c. Here we approach the third class of sympathetic rites; it is clear

that a remedy produces the contrary, when it cures the like; conversely, like by producing like expels its contrary.

Some statements of the law of sympathy suggest that it is absolute in its application. It is true that the current of magical power is sometimes held to be transmitted along lines indicated by the law of sympathy, without the intervention of any volition, human or otherwise; thus, the crow which carries stray hairs away to weave them into the structure of its nest is nowhere supposed to be engaged in a magical process; but it is commonly held that the person whose hair is thus used will suffer from headache or other maladies; this seems to indicate that the law of sympathy operates mechanically in certain directions, though the belief may also be explained as a secondary growth. In general the operation of these laws is limited in the extreme. For example, the medieval doctrine known as the Law of Signatures asserted that the effects of remedies were correlated to their external qualities; bear's grease is good for baldness, because the bear is a hairy animal. But the transference was held to terminate with the acquisition by the man of this single quality; in some magical books powdered mummy is recommended as a means of prolonging life, but it is simply the age of the remedy which is to benefit the patient; the magician who removes a patient's pains or diseases does not transfer them to himself; the child whose parents eat forbidden foods is held to be affected by their transgression, while they themselves come off unharmed. The magical effects are limited by exclusive attention and abstraction; and this is true not only of the kind of effect produced but also as to the direction in which it is held to be produced.

_The Magic of Names._--For primitive peoples the name is as much a part of the person as a limb; consequently the magical use of names is in some of its aspects assimilable to the processes dependent on the law of sympathy. In some cases the name must be withheld from any one who is likely to make a wrong use of it, and in some parts of the world people have secret names which are never used. Elsewhere the name must not be told by the bearer of it, but any other person may communicate it without giving an opening for the magical use of it. Not only human beings but also spirits can be coerced by the use of their names; hence the names of the dead are forbidden, lest the mention of them act as an evocation, unintentional though it be. Even among more advanced nations it has been the practice to conceal the real name of supreme gods; we may probably explain this as due to the fear that an enemy might by the use of them turn the gods away from those to whom they originally belonged. For the same reason ancient Rome had a secret name.

_Magical Rites._--The magic of names leads us up to the magic of the spoken word in general. The spell or incantation and the magical act together make up the rite. (a) The manual acts are very frequently symbolic or sympathetic in their nature; sometimes they are mere reversals of a religious rite; such is the marching against the sun (known as _widdershins_ or _deisul_); sometimes they are purificatory; and magic has its sacrifices just as much as religion. (b) There are many types of oral rites; some of the most curious consist in simply reciting the effect intended to be produced, describing the manual act, or, especially in Europe, telling a mythical narrative in which Christ or the apostles figure, and in which they are represented as producing a similar effect to the one desired; in other cases the "origin" of the disease or maleficent being is recited. Oral rites, which are termed spells or incantations, correspond in many cases to the oral rites of religion; they, like the manual rites, are a heterogeneous mass and hardly lend themselves to classification. Some formulae may be termed sympathetic; it suffices to name the result to be produced in order to produce it; but often an incantation is employed, not to produce a result directly, but to coerce a god or other being and compel him to fulfil the magician's will. The language of the incantations often differs from that of daily life; it may be a survival of archaic forms or may be a special creation for magical purposes. In many languages the word used to express the idea of magic means an act, a deed; and it may be assumed that few if any magical ceremonies consist of formulae only; on the other hand, it is certain that no manual act in magic stands absolutely alone without oral rite; if there is no spoken formula, there is at least an unspoken thought. It is in many cases difficult to discover the relative proportions and importance of manual and oral acts. Not only the words but also the tone are of importance in magic; in fact, the tone may be the more important. Rhythm and repetition are no less necessary in oral than in manual acts. (c) As preliminaries, more seldom as necessary sequels to the central feature of the rite, manual or oral, we usually find a certain number of accessory observances prescribed, which find their parallel in the sacrificial ritual. For example, it is laid down at what time of year, at what period of the month or week, at what hour of the day a rite must be performed; the waxing or waning of the moon must be noted; and certain days must be avoided altogether. Similarly, certain places may be prescribed for the performance of the ritual; often the altar of the god serves magical purposes also; but elsewhere it is precisely the impure sites which are devoted to magical operations--the cemeteries and the cross roads. The instruments of magic are in like manner often the remains of a sacrifice, or otherwise consecrated by religion; sometimes, especially when they belong to the animal or vegetable world, they must be sought at certain seasons, May Day, St George's Day, Midsummer Day, &c. The magician and his client must undergo rites of preparation and the exit may be marked by similar ceremonies.

_Magicians._--Most peoples know the professional worker of magic, or what is regarded as magic. (a) In most if not all societies magic, or certain sorts of it, may be performed by any one, so far as we can see, who has mastered the necessary ritual; in other cases the magician is a specialist who owes his position to an accident of birth (seventh son of a seventh son); to simple inheritance (families of magicians in modern India, rain-makers in New Caledonia); to revelation from the gods or the spirits of the dead (Malays), showing itself in the phenomena of possession; or to initiation by other magicians. (b) From a psychical point of view it may probably be said that the initiation of a magician corresponds to the "development" of the modern spiritualistic medium; that is to say, that it resolves itself into exercises and rites which have for their object the creation or evolution of a secondary personality. From this point of view it is important to notice that certain things are forbidden to magicians under pain of loss of their powers; thus, hot tea is taboo to the Arunta medicine man; and if this seems unlikely to cause the secondary personality to disappear, it must be remembered that to the physiological effects, if any, must be added the effects of suggestion. Of this duplication of personality various explanations are given; in Siberia the soul of the _shaman_ is said to wander into the other world, and this is a widely spread theory; where the magician is supposed to remain on earth, his soul is again believed to wander, but there is an alternative explanation which gives him two or more bodies. Here we reach a point at which the familiar makes its appearance; this is at times a secondary form of the magician, but more often is a sort of life index or animal helper (see LYCANTHROPY); in fact, the magician's power is sometimes held to depend on the presence--that is, the independence--of his animal auxiliary. Concurrent with this theory is the view that the magician must first enter into a trance before the animal makes its appearance, and this makes it a double of the magician, or, from the psychological point of view, a phase of secondary personality. (c) In many parts of the world magical powers are associated with the membership of secret societies, and elsewhere the magicians form a sort of corporation; in Siberia, for example, they are held to be united by a certain tie of kinship; where this is not the case, they are believed, as in Africa at the present day or in medieval Europe, to hold assemblies, so-called witches' Sabbaths; in Europe the meetings of heretics seem to be responsible for the prominence of the idea if not for its origin (see WITCHCRAFT). The magician is often regarded as possessed (see POSSESSION) either by an animal or by a human or super-human spirit. The relations of priest and magician are for various reasons complex; where the initiation of the magician is regarded as the work of the gods, the magician is for obvious reasons likely to develop into a priest, but he may at the same time remain a magician; where a religion has been superseded, the priests of the old cult are, for those who supersede them, one and all magicians; in the medieval church, priests were regarded as especially exposed to the assaults of demons, and were consequently often charged with working magic. The great magicians who are gods rather than men--e.g. kings of Fire and Water in Cambodia--enjoy a reverence and receive a cult which separates them from the common herd, and assimilates them to priests rather than to magicians. The function of the so-called magician is often said to be beneficent; in Africa the witch-doctor's business is to counteract evil magic; in Australia the magician has to protect his own tribe against the assaults of hostile magicians of other tribes; and in Europe "white magic" is the correlative of this beneficent power; but it may be questioned how far the beneficent virtue is regarded as magical outside Europe.

_Talismans and Amulets._--Inanimate objects as well as living beings are credited with stores of magical force; when they are regarded as bringing good, i.e. are positive in their action, they may be termed "talismans"; "amulets" are protective or negative in their action, and their function is to avert evil; a single object may serve both purposes. Broadly speaking, the fetish, whose "magical" properties are due to association with a spirit, tends to become a talisman or amulet. The "medicine" of the Red Indian, originally carried as means of union between him and his _manito_, is perhaps the prototype of many European charms. In other cases it is some specific quality of the object or animal which is desired; the boar's tusk is worn on the Papuan Gulf as a means of imparting courage to the wearer; the Lukungen Indians of Vancouver Island rub the ashes of wasps on the faces of their warriors, in order that they may be pugnacious. Some Bechuanas wear a ferret as a charm, in the belief that it will make them difficult to kill, the animal being very tenacious of life. Among amulets may be mentioned horns and crescents, eyes or their representations, and grotesque figures, all of which are supposed to be powerful against the Evil Eye (q.v.). Tylor has shown that the brass objects so often seen on harness were originally amuletic in purpose, and can be traced back to Roman times. Some amulets are supposed to protect from the evil eye simply by attracting the glance from the wearer to themselves, but, as a rule, magical power is ascribed to them.

_Evil Magic._--The object of "black" magic is to inflict injury, disease, or death on an enemy, and the various methods employed illustrate the general principles dealt with above and emphasize the conclusion that magic is not simply a matter of sympathetic rites, but involves a conception of magical force. (a) It has been mentioned that contagious magic makes use of portions of a person's body; the Cherokee magician follows his victim till he spits on the ground; collecting the spittle mingled with dust on the end of a stick, the magician puts it into a tube made of a poisonous plant together with seven earth worms, beaten into a paste, and splinters of a tree blasted by lightning; the whole is buried with seven yellow stones at the foot of a tree struck by lightning, and a fire is built over the spot; the magician fasts till the ceremony is over. Probably the worms are supposed to feed on the victim's soul, which is said to become "blue" when the charm works; the yellow stones are the emblem of trouble, and lightning-struck trees are reputed powerful in magic. If the charm does not work, the victim survives the critical seven days, and the magician and his employer are themselves in danger, for a charm gone wrong returns upon the head of him who sent it forth. (b) In homoeopathic magic the victim is represented by an image or other object. In the Malay Peninsula the magician makes an image like a corpse, a footstep long. "If you want to cause sickness, you pierce the eye and blindness results; or you pierce the waist and the stomach gets sick. If you want to cause death, you transfix the head with a palm twig; then you enshroud the image as you would a corpse and you pray over it as if you were praying over the dead; then you bury it in the middle of the path which leads to the place of the person whom you wish to charm, so that he may step over it." Sometimes the wizard repeats a form of words signifying that not he but the Archangel Gabriel is burying the victim; sometimes he exclaims, "It is not wax I slay but the liver, heart and spleen of So-and-so." Finally, the image is buried in front of the victim's doors. (c) Very widespread is the idea that a magician can influence his victim by charming a bone, stick or other object, and then projecting the magical influence from it. It is perhaps the commonest form of evil magic in Australia; in the Arunta tribe a man desirous of using one of these pointing sticks or bones goes away by himself into the bush, puts the bone on the ground and crouches over it, muttering a charm: "May your heart be rent asunder." After a time he brings the irna back to the camp and hides it; then one evening after dark he takes it and creeps near enough to see the features of his victim; he stoops down with the _irna_ in his hand and repeatedly jerks it over his shoulder, muttering curses all the time. The evil magic, _arungquiltha_, is said to go straight to the victim, who sickens and dies without apparent cause, unless some medicine-man can discover what is wrong and save him by removing the evil magic. The _irna_ is concealed after the ceremony, for the magician would at once be killed if it were known that he had used it. (d) Magicians are often said to be able to assume animal form or to have an animal familiar. They are said to suck the victim's blood or send a messenger to do so; sometimes they are said to steal his soul, thus causing sickness and eventually death. These beliefs bring the magician into close relation with the werwolf (see LYCANTHROPY).

_Rain-making._--In the lower stages of culture rain-making assumes rather the appearance of a religious ceremony, and even in higher stages the magical character is by no means invariably felt. It will, however, be well to notice some of the methods here. (a) Among the Dieri of Central Australia the whole tribe takes part in the ceremony; a hole is dug, and over this a hut is built, large enough for the old men; the women are called to look at it and then retire some five hundred yards. Two wizards have their arms bound at the shoulder, the old men huddle in the hut, and the principal wizard bleeds the two men selected by cutting them inside the arm below the elbow. The blood is made to flow on the old men, and the two men throw handfuls of down into the air. The blood symbolizes the rain; the down is the clouds. Then two large stones are placed in the middle of the hut; these two represent gathering clouds. The women are again summoned, and then the stones are placed high in a tree; other men pound gypsum and throw it into a water-hole; the ancestral spirits are supposed to see this and to send rain. Then the hut is knocked down, the men butting at it with their heads; this symbolizes the breaking of the clouds, and the fall of the hut is the rain, if no rain comes they say that another tribe has stopped their power or that the _Mura-mura_ (ancestors) are angry with them. (b) Rain-making ceremonies are far from uncommon in Europe. Sometimes water is poured on a stone; a row of stepping-stones runs into one of the tarns on Snowdon, and it is said that water thrown upon the last one will cause rain to fall before night. Sometimes the images of saints are carried to a river or a fountain and ducked or sprinkled with water in the belief that rain will follow; sometimes rain is said to ensue when the water of certain springs is troubled; perhaps the idea is that the rain-god is disturbed in his haunts. But perhaps the commonest method is to duck or drench a human figure or puppet, who represents in many instances the vegetation demon. The gipsies of Transylvania celebrate the festival of "Green George" at Easter or on St George's Day; a boy dressed up in leaves and blossoms is the principal figure; he throws grass to the cattle of the tribe, and after various other ceremonies a pretence is made of throwing him into the water; but in fact only a puppet is ducked in the stream.

_Negative Magic._--There is also a negative side to magic, which, together with ritual prohibitions of a religious nature, is often embraced under the name of taboo (q.v.); this extension of meaning is not justified, for taboo is only concerned with sacred things, and the mark of it is that its violation causes the taboo to be transmitted. All taboos are ritual prohibitions, but all ritual prohibitions are not taboos; they include also (a) interdictions of which the sanction is the wrath of a god; these may be termed religious interdictions; (b) interdictions, the violation of which will automatically cause some undesired magico-religious effect; to these the term negative magic should be restricted, and they might conveniently be called "bans"; they correspond in the main to positive rites and are largely based on the same principles.

(a) Certain prohibitions, such as those imposed on totem kins, seem to occupy an intermediate place; they depend on the sanctity of the totem animal without being taboos in the strict sense; to them no positive magical rites correspond, for the totemic prohibition is clearly religious, not magical.

(b) Among cases of negative magic may be mentioned (i.) the couvade, and prohibitions observed by parents and relatives generally; this is most common in the case of young children, but a sympathetic relation is held to exist in other cases also. In Madagascar a son may not eat fallen bananas, for the result would be to cause the death of his own father; the sympathy between father and son establishes a sympathy between the father and objects touched or eaten by the son, and, in addition, the fall of the bananas is equated with the death of a human being. Again, the wife of a Malagasy warrior may not be faithless to him when he is absent; if she is, he will be killed or wounded. Ownership, too, may create a sympathetic relation of this kind, for it is believed in parts of Europe that if a man kills a swallow his cows will give bloody milk. In some cases it is even harder to see how the sympathetic bond is established; some Indians of Brazil always hamstring animals before bringing them home, in the belief that by so doing they make it easier for themselves and their children to run down their enemies, who are then magically deprived of the use of their legs. These are all examples of negative magic with regard to persons, but things may be equally affected; thus in Borneo men who search for camphor abstain from washing their plates for fear the camphor, which is found crystallized in the crevices of trees, should dissolve and disappear. (ii.) Rules which regulate diet exist not only for the benefit of others but also for that of the eater. Some animals, such as the hare, are forbidden, just as others, like the lion, are prescribed; the one produces cowardice, while the other makes a man's heart bold. (iii.) Words may not be used; Scottish fishermen will not mention the pig at sea; the real names of certain animals, like the bear, may not be used; the names of the dead may not be mentioned; a sacred language must be used, e.g. camphor language in the Malay peninsula, or only words of good omen (cf. Gr. [Greek: euphêmeite]); or absolute silence must be preserved. Personal names are concealed; a man may not mention the names of certain relatives, &c. There are customs of avoidance not only as to (iv.) the names of relatives, but as to the persons themselves; the mother-in-law must avoid the son-in-law, and vice versa; sometimes they may converse at a distance, or in low tones, sometimes not at all, and sometimes they may not even meet. (v.) In addition to these few classes selected at random, we have prohibitions relating to numbers (cf. unlucky thirteen, which is, however, of recent date), the calendar (Friday as an unlucky day, May as an unlucky month for marriage), places, persons, orientation, &c.; but it is impossible to enumerate even the main classes. The individual origin of such beliefs, which with us form the superstitions of daily life but in a savage or semi-civilized community play a large part in regulating conduct, is often shrouded in darkness; the meaning of the positive rite is easily forgotten; the negative rite persists, but it is observed merely to avoid some unknown misfortune. Sometimes we can, however, guess at the meaning of our civilized notions of ill luck; it is perhaps as a survival of the savage belief that stepping over a person is injurious to him that many people regard going under a ladder as unlucky; in the one case the luck is taken away by the person stepping over, in the other left behind by the person passing under.

_History of Magic._--The subject is too vast and our data are too slight to make a general sketch of magic possible. Our knowledge of Assyrian magic, for example, hardly extends beyond the rites of exorcism; the magic of Africa is most inadequately known, and only in recent years have we well-analysed repertories of magical rituals from any part of the world. For certain departments of ancient magic, however, like the Pythagorean philosophy, there is no lack of illustrative material; it depended on mystical speculations based on numbers or analogous principles. The importance of numbers is recognized in the magic of America and other areas, but the science of the Mediterranean area, combined with the art of writing, was needed to develop such mystical ideas to their full extent. Among the neo-Platonists there was a strong tendency to magical speculation, and they sought to impress into their service the demons with which they peopled the universe. Alexandria was the home of many systems of theurgic magic, and gnostic gems afford evidence of the nature of their symbols. In the middle ages the respectable branches of magic, such as astrology and alchemy, included much of the real science of the period; the rise of Christianity introduced a new element, for the Church regarded all the religions of the heathen as dealings with demons and therefore magical (see WITCHCRAFT). In our own day the occult sciences still find devotees among the educated; certain elements have acquired a new interest, in so far as they are the subject matter of psychical research (q.v.) and spiritualism (q.v.). But it is only among what are regarded as the lower classes, and in England especially the rural population, that belief in its efficacy still prevails to any large extent.

_Psychology of Magic._--The same causes which operated to produce a belief in witchcraft (q.v.) aided the creed of magic in general. Fortuitous coincidences attract attention; the failures are disregarded or explained away. Probably the magician is never wholly an impostor, and frequently has a whole-hearted belief in himself; in this connexion may be noted the fact that juggling tricks have in all ages been passed off as magical; the name of "conjuring" (q.v.) survives in our own day, though the conjurer no longer claims that his mysterious results are produced by demons. It is interesting to note that magical leechcraft depended for its success on the power of suggestion (q.v.), which is to-day a recognized element in medicine; perhaps other elements may have been instrumental in producing a cure, for there are cases on record in which European patients have been cured by the apparently meaningless performances of medicine-men, but an adequate study of savage medicine is still a desideratum.

BIBLIOGRAPHY.--For a general discussion of magic with a list of selected works see Hubert and Mauss in _Année sociologique_, vii. 1-146; also A. Lehmann, _Aberglaube und Zauberei_; the article "Religion" in _La Grande encyclopédie_; K. T. Preuss in _Globus_, vols. 86, 87; Mauss, _L'Origine des pouvoirs magiques_, and Hubert, _La Réprésentation du temps_ (Reports of École pratique des hautes études, Paris). For general bibliographies see Hauck, _Realencyklopädie_, _s.v._ "Magie"; A. C. Haddon, _Magic and Fetishism_. J. G. T. Graesse's _Bibliotheca magica_ is an exhaustive list of early works dealing with magic and superstition. For Australia see Spencer and Gillen's works, and A. W. Howitt, _Native Tribes_. For America see _Reports of Bureau of Ethnology_, vii. xvii. For India see W. Caland, _Altindisches Zauber-ritual_; and W. Crooke, _Popular Religion_; also V. Henry, _La Magie_. For the Malays see W. W. Skeat, _Malay Magic_. For Babylonia and Assyria see L. W. King's works. For magic in Greece and Rome see Daremberg and Saglio, _s.v._ "Magia," "Amuletum," &c. For medieval magic see A. Maury, _La Magie_. For illustrations of magic see J. G. Frazer, _The Golden Bough_; E. S. Hartland, _Legend of Perseus_; E. B. Tylor, _Primitive Culture_; W. G. Black, _Folkmedicine_. For negative magic see the works of Frazer and Skeat cited above; also _Journ. Anthrop. Inst._ xxxvi. 92-103; _Zeitschrift für Ethnologie_ (Verhandlungen) (1905), 153-162; _Bulletin trimestriel de l'académie malgache_, iii. 105-159. See also bibliography to TABOO and WITCHCRAFT. (N. W. T.)

FOOTNOTE:

[1] For what is often called "magic," but is really trick-performance, see CONJURING.

MAGIC SQUARE, a square divided into equal squares, like a chess-board, in each of which is placed one of a series of consecutive numbers from 1 up to the square of the number of cells in a side, in such a manner that the sum of the numbers in each row or column and in each diagonal is constant.

From a very early period these squares engaged the attention of mathematicians, especially such as possessed a love of the marvellous, or sought to win for themselves a superstitious regard. They were then supposed to possess magical properties, and were worn, as in India at the present day, engraven in metal or stone, as amulets or talismans. According to the old astrologers, relations subsisted between these squares and the planets. In later times such squares ranked only as mathematical curiosities; till at last their mode of construction was systematically investigated. The earliest known writer on the subject was Emanuel Moscopulus, a Greek (4th or 5th century). Bernard Frenicle de Bessy constructed magic squares such that if one or more of the encircling bands of numbers be taken away the remaining central squares are still magical. Subsequently Poignard constructed squares with numbers in arithmetical progression, having the magical summations. The later researches of Phillipe de la Hire, recorded in the _Mémoires de l'Académie Royale_ in 1705, are interesting as giving general methods of construction. He has there collected the results of the labours of earlier pioneers; but the subject has now been fully systematized, and extended to cubes.

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FIG. 1.

Two interesting magical arrangements are said to have been given by Benjamin Franklin; these have been termed the "magic square of squares" and the "magic circle of circles." The first (fig. 1) is a square divided into 256 squares, i.e. 16 squares along a side, in Fig. 2. which are placed the numbers from 1 to 256. The chief properties of this square are (1) the sum of the 16 numbers in any row or column is 2056; (2) the sum of the 8 numbers in half of any row or column is 1028, i.e. one half of 2056; (3) the sum of the numbers in two half-diagonals equals 2056; (4) the sum of the four corner numbers of the great square and the four central numbers equals 1028; (5) the sum of the numbers in any 16 cells of the large square which themselves are disposed in a square is 2056. This square has other curious properties. The "magic circle of circles" (fig. 2) consists of eight annular rings and a central circle, each ring being divided into eight cells by radii drawn from the centre; there are therefore 65 cells. The number 12 is placed in the centre, and the consecutive numbers 13 to 75 are placed in the other cells. The properties of this figure include the following: (1) the sum of the eight numbers in any ring together with the central number 12 is 360, the number of degrees in a circle; (2) the sum of the eight numbers in any set of radial cells together with the central number is 360; (3) the sum of the numbers in any four adjoining cells, either annular, radial, or both radial and two annular, together with half the central number, is 180.

+---+---+---+---+---+-+---+---+---+---+---+ | | a |[e]| 5 | | | | |[e]| | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ | | 4 | b | |[d]| | | 4 | | |[d]| | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ | |[g]| | c | 3 | | |[g]| | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ | | | 2 |[b]| d | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ |1 e| | | | | | e | | | | | |[a]| | | | | | | | | | | +---+---+---+---+---+-+---+---+---+---+---+

FIG. 3.

_Construction of Magic Squares._--A square of 5 (fig. 3) has adjoining it one of the eight equal squares by which any square may be conceived to be surrounded, each of which has two sides resting on adjoining squares, while four have sides resting on the surrounded square, and four meet it only at its four angles. 1, 2, 3 are placed along the path of a knight in chess; 4, along the same path, would fall in a cell of the outer square, and is placed instead in the corresponding cell of the original square; 5 then falls within the square. a, b, c, d are placed diagonally in the square; but e enters the outer square, and is removed thence to the same cell of the square it had left. [alpha], [beta], [gamma], [delta], [epsilon] pursue another regular course; and the diagram shows how that course is recorded in the square they have twice left. Whichever of the eight surrounding squares may be entered, the corresponding cell of the central square is taken instead. The 1, 2, 3, ..., a, b, c, ..., [alpha], [beta], [gamma], ... are said to lie in "paths."

+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 1 | 4 | 2 | 5 | 3 | | 2 | 4 | 0 | 3 | 1 | |11 |24 | 2 |20 | 8 | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 4 | 2 | 5 | 3 | 1 | | 1 | 2 | 4 | 0 | 3 | | 9 |12 |25 | 3 |16 | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 2 | 5 | 3 | 1 | 4 | | 3 | 1 | 2 | 4 | 0 | |17 |10 |13 |21 | 4 | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 5 | 3 | 1 | 4 | 2 | | 0 | 3 | 1 | 2 | 4 | | 5 |18 | 6 |14 |22 | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 3 | 1 | 4 | 2 | 5 | | 4 | 0 | 3 | 1 | 2 | |23 | 1 |19 | 7 |15 | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+

FIG. 4. FIG. 5. FIG. 6.

+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 2 | 1 | 5 | 3 | 4 | |15 | 5 | 0 |20 |10 | |17 | 6 | 5 |23 |14 | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 3 | 4 | 2 | 1 | 5 | | 0 |20 |10 |15 | 5 | | 3 |24 |12 |16 |10 | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 1 | 5 | 3 | 4 | 2 | |10 |15 | 5 | 0 |20 | |11 |20 | 8 | 4 |22 | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 4 | 2 | 1 | 5 | 3 | | 5 | 0 |20 |10 |15 | | 9 | 2 |21 |15 |18 | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 5 | 3 | 4 | 2 | 1 | |20 |10 |15 | 5 | 0 | |25 |13 |19 | 7 | 1 | | | | | | | | | | | | | | | | | | | +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+

FIG. 7. FIG. 8. FIG. 9.

_Squares whose Roots are Odd._--Figs 4, 5, and 6 exhibit one of the earliest methods of constructing magic squares. Here the 3's in fig. 4 and 2's in fig. 5 are placed in opposite diagonals to secure the two diagonal summations; then each number in fig. 5 is multiplied by 5 and added to that in the corresponding square in fig. 4, which gives the square of fig. 6. Figs. 7, 8 and 9 give De la Hire's method; the squares of figs. 7 and 8, being combined, give the magic square of fig. 9. C. G. Bachet arranged the numbers as in fig. 10, where there are three numbers in each of four surrounding squares; these being placed in the corresponding cells of the central square, the square of fig. 11 is formed. He also constructed squares such that if one or more outer bands of numbers are removed the remaining central squares are magical. His method of forming them may be understood from a square of 5. Here each summation is 5 × 13; if therefore 13 is subtracted from each number, the summations will be zero, and the twenty-five cells will contain the series ± i, ± 2, ± 3, ... ± 12, the odd cell having 0. The central square of 3 is formed with four of the twelve numbers with + and - signs and zero in the middle; the band is filled up with the rest, as in fig. 12; then, 13 being added in each cell, the magic square of fig. 13 is obtained.

_Squares whose Roots are Even._--These were constructed in various ways, similar to that of 4 in figs. 14, 15 and 16. The numbers in fig. 15 being multiplied by 4, and the squares of figs. 14 and 15 being superimposed, give fig. 16. The application of this method to squares the half of whose roots are odd requires a complicated adjustment. Squares whose half root is a multiple of 4, and in which there are summations along all the diagonal paths, may be formed, by observing, as when the root is 4, that the series 1 to 16 may be changed into the series 15, 13, ... 3, 1, -1, -3, ... -13, -15, by multiplying each number by 2 and subtracting 17; and, vice versa, by adding 17 to each of the latter, and dividing by 2. The diagonal summations of a square, filled as in fig. 17, make zero; and, to obtain the same in the rows and columns, we must assign such values to the p's and q's as satisfy the equations p1 + p2 + a1 + a2 = 0, p3 + p4 + a3 + a4 = 0, p1 + p3 - a1 - a3 = 0, and p2 + p4 - a2 - a4 = 0,--a solution of which is readily obtained by inspection, as in fig. 18; this leads to the square, fig. 19. When the root is 8, the upper four subsidiary rows may at once be written, as in fig. 20; then, if 65 be added to each, and the sums halved, the square is completed. In such squares as these, the two opposite squares about the same diagonal (except that of 4) may be turned through any number of right angles, in the same direction, without altering the summations.

1 6 2 +--+--+--+--+--+ +--+--+--+--+--+ |11| | 7| | 3| |11|24| 7|20| 3| +--+--+--+--+--+ +--+--+--+--+--+ 16 | |12| | 8| | 4 | 4|12|25| 8|16| +--+--+--+--+--+ +--+--+--+--+--+ 21 |17| |13| | 9| 5 |17| 5|13|21| 9| +--+--+--+--+--+ +--+--+--+--+--+ 22 | |18| |14| |10 |10|18| 1|14|22| +--+--+--+--+--+ +--+--+--+--+--+ |23| |19| |15| |23| 6|19| 2|15| +--+--+--+--+--+ +--+--+--+--+--+ 24 20 25

FIG. 10. FIG. 11.

+---+---+---+---+---+ +---+---+---+---+---+ | -9| 12| 5| -3| -6| | 4| 25| 18| 11| 7 | +---+---+---+---+---+ +---+---+---+---+---+ | 1| 7|-11| 4| -1| | 14| 20| 2| 17| 13| +---+---+---+---+---+ +---+---+---+---+---+ | -8| -3| 0| 3| 8| | 5| 10| 13| 16| 21| +---+---+---+---+---+ +---+---+---+---+---+ | 10| -4| 11| -7|-10| | 23| 9| 24| 6| 3| +---+---+---+---+---+ +---+---+---+---+---+ | 6|-12| -5| 2| 9| | 19| 1| 8| 15| 22| +---+---+---+---+---+ +---+---+---+---+---+

FIG. 12. FIG. 13.

+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ | 1| 3| 2| 4| | 0| 3| 3| 0| | 1|15|14| 4| +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ | 4| 2| 3| 1| | 2| 1| 1| 2| |12| 6| 7| 9| +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ | 4| 2| 3| 1| | 1| 2| 2| 1| | 8|10|11| 5| +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ | 1| 3| 2| 4| | 3| 0| 0| 3| |13| 3| 2|16| +--+--+--+--+ +--+--+--+--+ +--+--+--+--+

FIG. 14. FIG. 15. FIG. 16.

+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | p1| p2| a1| a2| | 1| -3| 11| -9| | 9| 7| 14| 4| +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ | p3| p4| a3| a4| | -5| 7|-15| 13| | 6| 12| 1| 15| +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ |-a1|-a2|-p1|-p2| |-11| 9| -1| 3| | 3| 13| 8| 10| +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ |-a3|-a4|-p3|-p4| | 15|-13| 5| -7| | 16| 2| 11| 5| +---+---+---+---+ +---+---+---+---+ +---+---+---+---+

FIG. 17. FIG. 18. FIG. 19.

+---+---+---+---+---+---+---+---+ | -1| 3| 5| -7|-33| 35| 37|-39| +---+---+---+---+---+---+---+---+ | 9|-11|-13| 15| 41|-43|-45| 47| +---+---+---+---+---+---+---+---+ | 17|-19|-21| 23| 49|-51|-53| 55| +---+---+---+---+---+---+---+---+ |-25| 27| 29|-31|-57| 59| 61|-63| +---+---+---+---+---+---+---+---+

FIG. 20.

_Nasik Squares._--Squares that have many more summations than in rows, columns and diagonals were investigated by A. H. Frost (_Cambridge Math. Jour._, 1857), and called Nasik squares, from the town in India where he resided; and he extended the method to cubes, various sections of which have the same singular properties. In order to understand their construction it will be necessary to consider carefully fig. 21, which shows that, when the root is a prime, and not composite, number, as 7, eight letters a, b, ... h may proceed from any, the same, cell, suppose that marked 0, each letter being repeated in the cells along different paths. These eight paths are called "normal paths," their number being one more than the root. Observe here that, excepting the cells from which any two letters start, they do not occupy again the same cell, and that two letters, starting from any two different cells along different paths, will appear together in one and only one cell. Hence, if p1 be placed in the cells of one of the n + 1 normal paths, each of the remaining n normal paths will contain one, and only one, of these p1's. If now we fill each row with p2, p3, ... p_n in the same order, commencing from the p1 in that row, the p2's, p3's and p_n's will lie each in a path similar to that of p1, and each of the n normal paths will contain one, and only one, of the letters p1, p2,... p_n, whose sum will be [Sigma]p. Similarly, if q1 be placed along any of the normal paths, different from that of the p's, and each row filled as above with the letters q2, q3, ... q_n, the sum of the q's along any normal path different from that of the q1 will be [Sigma]q. The n² cells of the square will now be found to contain all the combinations of the p's and q's; and if the q's be multiplied by n, the p's made equal to 1, 2, ... n, and the q's to 0, 1, 2, ... (n - 1) in any order, the Nasik square of n will be obtained, and the summations along all the normal paths, except those traversed by the p's and q's, will be the constant [Sigma]nq + [Sigma]p. When the root is an odd composite number, as 9, 15, &c., it will be found that in some paths, different from the two along which the p1 and q1 were placed, instead of having each of the p's and q's, some will be wanting, while some are repeated. Thus, in the case of 9, the triplets, p1p4p7, p2p5p8, p3p6p9, and q1q4q7, q2q5q8, q3q6q9 occur, each triplet thrice, along paths whose summation should be--[Sigma]p 45 and [Sigma]r 36. But if we make p1, p2, ... p9, = 1, 3, 6, 5, 4, 7, 9, 8, 2, and the r1, r2, ... r9 = 0, 2, 5, 4, 3, 6, 8, 7, 1, thrice each of the above sets of triplets will equal [Sigma]p and [Sigma]q respectively. If now the q's are multiplied by 9, and added to the p's in their several cells, we shall have a Nasik square, with a constant summation along eight of its ten normal paths. In fig. 22 the numbers are in the nonary scale; that in the centre is the middle one of 1 to 9², and the sum of pair of numbers equidistant from and opposite to the central 45 is twice 45; and the sum of any number and the 8 numbers 3 from it, diagonally, and in its row and column, is the constant Nasical summation, e.g. 72 and 32, 22, 76, 77, 26, 37, 36, 27. The numbers in fig. 22 being kept in the nonary scale, it is not necessary to add any nine of them together in order to test the Nasical summation; for, taking the first column, the figures in the place of units are seen at once to form the series, 1, 2, 3, ... 9, and those in the other place three triplets of 6, 1, 5. For the squares of 15 the p's and q's may be respectively 1, 2, 10, 8, 6, 14, 15, 11, 4, 13, 9, 7, 3, 12, 5, and 0, 1, 9, 7, 5, 13, 14, 10, 3, 12, 8, 6, 2, 11, 4, where five times the sum of every third number and three times the sum of every fifth number makes [Sigma]p and [Sigma]q; then, if the q's are multiplied by 15, and added to the p's, the Nasik square of 15 is obtained. When the root is the multiple of 4, the same process gives us, for the square of 4, fig. 23. Here the columns give [Sigma]p, but alternately 2q1, 2q3, and 2q2, 2q4; and the rows give [Sigma]q, but alternately 2p1, 2p3, and 2p2, 2p4; the diagonals giving [Sigma]p and [Sigma]q. If p1, p2, p3, p4 and q1, q2, q3, q4 be 1, 2, 4, 3, and 0, 1, 3, 2, we have the Nasik square of fig. 24. A square like this is engraved in the Sanskrit character on the gate of the fort of Gwalior, in India. The squares of higher multiples of 4 are readily obtained by a similar adjustment.

+---+---+---+---+---+---+---+ +--+--+--+--+--+--+--+--+--+ | a | g | f | e | d | c | b | |63|88|74|13| 8|24|53|48|34| +---+---+---+---+---+---+---+ +--+--+--+--+--+--+--+--+--+ | a | d | g | c | f | b | e | |11| 9|25|51|49|35|61|89|75| +---+---+---+---+---+---+---+ +--+--+--+--+--+--+--+--+--+ | a | c | e | g | b | d | f | |52|47|36|62|87|76|12| 7|26| +---+---+---+---+---+---+---+ +--+--+--+--+--+--+--+--+--+ | a | f | d | b | g | e | c | |68|84|73|18| 4|23|58|44|33| +---+---+---+---+---+---+---+ +--+--+--+--+--+--+--+--+--+ | a | e | b | f | c | g | d | |19| 5|21|59|45|31|69|85|71| +---+---+---+---+---+---+---+ +--+--+--+--+--+--+--+--+--+ | a | b | c | d | e | f | g | |57|46|32|67|86|72|17| 6|22| +---+---+---+---+---+---+---+ +--+--+--+--+--+--+--+--+--+ | 0 | h | h | h | h | h | h | |64|83|78|14| 3|28|54|43|38| +---+---+---+---+---+---+---+ +--+--+--+--+--+--+--+--+--+ FIG. 21. |15| 1|29|55|41|39|65|81|79| +--+--+--+--+--+--+--+--+--+ |56|42|37|66|82|77|16| 2|27| +--+--+--+--+--+--+--+--+--+

FIG. 22.

+-----+-----+-----+-----+ +--+--+--+--+ | | | | | |15|10| 3| 6| |p4q3 |p2q4 |p4q1 |p2q2 | +--+--+--+--+ | | | | | | 4| 5|16| 9| +-----+-----+-----+-----+ +--+--+--+--+ | | | | | |14|11| 2| 7| |p3q1 |p1q2 |p3q3 |p1q4 | +--+--+--+--+ | | | | | | 1| 8|13|12| +-----+-----+-----+-----+ +--+--+--+--+ | | | | | FIG. 24. |p2q3 |p4q4 |p2q1 |p4q2 | | | | | | +-----+-----+-----+-----+ | | | | | |p1q1 |p3q2 |p1q3 |p3q4 | | | | | | +-----+-----+-----+-----+

FIG. 23.

. / \ . . / \ / \ . . . / \ / \ / \ . . . * . / \ / \ / \ / \ . . * . . . / \ / \ / \ / \ / \ . * . . . . . / \ / \ / \ / \ / \ / \ . . . . . . . / \ / \ / \ / \ / \ / \ / \ C . . . . . . * . . A |\ / \ / \ / \ / \ / \ / \ /| . . . . . * . . . . |\|\ / \ / \ / \ / \ / \ /|/| . . . . * . . . . . . |\|\|\ / \ / \ / \ / \ /|/|/| . . . . . . . . . . . |\|\|\|\ / \ / \ / \ /|/|/|/| .*. . . . . . . .*. . . |\|\|\|\|\ / \ / \ /|/|/|/|/| . . .*. . . . . . . .*. . |\|\|\|\|\|\ / \ /|/|/|/|/|/| . . . . .*. . * . . . . . .*. |\|\|\|\|\|\|\ /|/|/|/|/|/|/| . . . . . . .*O*. . . . . . . \|\|\|\|\|\|\|/|/|/|/|/|/|/ .*. . . . . . .*. . . . . \|\|\|\|\|\|/|/|/|/|/|/ . .*. . . . . .*. . . \|\|\|\|\|/|/|/|/|/ . . .*. . . . .*. \|\|\|\|/|/|/|/ . . . . . . . \|\|\|/|/|/ . . . . . \|\|/|/ . . . \|/ . B

FIG. 25--Nasik Cube.

+--+--+--+--+------------+--+--+--+--+ | 1| 8|29|28| |11|14|23|18| +--+--+--+--+ +--+--+--+--+ |30|27| 2| 7| |21|20| 9|16| +--+--+--+--+ +--+--+--+--+ | 4| 5|32|25| |10|15|22|19| +--+--+--+--+ +--+--+--+--+ |31|26| 3| 6| |24|17|12|13| +--+--+--+--+------------+--+--+--+--+

FIG. 26.

+--+--+--+--+--+ +--+--+--+--+--+--+ |23|18|11| 6|25| |30|21| 6|15|28|19| +--+--+--+--+--+ +--+--+--+--+--+--+ |10| 5|24|17|12| | 7|16|29|20| 5|14| +--+--+--+--+--+ +--+--+--+--+--+--+ |19|22|13| 4| 7| |22|31| 8|35|18|27| +--+--+--+--+--+ +--+--+--+--+--+--+ |14| 9| 2|21|16| | 9|26|17|26|13| 4| +--+--+--+--+--+ +--+--+--+--+--+--+ | 1|20|15| 8| 3| |32|23| 2|11|34|25| +--+--+--+--+--+ +--+--+--+--+--+--+ | 1|10|33|24| 3|12| FIG. 27. +--+--+--+--+--+--+

FIG. 28.

_Nasik Cubes._--A Nasik cube is composed of n³ small equal cubes, here called cubelets, in the centres of which the natural numbers from 1 to n³ are so placed that every section of the cube by planes perpendicular to an edge has the properties of a Nasik square; also sections by planes perpendicular to a face, and passing through the cubelet centres of any path of Nasical summation in that face. Fig. 25 shows by dots the way in which these cubes are constructed. A dot is here placed on three faces of a cubelet at the corner, showing that this cubelet belongs to each of the faces AOB, BOC, COA, of the cube. Dots are placed on the cubelets of some path of AOB (here the knight's path), beginning from O, also on the cubelets of a knight's path in BOC. Dots are now placed in the cubelets of similar paths to that on BOC in the other six sections parallel to BOC, starting from their dots in AOB. Forty-nine of the three hundred and forty-three cubelets will now contain a dot; and it will be observed that the dots in sections perpendicular to BO have arranged themselves in similar paths. In this manner, p1, q1, r1 being placed in the corner cubelet O, these letters are severally placed in the cubelets of three different paths of AOB, and again along any similar paths in the seven sections perpendicular to AO, starting from the letters' position in AOB. Next, p2q2r2, p3q3r3, ... p7q7r7 are placed in the other cubelets of the edge AO, and dispersed in the same manner as p1q1r1. Every cubelet will then be found to contain a different combination of the p's, q's and r's. If therefore the p's are made equal to 1, 2, ... 7, and the q's and r's to 0, 1, 2, ... 6, in any order, and the q's multiplied by 7, and the r's by 7², then, as in the case of the squares, the 7³ cubelets will contain the numbers from 1 to 7³, and the Nasical summations will be [Sigma]7²r + [Sigma]7q + p. If 2, 4, 5 be values of r, p, q, the number for that cubelet is written 245 in the septenary scale, and if all the cubelet numbers are kept thus, the paths along which summations are found can be seen without adding, as the seven numbers would contain 1, 2, 3, ... 7 in the unit place, and 0, 1, 2, ... 6 in each of the other places. In all Nasik cubes, if such values are given to the letters on the central cubelet that the number is the middle one of the series 1 to n³, the sum of all the pairs of numbers opposite to and equidistant from the middle number is the double of it. Also, if around a Nasik cube the twenty-six surrounding equal cubes be placed with their cells filled with the same numbers, and their corresponding faces looking the same way,--and if the surrounding space be conceived thus filled with similar cubes, and a straight line of unlimited length be drawn through any two cubelet centres, one in each of any two cubes,--the numbers along that line will be found to recur in groups of seven, which (except in the three cases where the same p, q or r recur in the group) together make the Nasical summation of the cube. Further, if we take n similarly filled Nasik cubes of n, n new letters, s1, s2, ... s_n, can be so placed, one in each of the n^4 cubelets of this group of n cubes, that each shall contain a different combination of the p's, q's, r's and s's. This is done by placing s1 on each of the n² cubelets of the first cube that contain p1, and on the n² cubelets of the 2d, 3d, ... and nth cube that contain p2, p3, ... p_n respectively. This process is repeated with s2, beginning with the cube at which we ended, and so on with the other s's; the n^4 cubelets, after multiplying the q's, r's, and s's by n, n², and n³ respectively, will now be filled with the numbers from 1 to n^4, and the constant summation will be [Sigma]n³s + [Sigma]n²r + [Sigma]nq + [Sigma]p. This process may be carried on without limit; for, if the n cubes are placed in a row with their faces resting on each other, and the corresponding faces looking the same way, n such parallelepipeds might be put side by side, and the n^5 cubelets of this solid square be Nasically filled by the introduction of a new letter t; while, by introducing another letter, the n^6 cubelets of the compound cube of n³ Nasik cubes might be filled by the numbers from 1 to n^6, and so _ad infinitum_. When the root is an odd composite number the values of the three groups of letters have to be adjusted as in squares, also in cubes of an even root. A similar process enables us to place successive numbers in the cells of several equal squares in which the Nasical summations are the same in each, as in fig. 26.

Among the many ingenious squares given by various writers, this article may justly close with two by L. Euler, in the _Histoire de l'académie royale des sciences_ (Berlin, 1759). In fig. 27 the natural numbers show the path of a knight that moves within an odd square in such a manner that the sum of pairs of numbers opposite to and equidistant from the middle figure is its double. In fig. 28 the knight returns to its starting cell in a square of 6, and the difference between the pairs of numbers opposite to and equidistant from the middle point is 18.

A model consisting of seven Nasik cubes, constructed by A. H. Frost, is in the South Kensington Museum. The centres of the cubes are placed at equal distances in a straight line, the similar faces looking the same way in a plane parallel to that line. Each of the cubes has seven parallel glass plates, to which, on one side, the seven numbers in the septenary scale are fixed, and behind each, on the other side, its value in the common scale. 1201, the middle number from 1 to 7^4 occupies the central cubelet of the middle cube. Besides each cube having separately the same Nasical summation, this is also obtained by adding the numbers in any seven similarly situated cubelets, one in each cube. Also, the sum of all pairs of numbers, in a straight line, through the central cube of the system, equidistant from it, in whatever cubes they are, is twice 1201. (A. H. F.)

_Fennell's Magic Ring._--It has been noticed that the numbers of magic squares, of which the extension by repeating the rows and columns of n numbers so as to form a square of 2n - 1 sides yields n² magic squares of n sides, are arranged as if they were all inscribed round a cylinder and also all inscribed on another cylinder at right angles to the first. C. A. M. Fennell explains this apparent anomaly by describing such magic squares as Mercator's projections, so to say, of "magic rings."

The surface of these magic rings is symmetrically divided into n² quadrangular compartments or cells by n equidistant zonal circles parallel to the circular axis of the ring and by n transverse circles which divide each of the n zones between any two neighbouring zonal circles into n equal quadrangular cells, while the zonal circles divide the sections between two neighbouring transverse circles into n unequal quadrangular cells. The diagonals of cells which follow each other passing once only through each zone and section, form similar and equal closed curves passing once quite round the circular axis of the ring and once quite round the centre of the ring. The position of each number is regarded as the intersection of two diagonals of its cell. The numbers are most easily seen if the smallest circle on the surface of the ring, which circle is concentric with the axis, be one of the zonal circles. In a perfect magic ring the sum of the numbers of the cells whose diagonals form any one of the 2n diagonal curves aforesaid is ½n(n² + 1) with or without increment, i.e. is the same sum as that of the numbers in each zone and each transverse section. But if n be 3 or a multiple of 3, only from 2 to n of the diagonal curves carry the sum in question, so that the magic rings are imperfect; and any set of numbers which can be arranged to make a perfect magic ring or magic square can also make an imperfect magic ring, e.g. the set 1 to 16 if the numbers 1, 6, 11, 16 lie thus on a diagonal curve instead of in the order 1, 6, 16, 11. From a perfect magic ring of n² cells containing one number each, n² distinct magic squares can be read off; as the four numbers round each intersection of a zonal circle and a transverse circle constitute corner numbers of a magic square. The shape of a magic ring gives it the function of an indefinite extension in all directions of each of the aforesaid n² magic squares. (C. A. M. F.)

See F. E. A. Lucas, _Récréations mathématiques_ (1891-1894); W. W. R. Ball, _Mathematical Recreations_ (1892); W. E. M. G. Ahrens, _Mathematische Unterhaltungen und Spiele_ (1901); H. C. H. Schubert, _Mathematische Mussestunden_ (1900). A very detailed work is B. Violle, _Traité complet des carrés magiques_ (3 vols., 1837-1838). The theory of "path nasiks" is dealt with in a pamphlet by C. Planck (1906).

MAGINN, WILLIAM (1793-1842), Irish poet and journalist, was born at Cork on the 10th of July 1793. The son of a schoolmaster, he graduated at Trinity College, Dublin, in 1811, and after his father's death in 1813 succeeded him in the school. In 1819 he began to contribute to the _Literary Gazette_ and to _Blackwood's Magazine_, writing as "R. T. Scott" and "Morgan O'Doherty." He first made his mark as a parodist and a writer of humorous Latin verse. In 1821 he visited Edinburgh, where he made acquaintance with the Blackwood circle. He is credited with having originated the idea of the _Noctes ambrosianae_, of which some of the most brilliant chapters were his. His connexion with Blackwood lasted, with a short interval, almost to the end of his life. His best story was "Bob Burke's Duel with Ensign Brady." In 1823 he removed to London. He was employed by John Murray on the short-lived _Representative_, and was for a short time joint-editor of the _Standard_. But his intemperate habits and his imperfect journalistic morality prevented any permanent success. In connexion with Hugh Fraser he established _Fraser's Magazine_ (1830), in which appeared his "Homeric Ballads." Maginn was the original of Captain Shandon in _Pendennis_. In spite of his inexhaustible wit and brilliant scholarship, most of his friends were eventually alienated by his obvious failings, and his persistent insolvency. He died at Walton-on-Thames on the 21st of August 1842.

His _Miscellanies_ were edited (5 vols., New York, 1855-1857) by R. Shelton Mackenzie and (2 vols., London, 1885) by R. W. Montagu [Johnson].

MAGISTRATE (Lat. _magistratus_, from _magister_, master, properly a public office, hence the person holding such an office), in general, one vested with authority to administer the law or one possessing large judicial or executive authority. In this broad sense the word is used in such phrases as "the first magistrate" of a king in a monarchy or "the chief magistrate" of the president of the United States. But it is more generally applied to minor or subordinate judicial officers, whether unpaid, as justices of the peace, or paid, as stipendiary magistrates. A stipendiary magistrate is appointed in London under the Metropolitan Police Courts Act 1839, in municipal boroughs under the Municipal Corporations Act 1882, and in particular districts under the Stipendiary Magistrates Act 1863 and special acts. In London and municipal boroughs a stipendiary magistrate must be a barrister of at least seven years' standing, while under the Stipendiary Magistrates Act 1863 he may be of five years' standing. A stipendiary magistrate may do alone all acts authorized to be done by two justices of the peace.

The term _magistratus_ in ancient Rome originally implied the office of _magister_ (master) of the Roman people, but was subsequently applied also to the holder of the office, thus becoming identical in sense with _magister_, and supplanting it in reference to any kind of public office. The fundamental conception of Roman magistracy is tenure of the _imperium_, the sovereignty which resides with the Roman people, but is by it conferred either upon a single ruler for life, as in the later monarchy, or upon a college of magistrates for a fixed term, as in the Republican period. The Roman theory of magistracy underwent little change when two consuls were substituted for the king; but the subdivision of magisterial powers which characterized the first centuries of the Republic, and resulted in the establishment of twenty annually elected magistrates of the people, implied some modification of this principle of the investiture of magistrates with supreme authority. For when the magistracies were multiplied a distinction was drawn between magistrates with _imperium_, namely consuls, praetors and occasionally dictators, and the remaining magistrates, who, although exercising independent magisterial authority and in no sense agents of the higher magistrates, were invested merely with an authority (_potestas_) to assist in the administration of the state. At the same time the actual authority of every magistrate was weakened not only by his colleagues' power of veto, but by the power possessed by any magistrate of quashing the act of an inferior, and by the tribune's right of putting his veto on the act of any magistrate except a dictator; and the subdivision of authority, which placed a great deal of business in the hands of young and inexperienced magistrates, further tended to increase the actual power as well as the influence of the senate at the expense of the magistracy.

In the developed Republic magistracies were divided into two classes: (a) magistrates of the whole people (_populi Romani_) and (b) magistrates of the _plebs_. The former class is again divided into two sections: ([alpha]) curule and ([beta]) non-curule, a distinction which rests mainly on dignity rather than on actual power, for it cuts across the division of magistrates according to their tenure or non-tenure of _imperium_.

a. The magistrates of the people--also known as patrician magistrates, probably because the older and more important of these magistracies could originally be held only by patricians (q.v.)--were: ([alpha]) Dictator, master of the horse (see DICTATOR), consuls, praetors, curule, aediles and censors (curule); and ([beta]) Quaestors, and the body of minor magistrates known as _xxvi. viri_ (non-curule). The dictatorship and consulship were as old as the Republic. The first praetor was appointed in 366 B.C., a second was added in 242 B.C., and the number was gradually increased for provincial government until Sulla brought it up to eight, and under the early principate it grew to eighteen. Censors were first instituted in 443 B.C., and the office continued unchanged until its abolition by Sulla, after which, though restored, it rapidly fell into abeyance. Curule aediles were instituted at the same time as the praetorship, and continued throughout the Republic. The quaestorship was at least as old as the Republic, but the number rose during the Republic from two to twenty. All these offices except the censorship continued for administrative purposes during the principate, though shorn of all important powers.

b. The plebeian magistrates had their origin in the secession of the _plebs_ to Mons Sacer in 494 B.C. (see ROME: _History_). In that year tribunes of the _plebs_ were instituted, and two aediles were given them as subordinate officials, who were afterwards known as plebeian aediles, to distinguish them from the curule magistrates of the same name. Both these offices were abolished during the decemvirate, but were restored in 449 B.C., and survived into the principate.

The powers possessed by all magistrates alike were two:--that of enforcing their enactments (_coercitio_) by the exercise of any punishment short of capital, and that of veto (_intercessio_) of any act of a colleague or minor magistrate. The right of summoning and presiding over an assembly of that body of citizens with whose powers the magistrate was invested lay with the higher magistrates only in each class, with the consuls and praetors, and with the tribunes of the _plebs_. Civil jurisdiction was always a magisterial prerogative at Rome, and criminal jurisdiction also, except in capital cases, the decision of which was vested in the people at least as early as the first year of the Republic, was wielded by magistrates until the establishment of the various _quaestiones perpetuae_ during the last century of the Republic. But in civil cases the magistrate, though controlling the trial and deciding matters of law, was quite distinct from the judge or body of judges who decided the question of fact; and the _quaestiones perpetuae_, which reduced the magistrate in criminal cases to a mere president of the court, gave him a position inferior to that of the praetor, who tried civil cases, only in so far as the praetor controlled the trial in some degree by his _formula_, under which the judges decided the question of fact.

Tenure of magistracy was always held to depend upon election by the body whose powers the magistrate wielded. Thus the magistrates of the _plebs_ were elected by the plebeian council, those of the people in the Comitia (q.v.). In every case the outgoing magistrate, as presiding officer of the elective assembly, exercised the important right of nominating his successor for election.

See A. H. J. Greenidge, _Roman Public Life_, 152 seq., 363 seq. (London, 1901); T. Mommsen, _Römisches Staatsrecht_, I. 11. i. (1887). (A. M. Cl.)

MAGLIABECHI, ANTONIO DA MARCO (1633-1714), Italian bibliophile, was born at Florence on the 28th of October 1633. He followed the trade of a goldsmith until 1673, when he received the appointment of librarian to the grand-duke of Tuscany, a post for which he had qualified himself by his vast stores of self-acquired learning. He died on the 4th of July 1714, bequeathing his large private library to the grand-duke, who in turn handed it over to the city.

MAGLIANI, AGOSTINO (1824-1891), Italian financier, was a native of Lanzino, near Salerno. He studied at Naples, and a book on the philosophy of law based on Liberal principles won for him a post in the Neapolitan treasury. He entered the Italian Senate in 1871, and had already secured a reputation as a financial expert before his _Questione monetaria_ appeared in 1874. In December 1877 he became minister of finance in the reconstructed Depretis ministry, and he subsequently held the same office in three other Liberal cabinets. In his second tenure he carried through (1880) the abolition of the grist tax, to take effect in 1884. Having to face an increased expenditure without offending the Radical electorate by unpopular taxes, he had recourse to unsound methods of finance, which seriously embarrassed Italian credit for some years after he finally laid down office in 1888. He died in Rome on the 22nd of February 1891. He was one of the founders of the anti-socialistic "Adam Smith Society" at Florence.

MAGNA CARTA, or the Great Charter, the name of the famous charter of liberties granted at Runnimede in June 1215 by King John to the English people. Although in later ages its importance was enormously magnified, it differs only in degree, not in kind, from other charters granted by the Norman and early Plantagenet kings. Its greater length, however, still more the exceptional circumstances attending its birth, gave to it a position absolutely unique in the minds of later generations of Englishmen. This feeling was fostered by its many confirmations, and in subsequent ages, especially during the time of the struggle between the Stuart kings and the parliament, it was regarded as something sacrosanct, embodying the very ideal of English liberties, which to some extent had been lost, but which must be regained. Its provisions, real and imaginary, formed the standard towards which Englishmen must strive.

The causes which led to the grant of Magna Carta are described in the article on _English History_. Briefly, they are to be found in the conditions of the time; the increasing insularity of the English barons, now no longer the holders of estates in Normandy; the substitution of an unpopular for a popular king, an active spur to the rising forces of discontent; and the unprecedented demands for money--demands followed, not by honour, but by dishonour, to the arms of England abroad. So much for the general causes. The actual crisis may be said to begin with the quarrel between John and Pope Innocent III. regarding the appointment of a new archbishop to the see of Canterbury. This was settled in May 1213, and in the new prelate, the papal nominee, Stephen Langton, who landed in England and absolved the king in the following July, the baronial party found an able and powerful ally. But before this event John had instituted a great inquiry, the inquest of service of June 1212, for the purpose of finding out how much he could exact from each of his vassals, a measure which naturally excited some alarm; and then, fearing a baronial rising, he had abandoned his proposed expedition into Wales, had taken hostages from the most prominent of his foes, and had sought safety in London.

His absolution followed, and then he took courage. Turning once more his attention to the recovery of Normandy, he asked the barons for assistance for this undertaking; in reply they, or a section of them, refused, and instead of crossing the seas the king marched northwards with the intention of taking vengeance on his disobedient vassals, who were chiefly barons of the north of England. Langton followed his sovereign to Northampton and persuaded him, at least for the present, to refrain from any serious measures of revenge. Before this interview a national council had met at St Albans at the beginning of August 1213, and this was followed by another council, held in St Paul's church, London, later in the same month; it was doubtless summoned by the archbishop, and was attended by many of the higher clergy and a certain number of the barons. Addressing the gathering, Langton referred to the laws of Edward the Confessor as "good laws," which the king ought to observe, and then mentioned the charter granted by Henry I. on his accession as a standard of good government. This event has such an important bearing on the issue of Magna Carta that it is not inappropriate to quote the actual words used by Matthew Paris in describing the incident. The chronicler represents the archbishop as saying "Inventa est quoque nunc carta quaedam Henrici primi regis Angliae per quam, si volueritis, libertates diu amissas poteritis ad statum pristinum revocare." Those present decided to contend to the death for their "long-lost liberties," and with this the meeting came to an end. Nothing, however, was done during the remainder of the year, and John, feeling his position had grown stronger, went abroad early in 1214, and remained for some months in France. With his mercenaries behind him he met with some small successes in his fight for Normandy, but on the 27th of July he and his ally, the emperor Otto IV., met with a crushing defeat at Bouvines at the hands of Philip Augustus, and even the king himself was compelled to recognise that his hopes of recovering Normandy were at an end.

Meanwhile in England, which was ruled by Peter des Roches as justiciar, the discontent had been increasing rather than diminishing, and its volume became much larger owing to an event of May 1214. Greatly needing money for his campaign, John ordered another scutage to be taken from his tenants; this, moreover, was to be at the unprecedented rate of three marks on the knight's fee, not as on previous occasions of two marks, although this latter sum had hitherto been regarded as a very high rate. The northern barons refused to pay, and the gathering forces of resistance received a powerful stimulus when a little later came the news of the king's humiliation at Bouvines. Then in October the beaten monarch returned to England, no course open to him but to bow before the storm. In November he met some of his nobles at Bury St Edmunds, but as they still refused to pay the scutage no agreement was reached. At once they took another step towards the goal. With due solemnity (_super majus altare_) they swore to withdraw their allegiance from the king and to make war upon him, unless within a stated time he restored to them their rightful laws and liberties. While they were collecting troops in order to enforce their threats, John on his part tried to divide his enemies by a concession to the clerical section. By a charter, dated the 21st of November 1214, he granted freedom of election to the church. However, this did not prevent the prelates from continuing to act to some extent with the barons, and early in January 1215 the malcontents asked the king to confirm the laws of Edward the Confessor and the other liberties of the kingdom. He evaded the request and secured a truce until Easter was passed. Energetically making use of this period of respite, he again issued the charter to the church, ordered his subjects to take a fresh oath of allegiance to him, and sent to the pope for aid; but neither these precautions, nor his expedient of taking the cross, deterred the barons from returning to the attack. In April they met in arms at Stamford, and as soon as the truce had expired they marched to Brackley, where they met the royal ministers and again presented their demands. These were carried to the king at Oxford, but angrily he refused to consider them. Then the storm burst. On the 5th of May the barons formally renounced their allegiance to John, and appointed Robert Fitzwalter as their leader. They marched towards London, while John made another attempt to delay the crisis, or to divide his foes, by granting a charter to the citizens of London (May 9, 1215), and then by offering to submit the quarrel to a court of arbitrators under the presidency of the pope. But neither the one nor the other expedient availed him. Arbitration under such conditions was contemptuously rejected, and after the king had ordered the sheriffs to seize the lands and goods of the revolting nobles, London opened its gates and peacefully welcomed the baronial army. Other towns showed also that their sympathies were with the insurgents, and John was forced to his knees. Promising to assent to their demands, he agreed to meet the barons, and the gathering was fixed for the 15th of June, and was to take place in a meadow between Staines and Windsor, called Runnimede.

At the famous conference, which lasted from Monday the 15th to Tuesday the 23rd of June, the hostile barons were present in large numbers; on the other hand John, who rode over each day from Windsor, was only attended by a few followers. At once the malcontents presented their demands in a document known popularly as the _Articles of the Barons_, more strictly as _Capitula quae barones petunt et dominus rex concedit_. Doubtless this had been drawn up beforehand, and was brought by the baronial leaders to Runnimede; possibly it was identical with the document presented to the royal ministers at Brackley a few weeks before. John accepted the Articles on the same day and at once the great seal was affixed to them. They are forty-eight in number, and on them Magna Carta was based, the work of converting them into a charter, which was regarded as a much more binding form of engagement, being taken in hand immediately. This duty occupied three days, negotiations between the two parties taking place over several disputed points, and it was completed by Friday the 19th, when several copies of the charter were sealed. All then took an oath to keep its terms, and orders were sent to the sheriffs to publish it, and to see that its provisions were observed, two or three days being taken up with making and sending out copies for this purpose. It should be mentioned that, although the charter was evidently not sealed until the 19th, the four existing copies of it are dated the 15th, the day on which John accepted the articles.

The days between Friday the 19th and the following Tuesday, when the conference came to an end, were occupied in providing, as far as possible, for the due execution of the reforms promised by the king in Magna Carta. The document itself provided for an elected committee of twenty-five barons, whose duty was to compel John, by force if necessary, to keep his promises; but this was evidently regarded as insufficient, and the matter was dealt with in a supplementary treaty (_Conventio facta inter regem Angliae et barones ejusdum regni_). As a guarantee of his good faith the king surrendered the city of London to his foes, while the Tower was entrusted to the neutral keeping of the archbishop of Canterbury. John then asked the barons for a charter that they on their part would keep the peace. This was refused, and although some of the bishops entered a mild protest, the question was allowed to drop. Regarding another matter also, the extent of the royal forests, the prelates made a protest. John and his friends feared lest the inquiry promised into the extent of the hated forest areas would be carried out too rigorously, and that these would be seriously curtailed, if not abolished altogether. Consequently, the two archbishops and their colleagues declared that the articles in the charter which provided for this inquiry, and for a remedy against abuses of the forest laws by the king, must not be interpreted in too harsh a spirit. The customs necessary for the preservation of the forests must remain in force.

No securities, however, could bind John. Even before Magna Carta was signed he had set to work to destroy it, and he now turned to this task with renewed vigour. He appealed to the pope, and hoped to crush his enemies by the aid of foreign troops, while the barons prepared for war, and the prelates strove to keep the peace. Help came first from the spiritual arm. On the 24th of August 1215 Innocent III. published a bull which declared Magna Carta null and void. It had been extorted from the king by force (_per vim et metum_), and in the words of the bull the pope said "compositionem hujusmodi reprobamus penitus et damnamus." He followed this up by excommunicating the barons who had obtained it, and in the autumn of 1215 the inevitable war began. Capturing Rochester castle, John met with some other successes, and the disheartened barons invited Louis, son of Philip Augustus of France and afterwards king as Louis VIII., to take the English crown. In spite of the veto of the pope Louis accepted the invitation, landed in England in May 1216, and occupied London and Winchester, the fortune of war having in the meantime turned against John. The "ablest and most ruthless of the Angevins," as J. R. Green calls this king, had not, however, given up the struggle, and he was still in the field when he was taken ill, dying in Newark castle on the 19th of October 1216.

In its original form the text of Magna Carta was not divided into chapters, but in later times a division of this kind was adopted. This has since been retained by all commentators, the number of chapters being 63.

The preamble states that the king has granted the charter on the advice of various prelates and barons, some of whom, including the archbishop of Canterbury, the papal legate Pandulf, and William Marshal, earl of Pembroke, are mentioned by name.