Blowing MACHINE. See BELLOWS. Boyleian MACHINE. Mr. Boyle's Air-Pump. Electrical MACHINE. See ELECTRICAL Machine. Wind MACHINE. See ANEMOMETER, and WIND Machine. Hydraulic or Water MACHINE, is used either to denote a simple machine, serving to conduct or raise water, as a sluice, pump, and the like; or several of these acting together, to produce some extraordinary effect; as the MACHINE of Marli. See FIRE-engine, STEA M-engine, and WATER-Works. tors of that university being desirous that he should supply the place of Mr. James Gregory, whose great age and infirmities had rendered him incapable of teaching. He had here some difficulties to encounter, arising from competitors, who had good interest with the patrons of the university, and also from the want of an additional fund for the new professor; which however at length were all surmounted, principally through the means of Sir Isaac Newton. Accordingly, in Nov. 1725, he was introduced into the university; after which the mathematical 'classes soon became very numerous, there being generally upwards of 100 students attending his lectures every year; who being of different standings and proficiency, he was obliged to divide them into four or five classes, in each of which be employed a full hour every day from the first of November to the first of June. In the first class he taught the first 6 books of Euclid's Elements, Plane Trigonometry, Practical Geometry, the Elements of Fortification, and an Introduction to Algebra. The second class studied Algebra, with the 11th and 12th books of Euclid, Spherical Trigonometry, Conic Sections, and the general Principles of Astronomy. The third studied Astronomy and Perspective, and read a part of Newton's Principia, he having performed a course of experiments for illustrating them: he afterwards read and demonstrated the Elements of Fluxions. Those in the fourth class read a System of Fluxions, the Doctrine of Chances, and the remainder of Newton's Principia. Military MACHINES, among the ancients, were of three kinds the first serving to throw arrows, as the scorpion; or javelins, as the catapult; or stones, as the balista; or fiery darts, as the pyrabolus: the 2d kind serving to beat down walls, as the battering-ram and terebra: and the 3d sort to shelter those who approach the enemy's wall, as the tortoise or testudo, the vinea, and the towers of wood. See the respective articles. The machines of war now in use, consist in artillery, including cannon, mortars, petards, &c. MACLAURIN (COLIN), a very eminent mathematician and philosopher, was the son of a clergyman, and born at Kilmoddan in Scotland, in the year 1698. He was sent to the university of Glasgow in 1709; where he continued 5 years, applying to his studies in a very intense manner, and particularly to the mathematics. His great genius for mathematical learning discovered itself so early as at 12 years of age; when, having accidentally met with a copy of Euclid's Elements in a friend's chamber, he became in a few days master of the first 6 books without any assistance: and it is certain, that in his 16th year he had invented many of the propositions which were afterwards published as part of his work, entitled Geometria Organica. In his 15th year he took the degree of master of arts; on which occasion he composed and publicly defended a thesis on the power of gravity, with great applause. After this he quitted the university, and retired to a country-seat of his uncle, who had the care of his education, his parents having died while he was very young. Here he spent two or three years in pursuing his favourite studies; but in 1717, at 19 years of age only, he offered himself a candidate for the professorship of mathematics in the Marischal College of Aberdeen, and obtained it after a ten days trial, against a very able competitor. In 1719, Mr. Maclaurin visited London, where he left his Geometria Organica to print, and where he became acquainted with Dr. Hoadley then bishop of Bangor, Dr. Clarke, Sir Isaac Newton, and other eminent men; at which time also he was admitted a member of the Royal Society and in another journey, in 1721, he contracted an intimacy with Martin Folkes, esq. the president of it, which continued during his life. In 1722, Lord Polwarth, plenipotentiary of the king of Great Britain at the congress of Cambray, engaged Maclaurin to go as a tutor and companion to his eldest son, who was then to set out on his travels. After a short stay at Paris, and visiting other towns in France, they fixed in Lorrain; where he wrote his piece, On the Percussion of Bodies, which gained him the prize of the Royal Academy of Sciences for the year 1724. But his pupil dying soon after at Montpelier, he returned immediately to his profession at Aberdeen. He was hardly settled here, when he received an invitation to Edinburgh; the cura In 1734, Dr. Berkeley, bishop of Cloyne, published a piece called the Analyst; in which he took occasion, from some disputes that had arisen concerning the grounds of the fluxionary method, to explode the method itself; and also to charge mathematicians in general with infidelity in religion. Maclaurin thought himself included in this charge, and began an answer to Berkeley's book: but other answers coming out, and as he proceeded, so many discoveries, so many new theories and problems occurred to him, that instead of a vindicatory pamphlet, he produced a Complete System of Fluxions, with their application to the most considerable problems in Geometry and Natural Philosophy. This work was published at Edinburgh in 1742, 2 vols. 4to; and as it cost him infinite pains, so it is the most considerable of all his works, and will do him immortal honour, being indeed the most complete treatise on that science that has yet appeared. In the mean time, he was continually presenting the public with some observation or performance of his own, several of which were published in the 5th and 6th volumes of the Medical Essays at Edinburgh. Many of them were likewise published in the Philosophical Transactions; as the following: 1. On the Construction and Measure of Curves, vol. 30.-2. A New Method of describing all kinds of Curves, vol. 30.-3. On Equations with Impossible Roots, vol. 34.-4. On the Roots of Equations, &c, vol. 34.-5. On the Description of Curve Lines, vol. 39. -6. Continuation of the same, vol. 39.-7. Observations on a Solar Eclipse, vol. 40.-S. A Rule for finding the Meridional Parts of a Spheroid with the same Exactness as in a Sphere, vol. 41.-9. An Account of the Treatise of Fluxions, vol. 42.-10. On the Bases of the Cells where the Bees deposit their Honey, vol. 42. In the midst of these studies, he was always ready to lend his assistance in contriving and promoting any scheme which might contribute to the public service. When the earl of Morton went, in 1739, to visit his estates in Orkney and Shetland, he requested Mr. Maclaurin to assist him in settling the geography of those countries, which was very erroneous in all the maps; to examine their natural history, to survey the coasts, and to take the measure of a degree of the meridian. Maclaurin's family affairs would not permit him to comply with this request: he however drew up a memorial of what he thought necessary to be observed, and furnished proper instruments for the work, recommending Mr. Short, the celebrated optician, as a proper operator for the management of them. Mr. Maclaurin had still another scheme for the improvement of geography and navigation, of a more extensive nature; which was the opening a passage from Green-land to the South Sea by the north pole. That such a passage might be found, he was so fully persuaded, that he used to say, if his situation could admit of such adventures, he would undertake the voyage, even at his own charge. But when schemes for finding it were laid before the parliament in 1741, and he was consulted by several persons of high rank concerning them, and before he could finish the memorials he proposed to send, the premium was limited to the discovery of a north-west passage: he regretted much that the word west was inserted, because he thought that passage, if at all to be found, must lie not far from the pole. In 1745, having been very active in fortifying the city of Edinburgh against the rebel army, he was obliged to fly from thence into England, where he was invited by Dr. Herring, archbishop of York, to reside with him during his stay in this country. In this expedition however, being exposed to cold and hardships, and naturally of a weak and tender constitution, which had been much more enfeebled by close application to study, he laid the foundation of an illness which put an end to his life, in June 1746, at 48 years of age, leaving his widow with two sons and three daughters. Mr. Maclaurin was a very good, as well as a very great man, and worthy of love as well as admiration. His peculiar merit as a philosopher was, that all his studies were accommodated to general utility; and we find, in many places of his works, an application even of the most abstruse theories, to the perfecting of mechanical arts. For the same purpose, he had resolved to compose a course of Practical Mathematics, and to rescue several useful branches of the science from the ill treatment they often met with in less skilful hands. These intentions however were prevented by his death; unless we may reckon, as a part of his intended work, the translation of Dr. David Gregory's Practical Geometry, which he revised, and published with additions, in 1745. In his lifetime, however, he had frequent opportunities of serving his friends and his country by his great talents. Whatever difficulty occurred concerning the constructing or perfecting of machines, the working of mines, the improving of manufactures, the conveying of water, or the execution of any public work, he was always ready to resolve it. He was employed to terminate some disputes of consequence that had arisen at Glasgow, concerning the gauging of vessels; and for that purpose presented to the commissioners of the excise two elaborate memorials, with their demonstrations, containing rules by which the officers now act. He made also calculations relating to the provision, now established by law, for the children and widows of the Scotch clergy, and of the professors in the universities, entitling them to certain annuities and sums, on the voluntary annual payment of a certain sum by the incumbent. In contriving and adjusting this wise and useful scheme, he bestowed a great deal of labour, and contributed not a little towards bringing it to perfection. Of his works, we have mentioned his Geometria Organica, in which he treats of the description of curve lines by continued motion; as also of his piece which gained the prize of the Royal Academy of Sciences in 1724. In 1740, he likewise shared the prize of the same academy, with the celebrated D. Bernoulli and Euler, for resolving the problem relating to the motion of the tides from the theory of gravity: a question which had been given out the former year, without receiving any solution. He had only 10 days to draw this paper up, and could not find leisure to transcribe a fair copy; so that the Paris edition of it is incorrect. He afterwards revised the whole, and inserted it in his Treatise of Fluxions; as he did also the substance of the former piece. These, with the Treatise of Fluxions, and the pieces printed in the Medical Essays and the Philosophical Transactions, a list of which is given above, are all the writings which our author lived to publish. Since his death, however, two more volumes have appeared; his Algebra, and his Account of Sir Isaac Newton's Philosophical Discoveries. The Algebra, though not finished by himself, is yet allowed to be excellent in its kind; containing, in a moderate volume, a complete elementary treatise on that science, as far as it had then been carried; besides some neat analytical papers on curve lines. His Account of Newton's Philosophy was occasioned in the following manner-Sir Isaac dying in the beginning of 1728, his nephew, Mr. Conduitt, proposed to publish an account of his life, and desired Mr. Maclaurin's assistance. The latter, out of gratitude to his great benefactor, cheerfully undertook, and soon finished the History of the Progress which Philosophy had made before Newton's time; and this was the first draught of the work in hand; which not going forward, on account of Mr. Conduitt's death, was returned to Mr. Maclaurin. To this he afterwards made great additions, and left it in the state in which it now appears. His main design seems to have been, to explain only those parts of Newton's Philosophy, which have been controverted; and this is supposed to be the reason why his grand discoveries concerning light and colours are but transiently and generally touched upon; for it is known, that whenever the experiments, on which his doctrine of light and colours is founded, had been repeated with due care, this doctrine had not been contested; while his accounting for the celestial motions, and the other great appearances of nature, from gravity, had been misunderstood, and even attempted to be ridiculed. MACULÆ, in Astronomy, are dark spots appearing on the luminous surfaces of the sun and moon, and even some of the planets. The solar maculæ are dark spots of an irregular and changeable figure, observed in the face of the sun. These are said to have been first observed in November and December of the year 1610, by Galileo in Italy, and Harriot in England, unknown to, and independent of each other, soon after the invention of telescopes. But Montucla, in his History of the Mathematics, says that the honour of this discovery is due to J. Fabricius, as appears from his work published at Wittenberg, in June 1611, entitled, De Maculis in Sole visis, et earum cum sole revolutione narratio. They were afterwards also observed by Scheiner, Hevelius, Flamsteed, Cassini, Kirch, and others. See MACULE, NEBULOUS, SPOTS, &c. MADRIER, in Artillery, is a thick plank, armed with plates of iron, and having a cavity sufficient to receive the mouth of a petard, with which it is applied against a gate, or any thing else intended to be broken down. This term is also applied to certain flat beams, fixed to the bottom of a moat, to support a wall.-There are also madriers lined with tin, and covered with earth; serving as defences against artificial fires, in lodgments, &c, where there is need of being covered overhead. MÆSTLIN (MICHAEL), in Latin Mæstlinus, a noted astronomer of Germany, was born in the duchy of Wittemberg; but spent his youth in Italy, where he made a speech in favour of Copernicus's system, which brought Galileo over from Aristotle and Ptolemy, to whom he was before wholly devoted. He afterwards returned to Germany, and became professor of mathematics at Tubingen; where, among his other scholars, he taught the celebrated Kepler, who has commended several of his ingenious inventions, in his Astronomia Optica. Mæstlin published many mathematical and astronomical works, and died in 1590.-Though Tycho Brahé did not assent to Mæstlin's opinion, yet he allowed him to be an extraordinary person, and deeply skilled in the science of astronomy. MAGAZINE, a place in which stores are kept, of arms, ammunition, provisions, &c. Artillery MAGAZINE, or the magazine to a field battery, is made about 25 or 30 yards behind the battery, towards the parallels, and at least 3 feet under ground, to receive the powder, loaded shells, port-fires, &c.-Its roof and sides should be well secured with boards, to prevent the earth from falling in: it has a door, and a double trench or passage sunk from the magazine to the battery, the one to enter, and the other to go out at, to prevent confusion. Sometimes traverses are made in the passages, to prevent ricochet shot from entering the magazine. Powder-MAGAZINE, is the place where powder is kept in large quantities. Authors differ very much with regard to the situation and construction of these magazines; but all agree, that they ought to be arched and bomb-proof. In fortifications, they were formerly placed in the rampart; but of late they have been built in different parts of the town. The first powder-magazines were made with Gothic arches: but M. Vauban thinking these too weak, constructed them of a semicircular form, the dimensions being 60 feet long within, and 25 feet broad; the foundations are 8 or 9 feet thick, and 8 feet high from the foundation to the spring of the arch; also the floor 2 feet from the ground, to preserve it from the damp. It is a constant observation, that after the centering of semicircular arches is struck, they settle at the crown, and rise up at the hances, even with a straight horizontal extrados; and still more so in powder-magazines, where the outside at top is formed, like the roof of a house, by inclined planes joining in an angle over the top of the arch, to give a proper descent to the rain; which effects are exactly what might be expected from the true theory of arches. Now, this shrinking of the arches, as it must be attended with very bad consequences, by breaking the texture of the cement, after it has in some degree been dried, and also by opening the joints of the vousoirs at one end, I have provided a remedy for this inconvenience, with regard to bridges, by the arch of equilibration, in my Tracts, vol. 1: but as the ill consequences of it are much c1; where n = 2718281828 the number whose hyp. log. is cy + A, and a = log. of a, also c + (c2 =log. of (a). Thus for example, in the C= a following figure, representing a transverse vertical section of the arch, if the span AB = 20, the pitch or height DQ 10, the thickness at top DK7, and the angle at top LKM = 112° 37'; then for every different value of PC, the last equation will give the following correspondent values of CI. That is, if ALKMB represent a vertical transverse section of the arch, the roof forming an angle LKM of 112° 37', also pc an ordinate parallel to the horizon taken in any part, and Ic perpendicular to the same, and the other dimensions as above; then for properly con structing the curve so as to be the strongest, or an arch of equilibration in all its parts, the corresponding values of PC and C1 will be as in the following table, where those numbers may denote any lengths whatever, either inclies, or feet, or half-yards. See my Tracts, vol. 1, pa. 57 &c. M MAGAZINE, or Powder-Room, on ship-board, is a close room or store-house, built in the fore or after part of the hold, in order to preserve the gunpowder for the use of the ship. This apartment is strongly secured against fire, and no person is allowed to enter it with a lamp or candle; it is therefore lighted, as occasion requires, by means of the candles or lamps in the light-room contiguous to it. MAGELLANIC CLOUDS, whitish appearances like clouds, seen in the heavens towards the south pole, and having the same apparent motion as the stars. They are three in number, two of them near each other. The largest lies far from the south pole; but the ther two are not many degrees more remote from it than the nearest con spicuous star, that is, about 11 degrees. Mr. Boyle con- is always the same number, viz 15. But if the same jectures that if these clouds were seen through a good - numbers be placed in their natural order, in form of a telescope, they would appear to be multitudes of small square, the first being 1, and the last of them a square stars, like the milky way. Dr. Herschel thinks rather number, they will form what is called a natural square, that nebula are often owing to a self-luminous fluid. See whose two diagonals, as also its middle column, and midPhilos. Trans. an. 1791, pa. 71, and an. 1811, pa. 269. dle horizontal line, will have the same sum as all the rows MAGIC LANTERN, an optical machine, by means of of the magic square, viz 15. which small painted images are represented on the wall of Natural Square. a dark room, magnified to any size at pleasure. This machine was contrived by Kircher (see his Ars Magna Lucis et Umbræ, pa.768); and it was so called, because the images were made to represent strange phantasms, and terrible apparitions, which have been taken for the effect of magic, by such as were ignorant of the secret. This machine is composed of a concave speculum, from 4 to 12 inches diameter, reflecting the light of a candle through the small hole of a tube, at the end of which is fixed a double convex lens of about 3 inches focus. Between the two are successively placed, many small plain glasses, painted with various figures, usually such as are the most formidable and terrifying to the spectators, when represented at large on the opposite wall. Thus (pl. 17, fig. 14) ABCD is a common tin lantern, to which is added a tube FG to draw out. In H is fixed the metallic concave speculum, from 4 to 12 inches diameter; or else, instead of it, near the extremity of the tube, there must be placed a convex lens, consisting of a segment of a small sphere, of but a few inches in diameter. The use of this lens is to throw a strong light upon the image; and sometimes a concave speculum is used with the lens, to render the image still more vivid. In the focus of the concave speculum or lens, is placed the lamp L; and within the tube, where it is soldered to the side of the lantern, is placed a small lens, convex on both sides, being a portion of a small sphere, having its focus about the distance of 3 inches. The extreme part of the tube FM is square, and has an aperture quite through, so as to receive an oblong frame NO passing into it; in which frame there are round holes, of an inch or two in diameter. Answering to the magnitude of these holes there are circles drawn on a plain thin glass; and in these circles are painted any figures, or images, at pleasure, with transparent watercolours. These images fitted into the frame, in an inverted position, at a small distance from the focus of the lens 1, will be projected on an opposite white wall of a dark room, in all their colours, greatly magnified, and in an erect position. By having the instrument so contrived, that the lens I may move on a slide, the focus may be made, and consequently the image appear distinct, at almost any, distance. Or thus: Every thing being managed as in the former case, into the sliding tube FG, insert another convex lens K, the segment of a sphere rather larger than 1. Now, if the picture be brought nearer to 1 than the distance of the focus, diverging rays will be propagated as if they proceeded from the object; therefore, if the lens K be so placed, as that the object be very near its focus, the image will be exhibited on the wall, greatly magnified. MAGIC Square, is a square figure, formed of a series of numbers in arithmetical progression, so disposed in parallel and equal ranks, as that the sums of each row, taken either perpendicularly, horizontally, or diagonally, are equal to one another. As the following square, formed of these nine numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, where the sum of the three figures in every row, in all the directions, Magic Square. 1 2 3 4 2 4 5 6 3 5 7 7 8 9 8 6 where every row and diagonal in the magic square makes just the sum 65, being the same as the two diagonals of the natural square, as well as of the middle row and middle column. It is probable that these magic squares were so called both because of this property in them, viz, that the ranks in every direction making the same sum, appeared extremely surprising, especially in the more ignorant ages, when mathematics passed for magic; and because also of the superstitious operations they were employed in, as the construction of talismans, &c; for, according to the childish philosophy of those days, which ascribed virtues to numbers, what might not be expected from numbers so seemingly wonderful! The magic square was held in great veneration among the Egyptians, and the Pythagoreans their disciples, who, to add more efficacy and virtue to this square, dedicated it to the then known 7 planets divers ways, and engraved it on a plate of the metal that was esteemed in sympathy with the planet.. The square thus dedicated, was inclosed by a regular polygon, inscribed in a circle, which was divided into as many equal parts as there were units in the side of the square; with the names of the angles of the planet, and the signs of the zodiac written upon the void spaces between the polygon and the circumference of the circumscribed circle. Such a talisman or metal they vainly imagined would, upon occasion, befriend the person who carried it about him. To Saturn they attributed the square of 9 places or cells, the side being 3, and the sum of the numbers in every row 15: to Jupiter the square of 16 places, the side being 4, and the amount of each row 34: to Mars the square of 25 places, the side being 5, and the amount of each row 65: to the sun the square with 36 places,. the side being 6, and the sum of each row 111: to Venus the square of 49 places, the side being 7, and the amount of each row 175: to Mercury the square with 64 places,. the side being 8, and the sum of each row 260: and to the moon the square of 81 places, the side being 9, and the amount of each row 369. Finally, they attributed to imperfect matter, the square with 4 divisions, having 2 for its side; and to God the square of only one cell, the side of which is also an unit, which multiplied by itself, undergoes no change. However, what was at first the vain practice of conjurers and makers of talismans, has since become the subject of a serious research among mathematicians. Not that they imagine it will lead them to any thing of solid use or advantage; but rather as it is a kind of play, in which the difficulty makes the merit, and it may chance to produce some new views or properties of numbers, which mathematicians might probably turn to some account. It would seem that Eman. Moschopulus, a Greek author of no high antiquity, was the first now known of, who has spoken of magic squares: he has left some rules for their construction; though, by the age in which he lived, there is reason to imagine he did not look upon them merely as a mathematician. In the treatise of Cornelius Agrippa, so much accused of magic, are found the squares of 7 numbers, viz, from 3 to 9 inclusive, disposed magically; and it is not to be supposed that those 7 numbers were preferred to all others without some good reason: indeed 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 Mer cury; 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 Moschopulus's work, 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 the squares of uneven numbers; for instance, 25, or 49, &c; but he could not succeed with those that have even roots. M. Frenicle next engaged in this subject. It was the opinion of some, that though the first 16 numbers might be disposed 20922789888000 different ways in a natural square, yet they could not be disposed more than 16 ways in a magic square; but M. Frenicle showed, that they might be thus disposed in 878 different ways. To this business he thought fit to add a difficulty that had not yet been considered; which was, to take away the marginal numbers quite around, or any other circumference at pleasure, or even several of such circumferences, and yet that the remainder should still be magical. 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 it often seems that he has no other guide but chance. is true, his book was not published by himself, nor did it appear till after his death, viz, in 1693. It In 1703, M. Poignard, canon of Brussels, published a treatise on sublime magic squares. Before his time there had been no magic squares made, but for series of natural numbers that formed a square; but M. Poignard made two very considerable improvements. 1st, 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 the side of the square, which in this case are 6; and these six numbers alone he disposes in such manner, in the 36 cells, that none of them occur 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; and this method M. Poignard calls repeated progressions. 2d, Instead of being confined to take these numbers according to the series and succession of the natural numbers, that is in arithmetical progression, he takes them likewise in a geometrical progression; and even in an har monical one, the numbers of all the ranks always following the same kind of progression: he makes squares of each of these three progressions repeated. M. Poignard's book gave occasion to M. Lahire to turn his thoughts to the same subject, which he did with such success, that he greatly extended the theory of magic squares, as well for even numbers as those that are uneven; as may be seen at large in the Memoirs of the Royal Academy of Sciences, for the years 1705, and in 1710 by M. Sauveur. See also Saunderson's Algebra, vol.1, pa. 354, &c; as also my Mathematical Recreations, translated from Ozanam and Montucla, giving the following easy method of filling up a magic square. To form a Magic Square of an Odd Number of Terms in the Arithmetic Progression 1, 2, 3, 4, &c. Place the least term 1 in the cell immediately under the middle, or central one, and the rest of the terms, in their natural order, in a descending diagonal direction, till they run off either at the bottom, or on the side: when the number runs off at the bottom, carry it to the uppermost cell that is not occupied, of the same column that it would have fallen in below, and then proceed descending diagonalwise again as far as you can, or till the numbers either run off at bottom or side, or are interrupted by coming at a cell already filled: now when any number runs off at the right-hand side, then bring it to the farthest cell on the left hand of the same row or line it would have fallen in towards the right hand and when the progress diagonalwise is interrupted by meeting with a cell already occupied by some other number, then descend diagonally to the left from this cell till an empty one is met with, where enter it; and thence proceed as before. Thus, to make : 22 47 16 4.1 10 35 4 30 5 23 48 17 42 11 29 6 24 49 18 36 12 13 31 7 25 43 19 37 38 14 32 26. 44 20 21 a magic square of the 49 numbers 1, 2, 3, 4, &c. First place the 1 next below the centre cell, and thence descend to the right till the 4 runs off at the bottom, which therefore carry to the top corner on the same column as it would have fallen in; but as it runs off at the side, bring it to the beginning of the second line, and thence descend to the right till you arrive at the cell occupied by 1; carry the therefore to the next diagonal cell to the left, and so proceed till 10 run off at the bottom, which carry there 8 333 |