« ՆախորդըՇարունակել »
of the convention that for u'?-2, the difference **'' is to be
understood to stand for u'. This being so, a single valued function
of u1, ... up without essential singularities for infinite or finite values of the variables can be shown, by induction, to be, as in the case of = 1, necessarily a rational function of the variables. A function aving no singularities for finite values of all the variables is as before called an integral function; it is expressible by a power series converging for all finite values of the variables; a single valued function having for finite values of the variables no singularities other than poles or polar points of indetermination is called a meromorphic function; as for f: 1 such a function can be expressed as a quotient of two integral functions having no common zero £ other than the points of indetermination of the function; ut the proof of this theorem is difficult. The single valued functions which occur, as explained above, in the inversion of algebraic integrals of the first kind, for p > 1, are meromorphic. They must also be periodic, unaffected that is when the variables u1, . . . we are simultaneously increased each by a proper constant, these being the additive constants of indeterminateness for the p integrals fri(x,y)dx arising when (x,y) makes a closed circuit, the same for each integral. The theory of such single valued meromorphic periodic functions is simpler than that of meromorphic functions of several variables in general, as it is sufficient to consider only finite values of the variables; it is the natural extension of the theory of doubly periodic functions previously discussed. It can be shown to reduce, though the proof of this requires considerable clevelopments of which we cannot speak, to the theory of a single integral function of u1, ... . up, called the Theta Function. This is expressible as a series of positive and negative integral powers of quantities exp (cital), exp (czuz), . . . exp (coup). wherein c1, . . . ce are proper constants; for p = 1 this theta function is essentially the same as that above given under a different form (see § 14, Doubl Periodic Functions), the function g(u). In the case of p = 1, all meromorphic functions periodic with the same two periods have been shown to be rational functions of two of them connected by a single algebraic cquation; in the same way all meromorphic functions of p variables, periodic with the same sets of simultaneous periods, 2p sets in all, can be shown to be expressible rationally in terms of +1 such periodic functions connected by a single algebraic equation. Act x1, ... x, y denote p-H1 such functions; then each of the partial derivatives dx, fou; will equally be a meromorphic function of the same periods, and so expressible rationally in terms of x1, ... x.p.); thus there will exist p cquations of the form
dx. : Rudui + • * * +Redup, and hence p equations of the form
du, = H, dx1+...+H,"dx, wherein H. are rational functions of x1, ... xi, y, these being connected by a fundamental algebraic (rational) equation,say f(x1,...,xn,v) to '', T. then is the generalized form of the corresponding equation
or p = I. § 26. Multiply-Periodic Functions and the Theory of Surfaces.—
The theory of algebraic integrals JR(x,y)dx, wherein x,y are connected by a rational equation f(x,y)=o, has developed concurrently with the theory of algebraic curves; in particular the existence of the number p invariant by all birational transformations is one result of an extensive theory in which curves capable of birational correspondence are regarded as equivalent; this point of view has made possible a general theory of what
might otherwise have remained a collection of isolated theorems.
In recent years developments have been made which point to a similar unity of conception as possible for surfaces, or indecd for algebraic constructs of any number of dimensions. These developments have been in two directions, at first followed independently, but now happily brought into the most intimate connexion. On the analytical side. E. Picard has considered the possibility of classifying integrals of the form f(Rds+Sdy), belonging to a surface f(x,y,z) =o, wherein R and S are rational functions of x, y, z, £ as they are (1) everywhere finite, (2) have poles, which then lie along curves upon the surface, or (3) have logarithmic infinities, also then lying along curves, and has brought, the theory to a high degree of perfection. On the geometrical side A. Clebsch and M. Noether, and more recently the Italian school, have considered the geometrical characteristics of a surface which are unaltered by birational transformation. It was first remarked that for surfaces of order n there are associated surfaces of order n-4, having properties in relation thereto amalogous to those of curves of order n-3 for a plane curve of order n; if such a surface f(x,y,z)=o have a double curve with triple points triple also for the surface, and p(x,y,z) =o be a surface of order n-4 passing through the double curve, the
algebraic constructs of any number of dimensions. The number of linearly independent surfaces of order n-4, possessing the requisite rticularity in regard to the singular lines and points of the surface, is thus a number invariant by birational transformation, and the equality of these numbers for two surfaces is a necessary condition of their being capable of such transformation. The number of surfaces of order m having the assigned particularity in regard to the singular points and lines of the fundamental surface can be given by a formula for a surface of given singularity; but the value of this formula for m=n-4 is not in all cases equal to the actual number of surfaces of order n-4 with the assigned particularity, and for a cone (or ruled surface) is in fact negative, being the negative of the deficiency of the plane section of the cone. Nevertheless this number for m=n-4 is also found to be invariant for birational transformation. This number, now denoted by p., is then a second invariant of birational transformation. The former number, of actual surfaces of order n-4 with the assigned particularity in regard to the singularities of the surface, is now denoted by ps. The difference pe-pe, which is never negative, is a most important characteristic of a surface. When it is zero, as in the case of the general surface of order n, and in a vast number of other ordinary cases, the surface is called regular. On a plane algebraical curve we may consider linear series of sets of points, obtained by the intersection with it of curves A4 +A141+ ... =o, wherein A, A1, . . . are variable coefficients; such a series consists of the sets of points where a rational function of given poles, belonging to the construct f(x,y) =o, has constant values. And we may consider series of sets of points determined by variable curves whose coefficients are algebraical functions, not necessarily rational. functions, of parameters. Similarly on a surface we may consider linear systems of curves, obtained by the intersection with the given surface of variable surfaces. At +Alti-H. : ... =o, and may consider algebraic systems, of which the individual curve is given by variable surfaces whose coefficients are algebraical, not necessarily rational, functions of parameters. Of a linear series upon a plane curve there are two numbers manifestly invariant in birational transformation, the order, which is the number of points forming a set of the series, and the dimension, which is the number of parameters),f\,X2/A, ..... entering linearly in the equation of the series. The series is complete when it is not contained in a series of the same order but of higher dimension. So for a lincar system of curves upon a surface, we have three invariants for birational transformation; the order, being in the number of variable intersections of two curves of the system, the dimension, being the number of linear parameters A1/A, X2/A, . . . in the equation for the system, and the deficiency of the individual curves of the system. Upon any curve of the linear system the other curves of £ system define a linear series, called the characteristic series; but even when the linear system is complete, that is, not contained in another linear system of the same order and higher dimension, it does not follow that the characteristic series is complete; it may be contained in a series whose dimension is greater by po-pa than its own dimension. When this is so it can be shown that the linear system of curves is contained in an algebraic system whose dimension is greater by p-pathan the dimension of the linear system. The extra p = pe—p. variable parameters so entering may be regarded as the independent co-ordinates of an algebraic construct f(y,x1, ... xi) =o; this construct has the roperty that its co-ordinates are single valued meromorphic unctions of # variables, which are periodic, possessing 2p systems of periods; the p variables are expressible in the forms
wherein R. (x,y) denotes a rational function of x1, . . . xr and y. The original surface has correspondingly p integrals of the form f(Rdx+Sdy), wherein R, S are rational in x, y, z, which are everywhere finite; and it can be shown that it has no other such integrals. From this point of view, then, the number p, = pe—pa is, for a surface, analogous to the deficiency of a plane curve; another analogy arises in the comparison of the theorems: for a plane curve of zero deficiency there exists no algebraic series of sets of points which does not consist of sets belonging to a linear series; for a surface for which ps—p. =9 there exists no algebraic system of curves not contained in a linear system.
But whereas for a £ curve of dehciency zcro, the co-ordinates of the points of the curve are rational functions of a single parameter, it is not necessarily the case that for a surface having pe—pa =o the co-ordinates of the points are rational functions of two parameters; it is necessary that p, -pa =o, but this is not sufficient. For surfaces, beside the pe linearly independent surfaces of order n–4 having a definite particularity at the singularitics of the surface, it is useful to consider surfaces of order k(n–4), also having each a definite particularity at the singularities, the number of these, not containing the original surface as component, which are linearly independent, is denoted by P*. It can then be stated that a sufficient condition for a surface to bf rational consists of the two conditions p. =o, P =o. ... More generally it becomes a problem to classify surfaces according to £ values of the various nurhbers which are invariant under birational transformation, and to determine for each the simplest form of surface to which it is birationally equivalent. Thus, for example, the hyperelliptic surface discussed £, of which the co-ordinates are meromorphic functions of two variables of the simplest kind, with four sets of periods, is characterized by p = 1, p.- -1; or again, any surface possessing a linear system of curves of which the order exceeds twice the deficiency of the individual curves diminished by two, is reducible by birational transformation to a ruled surface or is a rational surface. But beyond the general statement that much progress has already been finade in this direction, of great interest to the student of the theory of functions, nothing further can be added here. BiblioGRAPHY.-The learner will find a lucid introduction to the theory in E. Goursat, Cours d'analyse mathématique, t. ii. (Paris. # or, with much greater detail, in A. R. Forsyth, Theory of unctions of a Complex Variable (2nd ed., Cambridge, '' for logical rigour in the more difficult theorems, he £i consult # F. Osgood, Lehrbuch der Functionentheorie, 3d. i. (Leipzig, 1906– 1907); for greater precision in regard to the necessary quasigeometrical axioms, beside the indications attempted here, he should consult W. H. Young, The Theory of Sets of Points (Cambridge, 1906), chs, viii.-xiii., and C. Jordan, Cours d'analyse, t. i. (Paris, # chs. i., ii.; a comprehensive account of the Theory of Functions of Real Variables is by E. W. Hobson (Cambridge, 1907). Of the theory regarded as based after Weierstrass upon the theory of power series, there is J. Harkness and F. Morley, Introduction to the Theory of Analytic Functions (London, 1898), an clementary treatise; for the theory of the convergence of scrics there is also T. J. I’A. Bromwich, An Introduction to the Theory of Infinite Series (London, 1908); but the student should consult the collected works of Weicrstrass (Berlin, 1894 ff.), and the writings of Mittag-Leffler in thcearly volumes of the Acta mathematica; carlier cxpositions of the theory of functions on the basis of power scrics are in C. Méray, Leçons nouvelles sur l'analyse infinitésimale (Paris, 1894), and in Lagrange's books on the Theory of Functions. An account of the theory of potential in its applications to the present theory is found in most treatises; in particular consult E. Picard, Traité d'analyse, t. ii. (Paris, 1893). For clliptic functions there is an introductory book, P. Appell and E. Lacour, Principes de la théoriedes fonctionselliptiques et applications (Paris, 1897), beside the treatises of G. H. Halphen, Traité des fonctions elliptiques et de leurs # (three parts, Paris, 1886 ff.), and J. Tannery et J. Molk, Eléments de la théorie des fonctions elliptiques (Paris, 1893 ff.); a book, A. G. Greenhill, The Applications of Elliptic Functions (London, 1892), shows how the functions enter in problems of many kinds. For modular functions there is an cxtensive treatise, F. Klein and R. Fricke, Theorie der elliptischen Modulfunctionen (Leipzig, 1890); sce also the most interesting smaller volume, F. Klein, Uber das Ikosaeder (Leipzig, 1884) (also obtainable in English). For the theory of Riemann's surface; and algebraic integrals, an interesting, introduction is P. Appcil and E. Goursat, Théorie des fonctions algébriques et de leurs intégrales; for Abclian functions sce also H. Stahl, Theorie der Abel'schen Functionen # 1896), and H. F. Baker, An Introduction to the Theory of # y Periodic Functions (Cambridge, 1907), and H. F. Baker, Abel's Theorem and the Allied Theory, including the Theory of the Theta Functions (Cambridge, 1897); for theta f: of one variable a standard work is & G. £ Fundamenta nova, &c. # sberg, 1828); for the general theory of theta functions, consult W. irtinger, Untersuchungen über ThetaFunctionen (Leipzig, 1895). For a history of the-theory of algebraic functions consult A. Brill and M. Noether, Die Entwicklung der Theorie der algebraischen Functionen in alterer und neuerer Zeit, Bericht der deutschen Mathematiker- '# #): and for a special theory of algebraic functions, K. Hensel and G. Landsberg, heorie der algebraischen Function w.s.w.. (Leipzig, W£: The student will, # course, consult also Riemann's and Weierstrass's Ges. Werke. For the applications to geometry in general an important contribution, of permanent value, is E. Picard and G.Simart, Théorie des fonctions algébriques de deux variables indépendantes (Paris, 1897-1906). This work contains, as Note v. t. ii. p. 485, a valuable summary by MM. Castelnuovo and Enriques, Sur quelques résultats ncuveaux dans la théorie des surfaces algébriques, containing many references to the numerous memoirs to be found, for the most part, in the transactions of scientific societies and the mathematical journals of Italy. - Beside the books above enumerated there exists an unlimited number of individual memoirs, often of permanent, importance and only imperfectly, or too elaborately, reproduced in the pages of the volumes in which the student will find references to them. The German Encyclopaedia of Mathematics, and the Royal Society's Reference Catalogue of Current Scientific Literature, Pure Mathematics, published yearly, should also be consulted. (H. F. B.A.)
FUNDY, BAY OF, an inlet of the North Atlantic, separating New Brunswick from Nova Scotia. It is 145 m. long and 48 m. wide at the mouth, but gradually narrows towards the head, where it divides into Chignecto Bay to the north, which subdivides into Shepody Bay and Cumberland Basin (the French Beaubassin), and Minas Channel, leading into Minas Basin, to the east and south. Off its western shore opens Passamaquoddy Bay, a magnificent sheet of deep water with good anchorage,
receiving the waters of the St Croix river and forming part of the boundary between New Brunswick and the state of Maine The Bay of Fundy is remarkable for the great rise and fall of the tide, which at the head of the bay has been known to reach 62 ft. In Passamaquoddy Bay the rise and fall is about 25 ft., which gradually increases toward the narrow upper reaches. At spring tides the water in the Bay of Fundy is 19 ft. higher than it is in Bay Verte, in Northumberland Strait, only 15 m. distant. Though the bay is deep, navigation is rendered dangerous by the violence and rapidity of the tide, and in summer by frequent fogs. At low tide, at such points as Moncton or Amherst, only an expanse of red mud can be secn, and the tide rushes in a bore or crest from 3 to 6 ft. in height. Large areas of fertile marshes are situated at the head of the bay, and the remains of a submerged forest show that the land has subsided in the latest geological period at least 40 ft. The bay receives the waters of the St Croix and St John rivers, and has numerous harbours, of which the chief are St Andrews (on Passamaquoddy Bay) and St John in New Brunswick, and Digby and Annapolis (on an inlet known as Annapolis Basin) in Nova Scotia. It was first explored by the Sieur de Monts (d. c. 1628) in 1604 and named by him La Baye Française. FUNERAL RITES, the ceremonics associated with different methods of disposing of the dead. (See also BURIAL AND BURIAL ACTs, CEMETERY; and CREMATION.) In general we have little record, except in their tombs, of races which, in a past measured not merely by hundreds but by thousands of years, occupied the earth; and exploration of these often furnishes our only clue to the religions, opinions, customs, institutions and arts of long vanished societies. In the case of the great culture folks of antiquity, the Babylonians, Egyptians, Hindus, Persians, Greeks and Romans, we have, besides their monuments, the evidence of their literatures, and so can know nearly as much of their rites as we do of our own. The rites of modern savages not only help us to interpret prehistoric monuments, but explain peculiaritics in our own rituals and in those of the culture folks of the past of which the significance was lost or buried under ctiological myths. We must not then confine ourselves to the rites of a few leading' races, neglecting their less fortunate brethren who have never achieved civilization. It is better to try to classify the rites of all races alike according as they embody certain leading conceptions of death, certain fears, hopes, beliefs entertained about the dead, about their future, and their relations with the living. The main ideas, then, underlying funeral rites may roughly be enumgrated as follows: 1. The pollution or taboo attaching to a corpse. 2. Mourning. 3. The continued life of the dead as evinced in the housing and equipment of the dead, in the furnishing of food for them, and in the orientation and posture assigned to the 4. Communion with the dead in a £east and otherwisc. 5. Sacrifice for the dead and expiation of their sins. 6. Death witchery. 7. Protection of the dead from ghouls. 8. Fear of ghosts. 1. A dead body is unclean, and the uncleanness extends to things and persons which touch it. Hence the Jewish law (Num. v. 2) enacted that “whoever is unclean by the dead shall be put outside the camp, that they defile not the camp in the midst whereof the Lord dwells.” Such persons were unclean until the even, and might not eat of the holy things unless they bathed their flesh in water. A high priest might on no account “go in to any dead body” (Lev. xxi. 11). Why a cofpse is so widely tabooed is not certain; but it is natural to see one reason in the corruption which in warm climates soon sets in. The common experience that where one has died another is likely to do so may also have contributed, though, of course, there was no scientific idea of infection. The old Persian scriptures are full of this taboo. He who has touched a corpse is "powerless in mind, tongue and hand” (Zend Avesta in Sacred Books of the East, pt. i. p. 120), and the paralysis is inflicted by the innumerable drugs or evil spirits which invest a corpse, Fire and earth, being alike creations of the good and pure god Ahuramazda, a body must not be burned or buried, and so the ancient Persians and their descendants the Paksees build Dakmas or “towers of silence” on hill-tops far from human habitations. Inside these the corpses are laid on a flagged terrace which drains into a central pit. Twice a year the bones, picked clean by dogs and birds of prey, are collected in the pit, and when it is full another tower is built. In ancient times perhaps the bodies of the magi or priests alone were exposed at such expense; the common folk were covered with wax and laid in the earth, the wax saving the earth from pollution. In Rome and Greece the corpse was buried by night, lest it should pollute the sunlight; and a trough of water was set at the door of the house of death that men might purify themselves when they came out, before mixing in general society. Priests and magistrates in Rome mights not meet or look on a corpse, for they were thereby rendered unclean and incapable of fulfilling their official duties without undergoing troublesome rites of purification. At a Roman funeral, when the remains had been laid in the tomb, all present were sprinkled with lustral water from a branch of olive or laurel called aspergillum; and when they had gone home they were asperged afresh and stepped over a fire. The house was also swept out with a broom, probably lest the ghost of the dead should be lying about the floor. Many races, to avoid pollution, destroy the house and property of the deceased. Thus the Navahos pull down the hut in which he died, leaving its ruins on the ground; but if it be an expensive hut, a shanty is extemporized alongside, into which the dying man is transferred before death. No one will use the timbers of a hut so ruined. A burial custom of the Solomon Islands, noted by R. H. Codrington (The Mclanesians, p. 255), may be dictated by the same scruple. There “the mourners having hung up a dead man's arms on his house make great lamentations; all remains afterwards untouched, the house goes to ruin, mantled, as time goes on, with the vines of the growing yams, a picturesque and indeed, perhaps, a touching sight; for these things are not set up that they may in a ghostly manner accompany their former owner.” H. Oldenberg (Religion des Veda, p. 426) describes how Hindus shave themselves and cut off their nails after a death, at the same time that they wash, renew the hearth fire, and furnish themselves with new vessels. For the hair and nails may harbour pollution, just as the medieval Greeks believed that evil spirits could lurk in a man's beard (Leo Allatius, De opinionibus quorundam Graccorum). The dead man's body is shorn and the nails cut for a kindred reason; for it must be purified as much as can be before it is burned as an offering on the pyre and before he cnters on a new sphere of existence. 2. We are accustomed to regard mourning costume as primarily an outward sign of our grief. Originally, however, the special garb seems to have been intended to warn the general public that persons so attired were unclean. In ancient Rome mourners stayed at home and avoided all feasts and amusements; laying aside gold, purple and jewels, they wore black dresses called lugubria or even skins. They cut neither hair nor beard, nor lighted fire. Under the emperors women began to wear white. On the west coast of Africa negroes wear white, on the Gold Coast red. The Chinese wear hemp, which is cheap, for mourning dress must as a rule be destroyed when the season of grief is past to get rid of the taboo. Among the Aruntas of Australia the wives of a dead man smear themselves with white pipe-clay until the last ceremonies are finished, sometimes adding ashesthis not to conceal themselves from the ghost (which may partly be the aim of some mourning costumes), but to show the ghost that they are duly sorrowing for their loss. These widows must not talk except on their hands for a whole year. “Among the Maoris,” says Frazer (Golden Bough, i. 323), “anyone who had handled a corpse, helped to convey it to the grave, or touched a dead man's bones, was cut off from all intercourse and almost all communication with mankind. He could not enter any house, or come into contact with any person or thing, without utterly bedevilling them. He might not even touch food with his hands, which had become so frightfully tabooed or unclean as to be quite useless. Food would be set for him on the ground.
and he would thensit or kneel down, and, with his handscarefully held behind his back, would gnaw at it as best he could.” Often a degraded outcast was kept in a village to feed mourners. Such a taboo is strictly similar to those which surround a sacred chief or his property, a menstruous woman or a homicide, rendering them dangerous to themselves and to all who approach them. 3. Primitive folk cannot conceive of a man's soul surviving apart from his body, nor of another life as differing from this, and the dead must continue to enjoy what they had here. Accordingly the Patagonians kill horses at the grave that the dead may ride to Alhucmapu, or country of the dead. After a year they collect a chief's bones, arrange them, tie them together and dress them in his best garments with beads and feathers. Then they lay him with his weapons in a square pit, round which dead horses are placed set upright on their feet by stakes. As late as 1781 in Poland F. Casimir's horse was slain and buried with him. In the Caucasus a Christian lady's jewels are buried with her. The Hindus used to burn a man's widow on his pyre, because he could not do without her; and St Boniface commends the self-sacrifice of the Wend widows who in his day burned themselves alive on their husbands' pyres. The tumuli met with all over the north of Europe (in the Orkneys alone 20oo remain) are regular houses of the dead, models of those they occupied in life. The greater the dignity of the deceased, the loftier was his barrow. Silbury hill is 170 ft. high; the tomb of Alyattes, father of Croesus, was a fourth of a league round; the Pyramids are still the largest buildings in existence; at Oberea in Tahiti is a barrow 267 ft. long, 87 wide and 44 high. Some Eskimo just leave a dead man's body in his house, and shut it up, often leaving by his side a dog's head to guide him on his last journey, along with his tools and kayak. The Sea Dyaks set a chief adrift in his war canoe with his weapons. So in Norse story Hake “was laid wounded on a ship with the dead men and arms; the ship was taken out to sea and set on fire.” The Viking was regularly buried in his ship or boat under a great mound. He sailed after death to Valhalla. In the ship was laid a stone as anchor and the tools, clothes, weapons and treasures of the dead. The Egyptians, whose land was the gift of the river Nile, equally believed that the dead crossed over water, and fashioned the hearse in the form of a boat. Hence perhaps was derived the Greek myth of Charon and the Styx, and the custom, which still survives in parts of Europe, of placing a coin in the mouth of the dead with which to pay the ferryman. The Egyptians placed in the tomb books of a kind to guide the dead to the next world. The Copts in a later age did the same, and to this custom we owe the recovery in Egypt of much ancient literature. The Armenians till lately buried with a priest his missal or gospel. In Egyptian entombments of the XIIth to the XIVth dynasties were added above the sepulchres what Professor Petrie terms soulhouses, viz, small models of houses furnished with couch and table, &c., for the use of the ka or double whenever it might wish to come above ground and partake of meats and drinks. They recall, in point of size, the hut-urns of the Etruscans, but the latter had another use, for they contain incinerated remains. Etruscan tombs, like those of Egypt and Asia Minor, were made to resemble the dwelling-houses of the living, and furnished with coffered ceilings, panelled walls, couches, stools, easy chairs with footstools attached, all hewn out of the living rock (Dennis, Cities and Cemeteries of Etruric, vol i. p. lxx.). Of the old Peruvian mummies in the Kircherian Museum at Rome, several are of women with babies in their arms, whence it is evident that a mother had her suckling buried with her; it would console her in the next world and could hardly survive her in this. The practice of burying ornaments, tools and weapons with the dead characterizes the inhumations of the Quaternary epoch, as if in that dim and remote age death was already regarded as the portal of another life closely resembling this. The cups, tools, weapons, ornaments and other articles deposited with the dead are often carefully broken or turned upside down and inside out; for the soul or manes of objects is liberated by such fracture or inversion and so passes into the dead man's use and possession. For the same reason where the dead are burned, their properties are committed to the flames. The ghost of the warrior has a ghostly sword and buckler to fight with and a ghostly cup to drink from, and he is also nourished by the impalpable odour and reek of the animal victims sacrificed over his grave. Instead of valuable objects cheap images and models are often substituted; and why not, if the mere ghosts of the things are all that the wraith can enjoy 7 Thus Marco Polo (ii. 76) describes how in the land of Kinsay (Hang-chau) “the friends and relations make a great mourning for the deceased, and clothe themselves in hempen garments, and follow the corpse, playing on a variety of instruments and singing hymns to their idols. And when they come to the burning place they take representations of things cut out of parchment, such as caparisoned horses, male and female slaves, camels, armour, suits of cloth of gold (and money), in great quantitics, and these things they put on the fire along with the corpse so that they are all burned with it. And they tell you that the dead man shall have all these slaves and animals of which the effigies are burned, alive in flesh and blood, and the money in gold, at his disposal in the next world; and that the instruments which they have caused to be played at his funeral, and the idol hymns that have been chaunted shall also be produced again to welcome him in the next world.” The manufacture of such paper simulacra for consumption at funerals is still an important industry in Chinese cities. The ancient Egyptians, assured that a man's ka or double shall revivify his body, took pains to guard the flesh from corruption, steeping the corpse in natron and stuffing it with spices. A body so prepared is called a mummy (q.v.), and the custom was already of a hoary antiquity in 3200 B.C., when the oldest dated mummy we have was made. The bowels, removed in the process, were placed in jars over the corpse in the tomb, together with writing tablets, books, musical instruments, &c., of the dead. Cemeteries also remain full of mummies of crocodiles, cats, fish, cows and other sacred animals. The Greeks settled in Egypt learned to mummify their dead, but the custom was abhorrent to the Jews, although the Christian belief in the resurrection of the flesh must have been formed to a large extent under Egyptian influence. Half the superiority of the Jewish to other ancient religions lay in this, that it prescribed no funeral rites other than the simplest inhumation. The dead all over the world and from remote antiquity have been laid not anyhow in the earth, but with the feet and face towards the region in which their future will be spent; the Samoans and Fijians towards the far west whither their souls have preceded them; the Guarayos with head turned eastwards because their god Tamoi has in that quarter “his happy hunting grounds where the dead will meet again” (Tylor, Prim. Cult. ii. 422). The legend is that Christ was buried with His head to the west, and the church follows the custom, more ancient than itself, of laying the dead looking to the East, because that is the attitude of prayer, and because at the last trump they will hurry eastwards. So in Eusebius (Hist. Eccl. 430, 19) a martyr explains to his pagan judge that the heavenly Jerusalem, the fatherland of the pious, lay exactly in the east at the rising place of the sun. Where the body is laid out straight it is difficult to discern the presence of any other idea than that it is at rest. In Scandinavian barrows, e.g. in the one opened at Goldhavn in 1830, the skeletons have been found seated on a low stone bench round the wall of the grave chamber facing its opening, which always looks south or east, never north. Here the dead were continuing the drinking bouts they enjoyed on earth. The Peruvians mummified their dead and placed them jointed and huddled up with knees to chin, looking toward the sunset, with the hands held before the face. In the oldest prehistoric tombs along the Nile the bodies are doubled up in the same position. It would seem as if in these and numerous other similar cases the dead were deliberately given in their graves the attitude of a foetus in the womb, and, as Dr Budge remarks (Egyptian Ideas of the Future Life, London, 1899, p. 162), “we may perhaps be justified in secing in this custom the symbol of a hope that, as the child is born from this position into the
world, so might the deceased be born into the life beyond the grave.” The late Quaternary skeletons of the Mentone cave were laid in a layer of ferrugineous earth specially laid down for them, and have contracted a red colour therefrom. Many other prehistoric skeletons found in Italy have a reddish colour, perhaps for the same reason, or because, as often to-day, the bones were stripped of flesh and painted. Ambrose relates that the skeletons of the martyrs Gervasius and Protasius, which he found and deposited A.D. 386 under the altar of his new basilica in Milan, were mirae magnitudinis ut prisca actas ferebat, and were also coloured red. He imagined the red to be the remains of the martyrs' blood! Hic sanguis clamat coloris indicio. Salomon Reinach has rightly divined that what Ambrose really hit upon was a prehistoric tomb. Red earth was probably chosen as a medium in which to lay a corpse because demons flee from red. Sacred trees and stones are painted red, and for the most solemn of their rites savages bedaub themselves with red clay. It is a favourite taboo colour. 4. A feast is an essential feature of every primitive funeral, and in the Irish “wake" it still survives. A dead man's soul or double has to be fed at the tomb itself, perhaps to keep it from prowling about the homes of the survivors in search of victuals; and such food must also be supplied to the dead at stated intervals for months or years. Many races leave a narrow passage or tube open down to the cavity in which the corpse lies, and through it pour down drinks for the dead. Traces of such tubes are visible in the prehistoric tombs of the British Isles. However, such provision of food is not properly a funeral feast unless the survivors participate. In the Eastern churches and in Russia the departed are thus fed on the ninth, twelfth and fortieth days from death. “Ye appease the shades of the dead with wine and meals,” was the charge levelled at the Catholics by the 4th-century Manichaeans, and it has hardly ceased to be true even now after the lapse of sixteen centuries. The funeral feast proper, however, is either a meal of communion with or in the dead, which accompanies interment, or a banquet off the flesh of victims slain in atonement of the dead man's sins. Some anthropologists see in the common meal held at the grave “the pledge and witness of the unity of the kin, the chief means, if not of making, at least of repairing and renewing it.”" The flesh provided at these banquets is occasionally that of the dead man himself; Herodotus and Strabo in antiquity relate this of several half-civilized races in the East and West, and a similar story is told by Marco Polo of certain Tatars. Nor among modern savages are funeral feasts off the flesh of the dead unknown, and they seem to be intended to effect and renew a sacramental union or kinship of the living with the dead. The Uaupes in the Amazons incinerate a corpse a month after death, pound up the ashes, and mix them with their fermented drink. They believe that the virtues of the dead will thus be passed on to his survivors. The life of the tribe is kept inside the tribe and not lost. Such cannibal sacraments, however, are rare, and, except in a very few cases, the evidence for them weak. The slaying and eating of animal victims, however, at the tomb is universal and bears several meanings, separatcly or all at once. The animals may be slain in order that their ghosts may accompany the deceased in his new life. This significance we have already dwelt upon. Or it is believed that the shade feeds upon them, as the shades came up from Hades and lapped up out of a trench the blood of the animals slain by Ulysses. The survivors by eating the flesh of a victim, whose blood and soul the dead thus consume, sacramentally confirm the mystic tie of blood kinship with the dead. Or lastly, the victim may be offered for the sins of the dead. His sins are even supposed to be transferred into it and eaten by the priest Such expiatory sacrifices of animals for the dead survive in the Christian churches of Armenia, Syria and of the East generally. Their vicarious characteriscmphasized in the prayers which accompany them, but the popular understanding of them probably combines all the meanings above enumerated. It has been suggested by Robertson Smith (Religion of the Semites, 336) that the world-wide customs of * E. S. Hartland. Legend of Perseus (1895), ii. 278.
tearing the hair, rending the garments, and cutting and wounding the body were originally intended to establish a life-bond between the dead and the living. The survivors, he argues, in leaving portions of their hair and garments, and yet more by causing their own blood to stream over the corpse from self-inflicted wounds, by cutting off a finger and throwing it into the grave, leave what is eminently their own with the dead, so drawing closer their tie with him. Conversely, many savages daub themselves with the blood and other effluences of their dead kinsmen, and explain their custom by saying that in this way a portion of the dead is incorporated in themselves. Often the survivors, especially the widows, attach the bones or part of them to their persons and wear them, or at least keep them in their houses. The retention of the locks of the deceased and of parts of his dress is equally common. There is also another side to such customs. Having in their possession bits of the dead, and being so far in communion with him, the survivors are surer of his friendship. They have ensured themselves against ghosts who are apt to be by nature envious and mischievous. But whatever their original significance, the tearing of cheeks and hair and garments and cutting with knives are mostly expressions of real sorrow, and, as Robertson Smith remarks, of deprecation and supplication to an angry god or spirit. It must not be supposed that the savage or ancient man feels less than ourselves the poignancy of loss. 6. Death-witchery has close parallels in the witch and heretic hunts of the Christians, but, happily for us, only flourishes to-day among savages. Sixty #6 of the deaths which occur in West Africa are, according to Miss Mary Kingsley-acredible witness—believed to be due to witchcraft and sorcery. The blacks regard old age or effusion of blood as the sole legitimate causes of death. All ordinary diseases are in their opinion due to private magic on the part of neighbours, just as a widespread epidemic marks the active hatred “of some great outraged nature spirit, not of a mere human dabbler in devils.” Similarly in Christian countries an epidemic is set down to the wrath of a God offended by the presence of Jews, Arians and other heretics. The duty of an African witch-doctor is to find out who bewitched the deceased, just as it was of an inquisitor to discover the heretic. Every African post-mortem accordingly involves the murder of the person or persons who bewitched the dead man and caused him to die. The death-rate by these means is nearly doubled; but, since the use of poison against an obnoxious neighbour is common, the right person is occasionally executed. It is also well for neighbours not to quarrel, for, if they do and one of them dies of smallpox, the other is likely to be slain as a witch, and his lungs, liver and spleen impaled on a pole at the entrance of the village. It is the same case with the Australian blacks: “no such thing as natural death is realized by the native; a man who dies has of necessity been killed by some other man, or perhaps even by a woman, and sooner or later that man or woman will be attacked. In the normal condition of the tribe every death meant the killing of another individual.”? 7. Lastly, a primitive interment guards against the double risk of the ghost haunting the living and of ghouls or vampires taking possession of the corpse. The latter end is likely to be achieved if the body is cremated, for then there is no nidus to harbour the demon; but whether, in the remote antiquity to which belong many barrows containing incinerated remains, this motive worked, cannot be ascertained. The Indo-European race seems to have cremated at an early epoch, perhaps before the several races of East and West separated. In Christian funeral rites many prayers are for the protection of the body from violation by vampires, and it would seem as if such a motive dictated the architectural solidity of some ancient tombs. Christian graves were for protection regularly sealed with the cross; and the following is a characteristic prayer from the old Armenian rite for the burial of a layman: Mary Kingsley, West African Studies (1901), p. '. * B. Spencer and F. J. Gillen, The Native Tribes of Central Australia (1899), p. 48.
“Preserve, Almighty Lord, this man's # with all saints and with all lovers of Thy holy name. And do Thou seal and guard the sepulchre of Thy servant, Thou who shuttest up the depths and sealest them with Thy almighty right hand . . . so let the seal of Thy Lordship abide unmoved upon this man's dwelling-place and upon the shrine which guards Thy servant. And let not any filthy and unclean devil dare to approach him, such as assail the body and souls of the heathen, who possess not the birth of the holy font, and have not the dread seal laid upon their graves.”
A terrible and revolting picture of the superstitious belief in ghouls which violate Christian tombs is given by Leo Allatius (who held it) in his tract De opinionibus quorundam Graecorum (Paris, 1646). It was probably the fear of such demonic assaults on the dead that inspired the insanitary custom of burying the dead under the floors of churches, and as near as possible to the altar. In the Greek Church this practice was happily forbidden by the code of Justinian as well as by the older law in the case of churches consecrated with Encaenia and deposition of relics. In the Armenian Church the same rule holds, and Ephrem Syrus in his testament particularly forbade his body to be laid within a church. Such prohibitions, however, are a witness to the tendency in question. The custom of lighting candles round a dead body and watching at its side all night was originally due to the belief that a corpse, like a person asleep, is specially liable to the assaults of demons. The practice of tolling a bell at death must have had a similar origin, for it was a common medieval belief that the sound of a consecrated bell drives off the demons which when a man dies gather near in the air to waylay his fleeting soul. For a like reason the consecrated bread of the Eucharist was often buried with believers, and St Basil is said to have specially consecrated a Host to be placed in his coffin. 8. Some of the rites described under the previous heads may be really inspired by the fear of the dead haunting the living, but it must be kept in mind that the taboo attaching to a dead body is one thing and fear of a ghost another. A corpse is buried or burned, or scaffolded on a tree, a tower or a house-top, in order to get it out of the way and shield society from the dangerous infection of its taboo, but ghosts quá ghosts need not be feared and a kinsman's ghost usually is not. On the contrary, it is fed and consoled with everything it needs, is asked not to go away but to stay, is in a thousand ways assured of the sorrow and sympathy of the survivors. Even if the body be eaten, it is merely to keep the soul of the deceased inside the circle of kinsmen, and Strabo asserts that the ancient Irish and Massagetae regarded it as a high honour to be so consumed by relatives. In Santa Cruz in Melancsia they keep the bones for arrow heads and store a skull in a box and set food before it “saying that this is the man himself” (R. H. Codrington, The Mclanesians, p. 264), or the skull and jaw bone are kept and “are called mangate, which are saka, hot with spiritual power, and by means of which the help of the lio'a, the powerful ghost of the man whose relics these are, can be obtained ” (ibid. p. 262). Here we have the savage analogue to Christian relics. So the Australian natives make pointing sticks out of the small bones of the arm, with which to bewitch enemies. We may conclude then that in the most primitive societies, where blood-kinship is the only social tie and root of social custom it is the shades, not of kinsmen, but of strangers, who as such are enemies, that are dangerous and uncanny. In more developed societies, however, all ghosts alike are held to be so; and if a ghost walks it is because its body has not been properly interred or because its owner was a malefactor. Still, even allowing for this, it remains true that for a friendly ghost the proper place is the grave and not the homes of the living, and accordingly the Aruntas with cries of Wah / Wah / with wearing of fantastic head-dresses, wild dancing and beating of the air with hands and weapons “drive the spirit away from the old camp which it is supposed to haunt,” and which has been set fire to, and hunt it at a run into thc grave prepared, and there stamp it down into the earth. “The loud shouting of the men and women shows him
that they do not wish to be frightenca by him in his present
state, and that they will be angry with him if he does not rest.”