(Magia, Magia), in its ancient sense, the science or discipline and doctrine of the magi, or wise men of Persia. See MAGI.

The origin of magic and the magi is ascribed to Zoroaster. Salmasius derives the very name from Zoroaster, who, he says, was surnamed Mag, whence Magnus. Others, instead of making him the author of the Persian philosophy, make him only the restorer and improver thereof; alleging, that many of the Persian rites in use among the Magi were borrowed from the Zabii among the Chaldeans, who agreed in many things with the Magi of the Persians; whence some make the name magus common both to the Chaldeans and Persians. Thus Plutarch mentions, that Zoroaster instituted magi among the Chaldeans, in imitation whereof the Persians had theirs too.

a more modern sense, is a science which teaches to perform wonderful and surprising effects.

The word magic originally carried with it a very innocent, nay, laudable meaning; being used purely to signify the study of wisdom, and the more sublime parts of knowledge; but in regard the ancient magi engaged themselves in astrology, divination, sorcery, &c., the term magic in time became odious, and was only used to signify an unlawful and diabolical kind of science, depending on the assistance of the devil and departed souls.

If any wonder how so vain and deceitful a science should gain so much credit and authority over men's minds, Pliny gives the reason of it. It is, says he, because Magic because it has possessed itself of three sciences of the most esteem among men: taking from each all that is great and marvellous in it. Nobody doubts but it had its first origin in medicine; and that it insinuated itself into the minds of the people, under pretence of affording extraordinary remedies. To these fine promises it added every thing in religion that is pompous and splendid, and that appears calculated to blind and captivate mankind. Lastly, It mingled judicial astrology with the rest; persuading people, curious of futurity, that it saw every thing to come in the heavens.

Agrippa divides magic into three kinds; natural, celestial, and ceremonial or superstitious.

Natural Magic is no more than the application of natural active causes to passive subjects; by means whereof many surprising, but yet natural, effects are produced.

In this way many of our experiments in natural philosophy, especially those of electricity, optics, and magnetism, have a kind of magical appearance, and among the ignorant and credulous might easily pass for miracles. Such, without doubt, have been some of those miracles wrought by ancient magicians, whose knowledge of the various powers of nature, there is reason to believe, was much greater than modern vanity will sometimes allow*.

Baptista Porta has a treatise of natural magic, or of secrets for performing very extraordinary things by natural causes. The natural magic of the Chaldeans was nothing but the knowledge of the powers of simples and minerals. The magic which they called theurgia, consisted wholly in the knowledge of the ceremonies to be observed in the worship of the gods, in order to be acceptable. By virtue of these ceremonies they believed they could converse with spiritual beings, and cure diseases.

Celestial Magic borders nearly on judiciary astrology: it attributes to spirits a kind of rule or dominion over the planets, and to planets a dominion over men; and on these principles builds a ridiculous kind of system. See Astrology.

Superstitious or Goetic Magic consists in the invocation of devils. Its effects are usually evil and wicked, though very strange, and seemingly surpassing the powers of nature; supposed to be produced by virtue of some compact, either tacit or express, with evil spirits: but the truth is, these have not all the power that is usually imagined, nor do they produce those effects ordinarily ascribed to them.

This species of magic, there is every reason to believe, had its origin in Egypt, the native country of paganism. The first magicians mentioned in history were Egyptians; and that people so famed for early wisdom believed not only in the existence of demons, the great agents in magic (see Daemon), but also that different orders of those spirits presided over the elements of earth, air, fire, and water, as well as over the persons and affairs of men. Hence they ascribed every disease with which they were afflicted to the immediate agency of some evil daemon. When any person was seized with a fever, for instance, they did not think it necessary to search for any natural cause of the disease: it was immediately attributed to some daemon which had taken possession of the body of the patient, and which could not be ejected but by charms and incantations.

These superstitious notions, which had spread from Egypt over all the east, the Jews imbibed during their captivity in Babylon. Hence we find them in the writings of the New Testament attributing almost every disease to which they were incident to the immediate agency of devils (see Possession). Many of the same impious superstitions were brought from Egypt and Chaldea by Pythagoras, and transmitted by him and his followers to the Platonists in Greece. This is apparent from the writers of the life of Pythagoras. Jamblius, speaking of the followers of that philosopher, says expressly, that they cured certain diseases by incantations; and Porphyry adds, that they cured diseases both of the mind and of the body by songs and incantations. This was exactly the practice of the Egyptian priests, who were all supposed to keep up a constant intercourse with demons, and to have the power of controlling them by magical charms and sacred songs. Agreeably to this practice of his masters, we are told that Pythagoras directed certain diseases of the mind, doubtless those which he attributed to the agency of demons, to be cured partly by incantations, partly by magical hymns, and partly by music—καὶ τὰς ἐνυγμάτων διατροφὰς παρεδίδωσε τοὺς μὲν ἐπιβλέποντας καὶ μυστηρίων τοὺς δὲ μαντεύοντας.

That there are different orders of created spirits, whether called demons or angels, whose powers intellectual and active greatly surpass the powers of man, reason makes probable, and revelation certain. Now it was the universal belief of the ancient nations, says the learned Mosheim†, and especially of the orientals,† that certain sounds and words, for the most part bar. edition of Cudworth's Intellectual System.

Thereupon, were highly grateful, and that others were equally disagreeable, to these spirits. Hence, when they wished to render a daemon propitious, and to employ him on any particular office, the magicians composed their sacred songs of the words which were believed to be agreeable to him; and when it was their intention to drive him from themselves or others, they sung in a strain which they fancied a daemon could not hear but with horror. From the same persuasion arose the custom of suspending from the neck of a sick person, whose disease was supposed to be inflicted by a daemon, an amulet, sometimes made of gold and sometimes of parchment, on which was written one or more of those words which demons could not bear either to hear or to see: and in a didactic poem on the healing art still extant, we are taught by Serenus Sammonicus, that the word Abracadabra is an infallible remedy for a semiterminal fever or ague; and to banish grief of heart, Marcellinus thinks nothing more effectual than the word ξεσικενενε. In more modern times, as we are informed by Agrippa, the words used by those in compact with the devil, to invoke him, and to succeed in what they undertake, are, Dies, mie, Jesquet, benedict, douvima, entenma. There are a hundred other formulas of words composed at pleasure, or gathered from several different languages, or patched from the Hebrew or formed in imitation of it. And among the primitive Christians there was a superstitious custom, of which we suspect some remains may yet be found among the illiterate vulgar in different countries, of of fastening to the neck of a sick person, or to the bed on which he lay, some text from the New Testament, and especially the first two or three verses of the gospel of St John, as a charm undoubtedly efficacious to banish the disease.

That magicians who could thus cure the sick, were likewise believed to have the power of inflicting diseases, and of working miracles, by means of their subservient demons, need not be doubted. Ancient writers of good credit are full of the wonders which they performed. We shall mention a few of those which are best attested, and inquire whether they might not have been effected by other means than the interposition of demons.

The first magicians of whom we read are those who in Egypt opposed Moses. And we are told, that, when Aaron cast down his rod, and it became a serpent, they also did the like with their enchantments; "for they cast down every man his rod, and they became serpents." This was a phenomenon which, it must be confessed, had a very miraculous appearance; and yet there seems to have been nothing in it which might not have been effected by slight of hand. The Egyptians, and perhaps the inhabitants of every country where serpents abound, have the art of depriving them of their power to do mischief, so that they may be handled without danger. It was easy for the magicians, who were favoured by the court, to pretend that they changed their rods into serpents, by dexterously substituting one of these animals in place of the rod. In like manner they might pretend to change water into blood, and to produce frogs; for if Moses gave in these instances, as we know he did in others, any previous information of the nature of the miracles which were to be wrought, the magicians might easily provide themselves in a quantity of blood and number of frogs sufficient to answer their purpose of deceiving the people. Beyond this, however, their power could not go. It stopped where that of all workers in legerdemain must have stooped—at the failure of proper materials to work with. Egypt abounds with serpents; blood could be easily procured; and without difficulty they might have frogs from the river: But when Moses produced lice from the dust of the ground, the magicians, who had it not in their power to collect a sufficient quantity of these animals, were compelled to own this to be an effect of divine agency.

The appearance of Samuel to Saul at Endor is the next miracle, seemingly performed by the power of magic, which we shall consider. It was a common pretence of magicians, that they could raise up ghosts from below, or make dead persons appear unto them to declare future events; and the manner of their incantation is thus described by Horace:

Pallor utrasque Fecerat horrendas aspectu. Scalpere terram Unguibus, et pullam divellere mordicus agnam Coepérunt: crurum in fossam confusus, ut inde Manes elicerent, animas responsa daturas.

"With yellings dire they fill'd the place, And hideous pale was either's face. Soon with their nails they scrap'd the ground, And fill'd a magic trench profound

With a black lamb's thick-streaming gore, Whose members with their teeth they tore; That they might charm the sprights to tell Some curious anecdotes from hell."

Francis.

Whether the witch of Endor made use of such infernal charms as these, the sacred historian has not informed us; but Saul addressed her, as if he believed that by some form of incantation she could recall from the state of departed spirits the soul of the prophet who had been for some time dead. In the subsequent apparition, however, which was produced, some have thought there was nothing more than a trick, by which a cunning woman imposed upon Saul's credulity, making him believe that some confidant of her own was the ghost of Samuel. But had that been the case, she would undoubtedly have made the pretended Samuel's answer as pleasing to the king as possible, both to save her own life, which appears from the context to have been in danger, and likewise to have procured the larger reward. She would never have told her sovereign, she durst not have told him, that he himself should be shortly slain, and his sons with him; and that the host of Israel should be delivered into the hands of the Philistines. For this reason many critics, both Jewish and Christian, have supposed that the apparition was really a demon or evil angel, by whose assistance the woman was accustomed to work wonders, and to foretel future events. But it is surely very incredible, that one of the apostate spirits of hell should have upbraided Saul for applying to a sorceress, or should have accosted him in such words as these: "Why hast thou disquieted me, to bring me up? Wherefore dost thou ask of me, seeing the Lord is departed from thee, and is become thine enemy! For the Lord hath rent the kingdom out of thine hand, and given it to thy neighbour, even to David. Because thou obeyedst not the voice of the Lord, therefore the Lord hath done this thing to thee this day." It is to be observed farther, that what was here denounced against Saul was really prophetic, and that the event answered to the prophecy in every particular. Now, though we do not deny that there are created spirits of penetration vastly superior to that of the most enlarged human understanding; yet we dare maintain, that no finite intelligence could by its own mere capacity have ever found out the precise time of the two armies engaging, the success of the Philistines, the consequences of the victory, and the very names of the persons that were to fall in battle. Saul and his sons were indeed men of tried bravery, and therefore likely to expose themselves to the greatest danger; but after the menaces which he received from the apparition, he would have been impelled, one should think, by common prudence, either to chicane with the enemy, or to retire from the field without exposing himself, his sons, and the whole army, to certain and inevitable destruction; and his acting differently, with the consequences of his conduct, were events which no limited understanding could either foresee or certainly foretel. If to these circumstances we add the suddenness of Samuel's appearance, with the effect which it had upon the sorceress herself, we shall find reason to believe, that the apparition was that of no evil demon. There is not, we believe, upon record, another instance of any per- Magic. son's pretending to raise a ghost from below, without previously using some magical rites or some form of incantation. As nothing of that kind is mentioned in the case before us, it is probable that Samuel appeared before he was called. It is likewise evident from the narrative, that the apparition was not what the woman expected; for we are told, that "when she saw Samuel, she cried out for fear." And when the king exhorted her not to be afraid, and asked what she saw, "the woman said, I see gods (elohim) ascending out of the earth." Now, had she been accustomed to do such feats, and known that what she saw was only her subservient demon, it is not conceivable that she could have been so frightened, or have mistaken her familiar for elohim in any sense in which that word can be taken. We are therefore strongly inclined to adopt the opinion of those who hold that it was Samuel himself who appeared and prophesied, not called up by the wretched woman or her demons, but, to her utter confusion, and the disgrace of her art, sent by God to rebuke Saul's madness in a most affecting and mortifying way, and to deter all others from ever applying to magicians or demons for assistance when refused comfort from heaven. For though this hypothesis may to a superficial thinker seem to transgress the rule of Horace—Nec deus interit, &c.—which is as applicable to the interpretation of scripture, as to the introduction of supernatural agency in human compositions; yet he who has studied the theocratical constitution of Israel, the nature of the office which was there termed regal, and by what means the administration was in emergencies conducted, will have a different opinion; and at once perceive the dignus vindice nodus.

The sudden and wonderful destruction of the army of Brennus the Gaul, has likewise been attributed to magie, or, what in this inquiry amounts to the same thing, to the interposition of evil spirits, whom the priests of Apollo invoked as gods. Those barbarians had made an inroad into Greece, and invested the temple of Apollo at Delphi, with a view to plunder it of the sacred treasure. Their numbers and courage overpowered all opposition; and they were just upon the point of making themselves masters of the place, when, Justin informs us, that, to encourage the besieged, the priests and prophetess "advenisse deum clamant; eumque se vidisse desilientes in templum per culminis apera fastigia. Dum omnes opem dei suppliciter implorant, juvenem supra humanum modum insignis pulchritudinis, comitesque ei duas armatas virgines, ex propinquis duabus Diana Minerveoque adibus occurrisse, nec oculis tantum hae se perspexisse; audisse

(A) "Called aloud that the god had arrived: That they had seen him leap into the temple through the aperture in the roof: That whilst they were all humbly imploring his help, a youth of more than human beauty, accompanied by two virgins in armour, had run to their assistance from the neighbouring temples of Diana and Minerva; and that they had not only beheld these things with their eyes, but had also heard the whizzing of his bow and the clangor of his arms. They therefore earnestly exhorted the besiegéd not to neglect the heavenly signal, but to sally out upon their enemies, and partake with the divinities of the glory of the victory." With these words the soldiers being animated, eagerly rushed to battle: and were themselves quickly sensible of the presence of the god; for part of the rock being torn away by an earthquake, rolled down upon the Gauls; whose thickest battalions being thus thrown into confusion, fled, exposed to the weapons of their enemies. Soon afterwards a tempest arose, which by cold and the fall of hailstones cut off the wounded. Magic instead of walls. A large recess within assumed the form of a theatre; so that the shouts of soldiers, and the sounds of military instruments, re-echoing from rock to rock, and from cavern to cavern, increased the clamour to an immense degree; which, as the historian observes, could not but have great effects on ignorant and barbarous minds. The playing off these panic terrors was not indeed of itself sufficient to repulse and dissipate an host of fierce and hungry invaders, but it enabled the defenders to keep them at bay till a more solid entertainment was provided for them, in the explosion and fall of that portion of the rock at the foot of which the greater part of the army lay encamped.

"Among the caverns in the sacred rock, there was one which, from an intoxicating quality discovered in the steam which issued from it, was rendered very famous by being fitted to the recipient of the priestess of Apollo (b). Now, if we only suppose this, or any other of the vapours emitted from the numerous fissures, to be endowed with that unctuous, or otherwise inflammatory quality, which modern experience shows to be common in mines and subterraneous places, we can easily conceive how the priests of the temple might, without the agency of demons, be able to work the wonders which history speaks of as effected in this transaction. For the throwing down a lighted torch or two into a chasm whence such a vapour issued, would set the whole into a flame; which, by suddenly rarefying and dilating the air, would, like fired gunpowder, blow up all before it. That the priests, the guardians of the rock, could be long ignorant of such a quality, or that they would divulge it when discovered, cannot be supposed. Strabo relates, that one Onomarchus, with his companions, as they were attempting by night to dig their way through to rob the holy treasury, were frightened from their work by the violent shaking of the rock; and he adds, that the same phenomenon had defeated many other attempts of the like nature. Now, whether the tapers which Onomarchus and his companions were obliged to use while they were at work, inflamed the vapour, or whether the priest of Apollo heard them at it, and set fire to a countermine, it is certain a quality of this kind would always stand them in stead. Such then (presumes the learned prelate) was the expedient (c) they employed to dislodge this nest of hornets, which had settled at the foot of their sacred rock; for the storm of thunder, lightning, and hail, which followed, was the natural effect of the violent concussions given to the air by the explosion of the mine."

Two instances more of the power of ancient magic we shall just mention, not because there is any thing particular or important in the facts, but because some credit seems to have been given to the narration by the discerning Cudworth. Philostratus, in his life of Apollonius Tyaneus, informs us that a laughing demoniac at Athens was cured by that magician, who ejected the evil spirit by threats and menaces; and the biographer adds, that the demon, at his departure, is said to have overturned a statue which stood before the porch where the cure was performed. The other instance is of the same magician freeing the city of Ephesus from the plague, by stoning to death an old ragged beggar whom Apollonius called the plague, and who appeared to be a daemon by his changing himself into the form of a shagged dog.

That such tales as these should have been thought worthy of the slightest notice by the incomparable author of the Intellectual System, is indeed a wonderful phenomenon in the history of human nature. The whole story of Apollonius Tyaneus, as is now well known, is nothing better than a collection of the most extravagant fables*: but were the narrative such as that credit could be given to the facts here related, there appears no necessity in either case for calling in the agency of evil spirits by the power of magic.—The Athenians of that age were a superstitious people. Apollonius was a shrewd impostor, long practised in the art of deceiving the multitude. For such a man it was easy to persuade a friend and confidant to act the part of the laughing demoniac; and without much difficulty the statue might be so undermined as inevitably to tumble, upon a violent concussion being given to the ground at the time of the departure of the pretended daemon. If so, this feat of magic dwindles down into a very trifling trick performed by means both simple and natural. The other case of the poor man at Ephesus, who was stoned to death, is exactly similar to that of those innocent women in our own country, whom the vulgar in the last century were instigated to burn for the supposed crime of witchcraft. We have no reason to suppose that an Ephesian mob was less inflammable or credulous than a British mob, or that Apollonius played his part with less skill than a Christian demonologist; and as the spirits of our witches, who were sacrificed to folly and fanaticism, were often supposed to migrate from their dead bodies into the bodies of hares or cats accidentally passing by, so might this impostor at Ephesus persuade his cruel and credulous instruments, that the spirit of their victim had taken possession of the body of the shagged dog.

Still it may be said, that in magic and divination events have been produced out of the ordinary course of nature; and as we cannot suppose the Supreme Being

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(b) "In hoc rupis anfractu, media ferme montis altitudine, planities exigua est, atque in ea profundum terrae foramen, quod in oraculo patet, ex quo frigidus spiritus, vi quadam velet vento in sublimi expulsus, mentes vatum in vercordiam vertit, impetasque deo responsa consulentibus darc cogit." Just., lib. xxiv. c. 10.

(c) The learned author, by arguments too tedious to be here enumerated, confirms the reasoning which we have borrowed from him; and likewise shows from history, that the priests, before they came to extremities with the sacred rock, had entered into treaty with those barbarians, and paid them a large tribute to decamp and quit the country. This adds greatly to the probability of his account of the explosion; for nothing but the absolute impossibility of getting quit of their besiegers by any other means, could have induced the priests to hazard an experiment so big with danger to themselves as well as to their enemies. Magic. Being to have countenanced such abominable practices by the interposition of his power, we must necessarily attribute those effects to the agency of demons, or evil spirits. Thus, when Æneas consulted the Sibyl, the agency of the inspiring god changed her whole appearance:

Poseere fata Tempus," ait: "Deus, ece, Deus." Cui talia fanti Ante fores, subito non vultus, non color unus, Non comptae mansere comoe: sed pectus anhelum, Et rabie fera corda tument; majorque videri, Nec mortale sonans: afflata est numine quando Jam propiore Dei.

"Aloud she cries, "This is the time, inquire your destinies. He comes, behold, a god!" Thus while she said, And shivering at the sacred entry staid, Her colour chang'd, her face was not the same; And hollow groans from her deep spirit came; Her hair stood up; convulsive rage possess'd Her trembling limbs, and heav'd her lab'ring breast; Greater than human kind she seem'd to look, And with an accent more than mortal spoke. Her staring eyes with sparkling fury roll, When all the god came rushing on her soul."

DRYDEN.

In answer to this, it is to be observed, that the temple of Apollo at Cumæ was an immense excavation in a solid rock. The rock was probably of the same kind with that on which the temple of Delphi was built, full of fissures, out of which exhaled perpetually a poisonous kind of vapour. Over one of these fissures was the tripod placed, from which the priestess gave the oracle. Now we learn from St Chrysostom, that the priestess was a woman: "Quæ in tripodes sedent expansa malignum spiritum per internam immissum, et per genitalis partes subeuntem excitans, furore repleretur, ipsaque resolutis crinitibus baccaractur, ex ore spumam emittens, et sic furoris verba loquebatur." By comparing this account with that quoted above from Justin, which is confirmed both by Pausanias and by Strabo, it is evident, that what Chrysostom calls malignum spiritum was a particular kind of vapour blown forcibly through the fissure of the rock. But if there be a vapour of such a quality as, if received per partes genitalis, would make a woman furious, there is surely no necessity for calling into the scene at Cumæ the agency of a demon or evil spirit. Besides, it is to be remembered, that in all mystical and magical rites, such as this was, both the priests, and the persons consulting them, prepared themselves by particular kinds of food, and sometimes,

* Vide Lu-as there is reason to believe, by human sacrifices*, for the approach of the god or demon whose aid they invoked. On the present occasion, we know from the poet himself, that a cake was used which was composed of poppy-seed and honey; and Plutarch speaks of a shrub called teucropylus, used in the celebration of the mysteries of Hecate, which drives men into a kind of frenzy, and makes them confess all their wickedness which they had done or intended. This being the case, the illusions of fancy occasioned by poppy will sufficiently account for the change of the sibyl's appearance, even though the inhaled vapour should not have possessed that efficacy which Chrysostom and Justin attribute to it. Even some sorts of our ordinary food occasion strange dreams, for which onions in particular are remarkable. Excessive drunkenness, as is well known, produces a disorder named by the bacchanalians of this country the blue devils, which consists of an immense number of spectres, accompanied with extreme horror to the person who sees them. From these facts, which cannot be denied, there must arise a suspicion, that by using very unnatural food, such as human blood, the vilest of insects, serpents, and medicated cakes, by shutting themselves up in solitude and caves, and by devising every method to excite horrid and dreadful ideas or images in the fancy, the ancient magicians might by natural means produce every phenomenon which they attributed to their gods or demons. Add to this, that in ancient times magic was studied as a science. Now, as we cannot suppose that every one who studied it intended absolutely nothing, or that all who believed in it were wholly deceived; what can we infer, but that the science consisted in the knowledge of those drugs which produced the phantoms in the imagination, and of the method of preparing and properly employing them for that purpose? The celebrated Friar Bacon indeed, as far back as the 13th century, wrote a book de Nullitate Magia: but though we should allow that this book proved to demonstration, that in his time no such thing as magic existed, it never could prove that the case had always been so. At that time almost all the sciences were lost; and why not magic as well as others? It is likewise an undoubted fact, that magic at all times prevailed among the Asiatics and Africans more than among the Europeans. The reason doubtless was, that the former had the requisites for the art in much greater perfection than we. Human sacrifices were frequent among them; they had the most poisonous serpents, and the greatest variety of vegetable poisons, together with that powerful narcotic opium; all which were of essential use in mystical and magic rites. They had, besides, a burning sun, frightful deserts and solitudes; which, together with extreme fasting, were all called in to their assistance, and were sufficient to produce, by natural means, the most wonderful phenomena which have ever been attributed to magical incantations. Even in our own days, we have the testimony of two travellers, whom we cannot suspect to be either liars or enthusiasts, that both the Indians and Africans perform feats for which neither they nor the most enlightened Europeans can account. The one is Mr Grose, who visited the East Indies about the year 1762; and the other is Mr Bruce, who informs us, that the inhabitants of the western coast of Africa pretend to hold a communication with the devil, and verify their assertions in such a manner that neither he nor other travellers know what to make of it: but it does not from this follow, that Mr Bruce believed that communication to be real. We have all seen one of the most illiterate men that ever assumed the title of Doctor, perform feats very surprising, and such as even a philosopher would have been puzzled to account for, if he had not been previously let into the secret; and yet no man supposes that Katterfelto holds any communica- tion with the devil, although he has sometimes pretended it among people whose minds he supposed unenlightened.

Still it may be objected, that we have a vast number of histories of witches, who in the last century confessed, that they were present with the devil at certain meetings; that they were carried through the air, and saw many strange feats performed, too numerous and too ridiculous to be here mentioned. The best answer to this objection seems to be that given by Dr Ferriar in his essay on Popular Illusions*. "The solemn meeting of witches (says he) is supposed to be put beyond all doubt by the numerous confessions of criminals, who have described their ceremonies, named the times and places of their meetings, with the persons present, and who have agreed in their relations, though separately delivered. But I would observe, first, that the circumstances told of those festivals are in themselves ridiculous and incredible; for they are represented as gloomy and horrible, and yet with a mixture of childish and extravagant fancies, more likely to disgust and alienate than conciliate the minds of their guests. They have every appearance of uneasy dreams. Sometimes the devil and his subjects say mass; sometimes he preaches to them; more commonly he was seen in the form of a black goat, surrounded by imps in a thousand frightful shapes; but none of these forms are new, they all resemble known quadrupeds or reptiles. Secondly, I observe, that there is direct proof furnished even by demonologists, that all those supposed journeys and entertainments were nothing more than dreams. Persons accused of witchcraft have been repeatedly watched about the time they had fixed for their meeting: they have been seen to anoint themselves with soporific compositions; after which they fell into profound sleep; and on awaking several hours afterwards, they have related their journey through the air, with their amusement at the festival, and have named the persons whom they saw there." This is exactly conformable to the practice of the ancient magicians and diviners, and seems to be the true way of accounting, as well for many of the phenomena of magic, as for that extravagant and shameful superstition which prevailed so much during part of the last century, and by which such numbers of innocent men and women were cruelly put to death (c). We may indeed be assured, that the devil has it not in his power to reverse in a single instance the laws of nature without a divine permission; and we can conceive but one occasion (see Possession) on which such permission could be given consistently with the wisdom and the goodness of God. All the tales, therefore, of diabolical agency in magic and witchcraft must undoubtedly be false; for a power, which the devil is not himself at liberty to exert, he cannot communicate to a human creature. Were the case otherwise; were those powers, "which (according to Johnson) only the controul of Omnipotence re-strains from laying creation waste, subservient to the invocations of wicked mortals; were those spirits,

(c) For some farther account of popular illusions, see Animal Magnetism. Magic Square, a square figure, formed of a series of numbers in mathematical proportion; so disposed in parallel and equal ranks, that the sums of each row, taken either perpendicularly, horizontally, or diagonally, are equal.

Let the several numbers which compose any square number (for instance, 1, 2, 3, 4, 5, &c. to 25 inclusive, the square number) be disposed, in their natural order, after each other in a square figure of 25 cells, each in its cell; if now you change the order of these numbers, and dispose them in the cells in such manner, as that the five numbers which fill a horizontal rank of cells, being added together, shall make the same sum with the five numbers in any other rank of cells, whether horizontal or vertical, and even the same number with the five in each of the two diagonal ranks: this disposition of numbers is called a magic square, in opposition to the former disposition, which is called a natural square. See the figures following.

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

One would imagine that these magic squares had that name given them, in regard this property of all their ranks, which, taken any way, make always the same sum, appeared extremely surprising, especially in certain ignorant ages, when mathematics passed for magic: but there is a great deal of reason to suspect, that these squares merited their name still farther, by the superstitious operations they were employed in, as the construction of talismans, &c.; for according to the childish philosophy of those days, which attributed virtues to numbers, what virtue might not be expected from numbers so wonderful?

However, what was at first the vain practice of makers of talismans and conjurers, has since become the subject of a serious research among mathematicians; not that they imagine it will lead them to anything of solid use or advantage (magic squares savour too much of their original to be of much use); but only as it is a kind of play, where the difficulty makes the merit, and it may chance to produce some new views of numbers, which mathematicians will not lose the occasion of.

Eman. Moschopulus, a Greek author of no great antiquity, is the first that appears to have spoken of magic squares: and by the age wherein he lived, there is reason to imagine he did not look on them merely as a mathematician. However, he has left us some rules for their construction. In the treatise of Cor. Agrippa, so much accused of magic, we find the squares of seven numbers, viz. from three to nine inclusive, disposed magically; and it must not be supposed that those seven numbers were preferred to all the other without some very good reason: in effect, it is because their squares, according to the system of Agrippa and his followers, are planetary. The square of 3, for instance, belongs to Saturn; that of 4 to Jupiter; that of 5 to Mars; that of 6 to the Sun; that of 7 to Venus; that of 8 to Mercury; and that of 9 to the Moon. M. Bachet applied himself to the study of magic squares, on the hint he had taken from the planetary squares of Agrippa, as being unacquainted with the work of Moschopulus, which is only in manuscript in the French king's library; and, without the assistance of any author, he found out a new method for those squares whose root is uneven, for instance 25, 49, &c. but he could not make any thing of those whose root is even.

After him came M. Frenicle, who took the same subject in hand. A certain great algebraist was of opinion, that whereas the 16 numbers which compose the square might be disposed 29922789888000 different ways in a natural square (as from the rules of combination it is certain they may), they could not be disposed in a magic square above 16 different ways; but M. Frenicle showed, that they might be thus disposed 878 different ways: whence it appears how much his method exceeds the former, which only yielded the 55th part of magic squares of that of M. Frenicle.

To this inquiry he thought fit to add a difficulty that had not yet been considered: the magic square of 7, for instance, being constructed, and its 49 cells filled, if the two horizontal ranks of cells, and, at the same time, the two vertical ones, the most remote from the middle, be retrenched; that is, if the whole border or circumference of the square be taken away, there will remain a square whose root will be 5, and which will only consist of 25 cells. Now it is not at all surprising that the square should be no longer magical, because the ranks of the large ones were not intended to make the same sum, excepting when taken entire with all the seven numbers that fill their seven cells; so that being mutilated each of two cells, and having lost two of their numbers, it may be well expected, that their remainders will not any longer make the same sum. But M. Frenicle would not be satisfied, unless when the circumference or border of the magic square was taken away, and even any circumferences at pleasures, or, in fine, several circumferences at once, the remaining square was still magical: which last condition, no doubt, made these squares vastly more magical than ever.

Again, He inverted that condition, and required that any circumference taken at pleasure, or even several circumferences, should be inseparable from the square; that is, that it should cease to be magical when they were removed, and yet continue magical after the removal of any of the rest. M. Frenicle, however, gives no general demonstration of his methods, and frequently seems to have no other guide but chance. It is true, his book was not published by himself, not did it appear till after his death, viz. in 1693.

In 1703, M. Poignard, canon of Brussels, published a treatise of sublime magic squares. Before him there had been no magic squares made but for serieses of natural numbers that formed a square; but M. Poignard made two very considerable improvements. 1. Instead of taking all the numbers that fill a square, for instance the 36 successive numbers, which would fill all the cells of a natural square whose side is 6, he only takes as many successive numbers as there are units in in the side of the square, which, in this case, are six; and these six numbers alone he disposes in such manner in the 36 cells, that none of them are repeated twice in the same rank, whether it be horizontal, vertical, or diagonal; whence it follows, that all the ranks, taken all the ways possible, must always make the same sum, which M. Poignard calls repeated progression. 2. Instead of being confined to take these numbers according to the series and succession of the natural numbers, that is, in an arithmetical progression, he takes them likewise in a geometrical progression, and even in a harmonical progression. But with these two last progressions the magic must necessarily be different from what it was: in the squares filled with numbers in geometrical progression, it consists in this, that the products of all the ranks are equal; and in the harmonical progression, the numbers of all the ranks continually follow that progression: he makes squares of each of these three progressions repeated.

This book of M. Poignard gave occasion to M. de la Hire to turn his thoughts the same way, which he did with such success, that he seems to have well nigh completed the theory of magic squares. He first considers uneven squares: all his predecessors on the subject having found the construction of even ones by much the most difficult; for which reason M. de la Hire reserves those for the last. This excess of difficulty may arise partly from hence, that the numbers are taken in arithmetical progression. Now in that progression, if the number of terms be uneven, that in the middle has some properties, which may be of service; for instance, being multiplied by the number of terms in the progression, the product is equal to the sum of all the terms.

M. de la Hire proposes a general method for uneven squares, which has some similitude with the theory of compound motions, so useful and fertile in mechanics. As that consists in decomposing motions, and resolving them into others more simple; so does M. de la Hire's method consist in resolving the square that is to be constructed into two simple and primitive squares. It must be owned, however, it is not quite so easy to conceive these two simple and primitive squares in the compound or perfect square, as in an oblique motion to imagine a parallel and perpendicular one.

Suppose a square of cells, whose root is uneven, for instance 7; and that its 49 cells are to be filled magically with numbers, for instance the first 7; M. de la Hire, on the one side, takes the first 7 numbers, beginning with unity, and ending with the root 7; and on the other 7, and all its multiples to 49, exclusively; and as these only make six numbers, he adds 1, which makes this an arithmetical progression of 7 terms as well as the other; 1, 7, 14, 21, 28, 35, 42. This done, with the first progression repeated, he fills the square of the root magically: In order to this, he writes in the first seven cells of the first horizontal rank the seven numbers proposed in what order he pleases, for that is absolutely indifferent; and it is proper to observe here, that these seven numbers may be ranged in 5040 different manners in the same rank. The order in which they are placed in the first horizontal rank, be it what it will, is that which determines their order in all the rest. For the second horizontal rank, he places in its first cell, either the third, the fourth, the fifth, or the sixth number, from the first number of the first rank; and after that writes the six others in order as they follow. For the third horizontal rank, he observes the same method with regard to the second that he observed in the second with regard to the first, and so of the rest. For instance, suppose the first horizontal rank filled with the seven numbers in their natural order, 1, 2, 3, 4, 5, 6, 7; the second horizontal rank may either commence with 3, with 4, with 5, or with 6: but in this instance it commences with 3; the third rank therefore must commence with 5, the fourth with 7, the fifth with 2, the sixth with 4, and the seventh with 6.

The commencement of the ranks which follow the first being thus determined, the other numbers, as we have already observed, must be written down in the order wherein they stand in the first, going on to 5, 6, and 7, and returning to 1, 2, &c. till every number in the first rank be found in every rank underneath, according to the order arbitrarily pitched upon at first. By this means it is evident, that no number whatever can be repeated twice in the same rank; and by consequence, that the seven numbers 1, 2, 3, 4, 5, 6, 7, being in each rank, must of necessity make the same sum.

It appears, from this example, that the arrangement of the numbers in the first rank being chosen at pleasure, the other ranks may be continued in four different manners; and since the first rank may have 5040 different arrangements, there are no less than 20,160 different manners of constructing the magic square of seven numbers repeated.

The order of the numbers in the first rank being determined; if in beginning with the second rank, the second number 2, or the last number 7, should be pitched upon in one of these cases, and repeated; and in the other case, the other diagonal would be false unless the number repeated seven times should happen to be 4; for four times seven is equal to the sum of 1, 2, 3, 4, 5, 6, 7: and in general, in every square consisting of an unequal number of terms, in arithmetical progression, one of the diagonals would be false according to those two constructions, unless the term always repeated in that diagonal were the middle term of the progression. It is not, however, at all necessary to take the terms in an arithmetical progression; for, according to this method, one may construct a magic square of any numbers at pleasure, whether they be according to any certain progression or not. If they be in an arithmetical progression, it will be proper, out of the general method, to except those two constructions which produce a continual repetition of the same term in one of the two diagonals, and only to take in the case wherein that repetition would prevent the diagonal from being just; which case being absolutely disregarded when we computed that the square of 7 might have 20,160 different constructions, it is evident that by taking that case in, it must have vastly more.

To begin the second rank with any other number besides the second and the last, must not, however, be looked on as an universal rule: it holds good for the square of 7; but if the square of 9, for instance, were to be constructed, and the fourth figure of the first horizontal rank were pitched on for the first of the second, the consequence would be, that the fifth and eighth horizontal ranks would likewise commence with the same number, which would therefore be repeated three times in the same vertical rank, and occasion other repetitions in all the rest. The general rule, therefore, must be conceived thus: Let the number in the first rank pitched on, for the commencement of the second, have such an exponent for its quota; that is, let the order of its place be such, as that if an unit be taken from it, the remainder will not be any just quota part of the root of the square; that is, cannot divide it equally. If, for example, in the square of 7, the third number of the first horizontal rank be pitched on for the first of the second, such construction will be just; because the exponent of the place of that number, viz. 3, subtracting 1, that is, 2 cannot divide 7. Thus also might the fourth number of the same first rank be chosen, because $4 - 1$, viz. 3, cannot divide 7; and, for the same reason, the fifth or sixth number might be taken: but in the square of 9, the fourth number of the first rank must not be taken, because $4 - 1$, viz. 3, does divide 9. The reason of this rule will appear very evidently, by considering in what manner the returns of the same numbers do or do not happen, taking them always in the same manner in any given series. And hence it follows, that the fewer divisions the root of any square to be constructed has, the more different manners of constructing it there are; and that the prime numbers, i.e. those which have no divisions, as 5, 7, 11, 13, &c. are those whose squares will admit of the most variations in proportion to their quantities.

The squares constructed according to this method have some particular properties not required in the problem; for the numbers that compose any rank parallel to one of the two diagonals, are ranged in the same order with the numbers that compose the diagonal to which they are parallel. And as any rank parallel to a diagonal must necessarily be shorter, and have fewer cells than the diagonal itself, by adding to it the correspondent parallel, which has the number of cells by which the other falls short of the diagonal, the numbers of those two parallels, placed as it were end to end, still follow the same order with those of the diagonal: besides that their sums are likewise equal; so that they are magical on another account.

Instead of the squares which we have hitherto formed by horizontal ranks, one might also form them by vertical ones; the case is the same in both.

All we have hitherto said regards only the first primitive square, whose numbers, in the proposed example, were, 1, 2, 3, 4, 5, 6, 7; here still remains the second primitive, whose numbers are 0, 7, 14, 21, 28, 35, 42.

M. de la Hire proceeds in the same manner here as in the former; and this may likewise be constructed in 20,160 different manners, as containing the same number of terms with the first. Its construction being made, and of consequence all its ranks making the same sum, it is evident, that if we bring the two into one, by adding together the numbers of the two corresponding cells of the two squares, that is, the two numbers of the first of each, the two numbers of the second, of the third, &c. and dispose them in the 49 corresponding cells of a third square, it will likewise be magical in regard to its rank, formed by the addition of equal sums to equal sums, which must of necessity be equal among themselves. All that remains in doubt is, whether or no, by the addition of the corresponding cells of the two first squares, all the cells of the third will be filled in such manner, as that each not only contains one of the numbers of the progression from 1 to 49, but also that this number be different from any of the rest, which is the end and design of the whole operation.

As to this it must be observed, that if in the construction of the second primitive square care has been taken, in the commencement of the second horizontal rank, to observe an order with regard to the first, different from what was observed in the construction of the first square; for instance, if the second rank of the first square began with the third term of the first rank, and the second rank of the second square commence with the fourth of the first rank, as in the example it actually does; each number of the first square may be combined once, and only once, by addition with all the numbers of the second.

And as the numbers of the first are here 1, 2, 3, 4, 5, 6, 7, and those of the second, 0, 7, 14, 21, 18, 35, 42; by combining them in this manner we have all the numbers in the progression from 1 to 49, without having any of them repeated; which is the perfect magic square proposed.

The necessity of constructing the two primitive squares in a different manner does not at all hinder but that each of the 20,160 constructions of the one may be combined with all the 20,160 constructions of the other: of consequence, therefore, 20,160 multiplied by itself, which makes 406,425,600, is the number of different constructions that may be made of the perfect square, which here consists of the 49 numbers of the natural progression. But as we have already observed, that a primitive square of seven numbers repeated peated may have above 20,160 several constructions, the number 406,423,600 must come vastly short of expressing all the possible constructions of a perfect magic square of the 49 first numbers.

As to the even squares, he constructs them like the uneven ones, by two primitive squares; but the construction of primitives is different in general, and may be so a great number of ways; and those general differences admit of a great number of particular variations, which give as many different constructions of the same even square. It scarce seems possible to determine exactly, either how many general differences there may be between the construction of the primitive squares of an even square and an uneven one, nor how many particular variations each general difference may admit of; and, of consequence, we are still far from being able to determine the number of different constructions of all those that may be made by the primitive squares.

The ingenious Dr Franklin seems to have carried this curious speculation farther than any of his predecessors in the same way. He has constructed not only a magic square of squares, but likewise a magic circle of circles, of which we shall give some account for the amusement of our readers. The magic square of squares is formed by dividing the great square, as in Plate CCXCVIII. The great square is divided into 256 small squares, in which all the numbers from 1 to 256 are placed in 16 columns, which may be taken either horizontally or vertically. The properties are as follow:

1. The sum of the 16 numbers in each column, vertical and horizontal, is 2056. 2. Every half column, vertical and horizontal, makes 1028, or half of 2056. 3. Half a diagonal ascending added to half a diagonal descending, makes 2056; taking these half diagonals from the ends of any side of the square to the middle thereof; and so reckoning them either upward or downward, or sidewise from left to right hand, or from right to left. 4. The same, with all the parallels to the half diagonals, as many as can be drawn in the great square; for any two of them being directed upward and downward, from the place where they begin to that where they end, their sums will make 2056. The same downward and upward in like manner; or all the same if taken sidewise to the middle, and back to the same side again. N.B. One set of these half diagonals and their parallels is drawn in the same square upward and downward. Another such set may be drawn from any of the other three sides. 5. The four corner numbers in the great square, added to the four central numbers therein, make 1028; equal to the half sum of any vertical or horizontal column which contains 16 numbers; and equal to half a diagonal or its parallel. 6. If a square hole (equal in breadth to four of the little squares) be cut in a paper, through which any of the 16 little squares in the great square may be seen, and the paper be laid on the great square, the sum of all the 16 numbers, seen through the hole, is equal to the sum of the 16 numbers in any horizontal or vertical column, viz. to 2056.

The magic circle of circles, Plate CCXCVIII, is composed of a series of numbers from 12 to 75 inclusive, divided into eight concentric circular spaces, and ranged in eight radii of numbers, with the number 12 in the centre; which number, like the centre, is common to all these circular spaces, and to all the radii.

The numbers are so placed, that the sum of all those in either of the concentric circular spaces above mentioned, together with the central number 12, make 360; equal to the number of degrees in a circle.

The numbers in each radius also, together with the central number 12, make just 360.

The numbers in half of any of the above circular spaces, taken either above or below the double horizontal line, with half the central number 12, make 180; equal to the number of degrees in a semicircle.

If any four adjoining numbers be taken, as if in a square, in the radial divisions of these circular spaces, the sum of these, with half the central number, makes 180.

There are, moreover, included, four sets of other circular spaces, bounded by circles which are eccentric with respect to the common centre; each of these sets containing five spaces. The centres of the circles which bound them are at A, B, C, and D. The set whose centre is at A is bounded by dotted lines; the set whose centre is at C is bounded by lines of short unconnected strokes; and the set round D is bounded by lines of unconnected longer strokes, to distinguish them from one another. In drawing this figure by hand, the set of concentric circles should be drawn with black ink, and the four different sets of eccentric circles with four kinds of ink of different colours; as blue, red, yellow, and green, for distinguishing them readily from one another. These sets of eccentric circular spaces intersect those of the concentric, and each other; and yet the numbers contained in each of the eccentric spaces, taken all around through any of the 20 which are eccentric, make the same sum as those in the concentric, namely 360, when the central number 12 is added. Their halves also, taken above or below the double horizontal line, with half the central number, make 180.

Observe, that there is not one of the numbers but what belongs at least to two of the circular spaces, some to three, some to four, some to five; and yet they are all so placed as never to break the required number 360 in any of the 28 circular spaces within the primitive circle.

To bring these matters in view, all the numbers as above mentioned are taken out, and placed in separate columns as they stand around both the concentric and eccentric circular spaces, always beginning with the outermost and ending with the innermost of each set, and also the numbers as they stand in the eight radii, from the circumference to the centre: the common central number 12 being placed the lowest in each column. 1. In the eight concentric circular spaces.

| | | | | | | | |---|---|---|---|---|---|---| | 14 | 72 | 23 | 65 | 21 | 67 | 12 | | 25 | 63 | 16 | 70 | 18 | 68 | 27 | | 30 | 56 | 39 | 49 | 37 | 51 | 28 | | 41 | 47 | 32 | 54 | 34 | 52 | 43 | | 46 | 40 | 55 | 33 | 53 | 35 | 44 | | 57 | 31 | 48 | 38 | 50 | 36 | 59 | | 62 | 24 | 71 | 17 | 69 | 19 | 60 | | 73 | 15 | 64 | 22 | 66 | 20 | 75 | | 12 | 12 | 12 | 12 | 12 | 12 | 12 |

2. In the eight radii.

| | | | | | | | |---|---|---|---|---|---|---| | 14 | 25 | 30 | 41 | 46 | 57 | 62 | | 72 | 63 | 56 | 47 | 40 | 31 | 24 | | 23 | 16 | 39 | 32 | 55 | 48 | 71 | | 65 | 70 | 49 | 54 | 33 | 38 | 17 | | 31 | 18 | 37 | 34 | 53 | 50 | 69 | | 67 | 68 | 51 | 52 | 35 | 36 | 19 | | 12 | 27 | 28 | 43 | 44 | 59 | 60 | | 74 | 61 | 58 | 45 | 42 | 29 | 26 | | 12 | 12 | 12 | 12 | 12 | 12 | 12 |

3. In the five eccentric circular spaces whose centre is at A.

| | | | | | | |---|---|---|---|---|---| | 14 | 72 | 23 | 85 | 21 | | 63 | 16 | 70 | 18 | 68 | | 39 | 49 | 37 | 51 | 28 | | 54 | 34 | 52 | 43 | 45 | | 33 | 53 | 35 | 44 | 42 | | 48 | 38 | 50 | 36 | 39 | | 24 | 71 | 17 | 69 | 19 | | 73 | 15 | 64 | 22 | 66 | | 12 | 12 | 12 | 12 | 12 |

4. In the five eccentric circular spaces whose centre is at B.

| | | | | | | |---|---|---|---|---|---| | 30 | 56 | 39 | 49 | 37 | | 47 | 32 | 54 | 34 | 52 | | 55 | 33 | 53 | 35 | 44 | | 38 | 50 | 36 | 39 | 20 | | 17 | 69 | 19 | 60 | 26 | | 64 | 22 | 66 | 20 | 75 | | 72 | 23 | 65 | 21 | 67 | | 25 | 63 | 16 | 70 | 18 | | 12 | 12 | 12 | 12 | 12 |

5. In the five eccentric circular spaces whose centre is at C.

| | | | | | | |---|---|---|---|---|---| | 46 | 40 | 55 | 33 | 53 | | 31 | 48 | 38 | 50 | 36 | | 71 | 17 | 69 | 19 | 60 | | 22 | 66 | 20 | 75 | 13 | | 65 | 21 | 67 | 12 | 74 | | 16 | 70 | 18 | 68 | 27 | | 56 | 39 | 49 | 37 | 51 | | 41 | 47 | 32 | 54 | 34 | | 12 | 12 | 12 | 12 | 12 |

6. In the five eccentric circular spaces whose centre is at D.

| | | | | | | |---|---|---|---|---|---| | 62 | 24 | 71 | 17 | 69 | | 15 | 64 | 22 | 66 | 20 | | 24 | 65 | 21 | 67 | 12 | | 70 | 18 | 68 | 27 | 61 | | 49 | 37 | 51 | 28 | 58 | | 32 | 54 | 34 | 52 | 43 | | 40 | 55 | 33 | 53 | 35 | | 57 | 31 | 48 | 38 | 50 | | 12 | 12 | 12 | 12 | 12 |

If, now, we take any four numbers in a square form, either from No. 1 or No. 2 (we suppose from No. 1) as in the margin, and add half the central number 12 to them, the sum will be 180; equal to half the numbers in any circular space taken above or below the double horizontal line, and equal to the number of degrees in a semicircle. Thus, 14, 72, 25, 63, and 6, make 180.

Magic Lantern. See Dioptrics, art. x.