases (cf. Function).

An example of the necessity of this condition is afforded by the function f(x) which is given by the equation

(i.)

The sum of the series

(ii.)

is the same as that of the series

e−1 − x2 e−32 + x4 e−34 − ...

It is easy to prove that this is less than e−1 when x lies between 0 and 1, and also that f(x) is greater than e−l when x = 1/√3. Hence the sum of the series (i.) is not equal to the sum of the series (ii.).

The particular case of Taylor’s theorem in which a = 0 is often called Maclaurin’s theorem, because it was first explicitly stated by Colin Maclaurin in his Treatise of Fluxions (1742). Maclaurin like Taylor worked exclusively with the fluxional calculus.

Examples of expansions in series had been known for some time. The series for log (1 + x) was obtained by Nicolaus Mercator (1668) by expanding (1 + x)−1 by the method of algebraic division, and integrating the series term by term. He Expansions in power series. regarded his result as a “quadrature of the hyperbola.” Newton (1669) obtained the expansion of sin−1x by expanding (l − x2)−1/2 by the binomial theorem and integrating the series term by term. James Gregory (1671) gave the series for tan−1x. Newton also obtained the series for sin x, cos x, and ex by reversion of series (1669). The symbol e for the base of the Napierian logarithms was introduced by Euler (1739). All these series can be obtained at once by Taylor’s theorem. James Gregory found also the first few terms of the series for tan x and sec x; the terms of these series may be found successively by Taylor’s theorem, but the numerical coefficient of the general term cannot be obtained in this way.

Taylor’s theorem for the expansion of a function in a power series was the basis of Lagrange’s theory of functions, and it is fundamental also in the theory of analytic functions of a complex variable as developed later by Karl Weierstrass. It has also numerous applications to problems of maxima and minima and to analytical geometry. These matters are treated in the appropriate articles.

The forms of the coefficients in the series for tan x and sec x can be expressed most simply in terms of a set of numbers introduced by James Bernoulli in his treatise on probability entitled Ars Conjectandi (1713). These numbers B1, B2, ... called Bernoulli’s numbers, are the coefficients so denoted in the formula

and they are connected with the sums of powers of the reciprocals of the natural numbers by equations of the type

The function

has been called Bernoulli’s function of the mth order by J. L. Raabe (Crelle’s J. f. Math. Bd. xlii., 1851). Bernoulli’s numbers and functions are of especial importance in the calculus of finite differences (see the article by D. Seliwanoff in Ency. d. math. Wiss. Bd. i., E., 1901).

When x is given in terms of y by means of a power series of the form

x = y (C0 + C1y + C2y2 + ...)   (C0 ≠ 0) = yƒ0(y), say,

there arises the problem of expressing y as a power series in x. This problem is that of reversion of series. It can be shown that provided the absolute value of x is not too great,

To this problem is reducible that of expanding y in powers of x when x and y are connected by an equation of the form

y = a + xƒ(y),

for which problem Lagrange (1770) obtained the formula

For the history of the problem and the generalizations of Lagrange’s result reference may be made to O. Stolz, Grundzüge d. Diff. u. Int. Rechnung, T. 2 (Leipzig, 1896).

38. An important application of the theorem of intermediate value and its generalization can be made to the problem of evaluating certain limits. If two functions φ(x) and ψ(x) both vanish at x = a, the fraction φ(x)/ψ(x) may have a finite Indeterminate forms. limit at a. This limit is described as the limit of an “indeterminate form.” Such indeterminate forms were considered first by de l’Hospital (1696) to whom the problem of evaluating the limit presented itself in the form of tracing the curve y = φ(x)/ψ(x) near the ordinate x = a, when the curves y = φ(x) and y = ψ(x) both cross the axis of x at the same point as this ordinate. In fig. 10 PA and QA represent short arcs of the curves φ, ψ, chosen so that P and Q have the same abscissa. The value of the ordinate of the corresponding point R of the compound curve is given by the ratio of the ordinates PM, QM. De l’Hospital treated PM and QM as “infinitesimal,” so that the equations PM : AM =φ’(a) and QM : AM = ψ′(a) could be assumed to hold, and he arrived at the result that the “true value” of φ(a)/ψ(a) is φ′(a)/ψ′(a). It can be proved rigorously that, if ψ′(x) does not vanish at x = a, while φ(a) = 0 and ψ(a) = 0, then

It can be proved further if that φm(x) and ψn(x) are the differential coefficients of lowest order of φ(x) and ψ(x) which do not vanish at x = a, and if m = n, then

If m > n the limit is zero; but if m < n the function represented by the quotient φ(x)/ψ(x) “becomes infinite” at x = a. If the value of the function at x = a is not assigned by the definition of the function, the function does not exist at x = a, and the meaning of the statement that it “becomes infinite” is that it has no finite limit. The statement does not mean that the function has a value which we call infinity. There is no such value (see [Function]).

Such indeterminate forms as that described above are said to be of the form 0/0. Other indeterminate forms are presented in the form 0 × ∞, or 1∞, or ∞/∞, or ∞ − ∞. The most notable of the forms 1∞ is lim.x=0(1 + x)1/x, which is e. The case in which φ(x) and ψ(x) both tend to become infinite at x = a is reducible to the case in which both the functions tend to become infinite when x is increased indefinitely. If φ′(x) and ψ′(x) have determinate finite limits when x is increased indefinitely, while φ(x) and ψ(x) are determinately (positively or negatively) infinite, we have the result expressed by the equation

For the meaning of the statement that φ(x) and ψ(x) are determinately infinite reference may be made to the article [Function]. The evaluation of forms of the type ∞/∞ leads to a scale of increasing “infinities,” each being infinite in comparison with the preceding. Such a scale is

log x, ... x, x2, ... xn, ... ex, ... xx;

each of the limits expressed by such forms as lim.x=∞ φ(x)/ψ(x), where φ(x) precedes ψ(x) in the scale, is zero. The construction of such scales, along with the problem of constructing a complete scale was discussed in numerous writings by Paul du Bois-Reymond (see in particular, Math. Ann. Bd. xi., 1877). For the general problem of indeterminate forms reference may be made to the article by A. Pringsheim in Ency. d. math. Wiss. Bd. ii., A. 1 (1899). Forms of the type 0/0 presented themselves to early writers on analytical geometry in connexion with the determination of the tangents at a double point of a curve; forms of the type ∞/∞ presented themselves in like manner in connexion with the determination of asymptotes of curves. The evaluation of limits has innumerable applications in all parts of analysis. Cauchy’s Analyse algébrique (1821) was an epoch-making treatise on limits.

If a function φ(x) becomes infinite at x = a, and another function ψ(x) also becomes infinite at x = a in such a way that φ(x)/ψ(x) has a finite limit C, we say that φ(x) and ψ(x) become “infinite of the same order.” We may write φ(x) = Cψ(x) + φ1(x), where lim.x=aφ1(x)/ψ(x) = 0, and thus φ1(x) is of a lower order than φ(x); it may be finite or infinite at x = a. If it is finite, we describe Cψ(x) as the “infinite part” of φ(x). The resolution of a function which becomes infinite into an infinite part and a finite part can often be effected by taking the infinite part to be infinite of the same order as one of the functions in the scale written above, or in some more comprehensive scale. This resolution is the inverse of the process of evaluating an indeterminate form of the type ∞ − ∞.

For example lim.x=0{(ex − 1)−1 − x−1} is finite and equal to = ½, and the function (ex − 1)−1 − x−1 can be expanded in a power series in x.

39. Functions of several variables. The nature of a function of two or more variables, and the meaning to be attached to continuity and limits in respect of such functions, have been explained under [Function]. The theorems of differential calculus which relate to such functions are in general the same whether the number of variables is two or any greater number, and it will generally be convenient to state the theorems for two variables.

40. Let u or ƒ (x, y) denote a function of two variables x and y. If we regard y as constant, u or ƒ becomes a function of one variable x, and we may seek to differentiate it with respect to x. If the function of x is differentiable, the differential Partial differentiation. coefficient which is formed in this way is called the “partial differential coefficient” of u or ƒ with respect to x, and is denoted by ∂u/∂x or ∂ƒ/∂x. The symbol “∂” was appropriated for partial differentiation by C. G. J. Jacobi (1841). It had before been written indifferently with “d” as a symbol of differentiation. Euler had written (dƒ/dx) for the partial differential coefficient of ƒ with respect to x. Sometimes it is desirable to put in evidence the variable which is treated as constant, and then the partial differential coefficient is written “(dƒ/dx)y” or “(∂ƒ/∂x)y”. This course is often adopted by writers on Thermodynamics. Sometimes the symbols d or ∂ are dropped, and the partial differential coefficient is denoted by ux or ƒx. As a definition of the partial differential coefficient we have the formula

In the same way we may form the partial differential coefficient with respect to y by treating x as a constant.

The introduction of partial differential coefficients enables us to solve at once for a surface a problem analogous to the problem of tangents for a curve; and it also enables us to take the first step in the solution of the problem of maxima and minima for a function of several variables. If the equation of a surface is expressed in the form z = ƒ(x, y), the direction cosines of the normal to the surface at any point are in the ratios ∂ƒ/∂x : ∂ƒ/∂y : = 1. If f is a maximum or a minimum at (x, y), then ∂ƒ/∂x and ∂ƒ/∂y vanish at that point.

In applications of the differential calculus to mathematical physics we are in general concerned with functions of three variables x, y, z, which represent the coordinates of a point; and then considerable importance attaches to partial differential coefficients which are formed by a particular rule. Let F(x, y, z) be the function, P a point (x, y, z), P′ a neighbouring point (x + Δx, y + Δy, z + Δz), and let Δs be the length of PP′. The value of F(x, y, z) at P may be denoted shortly by F(P). A limit of the same nature as a partial differential coefficient is expressed by the formula

in which Δs is diminished indefinitely by bringing P′ up to P, and P′ is supposed to approach P along a straight line, for example, the tangent to a curve or the normal to a surface. The limit in question is denoted by ∂F/∂h, in which it is understood that h indicates a direction, that of PP′. If l, m, n are the direction cosines of the limiting direction of the line PP′, supposed drawn from P to P′, then

The operation of forming ∂F/∂h is called “differentiation with respect to an axis” or “vector differentiation.”

41. The most important theorem in regard to partial differential coefficients is the theorem of the total differential. We may write down the equation

ƒ(a + h, b + k) − ƒ(a, b) = ƒ(a + h, b + k) − ƒ(a, b + k) + ƒ(a, b + k) − ƒ(a, b).

If Theorem of the Total Differential. ƒx is a continuous function of x when x lies between a and a + h and y = b + k, and if further ƒy is a continuous function of y when y lies between b and d + k, there exist values of θ and η which lie between 0 and 1 and have the properties expressed by the equations

ƒ(a + h, b + k) − ƒ(a, b + k) = hƒx(a + θh, b + k),
ƒ(a, b + k) − ƒ(a, b) = kƒy(a, b + ηk).

Further, ƒx(a + θh, b + k) and ƒy(a, b + ηk) tend to the limits ƒx(a, b) and ƒy(a, b) when h and k tend to zero, provided the differential coefficients ƒx, ƒy, are continuous at the point (a, b). Hence in this case the above equation can be written

ƒ(a + h, b + k) − ƒ(a, b) = hƒx(a, b) + kƒy(a, b) + R,

where

In accordance with the notation of differentials this equation gives

Just as in the case of functions of one variable, dx and dy are arbitrary finite differences, and dƒ is not the difference of two values of ƒ, but is so much of this difference as need be retained for the purpose of forming differential coefficients.

The theorem of the total differential is immediately applicable to the differentiation of implicit functions. When y is a function of x which is given by an equation of the form ƒ(x, y) = 0, and it is either impossible or inconvenient to solve this equation so as to express y as an explicit function of x, the differential coefficient dy/dx can be formed without solving the equation. We have at once

This rule was known, in all essentials, to Fermat and de Sluse before the invention of the algorithm, of the differential calculus.

An important theorem, first proved by Euler, is immediately deducible from the theorem of the total differential. If ƒ(x, y) is a homogeneous function of degree n then

The theorem is applicable to functions of any number of variables and is generally known as Euler’s theorem of homogeneous functions.

42. Jacobians.Many problems in which partial differential coefficients occur are simplified by the introduction of certain determinants called “Jacobians” or “functional determinants.” They were introduced into Analysis by C. G. J. Jacobi (J. f. Math., Crelle, Bd. 22, 1841, p. 319). The Jacobian of u1, u2, ... un with respect to x1, x2, ... xn is the determinant

in which the constituents of the rth row are the n partial differential coefficients of ur, with respect to the n variables x. This determinant is expressed shortly by

Jacobians possess many properties analogous to those of ordinary differential coefficients, for example, the following:—

If n functions (u1, u2, ... un) of n variables (x1, x2, ..., xn) are not independent, but are connected by a relation ƒ(u1, u2, ... un) = 0, then

and, conversely, when this condition is satisfied identically the functions u1, u2 ..., un are not independent.

43. Partial differential coefficients of the second and higher Interchange of order of differentiations. orders can be formed in the same way as those of the first order. For example, when there are two variables x, y, the first partial derivatives ∂ƒ/∂x and ∂ƒ/∂y are functions of x and y, which we may seek to differentiate partially with respect to x or y. The most important theorem in relation to partial differential coefficients of orders higher than the first is the theorem that the values of such coefficients do not depend upon the order in which the differentiations are performed. For example, we have the equation

(i.)

This theorem is not true without limitation. The conditions for its validity have been investigated very completely by H. A. Schwarz (see his Ges. math. Abhandlungen, Bd. 2, Berlin, 1890, p. 275). It is a sufficient, though not a necessary, condition that all the differential coefficients concerned should be continuous functions of x, y. In consequence of the relation (i.) the differential coefficients expressed in the two members of this relation are written

The differential coefficient

in which p + g + r = n, is formed by differentiating p times with respect to x, q times with respect to y, r times with respect to z, the differentiations being performed in any order. Abbreviated notations are sometimes used in such forms as

Differentials of higher orders are introduced by the defining equation

in which the expression (dx·∂/∂x + dy·∂/∂y)n is developed by the binomial theorem in the same way as if dx·∂/∂x and dy·∂/∂y were numbers, and (∂/∂x)r·(∂/∂y)n−r ƒ is replaced by ∂nƒ/∂xr ∂yn−r. When there are more than two variables the multinomial theorem must be used instead of the binomial theorem.

The problem of forming the second and higher differential coefficients of implicit functions can be solved at once by means of partial differential coefficients, for example, if ƒ(x, y) = 0 is the equation defining y as a function of x, we have

The differential expression Xdx + Ydy, in which both X and Y are functions of the two variables x and y, is a total differential if there exists a function ƒ of x and y which is such that

∂ƒ/∂x = X,   ∂ƒ/∂y = Y.

When this is the case we have the relation

∂Y/∂x = ∂X/∂y.

(ii.)

Conversely, when this equation is satisfied there exists a function ƒ which is such that

dƒ= Xdx + Ydy.

The expression Xdx + Ydy in which X and Y are connected by the relation (ii.) is often described as a “perfect differential.” The theory of the perfect differential can be extended to functions of n variables, and in this case there are ½n(n − 1) such relations as (ii.).

In the case of a function of two variables x, y an abbreviated notation is often adopted for differential coefficients. The function being denoted by z, we write

Partial differential coefficients of the second order are important in geometry as expressing the curvature of surfaces. When a surface is given by an equation of the form z = ƒ(x, y), the lines of curvature are determined by the equation

{ (l + q2)s − pqt} (dy)2 + { (1 + q2)r − (1 + p2)t } dxdy − { (1 + p2)s − pqr} (dx)2 = 0,

and the principal radii of curvature are the values of R which satisfy the equation

R2(rt − s2) − R { (1 + q2)r − 2pqs + (1 + p2)t } √(1 + p2 + q2) + (1 + p2 + q2)2 = 0.

44. Change of variables.The problem of change of variables was first considered by Brook Taylor in his Methodus incrementorum. In the case considered by Taylor y is expressed as a function of z, and z as a function of x, and it is desired to express the differential coefficients of y with respect to x without eliminating z. The result can be obtained at once by the rules for differentiating a product and a function of a function. We have

The introduction of partial differential coefficients enables us to deal with more general cases of change of variables than that considered above. If u, v are new variables, and x, y are connected with them by equations of the type

x = ƒ1(u, v),   y = ƒ2(u, v),

(i.)

while y is either an explicit or an implicit function of x, we have the problem of expressing the differential coefficients of various orders of y with respect to x in terms of the differential coefficients of v with respect to u. We have

by the rule of the total differential. In the same way, by means of differentials of higher orders, we may express d2y/dx2, and so on.

Equations such as (i.) may be interpreted as effecting a transformation by which a point (u, v) is made to correspond to a point (x, y). The whole theory of transformations, and of functions, or differential expressions, which remain invariant under groups of transformations, has been studied exhaustively by Sophus Lie (see, in particular, his Theorie der Transformationsgruppen, Leipzig, 1888-1893). (See also [Differential Equations] and [Groups]).

A more general problem of change of variables is presented when it is desired to express the partial differential coefficients of a function V with respect to x, y, ... in terms of those with respect to u, v, ..., where u, v, ... are connected with x, y, ... by any functional relations. When there are two variables x, y, and u, v are given functions of x, y, we have

and the differential coefficients of higher orders are to be formed by repeated applications of the rule for differentiating a product and the rules of the type

When x, y are given functions of u, v, ... we have, instead of the above, such equations as

and ∂V/∂x, ∂V/∂y can be found by solving these equations, provided the Jacobian ∂(x, y)/∂(u, v) is not zero. The generalization of this method for the case of more than two variables need not detain us.

In cases like that here considered it is sometimes more convenient not to regard the equations connecting x, y with u, v as effecting a point transformation, but to consider the loci u = const., v = const. as two “families” of curves. Then in any region of the plane of (x, y) in which the Jacobian ∂(x, y)/∂(u, v) does not vanish or become infinite, any point (x, y) is uniquely determined by the values of u and v which belong to the curves of the two families that pass through the point. Such variables as u, v are then described as “curvilinear coordinates” of the point. This method is applicable to any number of variables. When the loci u = const., ... intersect each other at right angles, the variables are “orthogonal” curvilinear coordinates. Three-dimensional systems of such coordinates have important applications in mathematical physics. Reference may be made to G. Lamé, Leçons sur les coordonnées curvilignes (Paris, 1859), and to G. Darboux, Leçons sur les coordonnées curvilignes et systèmes orthogonaux (Paris, 1898).

When such a coordinate as u is connected with x and y by a functional relation of the form ƒ(x, y, u) = 0 the curves u = const. are a family of curves, and this family may be such that no two curves of the family have a common point. When this is not the case the points in which a curve ƒ(x, y, u) = 0 is intersected by a curve ƒ(x, y, u + Δu) = 0 tend to limiting positions as Δu is diminished indefinitely. The locus of these limiting positions is the “envelope” of the family, and in general it touches all the curves of the family. It is easy to see that, if u, v are the parameters of two families of curves which have envelopes, the Jacobian ∂(x, y)/∂(u, v) vanishes at all points on these envelopes. It is easy to see also that at any point where the reciprocal Jacobian ∂(u, v)/∂(x, y) vanishes, a curve of the family u touches a curve of the family v.

If three variables x, y, z are connected by a functional relation ƒ(x, y, z) = 0, one of them, z say, may be regarded as an implicit function of the other two, and the partial differential coefficients of z with respect to x and y can be formed by the rule of the total differential. We have

and there is no difficulty in proceeding to express the higher differential coefficients. There arises the problem of expressing the partial differential coefficients of x with respect to y and z in terms of those of z with respect to x and y. The problem is known as that of “changing the dependent variable.” It is solved by applying the rule of the total differential. Similar considerations are applicable to all cases in which n variables are connected by fewer than n equations.

45. Extension of Taylor’s theorem.Taylor’s theorem can be extended to functions of several variables. In the case of two variables the general formula, with a remainder after n terms, can be written most simply in the form

in which

and

The last expression is the remainder after n terms, and in it θ denotes some particular number between 0 and 1. The results for three or more variables can be written in the same form. The extension of Taylor’s theorem was given by Lagrange (1797); the form written above is due to Cauchy (1823). For the validity of the theorem in this form it is necessary that all the differential coefficients up to the nth should be continuous in a region bounded by x = a ± h, y = b ± k. When all the differential coefficients, no matter how high the order, are continuous in such a region, the theorem leads to an expansion of the function in a multiple power series. Such expansions are just as important in analysis, geometry and mechanics as expansions of functions of one variable. Among the problems which are solved by means of such expansions are the problem of maxima and minima for functions of more than one variable (see [Maxima] and [Minima]).

46. Plane curves.In treatises on the differential calculus much space is usually devoted to the differential geometry of curves and surfaces. A few remarks and results relating to the differential geometry of plane curves are set down here.

(i.) If ψ denotes the angle which the radius vector drawn from the origin makes with the tangent to a curve at a point whose polar coordinates are r, θ and if p denotes the perpendicular from the origin to the tangent, then

cos ψ = dr/ds,   sin ψ = rdθ/ds = p/r,

where ds denotes the element of arc. The curve may be determined by an equation connecting p with r.

(ii.) The locus of the foot of the perpendicular let fall from the origin upon the tangent to a curve at a point is called the pedal of the curve with respect to the origin. The angle ψ for the pedal is the same as the angle ψ for the curve. Hence the (p, r) equation of the pedal can be deduced. If the pedal is regarded as the primary curve, the curve of which it is the pedal is the “negative pedal” of the primary. We may have pedals of pedals and so on, also negative pedals of negative pedals and so on. Negative pedals are usually determined as envelopes.

(iii.) If φ denotes the angle which the tangent at any point makes with a fixed line, we have

r2 = p2 + (dp/dφ)2.

(iv.) The “average curvature” of the arc Δs of a curve between two points is measured by the quotient

where the upright lines denote, as usual, that the absolute value of the included expression is to be taken, and φ is the angle which the tangent makes with a fixed line, so that Δφ is the angle between the tangents (or normals) at the points. As one of the points moves up to coincidence with the other this average curvature tends to a limit which is the “curvature” of the curve at the point. It is denoted by

Sometimes the upright lines are omitted and a rule of signs is given:—Let the arc s of the curve be measured from some point along the curve in a chosen sense, and let the normal be drawn towards that side to which the curve is concave; if the normal is directed towards the left of an observer looking along the tangent in the chosen sense of description the curvature is reckoned positive, in the contrary case negative. The differential dφ is often called the “angle of contingence.” In the 14th century the size of the angle between a curve and its tangent seems to have been seriously debated, and the name “angle of contingence” was then given to the supposed angle.

(v.) The curvature of a curve at a point is the same as that of a certain circle which touches the curve at the point, and the “radius of curvature” ρ is the radius of this circle. We have 1/ρ = |dφ/ds|. The centre of the circle is called the “centre of curvature”; it is the limiting position of the point of intersection of the normal at the point and the normal at a neighbouring point, when the second point moves up to coincidence with the first. If a circle is described to intersect the curve at the point P and at two other points, and one of these two points is moved up to coincidence with P, the circle touches the curve at the point P and meets it in another point; the centre of the circle is then on the normal. As the third point now moves up to coincidence with P, the centre of the circle moves to the centre of curvature. The circle is then said to “osculate” the curve, or to have “contact of the second order” with it at P.

(vi.) The following are formulae for the radius of curvature:—

(vii.) The points at which the curvature vanishes are “points of inflection.” If P is a point of inflection and Q a neighbouring point, then, as Q moves up to coincidence with P, the distance from P to the point of intersection of the normals at P and Q becomes greater than any distance that can be assigned. The equation which gives the abscissae of the points in which a straight line meets the curve being expressed in the form ƒ(x) = 0, the function ƒ(x) has a factor (x − x0)3, where x0 is the abscissa of the point of inflection P, and the line is the tangent at P. When the factor (x − x0) occurs (n + 1) times in ƒ(x), the curve is said to have “contact of the nth order” with the line. There is an obvious modification when the line is parallel to the axis of y.

(viii.) The locus of the centres of curvature, or envelope of the normals, of a curve is called the “evolute.” A curve which has a given curve as evolute is called an “involute” of the given curve. All the involutes are “parallel” curves, that is to say, they are such that one is derived from another by marking off a constant distance along the normal. The involutes are “orthogonal trajectories” of the tangents to the common evolute.

(ix.) The equation of an algebraic curve of the nth degree can be expressed in the form u0 + u1 + u2 + ... + un = 0, where u0 is a constant, and ur is a homogeneous rational integral function of x, y of the rth degree. When the origin is on the curve, u0 vanishes, and u1 = 0 represents the tangent at the origin. If u1 also vanishes, the origin is a double point and u2 = o represents the tangents at the origin. If u2 has distinct factors, or is of the form a(y − p1x) (y − p2x), the value of y on either branch of the curve can be expressed (for points sufficiently near the origin) in a power series, which is either

p1x + ½ q1 x2 + ...,   or p2x + ½ q2 x2 + ...,

where q1, ... and q2, ... are determined without ambiguity. If p1 and p2 are real the two branches have radii of curvature ρ1, ρ2 determined by the formulae

When p1 and p2 are imaginary the origin is the real point of intersection of two imaginary branches. In the real figure of the curve it is an isolated point. If u2 is a square, a(y − px)2, the origin is a cusp, and in general there is not a series for y in integral powers of x, which is valid in the neighbourhood of the origin. The further investigation of cusps and multiple points belongs rather to analytical geometry and the theory of algebraic functions than to differential calculus.

(x.) When the equation of a curve is given in the form u0 + u1 + ... + un−1 + un = 0 where the notation is the same as that in (ix.), the factors of un determine the directions of the asymptotes. If these factors are all real and distinct, there is an asymptote corresponding to each factor. If un = L1L2 ... Ln, where L1, ... are linear in x, y, we may resolve un−1/un into partial fractions according to the formula

and then L1 + A1 = 0, L2 + A2 = 0, ... are the equations of the asymptotes. When a real factor of un is repeated we may have two parallel asymptotes or we may have a “parabolic asymptote.” Sometimes the parallel asymptotes coincide, as in the curve x2(x2 + y2 − a2) = a4, where x = 0 is the only real asymptote. The whole theory of asymptotes belongs properly to analytical geometry and the theory of algebraic functions.

47. The formal definition of an integral, the theorem of the existence of the integral for certain classes of functions, a list of Integral calculus. classes of “integrable” functions, extensions of the notion of integration to functions which become infinite or indeterminate, and to cases in which the limits of integration become infinite, the definitions of multiple integrals, and the possibility of defining functions by means of definite integrals—all these matters have been considered in [Function]. The definition of integration has been explained in § 5 above, and the results of some of the simplest integrations have been given in § 12. A few theorems relating to integrations have been noted in §§ 34, 35, 36 above.

48. Methods of integration.The chief methods for the evaluation of indefinite integrals are the method of integration by parts, and the introduction of new variables.

From the equation d(uv) = u dv + v du we deduce the equation

or, as it may be written

This is the rule of “integration by parts.”

As an example we have

When we introduce a new variable z in place of x, by means of an equation giving x in terms of z, we express ƒ(x) in terms of z. Let φ(z) denote the function of z into which ƒ(x) is transformed. Then from the equation

we deduce the equation

As an example, in the integral

∫ √(1 − x2)dx

put x = sin z; the integral becomes

∫ cos z · cos z dz = ∫ ½ (1 + cos 2z)dz = ½ (z + ½ sin 2z) = ½ (z + sin z cos z).

49. The indefinite integrals of certain classes of functions can be expressed by means of a finite number of operations of addition or multiplication in terms of the so-called “elementary” functions. The elementary functions are rational algebraic Integration in terms of elementary functions. functions, implicit algebraic functions, exponentials and logarithms, trigonometrical and inverse circular functions. The following are among the classes of functions whose integrals involve the elementary functions only: (i.) all rational functions; (ii.) all irrational functions of the form ƒ(x, y), where ƒ denotes a rational algebraic function of x and y, and y is connected with x by an algebraic equation of the second degree; (iii.) all rational functions of sin x and cos x; (iv.) all rational functions of ex; (v.) all rational integral functions of the variables x, eax, ebx, ... sin mx, cos mx, sin nx, cos nx, ... in which a, b, ... and m, n, ... are any constants. The integration of a rational function is generally effected by resolving the function into partial fractions, the function being first expressed as the quotient of two rational integral functions. Corresponding to any simple root of the denominator there is a logarithmic term in the integral. If any of the roots of the denominator are repeated there are rational algebraic terms in the integral. The operation of resolving a fraction into partial fractions requires a knowledge of the roots of the denominator, but the algebraic part of the integral can always be found without obtaining all the roots of the denominator. Reference may be made to C. Hermite, Cours d’analyse, Paris, 1873. The integration of other functions, which can be integrated in terms of the elementary functions, can usually be effected by transforming the functions into rational functions, possibly after preliminary integrations by parts. In the case of rational functions of x and a radical of the form √(ax2 + bx + c) the radical can be reduced by a linear substitution to one of the forms √(a2 − x2), √(x2 − a2), √(x2 + a2). The substitutions x = a sin θ, x = a sec θ, x = a tan θ are then effective in the three cases. By these substitutions the subject of integration becomes a rational function of sin θ and cos θ, and it can be reduced to a rational function of t by the substitution tan ½θ = t. There are many other substitutions by which such integrals can be determined. Sometimes we may have information as to the functional character of the integral without being able to determine it. For example, when the subject of integration is of the form (ax4 + bx3 + cx2 + dx + e)−1/2 the integral cannot be expressed explicitly in terms of elementary functions. Such integrals lead to new functions (see [Function]).

Methods of reduction and substitution for the evaluation of indefinite integrals occupy a considerable space in text-books of the integral calculus. In regard to the functional character of the integral reference may be made to G. H. Hardy’s tract, The Integration of Functions of a Single Variable (Cambridge, 1905), and to the memoirs there quoted. A few results are added here

(i.)

∫ (x2 + a) − ½ dx = log {x + (x2 + a)1/2 }.

(ii.)

can be evaluated by the substitution x − p = 1/z, and

can be deduced by differentiating (n − 1) times with respect to p.

(iii.)

can be reduced by the substitution y2 = (ax2 + 2bx + c)/(αx2 + 2βx + γ) to the form

where A and B are constants, and λ1 and λ2 are the two values of λ for which (a − λα)x2 + 2(b − λβ)x + c − λγ is a perfect square (see A. G. Greenhill, A Chapter in the Integral Calculus, London, 1888).

(iv.) ƒxm (axn + b)p dx, in which m, n, p are rational, can be reduced, by putting axn = bt, to depend upon ƒtq (1 + t)p dt. If p is an integer and q a fraction r/s, we put t = us. If q is an integer and p = r/s we put 1 + t = us. If p + q is an integer and p = r/s we put 1 + t = tus. These integrals, called “binomial integrals,” were investigated by Newton (De quadratura curvarum).

(v.)

(vi.)

(vii.) ∫ eax sin (bx + α) dx = (a2 + b2)−1 eax {a sin (bx + α) − b cos (bx + α) }.

(viii.) ∫ sinm x cosn x dx can be reduced by differentiating a function of the form sinp x cosq x;

Hence

(ix.)

(x.)

(xi.)

of which the first or the second is to be employed according as e < or > 1.

50. New transcendents.Among the integrals of transcendental functions which lead to new transcendental functions we may notice

called the “logarithmic integral,” and denoted by “Li x,” also the integrals

called the “sine integral” and the “cosine integral,” and denoted by “Si x” and “Ci x,” also the integral

∫x0 e−x2 dx

called the “error-function integral,” and denoted by “Erf x.” All these functions have been tabulated (see [Tables, Mathematical]).

51. Eulerian integrals.New functions can be introduced also by means of the definite integrals of functions of two or more variables with respect to one of the variables, the limits of integration being fixed. Prominent among such functions are the Beta and Gamma functions expressed by the equations

B(l, m) = ∫10 xl−1 (1 − x)m−1 dx,

Γ(n) = ∫∞0 e−t tn−1 dt.

When n is a positive integer Γ(n + 1) = n!. The Beta function (or “Eulerian integral of the first kind”) is expressible in terms of Gamma functions (or “Eulerian integrals of the second kind”) by the formula

B(l, m) · Γ(l + m) = Γ(l) · Γ(m).

The Gamma function satisfies the difference equation

Γ(x + 1) = x Γ(x),

and also the equation

Γ(x) · Γ(1 − x) = π/sin (xπ),

with the particular result

Γ(½)= √π.

The number

is called “Euler’s constant,” and is equal to the limit

its value to 15 decimal places is 0.577 215 664 901 532.

The function log Γ(1 + x) can be expanded in the series

− 1⁄3 (S3 − 1) x3 − 1⁄5 (S5 − 1) x5 − ...,

where

and the series for log Γ(1 + x) converges when x lies between −1 and 1.

52. Definite integrals.Definite integrals can sometimes be evaluated when the limits of integration are some particular numbers, although the corresponding indefinite integrals cannot be found. For example, we have the result

∫10 (1 − x2)−1/2 log x dx = −½ π log 2,

although the indefinite integral of (1 − x2)−1/2 log x cannot be found. Numbers of definite integrals are expressible in terms of the transcendental functions mentioned in § 50 or in terms of Gamma functions. For the calculation of definite integrals we have the following methods:—

The first three methods involve an interchange of the order of two limiting operations, and they are valid only when the functions satisfy certain conditions of continuity, or, in case the limits of integration are infinite, when the functions tend to zero at infinite distances in a sufficiently high order (see [Function]). The method of contour integration involves the introduction of complex variables (see [Function]: § Complex Variables).

A few results are added

(i.)

(ii.)

(iii.)

(iv.)

∫∞0 x2 · cos 2x · e−x2 dx = −1⁄4 e−1 √π,

(v.)

(vi.)

(vii.)

∫π0 log (1 − 2α cos x + α2) dx = 0 or 2π log α according as α < or > 1,

(viii.)

(ix.)

(x.)

(xi.)

(xii.)

(xiii.)

∫∞−∞ e−x2+2ax dx = √π · ea2,

(xiv.)

∫∞0 x−1/2 sin x dx = ∫∞0 x−1/2 cos x dx = √(½ π),

53. Multiple Integrals.The meaning of integration of a function of n variables through a domain of the same number of dimensions is explained in the article [Function]. In the case of two variables x, y we integrate a function ƒ(x, y) over an area; in the case of three variables x, y, z we integrate a function ƒ(x, y, z) through a volume. The integral of a function ƒ(x, y) over an area in the plane of (x, y) is denoted by

∫∫ ƒ(x, y) dx dy.

The notation refers to a method of evaluating the integral. We may suppose the area divided into a very large number of very small rectangles by lines parallel to the axes. Then we multiply the value of ƒ at any point within a rectangle by the measure of the area of the rectangle, sum for all the rectangles, and pass to a limit by increasing the number of rectangles indefinitely and diminishing all their sides indefinitely. The process is usually effected by summing first for all the rectangles which lie in a strip between two lines parallel to one axis, say the axis of y, and afterwards for all the strips. This process is equivalent to integrating ƒ(x, y) with respect to y, keeping x constant, and taking certain functions of x as the limits of integration for y, and then integrating the result with respect to x between constant limits. The integral obtained in this way may be written in such a form as

and is called a “repeated integral.” The identification of a surface integral, such as ∫∫ ƒ(x, y)dxdy, with a repeated integral cannot always be made, but implies that the function satisfies certain conditions of continuity. In the same way volume integrals are usually evaluated by regarding them as repeated integrals, and a volume integral is written in the form

∫∫∫ ƒ(x, y, z) dx dy dz.

Integrals such as surface and volume integrals are usually called “multiple integrals.” Thus we have “double” integrals, “triple” integrals, and so on. In contradistinction to multiple integrals the ordinary integral of a function of one variable with respect to that variable is called a “simple integral.”

A more general type of surface integral may be defined by taking an arbitrary surface, with or without an edge. We suppose in the first place that the surface is closed, or has no edge. We may mark a large number of points on the surface, and Surface Integrals. draw the tangent planes at all these points. These tangent planes form a polyhedron having a large number of faces, one to each marked point; and we may choose the marked points so that all the linear dimensions of any face are less than some arbitrarily chosen length. We may devise a rule for increasing the number of marked points indefinitely and decreasing the lengths of all the edges of the polyhedra indefinitely. If the sum of the areas of the faces tends to a limit, this limit is the area of the surface. If we multiply the value of a function ƒ at a point of the surface by the measure of the area of the corresponding face of the polyhedron, sum for all the faces, and pass to a limit as before, the result is a surface integral, and is written

∫∫∫ ƒ dS.

The Line Integrals.extension to the case of an open surface bounded by an edge presents no difficulty. A line integral taken along a curve is defined in a similar way, and is written

∫ ƒ ds

where ds is the element of arc of the curve (§ 33). The direction cosines of the tangent of a curve are dx/ds, dy/ds, dz/ds, and line integrals usually present themselves in the form

In like manner surface integrals usually present themselves in the form

∫∫ (lξ + mη + nζ) dS

where l, m, n are the direction cosines of the normal to the surface drawn in a specified sense.

The area of a bounded portion of the plane of (x, y) may be expressed either as

½ ∫ (x dy − y dx),

or as

∫∫ dx dy,

the former integral being a line integral taken round the boundary of the portion, and the latter a surface integral taken over the area within this boundary. In forming the line integral the boundary is supposed to be described in the positive sense, so that the included area is on the left hand.

53a. Theorems of Green and Stokes.We have two theorems of transformation connecting volume integrals with surface integrals and surface integrals with line integrals. The first theorem, called “Green’s theorem,” is expressed by the equation

where the volume integral on the left is taken through the volume within a closed surface S, and the surface integral on the right is taken over S, and l, m, n denote the direction cosines of the normal to S drawn outwards. There is a corresponding theorem for a closed curve in two dimensions, viz.,

the sense of description of s being the positive sense. This theorem is a particular case of a more general theorem called “Stokes’s theorem.” Let s denote the edge of an open surface S, and let S be covered with a network of curves so that the meshes of the network are nearly plane, then we can choose a sense of description of the edge of any mesh, and a corresponding sense for the normal to S at any point within the mesh, so that these senses are related like the directions of rotation and translation in a right-handed screw. This convention fixes the sense of the normal (l, m, n) at any point on S when the sense of description of s is chosen. If the axes of x, y, z are a right-handed system, we have Stokes’s theorem in the form

where the integral on the left is taken round the curve s in the chosen sense. When the axes are left-handed, we may either reverse the sense of l, m, n and maintain the formula, or retain the sense of l, m, n and change the sign of the right-hand member of the equation. For the validity of the theorems of Green and Stokes it is in general necessary that the functions involved should satisfy certain conditions of continuity. For example, in Green’s theorem the differential coefficients ∂ξ/∂x, ∂η/∂y, ∂ζ/∂z must be continuous within S. Further, there are restrictions upon the nature of the curves or surfaces involved. For example, Green’s theorem, as here stated, applies only to simply-connected regions of space. The correction for multiply-connected regions is important in several physical theories.

54. The process of changing the variables in a multiple integral, such as a surface or volume integral, is divisible into two stages. It Change of Variables in a Multiple Integral. is necessary in the first place to determine the differential element expressed by the product of the differentials of the first set of variables in terms of the differentials of the second set of variables. It is necessary in the second place to determine the limits of integration which must be employed when the integral in terms of the new variables is evaluated as a repeated integral. The first part of the problem is solved at once by the introduction of the Jacobian. If the variables of one set are denoted by x1, x2, ..., xn, and those of the other set by u1, u2, ..., un, we have the relation

In regard to the second stage of the process the limits of integration must be determined by the rule that the integration with respect to the second set of variables is to be taken through the same domain as the integration with respect to the first set.

For example, when we have to integrate a function ƒ(x, y) over the area within a circle given by x2 + y2 = a2, and we introduce polar coordinates so that x = r cos θ, y = r sin θ, we find that r is the value of the Jacobian, and that all points within or on the circle are given by a ≥ r ≥ 0, 2π ≥ θ ≥ 0, and we have

If we have to integrate over the area of a rectangle a ≥ x ≥ 0, b ≥ y ≥ 0, and we transform to polar coordinates, the integral becomes the sum of two integrals, as follows:—

∫a0 dx ∫b0 ƒ(x, y) dy = ∫0tan−1b/a dθ ∫0a sec θ ƒ(r cos θ, r sin θ) r dr

+ ∫1/2πtan−1b/a dθ ∫0b cosec θ ƒ(r cos θ, r sin θ) r dr.

55. A few additional results in relation to line integrals and multiple integrals are set down here.

(i.) Any simple integral can be regarded as a line-integral taken along a portion of the axis of x. When a change of Line Integrals and Multiple Integrals. variables is made, the limits of integration with respect to the new variable must be such that the domain of integration is the same as before. This condition may require the replacing of the original integral by the sum of two or more simple integrals.

(ii.) The line integral of a perfect differential of a one-valued function, taken along any closed curve, is zero.

(iii.) The area within any plane closed curve can be expressed by either of the formulae

∫ ½ r2 dθ or ∫ ½ p ds,

where r, θ are polar coordinates, and p is the perpendicular drawn from a fixed point to the tangent. The integrals are to be understood as line integrals taken along the curve. When the same integrals are taken between limits which correspond to two points of the curve, in the sense of line integrals along the arc between the points, they represent the area bounded by the arc and the terminal radii vectores.

(iv.) The volume enclosed by a surface which is generated by the revolution of a curve about the axis of x is expressed by the formula

π ∫ y2 dx,

and the area of the surface is expressed by the formula

2π ∫ y ds,

where ds is the differential element of arc of the curve. When the former integral is taken between assigned limits it represents the volume contained between the surface and two planes which cut the axis of x at right angles. The latter integral is to be understood as a line integral taken along the curve, and it represents the area of the portion of the curved surface which is contained between two planes at right angles to the axis of x.

(v.) When we use curvilinear coordinates ξ, η which are conjugate functions of x, y, that is to say are such that

∂ξ/∂x = ∂η/∂y and ∂ξ/∂y = −∂η/∂x,

the Jacobian ∂(ξ, η)/∂(x, v) can be expressed in the form

and in a number of equivalent forms. The area of any portion of the plane is represented by the double integral

∫∫ J−1 dξ dη,

where J denotes the above Jacobian, and the integration is taken through a suitable domain. When the boundary consists of portions of curves for which ξ = const., or η = const., the above is generally the simplest way of evaluating it.

(vi.) The problem of “rectifying” a plane curve, or finding its length, is solved by evaluating the integral

or, in polar coordinates, by evaluating the integral

In both cases the integrals are line integrals taken along the curve.

(vii.) When we use curvilinear coordinates ξ, η as in (v.) above, the length of any portion of a curve ξ = const. is given by the integral

∫ J−1/2 dη

taken between appropriate limits for η. There is a similar formula for the arc of a curve η = const.

(viii.) The area of a surface z = ƒ(x, y) can be expressed by the formula

When the coordinates of the points of a surface are expressed as functions of two parameters u, v, the area is expressed by the formula

When the surface is referred to three-dimensional polar coordinates r, θ, φ given by the equations

x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ,

and the equation of the surface is of the form r = ƒ(θ, φ), the area is expressed by the formula

The surface integral of a function of (θ, φ) over the surface of a sphere r = const. can be expressed in the form

∫2π0 dφ ∫π0 F (θ,φ) r2 sin θ dθ.

In every case the domain of integration must be chosen so as to include the whole surface.

(ix.) In three-dimensional polar coordinates the Jacobian

The volume integral of a function F (r, θ, φ) through the volume of a sphere r = a is

∫a0 dr ∫2π0 dφ ∫π0 F (r, θ, φ) r2 sin θ dθ.

(x.) Integrations of rational functions through the volume of an ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 are often effected by means of a general theorem due to Lejeune Dirichlet (1839), which is as follows: when the domain of integration is that given by the inequality

where the a’s and α’s are positive, the value of the integral

∫∫ ... x1 n1−1 · x2 n2−1 ... dx1 dx2 ...

is

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).

(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:—

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

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.

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.

The bacteriology of influenza is discussed in the article on [Parasitic Diseases]. The disease is often called “Russian” influenza, and its origin in 1889 suggests that the name may have some foundation in fact. A writer, who saw the epidemic break out in Bokhara, is quoted by him to the following effect:—“The summer of 1888 was exceptionally hot and dry, and was followed by a bitterly cold winter and a rainy spring. The dried-up earth was full of cracks and holes from drought and subsequent frost, so that the spring rains formed ponds in these holes, inundated the new railway cuttings, and turned the country into a perfect marsh. When the hot weather set in the water gave off poisonous exhalations, rendering malaria general.” On account of the severe winter, the people were enfeebled from lack of nourishment, and when influenza broke out suddenly they died in large numbers. Europeans were very severely affected. Russians, hurrying home, carried the disease westwards, and caravans passing eastwards took it into Siberia. There is a striking similarity in the conditions described to those observed in connexion with outbreaks of other diseases, particularly typhoid fever and diphtheria, which have occurred on the supervention of heavy rain after a dry period, causing cracks and fissures in the earth. Assuming the existence of a living poison in the ground, we can easily understand that under certain conditions, such as an exceptionally dry season, it may develop exceptional properties and then be driven out by the subsequent rains, causing a violent outbreak of illness. Some such explanation is required to account for the periodical occurrence of epidemic and pandemic diffusions starting from an endemic centre. We may suppose that a micro-organism of peculiar robustness and virulence is bred and brought into activity by a combination of favourable conditions, and is then disseminated more or less widely according to its “staying power,” by human agency. Whether central Asia is an endemic centre for influenza or not there is no evidence, but the disease seems to be more often prevalent in the Russian Empire than elsewhere. Extensive outbreaks occurred there in 1886 and 1887, and it is certain that the 1889 wave was active in Siberia at an earlier date than in Europe, and that it moved eastwards. The hypothesis that it originated in China is unsupported by evidence. But whatever may be the truth with regard to origin, the dissemination of influenza by human agency must be held to be proved. This is the most important addition to our knowledge of the subject contributed by recent research. The upshot of the inquiry by Dr Parsons was to negative all theories of atmospheric influence, and to establish the conclusion that the disease was “propagated mainly, perhaps entirely, by human intercourse.”

He found that it prevailed independently of climate, season and weather; that it moved in a contrary direction to the prevailing winds; that it travelled along the lines of human intercourse, and not faster than human beings can travel; that in 1889 it travelled much faster than in previous epidemics, when the means of locomotion were very inferior; that it appeared first in capital towns, seaports and frontier towns, and only affected country districts later; that it never commenced suddenly with a large number of cases in a place previously free from disease, but that epidemic manifestations were generally preceded for some days or weeks by scattered cases; that conveyance of infection by individuals and its introduction into fresh places had been observed in many instances; that persons brought much into contact with others were generally the first to suffer; that persons brought together in large numbers in enclosed spaces suffered more in proportion than others, and that the rapidity and extent of the outbreak in institutions corresponded with the massing together of the inmates.

These conclusions, based upon the 1889-1890 epidemic, have been confirmed by subsequent experience, especially in regard to the complete independence of season and weather shown by influenza. It has appeared and disappeared at all seasons and in all weathers and only popular ignorance continues to ascribe its behaviour to atmospheric conditions. In Europe, however, it has prevailed more often in winter than in summer, which may be due to the greater susceptibility of persons in winter, or, more probably, to the fact that they congregate more in buildings and are less in the open air during that part of the year. No doubt is any longer entertained of its infectious character, though the degree of infectivity appears to vary considerably. Many cases have been recorded of individuals introducing it into houses, and of all or most of the other inmates then taking it from the first case. Difficulties in preventing the spread of infection are due to (1) the shortness of the period of incubation, (2) the disease being infectious in the earliest stages before the nature of the illness is recognized, (3) the milder varieties being equally infectious with the severe attacks, and the patient going to work and spreading the infection, (4) the diagnosis often being difficult, influenza being possibly confused with ordinary catarrhal attacks, typhoid fever and other diseases. Domestic animals seem to be free from any suspicion of being liable to human influenza. Sanitary conditions, other than overcrowding, do not appear to exercise any influence on the spread of influenza.

Influenza has been shown to be an acute specific fever having nothing whatever to do with a “bad cold.” There may be some inflammation of the respiratory passages, and then symptoms of catarrh are present, but that is not necessarily the case, and in some epidemics such symptoms are quite exceptional. This had been recognized by various writers before the 1889 visitation, but it had not been generally realized, as it has been since, and some medical authorities, who persisted in regarding influenza as essentially a “catarrhal” affection, were chiefly to blame for a widespread and tenacious popular fallacy.

Leichtenstern, in his masterly article in Nothnagel’s Handbuch, divides the disease as follows:—(1) Epidemic influenza vera caused by Pfeiffer’s bacillus; (2) Endemic-epidemic influenza vera, which occurs several years after a pandemic and is caused by the same bacillus; (3) Endemic influenza nostras or eatarrhal fever, called la grippe, and bearing the same relation to true influenza as cholera nostras does to Asiatic cholera.

The “period of incubation” is one to four days. Susceptibility varies greatly, but the conditions that influence it are matters of conjecture only. It appears that the inhabitants of Great Britain are less susceptible than those of many other countries. Dr Parsons gives the following list, showing the proportion of the population estimated to have been attacked in the 1889-1890 epidemic in different localities:—

In and about London he reckoned roughly from a number of returns that the proportion was about 12 1/2% among those employed out of doors and 25% among those in offices, &c. The proportion among the troops in the Home District was 9.3%. The General Post Office made the highest return with 33.6%, which is accounted for partly by the enormous number of persons massed together in the same room in more than one department, and partly by the facilities for obtaining medical advice, which would tend to bring very light cases, unnoticed elsewhere, upon the record. No public service was seriously disorganized in England by sickness in the same manner as on the continent of Europe. Some individuals appear to be totally immune; others take the disease over and over again, deriving no immunity, but apparently greater susceptibility from previous attacks.

The symptoms were thus described by Dr Bruce Low from observations made in St Thomas’s Hospital, London, in January 1890:—

The invasion is sudden; the patients can generally tell the time when they developed the disease; e.g. acute pains in the back and loins came on quite suddenly while they were at work or walking in the street, or in the case of a medical student, while playing cards, rendering him unable to continue the game. A workman wheeling a barrow had to put it down and leave it; and an omnibus driver was unable to pull up his horses. This sudden onset is often accompanied by vertigo and nausea, and sometimes actual vomiting of bilious matter. There are pains in the limbs and general sense of aching all over; frontal headache of special severity; pains in the eyeballs, increased by the slightest movement of the eyes; shivering; general feeling of misery and weakness, and great depression of spirits, many patients, both men and women, giving way to weeping; nervous restlessness; inability to sleep, and occasionally delirium. In some cases catarrhal symptoms develop, such as running at the eyes, which are sometimes injected on the second day; sneezing and sore throat; and epistaxis, swelling of the parotid and submaxillary glands, tonsilitis, and spitting of bright blood from the pharynx may occur. There is a hard, dry cough of a paroxysmal kind, worst at night. There is often tenderness of the spleen, which is almost always found enlarged, and this persists after the acute symptoms have passed. The temperature is high at the onset of the disease. In the first twenty-four hours its range is from 100° F. in mild cases to 105° in severe cases.

Dr J. S. Bristowe gave the following description of the illness during the same epidemic:—

The chief symptoms of influenza are, coldness along the back, with shivering, which may continue off and on for two or three days; severe pain in the head and eyes, often with tenderness in the eyes and pain in moving them; pains in the ears; pains in the small of the back; pains in the limbs, for the most part in the fleshy portions, but also in the bones and joints, and even in the fingers and toes; and febrile temperature, which may in the early period rise to 104° or 105° F. At the same time the patient feels excessively ill and prostrate, is apt to suffer from nausea or sickness and diarrhoea, and is for the most part restless, though often (and especially in the case of children and those advanced in age) drowsy.... In ordinary mild cases the above symptoms are the only important ones which present themselves, and the patient may recover in the course of three or four days. He may even have it so mildly that, although feeling very ill, he is able to go about his ordinary work. In some cases the patients have additionally some dryness or soreness of the throat, or some stiffness and discharge from the nose, which may be accompanied by slight bleeding. And in some cases, for the most part in the course of a few days, and at a time when the patient seems to be convalescent, he begins to suffer from wheezing in the chest, cough, and perhaps a little shortness of breath, and before long spits mucus in which are contained pellets streaked or tinged with blood.... Another complication is diarrhoea. Another is a roseolous spotty rash.... Influenza is by no means necessarily attended with the catarrhal symptoms which the general public have been taught to regard as its distinctive signs, and in a very large proportion of cases no catarrhal condition whatever becomes developed at any time.

Several writers have distinguished four main varieties of the disease—namely, (1) nervous, (2)gastro-intestinal, (3)respiratory, (4) febrile, a form chiefly found in children. Clifford Allbutt says, “Influenza simulates other diseases.” Many forms are of typhoid or comatose types. Cardiac attacks are common, not from organic disease but from the direct poisoning of the heart muscle by influenza.

Perhaps the most marked feature of influenza, and certainly the one which victims have learned to dread most, is the prolonged debility and nervous depression that frequently follow an attack. It was remarked by Nothnagel that “Influenza produces a specific nervous toxin which by its action on the cortex produces psychoses.” In the Paris epidemic of 1890 the suicides increased 25%, a large proportion of the excess being attributed to nervous prostration caused by the disease. Dr Rawes, medical superintendent of St Luke’s hospital, says that of insanities traceable to influenza melancholia is twice as frequent as all other forms of insanity put together. Other common after-effects are neuralgia, dyspepsia, insomnia, weakness or loss of the special senses, particularly taste and smell, abdominal pains, sore throat, rheumatism and muscular weakness. The feature most dangerous to life is the special liability of patients to inflammation of the lungs. This affection must be regarded as a complication rather than an integral part of the illness. The following diagram gives the annual death-rate per million in England and Wales, and is taken from an article by Dr Arthur Newsholme in The Practitioner (January 1907).

The deaths directly attributed to influenza are few in proportion to the number of cases. In the milder forms it offers hardly any danger to life if reasonable care be taken, but in the severer forms it is a fairly fatal disease. In eight London hospitals the case-mortality among in-patients in the 1890 outbreak was 34.5 per 1000; among all patients treated it was 1.6 per 1000. In the army it was rather less.

The infectious character of influenza having been determined, suggestions were made for its administrative control on the familiar lines of notification, isolation and disinfection, but this has not hitherto been found practicable. In March 1895, however, the Local Government Board issued a memorandum recommending the adoption of the following precautions wherever they can be carried out:—

1. The sick should be separated from the healthy. This is especially important in the case of first attacks in a locality or a household.

2. The sputa of the sick should, especially in the acute stage of the disease, be received into vessels containing disinfectants. Infected articles and rooms should be cleansed and disinfected.

3. When influenza threatens, unnecessary assemblages of persons should be avoided.

4. Buildings and rooms in which many people necessarily congregate should be efficiently aerated and cleansed during the intervals of occupation.

There is no routine treatment for influenza except bed. In all cases bed is advisable, because of the danger of lung complications, and in mild ones it is sufficient. Severer ones must be treated according to the symptoms. Quinine has been much used. Modern “anti-pyretic” drugs have also been extensively employed, and when applied with discretion they may be useful, but patients are not advised to prescribe them for themselves.

Sir Wm. Broadbent in a note on the prophylaxis of influenza recommends quinine in a dose of two grains every morning, and remarks: “I have had opportunities of obtaining extraordinary evidence of its protective power. In a large public school it was ordered to be taken every morning. Some of the boys in the school were home boarders, and it was found that while the boarders at the school took the quinine in the presence of a master every morning, there were scarcely any cases of influenza among them, although the home boarders suffered nearly as much as before.” He continues, “In a large girls’ school near London the same thing was ordered, and the girls and mistresses took their morning dose but the servants were forgotten. The result was that scarcely any girl or mistress suffered while the servants were all down with influenza.”

The liability to contract influenza, and the danger of an attack if contracted, are increased by depressing conditions, such as exposure to cold and to fatigue, whether mental or physical. Attention should, therefore, be paid to all measures tending to the maintenance of health. Persons who are attacked by influenza should at once seek rest, warmth and medical treatment, and they should bear in mind that the risk of relapse, with serious complications, constitutes a chief danger of the disease.

In addition to the ordinary text-books, see the series of articles by experts on different aspects in The Practitioner (London) for January 1907.

IN FORMÂ PAUPERIS (Latin, “in the character of pauper”), the legal phrase for a method of bringing or defending a case in court on the part of persons without means. By an English statute of 1495 (11 Hen. VII. c. 12), any poor person having cause of action was entitled to have a writ according to the nature of the case, without paying the fees thereon. The statute of 1495 was repealed by the Statute Law Revision and Civil Procedure Act 1883, but its provisions, as well as the chancery practice were incorporated into one code and embodied in the rules of the Supreme Court (O. xvi. rr. 22-31). Now any person may be admitted to sue as a pauper, on proof that he is not worth £25, his wearing apparel and the subject matter of the cause or matter excepted. He must lay his case before counsel for opinion, and counsel’s opinion thereon, with an affidavit of the party suing that the case contains a full and true statement of all the material facts to the best of his knowledge and belief, must be produced before the proper officers to whom the application is made. A person who desires to defend as a pauper must enter an appearance to a writ in the ordinary way and afterwards apply for an order to defend as a pauper. Where a person is admitted to sue or defend as a pauper, counsel and solicitor may be assigned to him, and such counsel and solicitor are not at liberty to refuse assistance unless there is some good reason for refusing. If any person admitted to sue or defend as a pauper agrees to pay fees to any person for the conduct of his business he will be dispaupered. Costs ordered to be paid to a pauper are taxed as in other cases. Appeals to the House of Lords in formâ pauperis were regulated by the Appeal (Formâ Pauperis) Act 1893, which gave the House of Lords power to refuse a petition for leave to sue.

INFORMATION (from Lat. informare, to give shape or form to, to represent, describe), the communication of knowledge; in English law, a proceeding on behalf of the crown against a subject otherwise than by indictment. A criminal information is a proceeding in the King’s bench by the attorney-general without the intervention of a grand jury. The attorney-general, or, in his absence, the solicitor-general, has a right ex officio to file a criminal information in respect of any indictments, but not for treason, felonies or misprision of treason. It is, however, seldom exercised, except in cases which might be described as “enormous misdemeanours,” such as those peculiarly tending to disturb or endanger the king’s government, e.g. seditions, obstructing the king’s officers in the execution of their duties, &c. In the form of the proceedings the attorney-general is said to “come into the court of our lord the king before the king himself at Westminster, and gives the court there to understand and be informed that, &c.” Then follows the statement of the offence as in an indictment. The information is filed in the crown office without the leave of the court. An information may also be filed at the instance of a private prosecutor for misdemeanours not affecting the government, but being peculiarly flagrant and pernicious. Thus criminal informations have been granted for bribing or attempting to bribe public functionaries, and for aggravated libels on public or private persons. Leave to file an information is obtained after an application to show cause, founded on a sworn statement of the material facts of the case.

Certain suits might also be filed in Chancery by way of information in the name of the attorney-general, but this species of information was superseded by Order 1, rule 1 of the Rules of the Supreme Court, 1883, under which they are instituted in the ordinary way. Informations in the Court of Exchequer in revenue cases, also filed by the attorney-general, are still resorted to (see A.-G. v. Williamson, 1889, 60 L.T. 930).

INFORMER, in a general sense, one who communicates information. The term is applied to a person who prosecutes in any of the courts of law those who break any law or penal statute. Such a person is called a common informer when he furnishes evidence on criminal trials or prosecutes for breaches of penal laws solely for the purpose of obtaining the penalty recovered, or a share of it. An action by a common informer is termed a popular or qui tam action, because it is brought by a person qui tam pro domino rege quam pro se ipso sequitur. A suit by an informer must be brought within a year of the offence, unless a specific time is prescribed by the statute. The term informer is also used of an accomplice in crime who turns what is called “king’s evidence” (see [Accomplice]). In Scotland, informer is the term applied to the party who, in criminal proceedings, sets the lord advocate in motion.

INFUSORIA, the name given by Bütschli (following O.F. Ledermüller, 1763) to a group of Protozoa. The name arose from the procedure adopted by the older microscopists to obtain animalcules. Infusions of most varied organic substances were prepared (hay and pepper being perhaps the favourite ones), the method of obtaining them including maceration and decoction, as well as infusion in the strict sense; they were then allowed to decompose in the air, so that various living beings developed therein. As classified by C. G. Ehrenberg in his monumental Infusionstierchen als volkommene Organismen, they included (1) Desmids, Diatoms and Schizomycetes, now regarded as essentially Plant Protista or Protophytes; (2) Sarcodina (excluding Foraminifera, as well as Radiolaria, which were only as yet known by their skeletons, and termed Polycystina), and (3) Rotifers, as well as (4) Flagellates and Infusoria in our present sense. F. Dujardin in his Histoire des zoophytes (1841) gave nearly as liberal an interpretation to the name; while C. T. Van Siebold (1845) narrowed it to its present limits save for the admission of several Flagellate families. O. Bütschli limited the group by removing the Flagellata, Dinoflagellata and Cystoflagellata (q.v.) under the name of “Mastigophora” proposed earlier by R. M. Diesing (1865). We now define it thus:—Protozoa bounded by a permanent plasmic pellicle and consequently of definite form, never using pseudopodia for locomotion or ingestion, provided (at least in the young state) with numerous cilia or organs derived from cilia and equipped with a double nuclear apparatus: the larger (mega-) nucleus usually dividing by constriction, and disappearing during conjugation: the smaller (micro-) nucleus (sometimes multiple) dividing by mitosis, and entering into conjugation and giving rise to the cycle of nuclei both large and small of the race succeeding conjugation.

Fig. i. Ciliata.
1. Opalinopsis sepiolae, Foett.: a parasiticHolotrichous mouthless Ciliate fromthe liver of the Squid. a, branchedmeganucleus; b, vacuoles (non-contractile). 2. A similar specimen treated with picrocarmine,showing a remarkablybranched and twisted meganucleus(a), in place of several nuclei. 3. Anoplophrya naidos, Duj.; a mouthlessHolotrichous Ciliate parasitic inthe worm Nais. a, the large axialmeganucleus; b, contractile vacuoles. 4. Anoplophrya prolifera, C. and L.; fromthe intestine of Clitellio. Remarkablefor the adhesion of incompletefission-products in a metamericseries. a, meganucleus. 5. Amphileptus gigas, C. and L. (Gymnostomaceae).b, contractile vacuoles;c, trichocysts (see fig. 2); d, meganucleus;e. pharynx. 6, 7. Prorodon niveus, Ehr. (Gymnostomaceae).a, meganucleus; b,contractile vacuole; c, pharynx withhorny cuticular lining. 6. The fasciculate cuticle of the pharynxisolated.	8. Trachelius ovum, Ehr. (Gymnostomaceae);showing the reticulatearrangement of the endosarc, b,contractile vacuoles; c, the cuticle-linedpharynx. 9, 10, 11, 12. Icthyophthirius multifilius,Fouquet (Gymnostomaceae). Freeindividual and successive stagesof division to form spores. a, meganucleus;b, contractile vacuoles. 13. Didinium nasutum, Müll. (Gymnostomaceae).The pharynx is evertedand has seized a Paramecium asfood. a, meganucleus; b, contractilevacuole; c, everted pharynx. 14. Euplotes charon, Müll. (Hypotrichaceae);lateral view of the animalwhen using its great cirrhi, x, asambulatory organs. 15. Euplotes harpa, Stein (Hypotrichaceae);h, mouth; x, cirrhi. 16. Nyctotherus cordiformis, Stein (aHeterotriceae), parasitic in the intestineof the Frog; a, meganucleus;b, contractile vacuole; c, food particle;d, anus; e, heterotrichous bandof membranelles; f, g, mouth; h,pharynx; i, small cilia.

Thus defined, the Infusoria fall into two groups:—(1) Ciliata, with cilia or organs derived from cilia throughout their lives, provided with a single permanent mouth (absent in the parasitic Opalinopsidae) flush with the body or at the base of an oral depression, and taking in food by active swallowing or by ciliary action: (2) Suctoria, rarely ciliated except in the young state, and taking in their food by suction through protrusible hollow tentacles, usually numerous.

The pellicle of the Infusoria is stronger and more permanent than in many Protozoa, and sometimes assumes the character of a mail of hard plates, closely fitting; but even in this case it undergoes solution soon after death. It is continuous with a firm ectosarc, highly differentiated in the Ciliata, and in both groups free from coarse movable granules. The endosarc is semifluid and rich in granules mostly “reserve” in nature, often showing proteid or fat reactions. One or more contractile vacuoles are present in some of the marine and all the freshwater species, and open to the surface by pores of permanent position: a system of canals in the deeper layers of the ectoplasm is sometimes connected with the vacucle. The body is often provided with not-living external formations “stalk” and “theca” (or “lorica”).

The character of the nuclear apparatus excludes two groups both parasitic and mouthless: (1) the Trichonymphidae, with a single nucleus of Leidy, parasitic in Insects, especially Termites; (2) the Opalinidae, with several (often numerous) uniform nuclei, parasitic in the gut of Batrachia, &c., and producing 1-nuclear zoospores which conjugate. Both these families we unite into a group of Pseudociliata, which may be referred to the Flagellata (q.v.). Lankester in the last edition of this Encyclopaedia called attention to the doubtful position of Opalina, and Delage and Hérouard placed Trichonymphidae among Flagellates.

The theca or shell is present in some pelagic species (fig. iii. 3, 5) and in many of the attached species, notably among the Peritricha (fig. iii. 21, 22, 25, 26) and Suctoria (fig. viii. 11); and is found in some free-swimming forms (fig. iii. 3, 5): it is usually chitinous, and forms a cup into which the animal, protruded when at its utmost elongation, can retract itself. In Metacineta mystacina it has several distinct slits (pylomes) for the passage of tufts of tentacles. In Stentor it is gelatinous; and in the Dictyocystids it is beautifully latticed.

The stalk is usually solid, and expanded at the base into a disk in Suctoria. In Peritrichaceae (fig. iii. 8-22, 25, 26), the only ciliate group with a stalk, it grows for some time after its formation, and on fission two new stalks continue the old one, so as to form a branched colony (fig. iii. 18). In Vorticella (fig. iii. 11, 12, 14, &c.) the stalk is hollow and elastic, and attached to it along a spiral is a prolongation of the ectosarc containing a bundle of myonemes, so that by the contractions of the bundle the stalk is pulled down into a corkscrew spiral, and on the relaxation of the muscle the elasticity of the hollow stalk straightens it out.

On fission the stalk may become branched, as the solid one of Epistylis and Opercularia (fig. iii. 20); and the myoneme also in the tubular stem of Zoothaminum; or the branch-myoneme for the one offspring may be inserted laterally on that for the other in Carchesium (fig. iii. 18). In several tubicolous Peritrichaceae there is some arrangement for closing their tubes. In Thuricola (fig. iii. 25-26) there is a valve which opens by the pressure of the animal on its protrusion, and closes automatically by elasticity on retraction. In Lagenophrys the animal adheres to the cup a little below the opening, so that its withdrawal closes the cup: at the adherent part the body mass is hardened, and so differentiated as to suggest the frame of the mouth of a purse. In Pyxicola (fig. iii. 21-22) the animal bears some way down the body a hardened shield (“operculum”) which closes the mouth of the shell on retraction.

Fig. ii.
1, Surface view of Paramecium,showing the disposition ofthe cilia in longitudinalrows. 2, a, mega-; b, micronucleus;c, junction of ecto- and endosarc;D, pellicle; E, endosarc;f, cilia (much toonumerous and crowded);g, trichocysts; g′, samewith thread; h, discharged;i, pharynx, its undulatingmembrane not shown; k,food granules collecting intoa bolus; l, m, n, o, foodvacuoles, their contentsbeing digested as they passin the endosarc along thepath indicated by thearrows.	3, Outline showing contractilevacuoles in commencingdiastole, surrounded by fiveafferent canals. 4-7 Successive stages of diastoleof contractile vacuole.

The cytoplasm of the Infusoria is very susceptible to injuries; and when cut or torn, unless the pellicle contracts rapidly to enclose the wounded surface, the substance of the body swells up, becoming frothy, with bubbles which rapidly enlarge and finally burst; the cell thus disintegrates, leaving only a few granules to mark where it was. This phenomenon, observed by Dujardin, is called “diffluence.” The contractile vacuole appears to be one of the means by which diffluence is avoided in cells with no strong wall to resist the absorption of water in excess; for after growing in size for some time, its walls contract suddenly, and its contents are expelled to the outside by a pore, which is, like the anus, usually invisible, but permanent in position. The contractile vacuole may be single or multiple; it may receive the contents of a canal, or of a system of canals, which only become visible at the moment of the contraction of the vacuole (fig. ii. 4-7), giving liquid time to accumulate in them, or when the vacuole is acting sluggishly or imperfectly, as in the approach of asphyxia (fig. ii. 3). Besides this function, since the system passes a large quantity of water from without through the substance of the cell, it must needs act as a means of respiration and excretion. In all Peritrichaceae it opens to the vestibule, and in some of them it discharges through an intervening reservoir, curiously recalling the arrangements in the Flagellate Euglenaceae.

The nuclear apparatus consists of two parts, the meganucleus, and the micronucleus or micronuclei (fig. iii. 17d, iv. 1). The meganucleus alone regarded and described as “the nucleus” by older observers is always single, subject to a few reservations. It is most frequently oval, and then is indented by the micronucleus; but it may be lobed, the lobes lying far apart and connected by a slender bridge or moniliform, or horseshoe-shaped (Peritrichaceae). It often contains darker inclusions, like nucleoles.

It has been shown, more especially by Gruber, that many Ciliata are multinucleate, and do not possess merely a single meganucleus and a micronucleus. In Oxytricha the nuclei are large and numerous (about forty), scattered through the protoplasm, whilst in other cases the nucleus is so finely divided as to appear like a powder diffused uniformly through the medullary protoplasm (Trachelocerca). Carmine staining, after treatment with absolute alcohol, has led to this remarkable discovery. The condition described by Foettinger in his Opalinopsis (fig. i. 1, 2) is an example of this pulverization of the nucleus. The condition of pulverization had led in some cases to a total failure to detect any nucleus in the living animal, and it was only by the use of reagents that the actual state of the case was revealed. Before fission, whatever be its habitual character, it condenses, becomes oval, and divides by constriction; and though it usually is then fibrillated, only in a few cases does it approach the typical mitotic condition. The micronucleus described by older writers as the “nucleolus” or “paranucleus” (“endoplastule” of Huxley), may be single or multiple. When the meganucleus is bilobed there are always two micronuclei, and at least one is found next to every enlargement of the moniliform meganucleus. In the fission of the Infusoria, every micronucleus divides by a true mitotic process, during which, however, its wall remains intact. From their relative sizes the meganucleus would appear to discharge during cell-life, exclusively, the functions of the nucleus in ordinary cells. Since in conjugation, however, the meganucleus degenerates and is in great part either digested or excreted as waste matter, while the new nuclear apparatus in both exconjugates arises, as we shall see, from a conjugation-nucleus of exclusively micronuclear origin, we infer that the micronucleus has for its function the carrying on of the nuclear functions of the race from one fission cycle to the next from which the meganucleus is excluded.

Fission is the ordinary mode of reproduction in the Infusoria, and is usually transverse, but oblique in Stentor, &., as in Flagellata, longitudinal in Peritrichaceae; in some cases it is always more or less unequal owing to the differentiation of the body, and consequently it must be followed by a regeneration of the missing organs in either daughter-cell. In some cases it becomes very uneven, affording every transition to budding, which process assumes especial importance in the Suctoria. Multiple fission (brood-formation or sporulation) is exceptional in Infusoria, and when it occurs the broods rarely exceed four or eight—another difference from Flagellata. The nuclear processes during conjugation suggest the phylogenetic loss of a process of multiple fission into active gametes. As noted, in fission the meganucleus divides by direct constriction; each micronucleus by a mode of mitosis. The process of fission is subject in its activity to the influences of nutrition and temperature, slackening as the food supply becomes inadequate or as the temperature recedes from the optimum for the process. Moreover, if the descendants of a single animal be raised, it is found that the rapidity of fission, other conditions being the same, varies periodically, undergoing periods of depression, which may be followed by either (1) spontaneous recovery, (2) recovery under stimulating food, (3) recovery through conjugation, or (4) the death of the cycle, which would have ensued if 2 or 3 had been omitted at an earlier stage, but which ultimately seems inevitable, even the induction of conjugation failing to restore it. These physiological conditions were first studied by E. Maupas, librarian to the city of Algiers, in his pioneering work in the later ’eighties, and have been confirmed and extended by later observers, among whom we may especially cite G. N. Calkins.

Syngamy, usually termed conjugation or “karyogamy,” is of exceptional character in the majority of this group—the Peritrichaceae alone evincing an approximation to the usual typical process of the permanent fusion of two cells (pairing-cells or gametes), cytoplasm to cytoplasm, nucleus to nucleus, to form a new cell (coupled cell, zygote).

This process was elucidated by E. Maupas in 1889, and his results, eagerly questioned and repeatedly tested, have been confirmed in every fact and in every generalization of importance.

Previously all that had been definitely made out was that under certain undetermined conditions a fit of pairing two and two occurred among the animals of the same species in a culture or in a locality in the open; that after a union prolonged over hours, and sometimes even days, the mates separated; that during the union the meganucleus underwent changes of a degenerative character; and that the micronucleus underwent repeated divisions, and that from the offspring of the micronuclei the new nuclear apparatus was evolved for each mate. Maupas discovered the biological conditions leading to conjugation: (1) the presence of individuals belonging to distinct stocks; (2) their belonging to a generation sufficiently removed from previous conjugation, but not too far removed therefrom; (3) a deficiency of food. He also showed that during conjugation a “migratory” nucleus, the offspring of the divisions of the micronucleus, passes from either mate to the other, while its sister nucleus remains “stationary”; and that reciprocal fusion of the migratory nucleus of the one mate with the stationary nucleus of the other takes place to form a zygote nucleus in either mate; and that from these zygote nuclei in each by division, at least two nuclei are formed, the one of which enlarges to form a meganucleus, while the other remains small as the first micronucleus of the new reorganized animal, which now separates as an “exconjugate” (fig. iv). Moreover, if pairing be prevented, or be not induced, the individuals produced by successive fissions become gradually weaker, their nuclear apparatus degenerates, and finally they cannot be induced under suitable conditions to pair normally, so that the cycle becomes extinct by senile decay. In Peritrichaceae the gametes are of unequal sizes (fig. iii. 11, 12), the smaller being formed by brood fissions (4 or 8); syngamy is here permanent, not temporary, the smaller (male) being absorbed into the body of the larger (female); and there are only two nuclei that pair. Thus we have a derived binary sexual process, comparable to that of ordinary bisexual organisms.

Fig. iii.— Ciliata: 1, 2, Heterotrichaceae; 3-7, 23-24, Oligotrichaceae;8-22, 25, 26, Peritrichaceae.
1, Spirostomum ambiguum, Ehr.;on its left side oral grooveand wreath of membranellae;a, moniliform meganucleus;b, position of contractilevacuole. 2, Group of Stentor polymorphus,O. F. Müller; thetwisted end of the peristomeindicating the positionof the mouth. 3, Tintinnus lagenula, Cl. andL., in free shell. 4, Strombidium claparedii, S.Kent. 5, Shell of Codonella campanella,Haeck. 6, 7, Torquatella typica, Lank.(= Strombidium accordingto Bütschli); p, oral tubeseen through peristomialwreath of apparently coalescentmembranellae. 8. Basal, and 9, side (inverted)views of Trichodina pediculus,Ehr.; a, meganucleus;c, basal collarand ring of hooks; d,mouth; contractile vacuoleand oral tube seen bytransparency in 8. 10, Spirochona gammipara,Stein; a, meganucleus; g,bud. 11, 12, Vorticella microstoma,Ehr.; d, formation of abrood of 8 microgametes cby multiple fission; b,contr. vacuole. 13, Same sp. in binary fission;a, meganucleus.	14, V. nebulifera, Ehr.; budswimming away byposterior wreath, peristome contracted; e,peristomial disk; f, oral tube. 15, V. microstoma; b, contr.vacuole; c, d, two microgametes seeking to conjugate. 16, V. nebulifera, contracted, with body encysted. 17, Same sp. enlarged; c, myonemes converging posteriorly tomuscle of stalk; d, micronucleus. 18, Carchesium spectabile, Ehr.; (×50). 19, Nematocysts of Epistylis flavicans. Ehr. (afterGreeff). 20, Opercularia stenostoma, St.; (×200); a smallcolony showing upstanding (“opercular”) peristomial disk,protruded oral undulating membranejand cilia in oral tube. 21, 22, Pyxicola affinis, S.K., with stalk and theca; x,chitinous disk, or true “operculum” closing theca inretracted state. 23, 24, Caenomorpha medusula, Perty, (×250), with spiralperistomial wreath. 25, 26, Thuricola valvata, Str. Wright, in sessile theca, withinternal valve (v) to close tube, as in gastropod Clausilia;owing to recent fission two animals occupy one tube.

From Lankester’s Treatise on Zoology.
Fig. iv.—Diagrammatic Sketch of Changes during Conjugation inCiliata. (From Hickson after Delage and Maupas.)
1, Two individuals at commencementof conjugationshowing meganucleus(dotted) and micronucleus;successive stages of thedisintegration of the meganucleusshown in all figuresup to 9. 2, 3, First mitotic division ofmicronuclei. 4, 5, Second ditto. 6, One of the four nuclei resultingfrom the second divisionagain dividing to form thepairing-nuclei in eithermate, while the other 3nuclei degenerate.	7, Migration of the migratorynuclei. 8, 9, Fusion of the incomingmigratory with the stationarynucleus in either mate. 10, Fission of Zygote nucleusinto two, the new mega- andmicronucleus whosedifferentiation is shown in11, 12. The vertical dottedline indicates the separationof the mates.

Ciliata.—The Ciliate Infusoria represent the highest type of Protozoa. They are distinctly animal in function, and the Gymnostomaceae are active predaceous beings preying on other Infusoria or Flagellates. Some possess shells (fig. iii. 3, 5, 21, 22, 25, 26), most have a distinct swallowing apparatus, and in Dysteria there is a complex jaw—or tooth-apparatus, which needs new investigation. In the active Ciliata we find locomotive organs of most varied kinds: tail-springs, cirrhi for crawling and darting, cilia and membranellae for continuous swimming in the open or gliding over surfaces or waltzing on the substratum (Trichodina, fig. iii. 8) or for eddying in wild turns through the water (Strombidium, Tintinnus, Halteria). Their forms offer a most interesting variety, and the flexibility of many adds to their easy grace of movement, especially where the front of the body is produced and elongated like the neck of a swan (Amphileptus, fig. iii. 5; Lacrymaria).

The cytoplasm is very highly differentiated: especially the ectoplasm or ectosarc. This has always a distinct elastic “pellicle” or limiting layer, in a few cases hard, or even with local hardenings that affect the disposition of a coat of mail (Coleps) or a pair of valves (Dysteria); but is usually only marked into a rhomboidal network by intersecting depressions, with the cilia occupying the centres of the areas or meshes defined. The cytoplasm within is distinctly alveolated, and frequently contains tubular alveoli running along the length of the animal. Between these are dense fibrous thickenings, which from their double refraction, from their arrangement, and from their shortening in contracted animals are regarded as of muscular function and termed “myonemes.” Other threads running alongside of these, and not shortening but becoming wavy in the general contraction have been described in a few species as “neuronemes” and as possessing a nervous, conducting character. On this level, too, lie the dot-like granules at the bases of the cilia, which form definite groups in the case of such organs as are composed of fused cilia; in the deeper part of the ectoplasm the vacuoles or alveoli are more numerous, and reserve granules are also found; here too exist the canals, sometimes developed into a complex network, which open into the contractile vacuole.

The cilia themselves have a stiffer basal part, probably strengthened by an axial rod, and a distal flexible lash; when cilia are united by the outer plasmatic layer, they form (1) “Cirrhi,” stiff and either hook-like and pointed at the end, or brush-like, with a frayed apex; (2) membranelles, flattened organs composed of a number of cilia fused side by side, sometimes on a single row, sometimes on two rows approximated at either end so as to form a narrow oval, the membranelle thus being hollow; (3) the oral “undulating membrane,” merely a very elongated membranelle whose base may extend over a length nearly equal to the length of the animal; such membranes are present in the mouth oral depression and pharynx of all but Gymnostomaceae, and aid in ingestion; a second or third may be present, and behave like active lips; (4) in Peritrichaceae the cilia of the peristomial wreath are united below into a continuous undulating membrane, forming a spiral of more than one turn, and fray out distally into a fringe; (5) the dorsal cilia of Hypotrichaceae are slender and motionless, probably sensory.

Embedded in the ectosarc of many Ciliates are trichocysts, little elongated sacs at right angles to the surface, with a fine hair-like process projecting. On irritation these elongate into strong prominent threads, often with a more or less barb-like head, and may be ejected altogether from the body. Those over the surface of the body appear to be protective; but in the Gymnostomaceae specially strong ones surround the mouth. They can be injected into the prey pursued, and appear to have a distinctly poisonous effect on it. They are combined also into defensive batteries in the Gymnostome Loxophyllum. They are absent from most Heterotrichaceae and Hypotrichaceae, and from Peritrichaceae, except for a zone round the collar of the peristome.

The openings of the body are the mouth, absent in a few parasital species (Opalinopsis, fig. i. 1, 2), the anus and the pore of the contractile vacuole. The mouth is easily recognizable; in the most primitive forms of the Gymnostomaceae and some other groups, it is terminal, but it passes further and further back in more modified species, thereby defining a ventral, and correspondingly a dorsal surface; it usually lies on the left side. The anus is usually only visible during excretion, though its position is permanent; in a few genera it is always visible (e.g. Nyctotherus, fig. i. 16). The pore of the contractile vacuole might be described in the same terms.

The endoplasm has also an alveolar structure, and contains besides large food-vacuoles or digestive vacuoles, and shows movements of rotation within the ectoplasm, from which, however, it is not usually distinctly bounded. In Ophryoscolex and Didinium (fig. i. 13) a permanent cavity traverses it from mouth to anus.

Ingestion of food is of the same character in all the Hymenostomata. The ciliary current drives a powerful stream into the mouth, which impinges against the endosarc, carrying with it the food particles; these adhere and accumulate to form a pellet, which ultimately is pushed by an apparently sudden action into the substance of the endosarc which closes behind it (fig. ii. 2). In some of the Aspirotrichaceae accessory undulating membranes play the part of lips, and there is a closer approximation to true deglutition. The mouth is rarely terminal, more frequently at the bottom of a depression, the “vestibule,” which may be prolonged into a slender canal, sometimes called the “pharynx” or “oral tube,” ciliated as well as provided with a membrane, and extending deep down into the body in many Peritrichaceae.

In Spirostomaceae the “adoral wreath” of membranelles encloses more or less completely an anterior part of the body, the “peristome,” within which lies the vestibule. This area may be depressed, truncate, convex or produced into a short obconical disk or into one or more lobes, or finally form a funnel, or a twisted spiral like a paper cone. In most Peritrichaceae a collar-like rim surrounds the peristome, and marks out a gutter from which the vestibule opens; the peristome can be retracted, and the collar close over it. This rim forms a deep permanent spiral funnel in Spirochona (fig. iii. 10).

Movements of Ciliata.—H. S. Jennings has made a very detailed study of these movements, which resemble those of most minute free-swimming organisms. The following account applies practically to all active “Infusoria” in the widest sense.

The position of the free-swimming Infusoria, like that of Rotifers and other small swimming animals, is with the front end of the body inclined outward to the axis of advance, constantly changing its azimuth while preserving its angle constant or nearly so; if advance were ignored the body would thus rotate so as to trace out a cone, with the hinder end at the apex, and the front describing the base. On any irritation, (1) the motion is arrested, (2) the animal reverses its cilia and swims backwards, (3) it swerves outwards away from the axis so as to make a larger angle with it, and (4) then swims forwards along a new axis of progression, to which it is inclined at the same angle as to the previous axis (figs. vi., vii.). In this way it alters its axis of progression when it finds itself under conditions of stimulation. Thus a Paramecium coming into a region relatively too cold, too hot, or too poor in CO2 or in nutriment, alters its direction of swimming; in this way individuals come to assemble in crowds where food is abundant, or even where there is a slight excess of CO2. This reaction may lead to fatal results; if a solution of corrosive sublimate (Mercuric chloride) diffuses towards the hinder end of the animal faster than it progresses, the stimulus affecting the hinder end first, the axis of progression is altered so as to bring the animal after a few changes into a region where the solution is strong enough to kill it. This “motile reaction,” first noted by H. S. Jennings, is the explanation of the general reactions of minute swimming animals to most stimuli of whatever character, including light; the practical working out is, as he terms it, a method of “trial and error.” The action, however, of a current of electricity is distinctly and immediately directive; but such a stimulus is not to be found in nature. The motile reaction in the Hypotrichaceae which crawl or dart in a straight line is somewhat different, the swerve being a simple turn to the right hand—i.e. away from the mouth.

Parasitism in the Infusoria is by no means so important as among Flagellates. Ichthyophthirius alone causes epidemics among Fishes, and Balantidium coli has been observed in intestinal disease in Man. The Isotricheae, among Aspirotrichaceae and the Ophryoscolecidae among Heterotrichaceae are found in abundance in the stomachs of Ruminants, and are believed to play a part in the digestion of cellulose, and thus to be rather commensals than parasites. A large number of attached species are epizoic commensals, some very indifferent in choice of their host, others particular not only in the species they infest, but also in the special organs to which they adhere. This is notably the case with the shelled Peritrichaceae. Lichnophora and Trichodina (fig. iii. 8, 9) among Peritrichaceae are capable of locomotion by their permanent posterior wreath or of attaching themselves by the sucker which surrounds it; Kerona polyporum glides habitually over the body of Hydra, as does Trichodina pediculus.

Several Suctoria are endoparasitic in Ciliata, and their occurrence led to the view that they represented stages in the life-history of these. Again, we find in the endosarc of certain Ciliates green nucleated cells, which have a cellulose envelope and multiply by fission inside or outside the animal. They are symbiotic Algae, or possibly the resting state of a Chlamydomonadine Flagellate (Carteria?), and have received the name Zoochlorella. They are of constant occurrence in Paramecium bursaria, frequent in Stentor polymorphus and S. igneus, and Ophrydium versatile, and a few other species, which become infected by swallowing them.

Classification.

Order I.—Section A.—Gymnostomaceae. Mouth habitually closed; swallowing an active process; cilia (or membranelles) uniform, usually distributed evenly over the body; form variable, sometimes of circular transverse section.

Section B.—Trichostomata. Mouth permanently open against the endosarc, provided with 1 or 2 undulating membranes often prolonged into an inturned pharynx; ingestion by action of oral ciliary apparatus.

Order 2.—Subsection (a).—Aspirotrichaceae. Cilia nearly uniform, not associated with cirrhi or membranelles, nor forming a peristomial wreath. Form usually flattened, mouth unilateral. (N.B.—Orders 1, 2 are sometimes united into the single order Holotrichaceae.)

Subsection (b).—Spirotricha. Wreath of distinct membranelles—or of cilia fused at the base—enclosing a peristomial area and leading into the mouth.

§§ i.—Wreath of separate membranelles.

Order 3.—Heterotrichaceae; body covered with fine uniform cilia, usually circular in transverse section.

Order 4.—Oligotrichaceae; body covering partial or wholly absent; transverse section usually circular.

Order 5.—Hypotrichaceae; body flattened; body cilia represented chiefly by stiff cirrhi in ventral rows, and fine motionless dorsal sensory hairs.

Order 6.—§§ ii.—Peritrichaceae. Peristomial ciliary wreath, spiral, of cilia united at the base; posterior wreath circular of long membranelles; body circular in section, cylindrical, taper, or bell-shaped.

Illustrative Genera (selected).

1. Gymnostomaceae. (a) Ciliation general or not confined to one surface. Coleps Ehr., with pellicle locally hardened into mailed plates; Trachelocerca Ehr.; Prorodon Ehr. (fig. i. 6, 7); Trachelius Ehr., with branching endosarc (fig. i. 8); Lacrymaria Ehr. (fig. i. 5), body produced into a long neck with terminal mouth surrounded by offensive trichocysts; Dileptus Duj., of similar form, but anterior process, blind, preoral; Ichthyophthirius Fouquet (fig. i. 9-12), cilia represented by two girdles of membranellae; Didinium St. (fig. i. 13), cilia in tufts, surface with numerous tentacles each with a strong terminal trichocyst; Actinobolus Stein, body with one adoral tentacle; Ileonema Stokes. (b) Cilia confined to dorsal surface. Chilodon Ehr.; Loxodes Ehr., body flattened, ciliated on one side only, endosarc as in Trachelius; Dysteria Huxley, with the dorsal surface hardened and hinged along the median line into a bivalve shell, ciliated only on ventral surface, with a protrusible foot-like process, and a complex pharyngeal armature. (c) Cilia restricted to a single equatorial girdle, strong (probably membranelles); Mesodinium, mouth 4-lobed.

2. Aspirotrichaceae. Paramecium Hill (fig. ii. 1-3); Ophryoglena Ehr.; Colpoda O. F. Müller; Colpidium St.; Lembus Cohn, with posterior strong cilium for springing; Leucophrys St.; Urocentrum Nitsch, bare, with polar and equatorial zones and a posterior tuft of long cilia; Opalinopsis Foetlinger (fig. i. 1, 2); Anoplophyra St. (fig. i. 3, 4). (The last two parasitic mouthless genera are placed here doubtfully.)

3. Heterotrichaceae. (a) Wreath spiral; Stentor Oken. (fig. iii. 2), oval when free, trumpet-shaped when attached by pseudopods at apex, and then often secreting a gelatinous tube; Blepharisma Perty, sometimes parasitic in Heliozoa; Spirostomum Ehr., cylindrical, up to 1″ in length; (b) Wreath straight, often oblique; Nyctotherus Leidy, parasitic anus always visible; Balantidium Cl. and L., parasitic (B. coli in man); Bursaria, O.F.M., hollowed into an oval pouch, with the wreath inside.

4. Oligotrichaeceae. Tintinnus Schranck (fig. iii. 3); Trichodinopsis Cl. and L.; Codonella Haeck. (fig. iii. 5); Strombidium Cl. and L. (fig. iii. 4), including Torquatella Lank. (fig. iii. 6, 7), according to Bütschli; Halteria Duj., with an equatorial girdle of stiff bristle-like cilia; Caenomorpha Perty (fig. iii. 23, 24); Ophryoscolex St., with straight digestive cavity, and visible anus, parasitic in Ruminants.

5. Hypotrichaceae. Stylonychia Ehr.; Oxytricha Ehr.; Euplotes Ehr. (fig. i. 14, 15); Kerona Ehr. (epizoic on Hydra).