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ENCYCLOPAEDIA BEITANNICA

NINTH EDITION

THE

ENCYCLOPEDIA BRITANNICA

DICTIONAKY

OF

NINTH EDITION

VOLUME XIII

NEW YORK: CHARLES SCRIBNER'S SONS

MDCCCLXXXI

[ All Eights reserved. ]

fcdd'l

GIFT

PHIXTKD I1Y NEILL AND COMPANY, EDINBURGH.

BIOLOGY

ENCYCLOPEDIA BRITANNIOA.

INFANT

INFANT, in law, is a person under full age, and there fore subject to certain disabilities not affecting persons who have attained full age. The period of full age vanes widely in different systems, as do also the disabilities attaching to 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 is under the guardianship of a tutor, but several degrees of infancy are recognized. The first is absolute infancy in the literal sense — speechlessness ; after that, until the age of seven, a child is infantise proximus ; and from the eighth year to puberty he is 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 of course 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 ; and tutorship and curatorship were devices for protecting those who were free from the potestas, but unable by reason of infancy to control their own affairs. 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 of course for his own protection, and its general effect is to prevent him from binding him self absolutely by obligations. In the matter of contracts, the statement has generally passed current that an infant's contracts, except when they arc binding for special reasons, are either void or voidable, i.e., null, ab initio, or capable of being nullified by the infant at his choice. Contracts, for example, which cannot be beneficial to the infant are said to be absolutely void. A bond with a penalty is for this reason declared to be void. On the other hand, it is alleged by the more recent text-writers that the words void and voidable have not been carefully distinguished, and that a contract is often described as void when it is only meant that it is not binding. On this theory all the con tracts of an infant might bo described as voidable at his option except those few which are absolutely valid. On his voidable contracts an infant may sue if he chooses to j do so, but may not be sued. Of the contracts of an infant which are binding ab initio, the most important are those re lating to "necessaries." The word is used in an extended signification to cover " articles fit to maintain the person in the particular state, degree, and station in life in which he is." Whether a particular thing is necessary or not is a question of fact to be decided by a jury, but it is fur the judge to say whether it is prima facie of a descrip tion such that it may be a necessary. It has been ruled by judges, without consulting the jury, that the follow ing articles were not necessary : — expensive dinners sup plied to an undergraduate in his private rooms ; a pair of solitaire studs costing £25, and a goblet costing £15, for a baronet's son ; a chronometer worth .£'68, for a lieutenant in the navy ; ornaments to the value of £137. On the other hand, an undergraduate has been allowed a gold watch as a necessary ; and liveries for an officer's servant, money paid to release an infant from ejectment or distress, and necessaries for an infant's wife have .all been held to be necessaries of an infant. An object, in itself of a character to be pronounced a necessary, may in particular cases be declared not necessary, e.g., if the infant is already supplied with things of the kind. A sealskin waistcont may be a necessary to an infant of good fortune, but not if he has half a dozen such garments in his pOMCftrion

XIII. — i

INFANT

already. The person who supplies goods primci fade necessary to an infant must, it would seem, take the risk of their turning out to be unnecessary. An infant fraudu lently passing himself off as of full age and contracting on that footing will be hold bound in equity. 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. " Catch ing bargains," as they are called, throw on the persons claiming the benefit of them the burden of proving their substantial righteousness ; and, although the youth of the party charged is not an essential point, it is generally one of the facts relied on as showing undue influence.

At common law a bargain made by an infant might bo ratified by him after full age, and would then become in all respects 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 bo supplied (other than contracts for neces saries), 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 con sideration for such promise or ratification after full age." It has been held in a recent case that this action applies to promises of marriage, so that where an infant had promised marriage, and after attaining full age continued to recognize the promise, no action arose on the breach. But an actual 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 is unable to make a will. Infancy is in general a disqualification for public offices and professions, e.g., to be a member of parlia ment or an elector, a mayor or burgess, a priest or deacon, a barrister or solicitor, &c.

The custody of an infant belongs in the first place, and against all other persons, to the father, who is said to be " the guardian of his children by nature and nurture ;" and the father may by deed or will dispose of the custody or tuition of his children until the age of twenty^one. If the father is dead, and has appointed no testamentary guardian, themotheris recognized as "guardian by nature andnurture." But the children must be brought up in the father's religion, even when he has given no directions on the subject ; and it appears that no agreement between husband and wife to the contrary effect will be sustained. When, however, the father has in his lifetime suffered the children to be educated in their mother's religion, he may be held to have waived his rights. The Court of Chancery is unwilling to embarrass itself by departing from the strict rule, and an instance is recorded of a child which had been educated from eight to fifteen in the tenets of the Plymouth Brethren being ordered by the court to be educated in the religion of the Church of England. The right of the father to the custody of the child will be enforced, except where he has been guilty of gross immorality, by writ of habeas corpus, as long as a child is within the age of nurture, which for males at least may bo taken to be fourteen years. The infant then acquires a right of election. In two cases a boy over fourteen but under sixteen has been permitted by the court to choose, when the father had sued for the custody under a habeas corpus. In the case of female infants, tho courts have held that they do not acquire the right of election till sixteen, following the statute of Philip & Mary which punishes the abduction of maidens under

that age as a criminal offence. These rules do not apply to illegitimate children, as they are not under the legal guardianship of the putative father or the mother. The rights of the father or mother may be interfered with by the Court of Chancery under special circumstances, such as cruelty, immorality, &c. A recent Act (36 & 37 Viet. c. 12) gives power to the court to make orders for the mother of an infant under sixteen, to have or retain the custody of such infant, or to have access thereto, &c. The same statute legalizes agreements by the father to give up the custody and control of children to the wife. The Divorce Court has also very extensive powers of regulating the custody and maintenance of children, in exercising which it observes the same limits of age as the courts of law and chancery.

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 and the Elementary Educa tion Act. In the case, however, of debts contracted by a child for necessaries, the authority of the father would, to use a common phrase, be " easily implied."

Besides the natural guardianship of parents, children may in various other ways come under the authority of guardians. A recent author gives the following as a com plete list of guardians : — guardian in chivalry, in socage, in nature, by nurture, by election of the infant, by statute (4 Philip & Mary c. 8, 12 Chas. II. c. 24), by custom, by appointment of the ecclesiastical courts and of the Court of Chancery, foreign guardians, and guardians ad litem (Simpson's Law of Infants, London, 1875). Some of these have already been noticed, and others are obsolete or nearly so. The Act of Chas. II. enables the father to appoint a testamentary guardian to his children during infancy or any less period, who shall have the charge of the infant's real and personal estate. The Act is not to prejudice any customary guardianship, such as that of the City of London, where, according to ancient but now disused custom, the goods and lands of the orphans of freemen are in the custody of the lord mayor and aldermen in their court of orphans. By the custom of Kent, and by the special customs of certain manors, the lord has the right of appointing guardians to infant tenants. Guardian ship by socage extends only to lands of socage tenure, and belongs to the next of blood of the infant, other than those who might succeed to the estate when the infant dies. It ends when the infant reaches the age of fourteen ; after that age, or before if there was no guardian, infants were formerly allowed to elect a guardian, but that practice is now superseded by the action of the Court of Chancery which appoints guardians in all cases where it is for the benefit of the infants to do BO. An infant under a guardian appointed by the Court of Chancery is properly a " ward of court," but the term is also applied to infants brought under the authority of the court although no guardian be appointed. The office and duty of the guardian extend to the custody and control of the infant, to his education, maintenance, and advancement out of any property that may be available therefor, and to the prevention of im proper and disparaging marriages. The office of guardian is strictly a trust, the ordinary duties and responsibilities of a trustee with respect to property being accompanied by peculiar rights and duties with respect to the person of the cestui que trust. He must act in all cases for the benefit of the infant, and may not put himself into any position in which his interest would be hostile to that of the infant. The Court of Chancery has full jurisdiction over guardians of every kind, whether appointed by itself or not, and if it cannot actually dismiss a testamentary guardian, it may supersede him and entrust the charge of

I N F — I N F

3

the infant to some other person on proper cause being shown (see TRUSTEE).

An infant is liable to a civil action for torts and wrongful acts committed by him. But, as it is possible sometimes so to shape the pleadings in an action 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 commit ting 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 reduc tion 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 u contract in itself just and reasonable would not be lesion entitling to restitution. Deeds in nonage, other than those whi^h are absolutely null ab initio, must bo challenged within the quad- ricnnium utile, or four years after majority.

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. (E. R. )

INFANTICIDE. The history of infanticide as an archaic institution has already been referred to in the article FOUNDLING HOSPITALS (vol. ix. p. 481). Children of both sexes were sacrificed as religious offerings. Indeed, in some cases, e.g., in expiations for sacrilege, the boy, as being the nobler child, was preferred. But what may be called the normal infanticide of early society was probably confined to girls. The custom is in certain places and for certain periods explained by the system of exogamy ; but much more generally, as in China at the present day, it is simply an illustration of what Malthus would call a ."positive check," the reckless propagation of children far outrunning the means of subsistence which the energy of the parents can provide. Infanticide still survives among many savage races ; and, where the necessity for strong warriors is felt, a selection is sometimes made of the weaker children for destruction. But this political element seldom enters into the custom. It is because girls cannot provide for themselves that they are killed.

More complex were the leading forms of infanticide in India, now suppressed by the wise action of the British Government. The practice, though forbidden by both the Vedas and the Koran, prevailed among the Rajputs and certain of the aboriginal tribes. Among the aristocra tic Rajputs, for example, it was thought dishonour able 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 obviously most powerful with the highest castes, in which, accordingly, the disproportion of the sexes was painfully apparent. But, assuming marriage to be possible, it is ruinously expensive to the bride's father. He has to make gifts of money, clothes, jewels, and sweetmeats to the bride groom's relatives ; and when the marriage ceremony comes,

he has, chiefly owing to the exactions of the Brahmans and Bhats or minstrels, to face a lavish expenditure on feasts which in the case of some rajas has been known to reach more than £100,000. To avoid all this, the Rajput killed a certain proportion of his daughters, — sometimes in a very singular way. A pill of tobacco and bhang might be given to^the newborn child to swallow; or it was drowned in milk ; or the mother's breast was smeared with opium or the juice of the poisonous Datum. A common form 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 Kaudhs, 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. The 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 present century. One of its chief events, representing many minor events, was the Umritsur durbar of 1853, which was arranged by the late Lord Lawrence. At that great meeting the chiefs residing in the Punjab and the trans-Sutlej states signed an agreement engaging to expel from casto 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 Rajput clans keep their female offspring suspiciously close to the lowest average which secures them from surveillance.

The modern crime of infanticide shows no symptom of diminution in the leading nations of Europe. In all of them it is closely connected with illegitimacy in the class of farm and domestic servants. The crime is generally committed by the mother for the purpose of completing the concealment of her shame, and in other cases, where shame has not survived, in order to escape the burden of her child's support. The paramour sometimes aids in the crime, which is not confined to unmarried mothers. The ease with which affiliation orders are obtained in Great Britain must save the lives of many children. In France, where the inquiry into paternity is forbidden, a controversy still goes on as to the influence of hospitals for " assisted children," which are said to save life at the expense of morality. It seems certain that the great administrative change involved in closing the " tour " at these hospitals has not perceptibly affected infanticide in France. The laws of tho 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, the inexcusable killing of infants is theoretically murder. Tho infant must of course 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.' A 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.

I N F — I N P

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. There now is a presumption, said to be based on considerations of humanity, that every newborn child found dead is born dead until the contrary is very clearly shown. 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 has probably by this time reached the minds of the class to which most of the unhappy mothers belong. Partly to meet this complaint, it was suggested to the Royal Com mission 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.

It is a statutory offence, under 24 & 25 Viet. c. 100, to administer poison or any noxious thing to a woman with child with intent to procure her miscarriage, or to use any instrument for the same purpose, the maximum punish ment being penal servitude for life. The previous law had drawn the distinction of " quick with child," and in such cases had punished capitally. It was a very old con troversy among the philosophers and physicians of antiquity, when the foetus ceased to be pars viscerum matris and became "vital," or, as it was afterwards called, "ani mate." The law has not yet succeeded in putting down the degraded and wicked trade in abortion. There can be no doubt from the French and American treatises of Gallard and Storer that the crime prevails extensively, and even in classes of society in which infanticide proper would not be thought of without a shudder.

Under the same statute it is a misdemeanour punishable by two years' imprisonment with hard labour, as a maximum, to endeavour to conceal the birth of a child by any secret disposition of its dead body, whether the child died before, after, or at its birth. This does not apply to very premature births, where it was impossible that the foetus should be alive. But it does apply to all those numerous cases where the child's body, without being actually hidden, is placed where it is not likely to be found except by accident, or upon search. Lastly, under the same statute it is a misdemeanour punishable by five years' penal servitude, as a maximum, to abandon or expose a child under the age t)f two years, so as to endanger its life, or to inflict permanent injury, actual or probable, upon its health.

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 mere pur pose of obtaining the burial money from a benefit club.1 Lu 1871 the House of Commons found it necessary to appoint a select committee "to inquire as to the best means of preventing the destruction of the lives of infants put out to nurse for hire by their parents." The trials of

1 See Itejmrt on the Sanitary Condition of the Lnfiouring Classes, "Supplementary Report on Interment in Towns," by Edwin Chad- wick (Parl. Papers, 1843, xii. 395); and The Social Condition and Education of the People, by Joseph Kay, 1850.

Margaret Waters and Mary Hall called attention to the infamous relations between the lying-in houses and the baby-farming houses of London. The form was gone through of paying a ridiculously insufficient sum for the maintenance of the child. " Improper and insufficient food," said the committee, " opiates, drugs, crowded rooms, bad air, want of cleanliness, and wilful neglect are sure to be followed in a few months by diarrhoea, convulsions, and wasting away." These unfortunate children were nearly all illegitimate, and the mere fact of their being hand-nursed, and not breast-nursed, goes some way (according to the experience of the Foundling Hospital and the Magdalene Home) to explain the great mortality among them. Such children, when nursed by their mothers in the workhouse, generally live. The practical result of the committee of 1871 was the Act of 1872, 35 & 36 Viet. c. 38, which provides for the compulsory registration of all houses in which more than one child under the age of one year arc received for a longer period than twenty-four hours. No licence is granted by the justices of the peace, unless the house is suitable for the purpose, and its owner a person of good character and able to maintain the children. Offences against the Act, including wilful neglect of the children even in a suitable house, are punishable by a fine of £5 or six months' imprisonment with or without hard labour.

The law of Scotland also treats the unlawful killing of completely born infants as murder. In such cases a verdict of culpable homicide is usually returned, the punishment being entirely in the discretion of the court. Still more commonly the charge of concealment of pregnancy is made under the Act 49 Geo. III. c. 14, the maximum punishment being two years' imprisonment. It must be shown that the woman concealed her condition during the whole period of pregnancy, and did not call for help at the birth. Unlawfully procuring abortion, whether by drugs or instruments, is also a crime known to the common law of Scotland, the punishment being penal servitude or imprisonment according to circumstances. In a variety of cases, which do not admit of general statement, convictions have also been obtained against parents of exposing and deserting children or placing them in danger, and of cruel and unnatural treatment and neglect.

Infanticide will have to be further considered under the heading MEDICAL JURISPRUDENCE. For that branch of the subject the works of Tardieu and Taylor are the most important and recent authorities. See also White- head On Abortion and Sterility, and the works of Gallard and Storer already referred to.

Bibliography. — Besides a very large number of theses and special dissertations, and the chapters on the subject in the leading works in medical jurisprudence, the following are the most important works on infanticide. Ploucquet, Commcntarius Medicus inproccs- sus criminates super homicidio, infanticidio, &c., 1736; W. Hun ter, Uncertainty of the Signs of Murder in Bastard Children ; Olivard, DC I' infanticide ct des moyens que Von employe pour le con- stater, Paris, 1802 ; Malion, An Essay on the Signs of Murder in Ncw-Born Children, translated by Johnson, Lancaster, 1813; -Arrowsmith, Medico-legal Essay on Infanticide, Edin., 1828 ; Cummin, Proofs of Infanticide Considered, London, 1836; Ryan, Child Murder in its Sanitary and Social Bearings, 1858, and Infan ticide, its Law, Prevalence, Prevention, and History, London, 1862; Kunzo, Der Kindermord, historisch u. kritisch dargcstcllt, Leipsic, 1860 ; Greaves, Observations on some of the causes of Infanticide, Manchester, 1863, and Observations on the Laws referring to Child Murder and Criminal Abortion, Manchester, 1864 ; Storer and Heard, Criminal Abortion, its Nature, Evidence, and Law, Boston, 1868 ; Tardieu, Etude med..-leg. et clinique sur Vavortement, Paris, 1864, and Etude med.-Ug. sur I' infanticide,, Paris, 1880; Toul- mouche, Etudes sur I'infanticidc et la grossesse cachee ou simuUe Paris, 1875 ; Gallard, Do I'avortcmcnt au point de vue med.-lcg., Paris, 1878. There are several works describing Indian infanticide. The best known is Infanticide., its Origin, Progress, and Suppression, London, 1857, by J. Cave Browne. See also the works of Moore, Cormack, and Wilson. (W. C. S. )

INFANTRY. See ARMY.

INFINITESIMAL CALCULUS

T

HISTORICAL INTRODUCTION.

HE mathematical and physical sciences owe their present great development to the introduction of the infinitesimal calculus. The power, for example, of that calculus as an instrument of analysis has vastly extended the science of geometry, so that the investigations of the ancient Greeks go but a short way into the field of know ledge which has been laid open by the modern method; the discoveries of Archimedes and Apollonius are now easy deductions from its more extended results.

So long as the early geometers confined their speculations to the comparison of the areas of rectilinear figures they encountered little difficulty. They readily showed that the determination of the area of any such figure can be reduced to that of a rectangle, or of a square, and thus be completely effected. This process of finding areas was named the " method of quadratures." It failed, however, when they attempted to determine the areas bounded by curved lines, or the surfaces of the elementary solids such as the right cone and the sphere. In treating of these the ancients found it necessary to introduce new notions and modes of demonstration into geometry, and the diffi culty of comparing the areas of curvilinear with those of rectilinear figures gave rise to the " method of exhaus tions." The fundamental principle of this method con sists in conceiving the continual approach of two varying magnitudes to a fixed intermediate magnitude, with which they never become identical, though they may approach it to within less than any assignable difference. For example, a polygon may be inscribed in a circle, and another cir cumscribed to it, each differing from it by less than any assignable area ; hence the ancients may have concluded that areas of circles have to each other the same ratio as the similar polygons inscribed in or circumscribed to them, — that is, the ratio of the squares of the radii. But, as this kind of proof was of a different nature from that by which the more elementary doctrines were established, the Greek geometers fortified it by a reductio ad absurdum, — proving, in the above example, that the square of the radius of one circle is to that of another as the area of the former is to a space which is neither less nor greater than the latter, and therefore exactly •equal to it.

rchi- By the aid of this method Archimedes arrived at his edes. great geometrical discoveries. He determined that the ratio of the circumference to the diameter of a circle lies between 3| and 3|£, by considering the regular polygons of ninety-six sides which may be inscribed in or circum scribed to the circle. He proved that the area of a segment of a parabola cut off by any chord equals two- thirds of a parallelogram included between the chord and the parallel tangent to the curve. He determined the f quadrature of the ellipse. In the curves named after him the " spirals of Archimedes," he showed how to draw a tangent at any point, and also determined the area of any portion.

In space of three dimensions, Archimedes proved that the surface of a sphere equals four times that of one of its great circles, that the surface of a spherical cap is equal to the area of a circle the length of whose radius is the distance from the vertex of the cap to any point on its bounding circle; that a sphere has a volume which is two-thirds of that of a cylinder circumscribed to it, and that their surfaces are in the same ratio. Further, the same method of exhaustions furnished Archimedes with the cubature of conoids and spheroids, as he termed

surfaces generated by the revolution of the parabola, the hyperbola, and the ellipse.

During nearly two thousand years no new method Kepler, enabled mathematicians to rise to a higher generality than that attained in the works of the great Greek geometers. The celebrated Kepler was the first to extend the results of Archimedes. In his treatise entitled Nova Stereometria Dolionmi; accessit stereometric Archimedeae supplementum (1G15),1 he discussed a number of solids of revolution, — for example, those formed by the revolution of a conic section about any ordinate, or a tangent at the vertex, or any line within or without the curve. Thus ho con sidered some ninety new solids, and proposed problems concerning them; of these problems he resolved only a few of the most simple. In this treatise he introduced for the first time the name and notion of " infinity " into the language of geometry. Thus, ho considered a circle as composed of an "infinite" number of triangles, having their common vertex at the centre, and forming the circumference by their bases. In like manner lie regarded a cone as composed of an infinite number of pyramids, having their vertices at its vertex, and stand ing on an infinite number of triangular bases, bounded by the circular base of the cone. It may also be noted that Kepler was the first to observe that the increment of a variable — the ordinate of a curve, for example — is evan escent for values infinitely near a maximum or minimum value of the variable. This remark contains the germ of the rule for determining "maxima" and "minima," given by Format about twenty years subsequently.

Several years after Kepler had given his method of Cava- determining volumes of revolution, another celebrated lieri- theory, of a similar kind, — the " geometry of indivisibles " (1635) of Cavalieri, professor of geometry at Bologna, — marks an epoch in the progress which science has made in modern times. In this work lines were considered as composed of an infinite number of points, surfaces of lines, and solids of surfaces. For example, if the perpendicular of a triangle be divided into an infinite number of equal parts, and through each point of division a line be drawn parallel to the base and terminated by the sides, then, according to Cavalieri, we may consider the area of the triangle as the sum of all these parallel lines, regarded as its elements. Again, as these parallels form a series in arithmetical progression, of which the first term is zero, this sum is represented by half the product of the last term and the number of terms. Now the base js the last term, and the altitude measures the number of terms ; thus he deduced the ordinary expression for the area of a triangle. Cavalieri applied his method to a number of problems, such as finding the volumes of pyramids, the areas of certain simple curves of the parabolic species, the determination of centres of gravity, &c. ; and it is remarkable that he was the first who gave an accurate demonstration of the well- known properties of the centre of gravity, originally announced by Pappus, but commonly called Guldin's theorems. It is accordingly to Cavalieri, and not to Guldin, that the credit is due of having made the first advance beyond Pappus. Cavalieri's method is analogous to that employed in the integral calculus, the " indivisible " being that which has since been styled the " differential element " of the integral.

J This work is enlarged from his earlier Stereometria Dolionim Vinariorum (1605), which originated in a dispute with a eller ol wine as to the proper method of gauging the contents , accounts for its strange title.

6

INFINITESIMAL CALCULUS

The method of Cavalieri was severely criticized "by some of his contemporaries, more especially by Guldin. They alleged that, since a line has no breadth, no number of right lines, however great, when taken together, could make up a plane area. This objection was answered by Cavalieri ; but the reply was stated in

Pascal, the clearest form by Pascal, who observed (letter to M. do Carcavi, 1658) that the method of indivisibles possessed all the rigour of that of exhaustions, from which it differed only in the manner of expression, and that, when we con ceive an area as a sum of a system of parallel ordinates, we mean in reality an indefinite number of rectangles under the several ordinates, and the small equal portions into which we conceive the common perpendicular to these ordinates to be divided. This passage is remarkable — as was well observed by Carnot — as it shows that the notion of mathematical infinity, as now employed, was not strange to the geometers of that time ; for it is clear that Pascal employed the word " indefinite " in the same signification as we now attach to the word " infinite," and that he called " small" that which is now called "infinitely small," also that he neglected these small quantities in comparison with finite quantities — thus he regarded as simple rectangles the small portions of the area of the curve comprised between two consecutive ordinates, neglecting the small triangles which have for their bases the differences of these ordinates. Carnot adds that no person attempted to reproach Pascal with want of rigour in his demonstrations.

Pascal applied the method of Cavalieri with eminent success to the investigation of properties of the cycloid, and other problems. His researches, according to D'Alembert, closely approach to the integral calculus, and form the con nexion between the methods of Archimedes and of Newton.

Wallis. The most important application, however, of Cavalieri's method was that of Wallis, Savilian professor at Oxford, who, in 1G55, gave an admirable specimen of this method in his Arithmetical, Infinitorum, sive nova Methodus in- quirendi in Curvilineariim Quadraturam. Pursuing Cava lieri's views, he reduced the problem of finding the areas of a large number of curves, and the volumes of solids of revolution, to the summation of the powers of the terms of arithmetical series, consisting of an infinite number of terms, — or rather to the determination of the ratio of the arithmetical mean of all such powers of the terms to the like power of the last term.

For example, in the series of square numbers 0, 1, 4, 9, 16, &c., the ratio of the mean to the last is, for the first

,,

three terms,

0 + 1 + 4

4 + 4 + 4 = 15 + T« > f °r ^

JTJ-; for the first four, 0 + 1+4 + 9 + 16

0+1+4+9 9+9+9+9

J . = 1 4. _i •

'16 + 16 + 16 + 16 + 16 » aT; in like manner the next fraction is ^ + -3V- Hence Wallis noticed that the fractions approach nearer and nearer to ^ ; and, as the denominators in the fractions -jV, Jg-, ^ -i form an arithmetical series, with a common difference 6,. it follows that, when the number of terms is indefinitely increased, the resulting fraction becomes ultimately ^.

Wallis applied the same method to the series 0, I3, 23, 33, &c., and found without difficulty that the aforesaid ratio is ^ in this case ; and so generally. He also introduced into analysis the notation of fractional indices instead of radicals, and extended his method of summation to series proceeding by fractional powers of the natural numbers 1, 2, 3, &c. Wallis was enabled by these principles to obtain the areas of many curves, and the volumes of solids which had not been ^previously found. _ He also, by aid of this method, combined with the principles of " interpolation," arrived at bis well-known expression for TT, viz. : —

•K _2^ 4j6 6-8

•f ^3-3 ' 5-fl ' 77 ' '

Again, in his treatise De Curv. rectif. (1G59), Wallis showed that certain curves were capable of being " recti fied," or that straight lines might be found to which they were exactly equal, a remark which was Very soon verified by