MAGIC Lantern
, an optical machine, by means of which small painted images are represented on the wall of a dark room, magnified to any size at pleasure. This machine was contrived by Kircher, (see his Ars Magna Lucis and 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. 13, 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 drawn circles on a plain thin glass; and in these circles are painted any figures, or images, at pleasure, with transparent water colours. These images fitted into the frame, in an inverted position, at a small distance from the focus of the lens I, will be projected on an opposite white wall of a dark room, in all their colours, greatly magnisied, and in an erect position. By having the instrument so contrived, as 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 I. Now, if the picture be brought nearer to I than the distance of the focus, diverging rays will be propagated as if they proceeded from the object; wherefore, 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 magnisied.
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 annexed square, form- ed 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 directions, is always the same number, viz 15. But if the same numbers be placed in this natural order, the first being 1, and the last of them a square number, they will form what is called a natural square. As in the first 25 numbers, viz, 1, 2, 3, 4, 5, &c to 25.
| 4 | 9 | 2 |
| 3 | 5 | 7 |
| 8 | 1 | 6 |
| Natural Square. | ||||
| 1 | 2 | 3 | 4 | 5 |
| 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 |
| 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 |
| Magic Square. | ||||
| 16 | 14 | 8 | 2 | 25 |
| 3 | 22 | 20 | 11 | 9 |
| 15 | 6 | 4 | 23 | 17 |
| 24 | 18 | 12 | 10 | 1 |
| 7 | 5 | 21 | 19 | 13 |
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.
It is probable that these magic squares were so called, both because of this property in them, viz, that the ranks in every direction make 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 seven planets divers ways, and engraved it upon 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 angels 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 of numbers, which mathematicians will not lose the occasion of.
It would seem that Eman. Moschopulus, a Greek author of no high antiquity, is 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 seven numbers, viz, from 3 to 9 inclusive, disposed magically; and it is not to be supposed that those seven 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 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 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 although 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 shewed, 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. It is true, his book was not published by himself, nor did it appear till after his death, viz, in 1693.
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 serieses 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 sill 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 harmonical progression, 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. de la Hire 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 1710. See also Saunderson's Algebra, vol. 1, pa. 354, &c; as also Ozanam's Mathematical Recreations, who lays down 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 righthand: 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.
| 22 | 47 | 16 | 41 | 10 | 35 | 4 |
| 5 | 23 | 48 | 17 | 42 | 11 | 29 |
| 30 | 6 | 24 | 49 | 18 | 36 | 12 |
| 13 | 31 | 7 | 25 | 43 | 19 | 37 |
| 38 | 14 | 32 | 1 | 26 | 44 | 20 |
| 21 | 39 | 8 | 33 | 2 | 27 | 45 |
| 46 | 15 | 40 | 9 | 34 | 3 | 28 |
Thus, to make 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 runs off at the side, bring it to the beginning of the second line,| and thence descend to the right till they arrive at the cell occupied by 1; carry the 8 therefore to the next diagonal cell to the left, and so proceed till 10 run off at the bottom, which carry therefore to the top of its column, and so proceed till 13 runs off at the side, which therefore bring to the beginning of the same line, and thence proceed till 15 arrives at the cell occupied by 8; from this therefore descend diagonally to the left; but as 16 runs off at the bottom, carry it to the top of its proper column, and thence descend till 21 run off at the side, which is therefore brought to the beginning of its proper line; but as 22 arrives at the cell occupied by 15, descend diagonally to the left, which brings it into the 1st column, but off at the bottom, and therefore it is carried to the top of that column; thence descending till 29 runs off both at bottom and side, which therefore carry to the highest unoccupied cell in the last column; and here, as 30 runs off at the side, bring it to the beginning of its proper column, and thence descend till 35 runs off at the bottom, which therefore carry to the beginning or top of its own column; and here, as 36 meets with the cell occupied by 29, it is brought from thence diagonally to the left; thence descending, 38 runs off at the side, and therefore it is brought to the beginning of its proper line; thence descending, 41 runs off at the bottom, which therefore is carried to the beginning or top of its column; from whence descending, 43 arrives at the cell occupied by 36, and therefore it is brought down from thence to the left; thence descending, 46 runs off at the side, which therefore is brought to the beginning of its line; but here, as 47 runs off at the bottom, it is carried to the beginning or top of its column, from whence descending with 48 and 49, the square is completed, the sum of every row and column and diagonal making just 175.
There are many other ways of filling up such squares, but none that are easier than the above one.
It was observed before, that the sum of the numbers in the rows, columns and diagonals, was 15 in the square of 9 numbers, 34 in a square of 16, 65 in a square of 25, &c; hence then is derived a method of finding the sums of the numbers in any other square, viz, by taking the successive differences till they become equal, and then adding them successively to produce or find out the amount of the following sums. Thus,
| Side | Cells | Sums | Diffs. | ||
| 0 | 0 | 0 | 0 | ||
| 1 | 3 | ||||
| 1 | 1 | 1 | 3 | ||
| 4 | 3 | ||||
| 2 | 4 | 5 | 6 | ||
| 10 | 3 | ||||
| 3 | 9 | 15 | 9 | ||
| 19 | 3 | ||||
| 4 | 16 | 34 | 12 | ||
| 31 | 3 | ||||
| 5 | 25 | 65 | 15 | ||
| 46 | 3 | ||||
| 6 | 36 | 111 | 18 | ||
| 64 | 3 | ||||
| 7 | 49 | 175 | 21 | ||
| 85 | 3 | ||||
| 8 | 64 | 260 | 24 | ||
| 109 | 3 | ||||
| 9 | 81 | 369 | 27 | ||
| 136 | 3 | ||||
| 10 | 100 | 505 | 30 |
Again, like as the terms of an arithmetical prog<*>ession arranged magically, give
| 8 | 256 | 2 |
| 4 | 16 | 64 |
| 128 | 1 | 32 |
| 1260 | 840 | 630 |
| 504 | 420 | 360 |
| 315 | 280 | 252 |
Also, the terms of an harmonical progression being ranged magically, as in the margin, have the terms in each row &c in harmonical progression.
The ingenious Dr. Franklin, it seems, carried this curious speculation farther than any of his predecessors in the same way. He constructed both a magic square of squares, and a magic circle of circles, the description of which is as follows. The magic square of squares is formed by dividing the great square as in fig. 1, Pl. 15. The great square is divided into 256 little squares, in which all the numbers from 1 to 256, or the square of 16, are placed, in 16 columns, which may be taken either horizontally or vertically. Their chief properties are as follow:
1. The sum of the 16 numbers in each column or row, vertical or horizontal, is 2056.
2. Every half column, vertical and horizontal, makes 1028, or just one half of the same sum 2056.
3. Half a diagonal ascending, added to half a diagonal descending, makes also the same sum 2056; taking these half diagonals from the ends of any side of the square to the middle of it; and so reckoning them either upward or downward; or sideways from right to left, or from left to right.
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 still make the same 2056. Also the same holds true downward and upward; as well as if taken sideways to the middle, and back to the same side again. Only one set of these half diagonals and their parallels, is drawn in the same square upward and downward; but another 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 in it, make 1028, the| half sum of any vertical or horizontal column, which contains 16 numbers; and also equal to half a diagonal or its parallel.
6. If a square hole, equal in breadth to four of the little squares or cells, be cut in a paper, through which any of the <*>6 little cells in the great square may be seen, and the paper be laid upon the great square; the sum of all the 16 numbers, seen through the hole, is always equal to 2056, the sum of the 16 numbers in any horizontal or vertical column.
The Magic Circle of Circles, fig. 2, pl. 15, by the same author, is composed of a series of numbers, from 12 to 75 inclusive, divided into 8 concentric circular spaces, and ranged in 8 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 1st, the sum of all those in either of the concentric circular spaces above mentioned, together with the central number 12, amount to 360, the same as the number of degrees in a circle.
2. The numbers in each radius also, together with the central number 12, make just 360.
3. 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 just 180, or half the degrees in a circle.
4. 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, make also the same 180.
5. There are also included four sets of other circular spaces, bounded by circles that are excentric with regard to the common centre; each of these sets containing five spaces; and the centres of them being at A, B, C, D. For distinction, these circles are drawn with different marks, some dotted, others by short unconnected lines, &c; or still better with inks of divers colours, as blue, red, green, yellow.
These sets of excentric circular spaces intersect those of the concentric, and each other; and yet, the numbers contained in each of the excentric spaces, taken all around through any of the 20, which are excentric, 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 up 180.
It is observable, 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. They have also other properties. See Franklin's Exp. and Obs. pa. 350, edit. 4to, 1769; or Ferguson's Tables and Tracts, 1771, pa. 318.
MAGICAL Picture, in Electricity, was first contrived by Mr. Kinnersley, and is thus made: Having a large mezzotinto with a frame and glass, as of the king for instance, take out the print, and cut a pannel out of it, near two inches distant from the frame all around; then with thin paste or gum-water, six the border that is cut off on the inside of the glass, pressing it smooth and close; then sill up the vacancy by gilding the glass well with leaf gold or brass. Gild likewise the inner edge of the back of the frame all round, except the top part, and form a communication between that gilding and the gilding behind the glass; then put in the boardy and that side is finished. Next turn up the glass, and gild the foreside exactly over the back gilding, and when it is dry, cover it by pasting on the pannel of the picture that has been cut out, observing to bring the corresponding parts of the border and picture together, by which means the picture will appear entire, as at first, only part behind the glass, and part before.
Hold the picture horizoutally by the top, and place a small moveable gilt crown on the king's head. If now the picture be moderately electrified, and another person take hold of the frame with one hand, so that his fingers touch its inside gilding, and with the other hand endeavour to take off the crown, he will receive a violent blow, and fail in the attempt. If the picture were highly charged, the consequence might be as fatal as that of high treason. The operator, who holds the picture by the upper end, where the inside of the frame is not gilt, to prevent its falling, feels nothing of the shock, and may touch the face of the picture without danger. And if a ring of persons take the shock among them, the experiment is called the conspirators. See Franklin's Exper. and Observ. pa. 30.
