The word magic is still· used, as in the ancient world, to include a confused mass of beliefs and practices, hardly agreeing except in being beyond those ordinary actions of cause and effect which men accustomed to their regularity have come to regard as merely natural. Thus magical rites are difficult even to arrange in systematic order. A large proportion of them belong properly to the general theory of religion, inasmuch as their efficacy is ascribed to the intervention of spiritual beings. Thus the ghosts of the dead are called up by the necromancer to give oracles or discover hidden treasures, or sent to enter men’s bodies and afflict them with diseases or to cure them, or in a score of ways to do the behests of the magician, whose spells or incantations are thought powerful enough to control the will even of such divine beings as can drive the winds and give or withhold the rain. It must be noticed, on the other hand, that many magical arts show no connexion with spirits at all, or, even if ghosts or demons or gods have to do with them, the nature of these beings does not of itself account for the processes employed or the effects believed to result. This non-spiritual element in magic depends on imagined powers and correspondences in nature, of which the adepts avail themselves in order to discover hidden knowledge, and to act on the world around them by means beyond the ordinary capabilities of men. Thus by mere effort of will, by traditional formulas and rites, or by working on symbolic fancies, the sorcerer believes he can bewitch others to sickness or death, the astrologer reads the future in the aspects of the stars, the augur attends to omens from the cries of birds and beasts, the haruspex prophesies by the heart or liver of a slaughtered animal, and other classes of diviners judge of the hidden past and the yet more hidden future by the falling of lots or dice, the twitching of their own fingers or the tingling of their ears, and a host of other facts of nature which, as the educated world has now found out, have no practical connexion with the magical meanings or effects assigned to them. The great characteristic of magic is its unreality. Its methods have often an ideal coherence which may be plainly traced, but practical effect they have none, and so they may be altered or transposed without being made worse or better.

One remarkable consequence of this is the fixity with which some magical formulas framed thousands of years ago hold on almost unchanged to this day. To understand this, it must be borne in mind that, if there were any practical use in such rules as those for divining by the cries of animals, the old rules would have been improved by experience into new shapes. But, they being worthless and incapable of improvement, this motive of change is absent, and the old precepts have held their ground, handed on by faithful but stupid tradition, from age to age. When the test of practical efficacy comes in upon the magic art, it is apt either to destroy it utterly or to transform it into something more rational, which passes from supernatural into natural science.

Magic is to be reckoned among the’earliest growths of human thought. The evidence for its remote antiquity lies partly in its presence among all races of mankind, the ruder tribes especially showing it in such intelligible shapes that the beginnings of magical crafts may be fairly supposed to have arisen in the oldest and lowest periods of culture. An example may be taken from the wild natives of Australia, whose whole life is pervaded by the belief, and embittered by the terror, of sorcery. They imagine the sorcerers, armed with their mysterious power called boyl-ya, to come moving along in the sky, invisible except to other sorcerers; they enter the bodies of men, and feed stealthily on them, not eating the bones, but consuming· the flesh; the native feels the pain as the boyl-ya enters him like a bit of pointed quartz, and in this shape of quartz crystal the evil can be extracted by another sorcerer. The sorcerer has other means of attacking his victim: he can creep near to him when asleep, and bewitch him to death by merely pointing at him a leg bone of a kangaroo; or he can steal away his kidney fat, where, as the natives believe, a man’s power dwells; or he can call in the aid of a malignant demon to strike the poor wretch with his club behind the neck; or he can get a lock of hair, and roast it with fat over the fire till its former owner pines away too, and dies. The Australians, like other low tribes in the world whose minds are thus set on imaginary causes of death, hardly believe a man can die unless by being slain or bewitched. When a native dies what we call a natural death, they ascribe it to magic. Then other magic must reveal the hostile sorcerer who has done him to death: either the corpse itself will seem to push its bearers in the direction of the murderer, or the flames of the grave-fire are seen to flicker towards where he is, or some insect will be seen creeping towards his home; and, when the next of kin thus discover the magic enemy, they set off to take vengeance with earthly weapons. The sorcerer has kindlier duties when he sits by a sick man and charms and charms till he recovers, or sucks out the disease from his body in the shape of a stone spear-head or a fish bone, or brings out the ailment along a string, the other end of which he draws between his own lips till it is covered with blood, telling the bystanders (who believe it) that this blood came along the string out of the sick man. Not disease and cure only, but other events of life, come within the scope of native magic. Storm and thunder are the work of the sorcerers; they can bring rain and make the rivers swell, or burn up the land with drought. Shooting stars and comets are to the natives omens of disaster; the great hawk’s cry in the night portends the death of a child, whose soul the bird is carrying off; but when a man’s finger-joints crack he stretches out his arm, for in that direction some one is doing him a kindness.

Taken together, such a repertory of the demonology and witchcraft of a special group of savage tribes shows remarkable correspondence in principle with the magic which once flourished in the civilized world, and which still lingers in peasant folklore. The very details often agree so much as to raise the question whether the magic of savages may sometimes have been borrowed from the lower class of colonists. The superstitions of the peasant are in fact what the savage would readily assimilate, as belonging to a state of mind like his own, and there is even évidence of European charms and omens having been sometimes borrowed by native tribes of Australia or America. It was necessary to mention this, if merely to point out that such borrowing has been only slight and superficial. It in no way upsets the genera] principle that the magic of the lower races was developed among them, fitting as it does with their low level of knowledge. Every book of travels in savage and barbaric countries shows the influence of the native magician, who, often at once sorcerer and established priest, and sometimes even chief of his tribe, by the aid of spirits and other supernatural means interferes in every act of life. Thus in the Pacific islands the Europeans found a whole class of sorcerers living by making diseases, their method being the familiar one of burning or otherwise practising on some morsel of hair or remnant of food, so as to send disease into its owner, by a malignant spirit tying knots in his inside till he writhed with agony. Every sick man was a source of profit to the sorcerer who was believed to have brought on the disease by burning his rubbish, and of course had to be bought off by liberal presents. In these Pacific islands a fact most important in the theory of magic everywhere comes into view with particular distinctness—that such magical arts prove effective through the patient’s own imagination; when he knows or fancies that he has been bewitched he will fall ill, and he will actually die unless he can be persuaded that he has been cured. Thus, wherever sorcery is practised with the belief of its victims, some system of exorcism or some protective magical art becomes, not only necessary, but actually effective, a mental disease being met by a mental remedy to match it. At the discovery of America, the Spaniards found the native sorcerers throwing themselves into delirious ecstasy by snuffing a narcotic powder, their ravings in this state being held to be conversation with departed souls, through whose help they were able to cure the sick by expelling the disease. The class to which these sorcerers belong extends over South America, and is generally known under the name of paye (or allied terms). The sorcerer is described as being initiated by living in some wild spot till by fasting and self-torture he attains his supernatural craft, becoming able to see spirits, to consecrate bits of bone or stone into powerful amulets, to make good or bad weather, to gain mystic powers over familiar birds and beasts, to take omens from their cries or from the itching of his own skin, which latter symptom an Abipone diviner declared to portend an attack from a tribe of enemies, in spite of the missionary, who irreverently set it down to fleas. The old arts of the payés, their malicious witchcraft with herbs and hair, the use of narcotics to produce ecstasy, and their mental excitement by drumming, rattling, and dancing are still to be met with in the wild districts of Guiana and Brazil. In North America practitioners of the same kind are generally known as “medicine-men,” from the French colonists calling them médecins, as being the native doctors; the term is really appropriate to barbaric magicians in all parts of the world, whose arts of causing and curing disease generally include considerable knowledge of herbs powerful as poisons and remedies, of simple stopping of wounds and bandaging hurt limbs, in fact of medicine in its elementary state, as yet not separated from the magic with which it was at first inextricably mixed up. The medicine-man’s apparatus includes the sorcerer’s usual music, the rattle and the drum, simple and primitive instruments whose constant association with the lower magic bears witness to the beginnings of music and magic having been associated together when civilization was yet in its low stages of development. The American sorcerer carries a “medicinebag” made with the skin of his guardian animal, which protects him in fight, cures the bites of serpents, and strikes at a distance as a spiritual weapon. Be knows magic chants of power over the elements; he can bysucking and blowing extract disease animals from the sick; he can make pictures and images and pierce them with thorns so as to kill the men or animals they represent; and he can compel love by practising on the heart of the picture of the beloved one. In Africa the native sorcerer bears the name of mganga among the west and central negro tribes, nyanya among the Zulus of the south. He is the rain-maker, an office of the utmost importance among tribes who may perish of famine and disease after a long drought. In his craft a principal part is played by what the English in Africa (using the Portuguese word feitiςo, charm or amulet) call “fetiches,” which are claws, fangs, roots, stones, and any other odds and ends fancied to be inhabited by spirits or invested with superhuman power. These fetiches the negroes trust in for good and against evil fortune, with a confidence which no failure can shake further than to cause the unlucky bearer to discard a particular fetich which has failed, and to replace it by a more successful one. The African mganga has intercourse with demons; and, being called on every day to predict the fortune of a fight or a bargain, or to discover lost or stolen cattle, he professes to gain information from the spirits, or uses his various modes of divination, such as taking omens from the cries of the eagle or the owl, the swimming of berries, or the moving of sticks in his own hands as they twitch spasmodically in nervous excitement. As with magicians everywhere, his trade is profitable but dangerous, for if his arts of killing have been successful beyond bearing, or still worse if public opinion decides that he has wilfully withheld the rain, he may be drowned or burned as miserably as one of the many victims he has done to death. ® These instances are selected to give an idea of the sorcerer» of the lower races and their modes of working, which are remarkable for their uniformity in the most distant regions, among tribes who can have had no communication or connexion since remote ages. Where, however, such races as the African negroes come in contact with such foreigners as the Arabs, who though more civilized than themselves hâve not outgrown the magical stage, they borrow their more cultured magical arts, such as divination by lots. In this way the natives of Madagascar appear to have borrowed from the Arabs a system of lucky and unlucky days of birth, which, carried out with stupid ferocity, has cost the lives of thousands of children, born truly in an evil hour, for when the magician declares their birth ill-omened their fate is settled at once by putting them to death.

Turning now to the cultured nations of antiquity, among whom the art of writing consolidated and developed false as well as true science, we find magic in full vogue, hardly differing in principle from that of the illiterate barbarians, but worked into more elaborate system and ritual. Of ancient Egyptian magic various original documents have been preserved, containing formulas, mostly of religious magic,—that is, acting through the aid of deities invoked. For instance, there are hymns against dangerous animals in the water, and spells for remaining in the country; and the power ascribed to such formulas appears from passages like the following :—“I confide in the efficacy of that excellent written book given to-day into my hand, which repels lions through fascination, disables men, which muzzles the mouths of lions, hyaenas, wolves, the mouth of all men who have bad faces, so as to paralyse their limbs,” &c. Ancient as Egyptian magic is, it has evidently grown up from still earlier forms, as is shown by that plainest symptom of old traditional lore, the relying on ancient or foreign epithets as words of power over the gods. This practice appears in the ancient papyri, and goes on to later ages, when the god Set is invoked by other mystically powerful names which he must obey, such as “Joerbeth.” The medical art in ancient Egypt shows an interesting combination of practical and magical remedies. The practical recipe might contain nitre or cedar chips, or deer horn, or various other ingredients administered in ointment or drunk in beer, but with this the magical formula was also required to deal with the demon-cause of the ailment. Thus an emetic was given with the following formula, “O demon who art lodged in the stomach of M., son of N., thou whose father is called Head-smiter, whose name is Death, whose name is cursed for ever !” &c. It must be remembered that such formulas, foolish as they seem to modern education, had and still have great efficacy in relieving the mind of the superstitious patient, and giving a fair chance to diet and medicaments. Their appearance in medicine so ancient as that of Egypt is good historical evidence how the old magical treatment was encroached upon by natural remedies, though then and for many ages-afterwards the physicians, wise in their generation, thought it best not to discard the supernatural charm. The Egyptians divided out the limbs and organs of the human body, putting each-under the special care of a god, a system which, like many other details of their magic, has lasted on into the modern world. From the astrological point of view they made a calendar of lucky and unlucky days, according to which for instance on the 19th of the month Athor one must not embark on the Nile, while a child born on the 5th of the month Paopi will be killed by a bull; traces of this set of precepts may be discerned still in the modern Egyptian almanac. Another point deserving attention is the appearance in early Egypt of the distinction between good and bad magic. Magical curative arts were practised by learned scribes or priests, and were doubtless in high esteem, but when it came to attracting love by charms or philtres, or paralysing men by secret arts, this was held to be a crime. As long ago as the time of Rameses III. it is recorded that one Hai was accused of making images and paralysing a man’s hand, for which he was condemned to death; this was doubtless the ordinary bewitching by an image or picture, here already mentioned among the lower races, and to be mentioned again as not forgotten among ourselves.

Still more prominent among the ancient nations who brought magic into its pseudo-scientific stage were the Babylonians, whose supernatural arts were adopted and continued among the Assyrians. No savage tribe ever filled their world with more swarming hosts of nature spirits and demons; only these more cultured nations dealt systematically with them by set formulas of propitiation and expulsion. The cuneiform writings preserve numerous documents of this kind, such as “From the burning spirit of the entrails which devours the man, from the spirit of the entrails which works evil, may the king of heaven preserve, may the king of earth preserve!”—“The god . . . . shall stand by his bedside. Those seven evil spirits he shall root out, and shall expel them from his body; and those seven shall never return to the sick man again.” The magic power believed to reside in the secret names of the gods was recognized by the Babylonians, one of whose famous myths relates how by the utterance of these mystic names the goddess Ishtar was delivered from Hades. In the rites of the magician priests, this kind of supernatural power resided in sacred texts, whether chanted or tied on as phylacteries. In divinatory magic the Babylonians had elaborate codes of rules, of which many have been preserved. Thus omens were drawn from prodigies, such as “when a woman bears a child and at the time of birth its teeth are cut, the days of the prince will be long.” So with omens from animals: “if a dog goes to the palace and lies down on a throne, that palace will be burned.” ,A remarkable passage, Ezekiel xxi. 21, mentions three modes of divination practised by the king of Babylon as he stood at the head of the two ways: “he shuffled arrows, he consulted teraphim, he looked in the liver.” The arrowdivination or belomancy here mentioned was done with pointless arrows marked and drawn äs lots. They are often represented on Babylonian and Assyrian cylinders, and their use was kept up among the Arabs till the time of Mohammed. The Babylonian rules of haruspication,...

The magic lantern in its simplest form is represented in fig. 1. A is a dark box surmounted by a suitable chimney for carrying off the products of combustion from the source of light I., which is placed in the focus of a spherical reflector. On the side opposite the reflector the box is pierced by a round hole, into which is fitted the metal tube D, which may either be cylindrical or slightly tapered. To the inner end of the tube D is fitted a lens or combination of lenses C called the “condenser.” Immediately in front of the condenser, and at right angles to the axis of the tube, is a vertical slit S for receiving the transparent pictures usually called “slides.” To the outer end of D is fitted a lens or combination of lenses 0 called the “objective” or “projecting lens.” At a suitable distance from O is placed the screen for receiving the magnified picture —the screen and slit being in the conjugate foci of the lens O. Since the objective reverses, the picture must be inserted in the slit upside down. The objective can be moved backwards and forwards in the tube D by means of a rack and pinion. Since the screen and slide are in the conjugate foci of O the diameter of the picture on the screen has the same ratio to the screen’s distance from O that the diameter of the picture on the slide has to the slide’s distance from O. The ratio adopted is generally 1 to 3 or 4.

Source of Light.— Almost any good source of light can be used in the magic lantern. In the earlier forms a simple oil light was used; and in the toy forms either an oil light or simple gasflame is still employed. Better effects are obtained from the Argand fountain lamp (fig. 2) or from the common Argand gas burner with a glass chimney. In the sciopticon (fig. 3)—a handy and excellent form recently introduced—a specially constructed paraffin lamp is employed with three parallel flat wicks set edgeways to the condenser. With this lamp a clear, well-defined picture varying from 6 to 10 feet in diameter can be readily thrown on the screen. For the best effects, however, recourse must be had to the oxy-calcium light (fig. 4), in which a small cylinder of lime is heated to intense luminosity in the flame of a spirit lamp through which a jet of oxygen plays; the Oxy-hydrogen Light

(ç.v.); the magnesium light, in which two narrow ribands of magnesium are put through slits bya clock-work arrangement and burn as they advance; or, best of all,—unless when sunlight can be used with the aid of a heliostat (see Heliography), —the electric light.

Condenser and Objective.— The object of the condenser (fig. 5, C) is to collect as much light as possible from the source, and pass it through the transparent picture in the slit. For this purpose the condenser should subtend as large an angle as possible at the source of light. To secure this, the condenser should be tolerably large, and its distance from the light, that is, its focal length, small. Since effective single lenses of large diameter are necessarily of long focus, a really good condenser of considerable diameter and yet of short focus must be a combination of two or more lenses.

Let f 1 and ∕ 2 be the focal lengths of two lenses, and f the focal length of their combination. Then, neglecting the thickness of the lenses, we have when the two are close toegther

7"⅛ + ⅛

From which we get/-a fraction which ie always less than either f 1 or∕ 2 provided these are not zero. Suppose, for example, two lenses of 6 and 10 inches focal length respectively; the focal length of their combination will be θ^θ =3f inches.

In the earlier lanterns, as still in the cheaper forms, only a single plano-convex lens or bull’s-eye was employed as a condenser. Better effects are produced by two such lenses. Perhaps the best condenser for ordinary work is that proposed by Herschel and represented in fig. 5, C, consisting of a biconvex lens and a meniscus mounted together with the concave side of the meniscus next the light. The diameter of such a condenser is about 6 inches, and its focal length a little over 3 inches. The focus must not be so short as to bring the lens too near the light, and render it liable to crack from the intense heat. In some lanterns this is guarded against by placing a plate of thin glass between the condenser and the light. In the sciopticon (fig. 3) the condenser consists of two plane-convex lenses, each about 4 inches in diameter. Condensers of large diameter are not so essential now as formerly, seeing that small pictures can easily be produced by photography.

The function of the objective (fig. 5, 0) is to produce a magnified inverted image of the picture on the screen. In toy lanterns it is a simple double-convex lens of short focus. This, however, can only produce a small picture, and that not very distinct at the edges. The best objective is the portrait combination lens used in ordinary photographic cameras. These are carefully corrected both for spherical and chromatic aberration, which is absolutely essential in the objective, although not so necessary in the condenser. It is essential, however, that the condenser be free from cracks or flaws, as these would appear and mar the picture.

Slides.— These are pictures, painted with transparent water or oil colours, or photographed on pieces of glass. The pieces of glass are usually fixed in small wooden frames for insertion in the slit. If parts of the picture are to be movable, two disks of glass are employed, the one movable in front of the other, the fixed part of the picture being painted on the fixed disk and the movable part on the other. By means of a lever the latter disk is moved in its own plane and in this way a cow for instance can be represented drinking, or a donkey cutting amusing capers. A lever slide is represented in fig. 6. In the chromatrope elide (fig. 7) two circular disks of glass are placed face to face, each containing a design radiating from the centre, and painted with brilliant transparent colours. By a small pinion gearing in toothed wheels or endless bands the disks are made to move in opposite directions in their own plane.

The effect produced is a singularly beautiful change of design and colour. In astronomical slides the motions of the heavenly bodies, eclipses, the phases of the moon, or the like are similarly represented by mechanical means. »Slides can also be made from narrow glass tanks with parallel sides. When these are filled with water containing delicate living organisms the forms and movements of the latter are beautifully seen. Such tanks can also be employed to show such phenomena as the gradual growth of crystals, the electrolysis of water between platinum electrodes, &c. A great variety of physical and chemical experiments can be shown in this way.

Dissolving Views.— For this purpose two magic lanterns are necessary, arranged either side by side or the one on the top of the other. The fronts of the lanterns are slightly inclined to each other so as to make the illuminated disks on the screen due to each lantern coincide. By means of a pair of thin metallic shutters terminating in comb-like teeth, and movable by a rack or lever, the light from either lantern can be gradually cut off at the same time that the light from the other is allowed gradually to fall on the screen. In this way one view appears to melt or dissolve into another. This arrangement was first adopted by Childe in 1811.

Phantasmagoria.— In this arrangement the pictures on the screen appear gradually to increase or diminish in size and brightness. To effect this a semi-transparent screen of cotton or other material is used, the lantern being behind and the audience in front. The lantern is mounted on wheels so that it can be rapidly moved up to or withdrawn from the screen; and an automatic arrangement is provided whereby simultaneously with this the objective is made to approach or recede from the slide so as to focus the picture on the screen in any position of the lantern. In this way a very small picture appears gradually to grow to enormous dimensions.

Lantern Polariscope.— This, perhaps the most beautiful modification of the magic lantern for scientific purposes, consists of an elbowshaped tube, containing mirrors, lenses, &c., and attached to the front of the lantern in place of the tube containing the objective. It is represented in section in fig. 8. C' is the usual condenser belonging to the lantern. G is a set of thin glass plates inclined at the polarizing angle 56° 45' to the axis of the tube. The beam of polarized light from I. reflected from G passes through the lenses

F and the analysing Nicol’s prism P, and falls on the screen. The objects to be examined by the polarized light are placed in the transverse slit 0. When thin plates of selenite or other doubly refracting crystals are placed in 0, a most beautiful display of complementary colours is produced on the screen by rotating the N icol’s prism. Almost all the experiments on polarized light can be well shown by this arrangement.

The sdopticon (fig. 3) is an excellent and convenient lantern, very suitable for all the requirements of the lecturer, as well as for school use in teaching geography, &c.

See Brewster's optics; Ganot's Physics; and Chadwick’s Manual of the Magic Lantern.

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 mystic imaginings of the old astrologers relations subsisted between these squares and the planets: a square with only one cell, containing 1, symbolized the unity of the deity; a square of two, containing the four elements, was the symbol of matter; while those of 3, 4, 5, 6, 7, 8 were consecrated respectively to Saturn, Jupiter, Mars, the Sun, Venus, and Mercury. In later times such squares ranked only as mathematical curiosities; till at last their mode of construction was systematically investigated. These squares were at first mere triumphs of the same dogged perseverance as was in later times exhibited by the Dutchman, Ludolph van Ceulen, who, after calculating τr to 35 places of decimals, directed, like Archimedes, that it should be engraven on his tomb, though his industry was surpassed by M. de Lagry, who continued the decimal to 127 places. The earliest known writer on the subject was Emanuel Moscopulus, a Greek, who lived in the 4th or 5th century, and whose manuscript is preserved in the National Library at Paris. After him Frenicle 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 M. Poignard constructed squares with numbers in arithmetical progression, having the magical summations. The later researches of M. de la Hire, recorded in the Mémoires de lj Academie 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.

In order to understand the rest of this article diagram A should be carefully examined. A square of 5 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 ones, 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. α, δ, c, d are placed diagonally in the square; but e enters the outer [table]

square, and is removed thence to the same cell of the square it had left, σ, β, γ, δ, e pursue another, but 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, i>, c, . . . . , α, β, γ, . . . . are said to lie in paths. Squares whose Poots are Odel.— Diagrams B, C, D exhibit one of the earliest methods of constructing magic squares. Here the 3’s in B and 2’s in C are placed in [table]

opposite diagonals to secure the two diagonal summations; then each number in C is multiplied by 5 and added to [table]

that in the corresponding number in B, which gives the square D. Diagrams E, F, G give M. de la Hire’s method; [table]

the squares E, F, being combined as above, give the magic square G. Μ. Bachet arranged the numbers as in H, where there are three numbers in each of four surrounding squares; these being placed in the corresponding cells of the central square, the square I 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 5x13; if therefore 13 is subtracted from each number, the summa [table]

tions will be zero, and the twenty-five cells will contain the series ± 1, ÷ 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 diagram J; then, 13 being added in each cell, the magic square K is obtained.

Squares whose Poots are Even.— These were constructed in various ways, similar to that of 4 in diagrams I., M, N. The numbers in M being multiplied by 4, and the squares I., M being superimposed, give N. The application of [table]

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 [table]

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 diagram O, make zero; and, to obtain the same [table]

in the rows and columns, we must assign such values to the ρ ’ Α and <fs as satisfy the equations jo 1 +jo 2 + α 1 -t-α 2 = O, p 3 + p i + a 3 + a 4 = 0, p 1 +p 3 ~α 1 -α 3 = 0, and p 2 + p i - α 2 - a 4 = Ο,—a solution of which is readily obtained by inspection, as in diagram P; this leads to the square, diagram Q. When the root is 8, the upper four subsidiary rows may at once be written, as in diagram R; then, if the square be completed, 65 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.

N asile Squares.— Squares that have many more summations than in rows, columns, and diagonals have been investigated by the Rev. A. H. Frost ^Cambridge Math. Jour., 1857), and called Nasik squares, from the town in India where he resided; and he has 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 diagram S, 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 ρ x be placed in the cells of one of the n + 1 normal paths, each of the remaining η normal paths will contain one, and only one, of these p 1 ⅛. If now we fill each row with ρ 2 , p 3 , ... p η in the same order, commencing from the ρ λ in that row, the p 2 ’ s, jd 3 , s, and pfs will lie each in a path similar to that of ρ 1 , and each of the n normal paths will contain one, and only one, of the letters ρ 1 , ρ 2 , . . . p n , whose sum will be '⅜p. Similarly, if q χ be placed along any of the normal paths, different from that of the ∕>⅛, and each row filled as above with the letters g 2 , q 3 , . . . q n , the sum of the 7’3 along any normal path different from that of the q l will be 5⅛. The n 2 cells of the square will now be found to contain all the combinations [table]

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 fs 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 ‰q + ⅜p. When the root is an odd composite number, as 9, 15, <fcc., it will be found that in some paths, different from the two along which the p χ and q x 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 p l p i p 7 , p 2 p 3 p 8 , p 3 p 6 p 3 , and q χ q i q 7 , q 2 q b q 6 , q 3 q 3 q 9 occur, each triplet thrice, along paths whose summation should be— ⅜p, 45, and Sr, 36. But if we make p x , p 2 ,. . . p Q = 1, 3, 6, 5, 4, 7, 9,8, 2, and the r 1 , r 2 , • . . r 9 = 0, 2, 5, 4, 3, 6, 8, 7, 1, thrice each of the above sets of triplets will equal Sp and ¾ 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 diagram T, the numbers are in the nonary scale; that in the centre is the middle one of 1 to 9 2 , and the sum of pairs 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 T 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 g ,, s may be respectively 1, 2, 10, 8, 6, 14, 15, 11, 4, 13, 9, 7,3, 12, 5, and0,1,9, 7,5,13,14, 10, 3, 12, 8, 6, 2, 11, 4, where five times the sum of every third number [table]

and three times the sum of every fifth number makes ⅜p and 2≥⅛; 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 a multiple of 4, the same process gives us, for the square of 4, the diagram U. Here the columns give ⅛p, but alternately 2<2 1 , 2ç 3 , and 2g 2 , 2g ’ 4 ; and the rows give ¾, but alternately 2p 1 , 2p 3 , and 2p 2 , 2p i ; the diagonals giving ¾> and ⅛q. If p χ , p 2f p. i , p i and g 1 , q 2 , q 3 , q i be 1, 2, 4, 3, and 0, 1, 3, 2, we have the Nasik square of diagram V. A square like this is engraven 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. [table] [table]

A Nasik cube is composed of n 3 small equal cubes, here called cubelets, in the centres of which the natural numbers from 1 to n 3 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. Diagram W 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, BOG, 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 BOG. Dots are now placed in the cubelets of similar paths to that on BOG in the other six sections parallel to BOO, 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, p x , q x , r x being placed in the comer cubelet 0, 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,p 2 g ’ 2 r 2 , pgq 3 r 3 , . . . prq∙flrι are placed in the other cubelets of the edge AO, and dispersed in the same manner as PN, ∖ r ∖ ∙ Every cubelet will then be found to contain a different combination of then’s, g ,, s, and r’s. If therefore the p’s are made equal to 1, 2, . . . 7, and the fs and r’s to 0, 1, 2, . . . 6, in any order, and the g ,, s multiplied by 7, and the r’s by 7 2 , then, as in the case of the squares, the 7 3 cubelets will contain the numbers from 1 to 7 3 , and the Nasical summations will be 27 2 r + S7g ’ +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 n s , 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 η, n new letters, s,, s, 1 , . . . 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 s 1 on each of the ri 2 cubelets of the first cube that contain p 1 , and on the n 2 cubelets of the 2d, 3d, . . . and nth cube that contain p 2 , p 3 , . . . p n respectively. This process is repeated with s 2 , 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, v?, and n i respectively, [table]

will now be filled with the numbers from 1 to n i , and the constant summation will be '2, ∣ ι 3 s + Sn 2 r + ~2,nq + ^2,ρ. 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 β cubelets of the compound cube of n s Nasik cubes [table] [table]

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 diagrams X.

Among the many ingenious squares given by various writers, this article may justly close with two by Euler, in the Histoire de Γ Académie Royale des Sciences, Berlin, 1759. In diagram Y 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 diagram Z 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 Mr Frost, can be seen 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 very complete bibliographical index of writers on this subject is given in Professor Lucas’s Récréations Mathématiques, Pans, 1882.