Chapter X: Front Matter (10)

_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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| 55| 42| 23| 10|247|234|215|202|183|170|151|138|119|106| 87| 74|
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|203|214|235|246| 11| 22| 43| 54| 75| 86|107|118|139|150|171|182|
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| 53| 44| 21| 12|245|236|213|204|181|172|149|140|117|108| 85| 76|
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|207|210|239|242| 15| 18| 47| 50 79| 82|111 114|143|146|175|178|
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| 49| 48| 17| 16|241|240|209|208 177|176|145 144|113|112| 81| 80|
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|196|221|228|253| 4| 29| 36| 61 68| 93|100 125|132|157|164|189|
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| 62| 35| 30| 3|254|227|222|195 190|163|158 131|126| 99| 94| 67|
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| 64| 33| 32| 1|256|225|224 193|192|161 160|129|128| 97| 96| 65|
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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.