Chapter I: II. III. IV (3)

The problem respecting the placing the knight on any given square, and moving him from that square to any house on the board, has not been thought unworthy the attention of the first mathematicians. Euler, Ozanam, De Montmart, De Moivre, De Majron, and others, have all given methods by which this feat might be accomplished. It was reserved, however, for the present century to lay this down on a general plan; and the only English writer who has noticed this is Mr. George Walker, in his _Treatise on Chess_. The plan is this: Let the knight be placed on any square, and move him from square to square, on the principle of always playing him to that point, from which, in actual play, he would command the fewest other squares; observing, that in reckoning the squares commanded by him you must omit such as he has already covered. If, too, there are two squares, on both of which his powers would be equal, you may move him to either. Try this on the board, with some counters or wafers, placing one on every square; and, when you clearly understand it, you may astonish your friends by inviting them to station the knight on any square they like, and engaging to play him, from that square, over the remaining sixty-three in sixty-three moves. When the automaton Chessplayer was last exhibited in England, this was made part of the wonders he accomplished, though as the above plan was not then known here, he could not adopt it, but used something like the method laid down by Euler, and which we subjoin.

Our young Chess-players must remember that it does not matter on which square the knight is placed at starting; as, by acquiring the plan by heart, which is soon done, he can play him over all the squares from any given point, his last square being at the distance of a knight's move from his first. It is obvious that this route may be varied many ways, and we have often amused ourselves by trying to work it on a slate.

ANOTHER METHOD.

The problem of the knight's covering successively each square of the board, has, in all ages, attracted the attention of the first mathematicians; it is only lately, however, that this very ingenious system for performing the feat without seeing the board, has been invented by an Edinburgh gentleman. We well recollect the surprise occasioned among chess-amateurs when it was first performed; indeed it was generally considered a greater mental effort than that of playing three games of chess at the same time, without seeing the board.

The general rule for moving the Knight upon all the squares of the board, is to commence by moving him to that square which commands the fewest points of attack, and by continuing this principle he will occupy all the squares in rotation, observing, that if on any two or more squares his power would be equal, he may be placed indifferently on either of such squares. Thus we see, that there are different routes which the Knight-errant may take in his progress over all the board; still, whichever of these routes for covering the sixty-four squares may be adapted, each move forms, if we may so express ourselves, a link in an endless chain, so that whatever square we start from, by taking one known route, we are sure to arrive at a square, the last link of the chain, a Knight's move distant from the square of our departure. Consequently, if any person could commit to memory the consecutive moves of any one route over the board, he would be able to start from any one square in that route, in the same manner that any of us, if required to mention the numerals up to sixty-four, could as easily start at thirty and end at twenty-nine, as if we started at one and ended at sixty-four.

+------+------+------+------+------+------+------+------+
| | | | | | | | |
| Met. | Let. | Ket. | Het. | Get. | Fet. | Det. | Bet. |
+------+------+------+------+------+------+------+------+
| | | | | | | | |
| Men. | Len. | Ken. | Hen. | Gen. | Fen. | Den. | Ben. |
+------+------+------+------+------+------+------+------+
| | | | | | | | |
| Mix. | Lix. | Kix. | Hix. | Gix. | Fix. | Dix. | Bix. |
+------+------+------+------+------+------+------+------+
| | | | | | | | |
| Miv. | Liv. | Kiv. | Hiv. | Giv. | Fiv. | Div. | Biv. |
+------+------+------+------+------+------+------+------+
| | | | | | | | |
| Mor. | Lor. | Kor. | Hor. | Gor. | For. | Dor. | Bor. |
+------+------+------+------+------+------+------+------+
| | | | | | | | |
| Mee. | Lee. | Kee. | Hee. | Gee. | Fee. | Dee. | Bee. |
+------+------+------+------+------+------+------+------+
| | | | | | | | |
| Moo. | Loo. | Koo. | Hoo. | Goo. | Foo. | Doo. | Boo. |
+------+------+------+------+------+------+------+------+
| | | | | | | | |
| Mun. | Lun. | Kun. | Hun. | Gun. | Fun. | Dan. | Bun. |
+------+------+------+------+------+------+------+------+
M L K H G F D B

These considerations greatly reduce the apparent impossibility of performing the feat; but the reader will exclaim, "What an immense undertaking it would be, to commit to memory the moves forming a Knight's route over the sixty-four squares!" and we reply, "Certainly it would be, if we used the language of Chess to designate the squares;" and herein lies the beauty of the invention. A set of names, whose application can be understood at a glance, are invented for the squares, and the performer of the feat, having learned a route of the Knight, expressed by these invented names, thinks in the new language which he directs the moves in the terms of chess--just as many of us _think_ in English, when we are writing or speaking French.

The diagram given above represents the chess-board; the distinction of white and black squares is not necessary for our purpose. The files, commencing from the right hand are distinguished by the consonants in alphabetical succession (C and J are, for obvious reasons, omitted.) Thus, the King's rook's file is known as B, the King's Knight's as D, the King's Bishops as F, the King's as G, the Queen's as H, the Queen's Bishop's as K, the Queen's Knight's as L, and the Queen's Rook's as M. This is all that has to be learned, in this system of Chess notation; for the lines of squares tell their own numbers--one being _un_, two _oo_, three _ee_, four _or_, six _ix_, seven _en_, eight _et_--being, in fact, the terminal sounds of the first eight numerals. Bun being B _one_, or King's Rook's square; Gix, G _six_, or King's sixth square. We consider it quite unnecessary to say another word in explanation of this system; its ingenious simplicity causes it to be understood and learned at a glance. All that is required now is, to select a Knight's route over all the squares of the board, and commit it to memory, not in the complicated terms of Chess, but in these simple equivalents. Suppose we start from the Queen's Knights seventh square, _len_, the route will be as follows:

Len het fen bet. Dix bor doo gun
Koo mun lee kun. Moo kee goo dun.
Bee div ben fet. Hen let mix lor.
Hee kiv gor hix. Liv men ket gen.
Kix giv fee hor. Gix for hiv gee.
Fiv den biv dee. Bun foo hun loo.
Mor lix met ken. Get fix det bix.
Dor boo fun hoo. Lun mee kor miv

The only trouble is to commit this cabalistical-looking table to memory, which may be all accomplished in half an hour; the process will be greatly facilitated by the learner frequently playing the route over on the chess-board. He will be amply rewarded by the astonishment he will cause to the _natives_ of his locality, who may have the great misfortune of being unacquainted with our book's enlightening pages; and, if not quite a first-rate player, he will acquire an intimate knowledge of the peculiar powers and perplexing peregrinations of the eccentric _Caballeros_, who

"----fiery coursers guide
With headlong speed throng wars empurpled tide;
Alert and brave they spring amidst the fight,
From white to black, from black to candid white."

ANOTHER METHOD.

Let Black Queen's Rook's Square count 1, (as in the diagram,) Black King's Rook 8, and count all the other Squares in the same way from 9 to 64. Place the Knight upon Black King's Rook's Square, 8, and move as follows: 23, 40, 55, 61, 51, 57, 42, 25, 10, 4, 14, 24, 39, 56, 62, 52, 58, 41, 26, 9, 3, 13, 7, 22, 32, 47, 64, 54, 60, 50, 33, 18, 1, 11, 5, 15, 21, 6, 16, 31, 48, 63, 53, 59, 49, 34, 17, 2, 12, 27, 44, 38, 28, 43, 37, 20, 35, 45, 30, 36, 18, 29, and 46. It may be well to chalk the figures on the board, as a guide, until the feat is well understood.

47. ROSAMOND'S BOWER.

The subjoined cut represents, it is said, the Maze at Woodstock, in which King Henry placed Fair Rosamond to protect her from the Queen. It certainly is a most ingenious contrivance, and may be made productive of much amusement. The puzzle consists in getting, from one of the numerous outlets, to the bower in the center, without crossing any of the lines.

ROSAMOND'S BOWER.

48. A MAZE OR LABYRINTH.

This maze is a correct ground-plan of one in the gardens of the Palace of Hampton Court. No legendary tale is attached to it, of which we are aware, but its labyrinthine walks occasion much amusement to the numerous holiday parties who frequent the palace grounds. The puzzle is to get into the center, where seats are placed under two lofty trees; and many are the disappointments experienced before the end is attained; and even then, the trouble is not over, it being quite as difficult to get _out_ as to get _in_.

49. THE CHINESE PUZZLE.

This puzzle, being one for the purpose of constructing different figures by arranging variously-shaped pieces of card or wood in certain ways, requires no separate explanation. Cut out of very stiff cardboard, or thin mahogany, which is decidedly preferable, seven pieces, in shape like the annexed figures and bearing the same proportion to each other; one piece must be made in the shape of figure 1, one of figure 2, and one of figure 3, and two of each of the other figures. The combinations of which these figures are susceptible, are almost infinite; and we subjoin a representation of a few of the most curious. It is to be borne in mind, that all the pieces of which the puzzle consists, must be employed to form each figure.

50. TROUBLE-WIT.

Take a sheet of stiff paper, fold it down the middle of the sheet, longways; then turn down the edge of each fold outward, the breadth of a penny; measure it as it is folded, into three equal parts, with compasses, which make six divisions in the sheet; let each third part be turned outward, and the other, of course, will fall right; then pinch it a quarter of an inch deep, in plaits, like a ruff, so that, when the paper lies pinched in its form, it is in the fashion represented by A; when closed together, it will be like B; unclose it again, shuffle it with each hand, and it will resemble the shuffling of a pack of cards; close it and turn each corner inward with your fore finger and thumb, it will appear as a rosette for a lady's shoe, as C; stretch it forth, and it will resemble a cover for an Italian couch, as D; let go your fore finger at the lower end, and it will resemble a wicket, as E; close it again, and pinch it at the bottom, spreading the top, and it will represent a fan, as F; pinch it half way, and open the top, and it will appear in the form shown by G; hold it in that form, and with the thumb of your left hand turn out the next fold, and it will be as H.

In fact, by a little ingenuity and practice, Trouble-wit may be made to assume an infinite variety of forms, and be productive of very considerable amusement.

ANSWERS TO PRACTICAL PUZZLES.

1. THE CHINESE CROSS ANSWER.

Place Nos. 1 and 2 close together, as in Fig. 1; then hold them together with the finger and thumb of the left hand horizontally and with the square hole to the right. Push No. 3--placed in the same position _facing you_ (_a_) in No. 4--through the opening at K, and slide it to the left at A, so that the profile of the pieces should be as in Fig. 2. Now push No. 4 _partially_ through the space from below upwards, as seen in f, Fig. 2. Place No. 5 cross-ways upon the part Y, so that the point R is directed upwards to the right hand side; then push No. 4 quite through, and it will be in the position shown by the dotted lines in Fig. 2. All that now remains is to push No. 6--which is the key--through the opening M and the cross is completed as in Fig. 3.

Fig 1 Fig 2 Fig 3
]

2. ANSWER TO THE "PARALLELOGRAM."

Divide the piece of card into five steps, and by shifting the position of the pieces, the desired figures may be obtained.

3. THE DIVIDED GARDEN ANSWER.

4. ANSWER TO THE ENDLESS STRING.

The string must be put through the armhole, and over the head, then through the opposite armhole; then the hand must be put up underneath the waistcoat, and the string drawn down around the body until the former drops down about the waist, when the experimenter may jump out of it and claim his coat.

5. ANSWER TO THE CHINESE MAZE.

KOONG-SEE'S WHISPERS.

A Why linger near the fence? a word or two
Would kindle up a flame for ever true.
B Beware of rivals--mischief hovers near;
Or, worse mischance, parental frowns appear.
C Favored indeed, the open door to gain--
Let no dishonor now your conduct stain.
E The ground is rough, and difficult the road;
But, faint not, thou shalt reach thy love's abode!
F Against thy course runs the opposing tide,
And waves of trouble cast thy hopes aside.
G A modest competence thy lot will be;
But richer joys than wealth are stored for thee.
A Take heed! take heed! a strange transforming doom
May fix thy love, but never let it bloom.
J Be not too rash--nor leap the Bridge of Love,
Leaving fond eyelids, moist with tears, above.
K What dost thou on the house top? do not steal
Thy love, but win by dutiful appeal!
L A barren path this way thy footsteps tread;
Thy heart will soon grow cold, thy love be fled.
M Thou hast a friend can help thy onward way--
And such a friend will ne'er thy trust betray.
D Joy! thou hast reached, at length, the wedding ring;
Let white-robed maidens orange blossoms bring;
Oh may your years of happy wedlock be
Bright as your hopes, and from misgiving free.

6. ANSWER TO VERTICAL LINE PUZZLE.

|\ | | |\ | +-----
| \ | | | \ | |
| \ | | | \ | +-----
| \ | | | \ | |
| \| | | \| +-----
]

7. THE THREE RABBITS, ANSWER.

8. THE ACCOMMODATING SQUARE.

9. ANSWER TO THE CIRCLE PUZZLE.

Thrust it upwards from the other side.

10. ANSWER TO THE CUT CARD PUZZLE.

Double the cardboard or leather lengthways down the middle, and then cut first to the right, nearly to the end, (the narrow way,) and then to the left, and so on to the end of the card; then open it and cut down the middle, except the two ends. The diagram shows the proper cuttings. By opening the card or leather, a person may pass through it. A laurel leaf may be treated in the same manner.

11. ANSWER TO THE BUTTON PUZZLE.

Draw the narrow slip of the leather through the hole, and the string and buttons may be easily released.

12. ANSWER TO THE QUARTO PUZZLE.

Divide the figure in the direction shown by the lines, and you will have four pieces of the same size and shape.

13. ANSWER TO THE PUZZLE OF FOURTEEN.

14. ANSWER TO THE SQUARE AND CIRCLE PUZZLE.

+---------------+
|o o | o|
|-------+ | |
| o|o | |
|o +---+---+ o|
| | o|o |
| | +-------|
|o | o o|
+---------------+
]

15. THE SCALE AND RING PUZZLE.

The puzzle consists in releasing the ring; to effect which, you have only to reverse the former process, by passing the loop through the holes D, C, B, and A, in the manner before described.

16. ANSWER TO THE HEART PUZZLE.

Loosen the string, and draw the loop through the hole No. 2; pass it behind, and bring it through No. 1, and slip it over the smaller heart; then the string may be easily drawn out.

17. ANSWER TO THE CROSS PUZZLE.

18. ANSWER 10 THE YANKEE SQUARE.

Arrange the pieces as shown in the figure above.

19. ANSWER TO THE CARD PUZZLE.

In order to take the pipe off, the card must be doubled (as in Fig. 2), the slip passed through it, until there is sufficient of the loop below the pipe to allow one of the square ends of the slip (Fig. 3) being passed through it. Fig. 3 is then to be taken away, and the pipe slipped off. The card for this puzzle must be cut very neatly, the puzzle handled gently, and great care taken that, in doubling the card to put on the pipe no creases are made in it, as they would in all probability spoil your puzzle, by betraying to an acute spectator the mode of operation.

20. ANSWER TO THE THREE SQUARE PUZZLE.

Takeaway the pieces numbered 8, 10, 1, 3, 13, and three squares only will remain.

21. ANSWER TO THE CYLINDER PUZZLE.

Take a round cylinder of the diameter of the circular hole, and of the height of the square hole. Having drawn a straight line across the end, dividing it into two equal parts, cut an equal section from either side to the edge of the circular base, a figure like that represented by the woodcut in the margin would then be produced, which would fulfill the required conditions.

22. ANSWER TO THE FOUR TENANTS.

My ground is divided,
My tenants at work,
And he'll profit most
Who does not labor shirk
So let them toil on
Till cabbages rise,
And carrots and turnips
To gladden their eyes.
Gooseberries and currants,
And raspberries too,
Shall amply repay
The work they may do.

23. THE PUZZLE WALL.

24. ANSWER TO THE NUN'S PUZZLE.

25. HORSE SHOE PUZZLE.

By cutting off the upper circular part containing two of the pins, and by changing the position of the pieces, another cut will divide the horse shoe into six portions, each containing one pin.

26. ANSWER TO CARD SQUARE PUZZLE.

27. THE DOGS PUZZLE ANSWERED SEE DOTTED LINES.

28. PUZZLE OF THE TWO FATHERS.

The first father divided the land in this way:

The second father divided the land in the following manner:

The different colors represent the several sons' portions.

29. THE TRIANGLE PUZZLE.

The solution of this puzzle may be easily acquired by observing the dotted lines in the engraving; by which it will be seen that four triangles are to be placed at the corners, and a small square made in the center. When this is done, the rest of the square may be quickly formed.

30. ANSWER TO CUTTING OUT A CROSS PUZZLE.

Take a piece of writing paper about three times as long as it is broad, say six inches long and two wide. Fold the upper corner down, as shown in Fig. 1; then fold the other upper corner over the first, and it will appear as in Fig. 2; you next fold the paper in half lengthwise, and it will appear as in Fig. 3. Then the last fold is made lengthwise also, in the middle of the paper, and it will exhibit the form of Fig. 4, which, when cut through with the scissors in the direction of the dotted line, will give all the forms mentioned.

31. ANSWER TO ANOTHER CROSS PUZZLE.

32. ANSWER TO THE FOUNTAIN PUZZLE.

33. ANSWER TO THE STAR PUZZLE.

Good-tempered friends! here _nine_ stars see:
_Ten_ rows there are, in each row _three!_

34. THE COUNTER PUZZLE ANSWER.

Place 4 on 7, 6 on 2, 1 on 3, and 8 on 5; _or_, 5 on 2, 3 on 7, 8 on 6, 4 on 1, &c.

35. ANSWER TO THE JAPAN SQUARE.

36. ANSWER TO THE CABINET MAKERS' PUZZLE.

The cabinet-maker must find the center of the circle, and strike another circle, half the diameter of the first, and having the same center. Then cut the whole into four parts, by means of two lines drawn at right angles to each other, then cut along the inner circle, and put the pieces together as in the above diagram.

37. ANSWER TO THE STRING AND BALLS PUZZLE.

Draw the loop well down, slipping either ball through it. Push it through the hole at the extremities, pass it over the knot, and draw it through again. The same process must be repeated with the other ball; the loop can then be drawn through the hole in the center, and the ball will slide along the cord until it reaches the other side. The string is then replaced, having both balls on the same side.

There is another and perhaps a neater way of performing this trick. Draw the loop through the central hole, and bring it through far enough to pass one of the balls through. Having done this, draw the string back, and both balls will be found on the same side.

38. ANSWER TO THE DOUBLE-HEADED PUZZLE.

Arrange them side by side in the short arms of the cross, draw out the center piece, and the rest will follow easily. The reversal of the same process will put them back again.

39. ARITHMETICAL PUZZLE.

The four figures are 8888, which being divided by a line drawn through the middle, become 0000/0000, the sum of which is eight 0s, or nothing.

40. GRAMMATICAL PUZZLE.

Take away L in the subjunctive "Let" at the beginning of the first line, and substitute S, and so turn it into the imperative "Set," when the changes which necessarily follow will be immediately apparent.

41. ANSWER TO THE TREE PUZZLE.

42. ANSWER TO AN EPITAPH ON ELLINOR BACHELLOR, AN OLD PIE WOMAN.

Beneath in the dust
The mouldy old crust
Of Nell Bachellor lately was shoven:
Who was skilled in the arts
Of pies, custards, and tarts,
And knew every use of the oven.
When she'd liv'd long enough,
She made her last puff,
A puff by her husband much prais'd:
Now here she doth lie,
To make a dirt pie,
In hopes that her crust will be rais'd.

43. ANSWER TO A CURIOUS LETTER.

"Sir, between friends, I understand your overbearing disposition; a man even with the world is above contempt, whilst the ambitious are beneath ridicule."

44. ANSWER TO THE PUZZLE INSCRIPTION.

By the use of the single vowel E, the following couplet was formed,

PERSEVERE YE PERFECT MEN,
EVER KEEP THESE PRECEPTS TEN.

THE MAGIC OF ART.

"Tired at first sight with what the Muse imparts,
In fearless youth we tempt the height of arts."

An almost endless source of amusement, combining at the same time a considerable amount of instruction, may be obtained in the following manner. Take a card or piece of pasteboard, or even stiff paper, and draw upon it the form of an egg--an oval in outline. The dimensions of the oval are immaterial, and the experimenter may suit his own fancy in this respect. With a stout needle, or tracing point, prick quite through the outline, for the purposes of tracing. Some of our readers may be unacquainted with the mode of tracing an outline, and it may be advisable to particularize one method among many. Having pricked out the oval upon the card, get a little red or black lead, powdered, and placing the card upon apiece of drawing paper--any white paper will however do--rub it over the pricked-out oval, which will be found to be transferred to the white paper beneath, thus:

The powder may be applied either with a piece of wool or wadding, or by means of a dry camel's-hair pencil; care should be taken not to let the tracing-powder get beyond the edge of the pricked card, as in that case a soiled, dirty appearance is given to the tracing. The pierced card will serve, if carefully done, for hundreds of tracings, and it is obviously the best plan to take a little extra pains with that in the first instance.

With this traced oval for a basis, (a little further on we shall speak of other figures, to be used singly or in combination with each other) any one with a very little skill will be able to form an infinite number of objects.

The best drawing tool will be found to be an ordinary black lead pencil.

Figs. 1, 2, 3, 4, 5, 6 are very easy results, suggestive also of others. The rules of procedure are the same in all. Leaving the traced-out oval at first in its dotted form, with the pencil you draw a horizontal line as the basis of your figure. Let this and the other lines, which serve merely as the scaffolding of your figure, be done faintly or in dots. Next, draw a line through the center of the oval and perpendicular to the first. These will ensure your making the object square and properly balanced. After this you may draw lines parallel to the others: but these are not so material, although they serve as guides.

Now the imagination and fancy may step in to produce forms having the oval for a foundation; and not only is a very rational source of amusement opened out, but the opportunity is given to a cultivation of the noble art of design, whether as applied to utility or ornament.

It is obvious to remark that the hand of many an amateur artist will readily be able to form the oval without having recourse to the pierced card; but as this portion of our work is intended for _all_, we have suggested the above mode as sure to succeed under every circumstance.

Following the same plan in every particular, we subjoin some examples of what may be done with the square.

The dotted lines (_figs._ 7, 8) represent the traced or sketched square and plan lines; the firmer lines suggest objects formed upon that figure. In the same way the thin square outline (_fig._ 9) suggests the inner sketch of a church.

I stated before that the size of the fundamental oval or square made little difference; but I would recommend my younger readers to get these as large as possible, or convenient. If a large black board, such as is used in most schools, could be obtained, and the tracings prepared proportionably large (pounded chalk being used instead of the black or red powder in transferring the forms thereto), and the designs made upon these with a piece of chalk, so much the better. However, this matters little; and each one will suit his or her own taste in that respect. I now proceed to submit some examples of what may be done with other rudimentary forms.

Following the instructions previously given, in place of the square suggested in Figs. 7, 8, 9, describe a circle. This may be done with a pair of compasses, or simply sketched or traced by means of any round object, such as a coin laid flat upon the paper. Figs. 10, 11, 12, 13, 14, 15, are given merely as suggestions, the circle forming an important part of their figure. The mind of the experimenter will immediately revert to other objects--thousands such are to be met with around us--having the circle or the sphere for their basis. And it will be no mean result of my labors, if any number of my younger readers are led thereby to a habit of observation, whereby they will not fail to notice that nearly all natural objects have the curved line for a basis, if they are not actually distinguishable thereby from those that are artificial.

Fig. 16 is drawn upon two circles in combination with each other. The dotted lines of the plan will be readily perceived; but lest there should be any difficulty, they have been drawn separately in Fig. 17. With this duplex figure little skill will be required to present the lord of the farm yard. The three outlines, Figs. 18, 19, 20, are based upon the square turned diamond-wise, and will need no further remark: examples upon this plan may be multiplied easily. Those given will serve as hints in the several directions of flowers, foliage, and landscapes generally.

Before proceeding to show what may easily be done by a simple combination of the figures we have constructed, _i. e._, the oval, square, and circle, let me introduce another, which enters, by a kind of natural law, into almost all forms or groups of forms, namely, the triangle. Observe in the annexed cut, Fig. 21, how naturally, although unconsciously, the girl seats herself within one.

A moment's reflection will show, that from the little nymph in the cut to the great pyramid, everything that rests solidly upon the earth must take the form, more or less, of this broad-based tapering figure. Roofs of houses, churches, and towers, are all triangular in their form, as are all great trees, differing from each other only in the width of their angles.

Construct a triangle,[14] and trace it according to former directions, and from the examples, Figs. 22, 23, 24, look around you for others, and make various exercises upon this foundation.

Now, to proceed to something more complicated. Suppose you had either in your mind, or sketched out upon paper, the plan of a garden; that is to say, suppose you had the dimensions of a piece of ground, and intended to lay it out as a garden, allotting so much space to this and that bed, so much to gravel walks, and wanted to see how such an arrangement would look in perspective,--in other words, in reality, for perspective, however alarming it may look in books, with its net-work of lines, cross and across, like an insoluble riddle or a monster cobweb, is nothing more than the actual representation of things as they meet the eye.

Let your plan be what is shown in the square portion of Fig. 25; at the top of this plan place your triangle, draw a line through the center of the square upwards, until it meets the top A of the triangle. Next draw lines from the corner of the beds parallel to the center line until they meet the base line of the triangle. From thence continue all these lines to the point A. These give you the width of the beds _in perspective_. The other sides of their figure may be easily enough found. Fig. 26 is the perspective view sought, and is what your experimental drawing would be if, having done the plan and guide-lines in pencil and the rest in pen and ink, you had erased the former with a piece of india rubber.

I do not know whether my readers regard the matter in the same light, but it appears to the present writer that this little figure--the triangle--is capable of working wonders in the hands of an amateur draughtsman, if only properly used. Of course, those regularly educated, or submitted to a long course of training as artists, are not referred to, but only the general public, which by the by, means nineteen out of every twenty individuals. I ask whether the preceding cut is any exaggeration on the average sort of result attained, not only amongst very juvenile experimenters, but those of maturer age?

Everybody possessed of vision can tell, ordinarily, whether a building or other object is upright, or in the position proper to it, or necessary to its stability. By accustoming the hand to form lines, ovals, circles, squares, and triangles, and by habituating the mind to form comparisons between objects, and these and other figures, a person is put imperceptibly, as it were, in the way of depicting them with accuracy.

To proceed--let us take the above misrepresented country residence, and applying to it the previously given rules, see what we can make of it. We would first draw or trace the parallelogram shown in dotted lines; over this, we place a triangle; then drawing an upright line through the center of both, make that the base of another and lengthened triangle, as shown (see Fig. 28). Thus we get the three lines of the side and roofs; and if we knew the proportionate height of the side window, by marking the same at a, b, and carrying the lines from those points to the apex of the triangle, we get its true perspective dimensions.

The difference between the two results is as great as possible.

In Fig. 29 the triangle placed at the side of the soldier in front gives the perspective of the whole line.

Fig. 30 shows how two parallelograms in combination assist in giving the perspective of a block of stone or bale of goods.

Fig. 31 exhibits the parallelogram and triangle in combination.

Perhaps nothing is more puzzling to the tyro in sketching than the interior of rooms and halls. In Fig. 32 a very easy method is given. Trace the outer parallelogram, and within it, a smaller one; then connect the corners of the two as shown in the cut.

Fig. 33 is the application of the preceding.

By the present paper I intend to let you into a great secret, the secret, namely, of Comic or "Funny" Drawing--a method, in fact, which is at the bottom of all humorous, or caricature sketching. Don't let any one be alarmed, and suppose that it is intended to set you quizzing and caricaturing your friends. Far from it.

Draw the oval, Fig. 34. Divide it by transverse lines into about equal portions. You have now the basis for a face. Let the central line (across) mark the position of the eyes, the line above that the top of the forehead, the one below the bottom of the nose. By Fig. 35 you will see this worked out, and have what is considered a well proportioned face.

Now oddity of feature or expression is simply the result of a deviation from this regularity; and if, as you will perceive by the other Figs., 36, 37 and 38, these lines are placed higher or lower, or out of their, strictly speaking, _proper places_, you have, as a necessary result of such disarrangement, oddity, or comicality, which is founded upon irregularity or incongruity in things.

I shall carry out this hint more fully, at present merely pointing out, in reference to the next two figures, how the end is attained by placing a pair of dark spectacles upon a regularly-featured face, or adding a little flesh to the lower portion of that at Fig. 39.

But not to forget the "Art" in the "Sport," let me add, that by sketching the plain oval, and remarking whereabout the lines of their features would cut it, you may, without difficulty, attempt likenesses of your friends and companions.

Now fill your slates or sketch-books with ovals, and try the effects of which the above are but indications. Your imaginations will furnish an endless variety of subjects. The omission of one eye, or its being covered by a shade, or closed while the other stares; the nose slightly on one side, the mouth a little wider than usual--these are all sources of the humorous, which, however, is far from being heightened by _ugliness_. Indeed, it should be borne in mind, that great distortion or hideousness, so far from contributing to humor, destroys it by raising painful images in the mind. True humor is closely allied to kindness.

Now let us take the simplest elements of the profile or side face. This is also formed upon the oval, with a slight variation. And here we must go a little more into the "Art" than at first sight the "Sport" seems to warrant. You will perceive by Fig. 41 that the oval used for profile purposes is divided as before into four about equal portions, which are appropriated in the same manner. That is to say, the central line across is for the eye, and the other two for the limit of the hair and the bottom of the nose.

But take notice that portions are cut off--_e. g._, at the back where the neck is inserted; a little has to be added for forehead, chin, and hair; and some modification takes place about the region of the eye.

Suffice it that the oval forms essentially the basis of the structure of a well proportioned face, such as is shown in the Fig. (41). Draw for yourself, or trace from Fig. 41, a figure for your basis. Next make a number of these tracings upon a clean sheet of drawing paper, and marking them in very lightly, in pencil, proceed as directed in the case of the front face in the last lesson; altering the feature lines, lengthening or shortening the chin, nose, and forehead according to your fancy. This will be a sufficient guide for you, and illustrations of this are accordingly omitted here.

Let us proceed a step further. The last hint only dealt with the depth relatively of the several parts of the face. Now, as to their prominence. How very easily, by means of a few magic touches, which, by this time, you are magicians enough to impart, may you summon up our ancient acquaintance Mother Hubbard, or the modern hero Punch. (See Figs. 42, 43.)

Observe that the peculiarity of these comic physiognomies consists merely in their deviation from the regularly formed head of Fig. 41. They are constructed upon that figure, which may be seen underneath in dotted lines. The variety of ways in which this exercise may be worked is infinite. Subjoined are a few. In Figs. 44 and 45, beards, mustaches, eyebrows, the hair cut absurdly short, or left redundant, joined to the sinking in of the facial angle, produce the effect of comicality. In Figs. 46, 47, the same end is attained by the simplest means, and with even less exaggeration. And here I again repeat, that the less deviation there is from the proper proportions the better.

As a pendant to the comical landscape given No. 27 I give you the annexed (Fig. 48).

Every one will recognize it as a model drawing--such as is to be found upon walls, and occasionally upon the margins of school-books. This the artist (!) intends from a comic drawing. Of course it is no such thing.

We will new take up the grandest object of art the Human Figure. In designing the human figure, there are three principal rules to be observed:

First, the standard height of the human body may be reckoned as eight times the length of the face. Dividing the entire length by eight, as shown in the annexed diagram (Fig. 49), it will be perceived that the face comprises one of the spaces: the second reaching to the chest: the third, to above the hips; the fourth cuts the entire length into two equal parts; the fifth extends to the center of the thigh: the sixth, to the knee joint; the seventh to half way down the leg; and the eighth, to the sole of the foot.

The second rule is, that no part of the body, viewed laterally, is more than twice the thickness of the head. In very young children, however, the rule is, that where the head will go, any part of the body will follow, as the experience of most people has tested.

The third rule concerns the center of gravity. By reference to the fig. (49), a vertical line will be perceived, drawn through the center of the figure. Whenever the body is at rest upon its legs, standing at ease, as one may say, this imaginary line must always pass through its center.

We shall see more about this hereafter, at present confining ourselves to the consideration of the first two rules These must be considered as only generally true.

They have, however, to be well considered in connection with our present subject; for as we said with the face, any great deviation from them leads to oddity, and is at the root of caricature drawing.

Trace out, or sketch out any size, the figures (Fig. 49 and 50), or any others for yourselves: or taking any well drawn figure in a print which may not be too costly to use so, draw with a black lead pencil upon the print similar lines to those in the figures, that is to say, divide its length into eight parts, first dropping a central line perpendicular to the ground.

You will thus test the accuracy of the rule, and familiarize yourself with the proportions of the figure. Then, for the purpose of comic drawing, you will vary these proportions. A face too long or too short, a body too large or too small for the legs, or legs otherwise disproportionate to the rest of the body, will yield the desired results. It will be seen by the Figs. 51 and 52 that their oddity has been arrived at simply by this rule, or by the deviation from the strict rule of proportion.

Fig. 53 is given in illustration of the remarks upon the second rule. The form is correct enough as regards height, and deviates in the matter of lateral proportion.

We now come to consider the third principle--that of the center of gravity.

Observe in the annexed figures how the first (Fig. 54) being at rest, commends itself to the reason like a mathematical demonstration. The next diagram shows a partial deviation from the center of gravity, is in a false position, and we begin either to pity or to laugh--poor diagram! The two following figures are other cases of the same sort--we feel instinctively for them--they are very far gone. Try this rule upon your slates or sketching blocks; and after that we will go on to the next subject. In the diagrams (Figs. 58, 59), the same principle is enforced. The first is at rest, because the line passing through the center of the figure is a vertical line. In the next figure, that line being out of the vertical, the balance is disturbed, and the figure topples; so with the next. This is so plain that argument is not needed to demonstrate it.

Try this also for yourselves as before. Nor need we confine our experiments to figures comparatively at rest: forms in every variety of action come under the same rule--it is a law of nature. There is a central line drawn through the whole system of the universe, through every tree, and plant, and stone, and every upright thing, could we but see it.

The first of the following figures is in full action, but it may go on for ever, as its balance is not in any way disturbed. The second is fast hastening to its fall. The third is much nearer still to that consummation.

It will suggest itself to every reader to apply the rule to other objects than the human figure. Trees in the positions of Figs. 64 and 65 are never seen unless through some violent accident; they may bend, and twist, and meander, but taking the objects as a whole, a central line, vertical to the horizon, will be detected, as shown in Fig. 65.

If we turn our attention in any direction upon natural objects, the clouds, the earth, the sea, flowers, trees, or animal bodies, we cannot fail to see that _a curved line_ is always to be made out in their forms. Indeed, just so far as they are graceful and pleasing objects to the eye, this curved line is distinguishable. On the contrary, square lines offend the eye when met with under such circumstances. It is almost impossible indeed to imagine a square cloud, a square flower, or a square horse. When we see a square headed man, we are not impressed in his favor. We may have met with representations of natural objects, such as rocks, hill tops, mountain precipices, and the like, which had a square or nearly square appearance; but such things are almost always presented to our view as phenomena--_i. e._ things violating the regular order or general rule of nature. This curved line, which is the line of beauty, must pervade all nature; it is the natural law; and we cannot sufficiently admire the truth that that which is most necessary is also most beautiful.

Does any one ask what particular reference these observations have to "Art in Sport?" Let us say that they alone can properly understand what is comic who have learned to appreciate what is _not_ comic. The distance between the sublime and the ridiculous is said to be very small--only one step. At any rate, the student who best understands the first will best appreciate the second. Socrates did not disdain to write an essay upon this subject, insisting that the very same qualities were essential in the comic and the tragic artist. But this is digressive.

Let us resume: In the annexed figure (67) you perceive the curved line. In proportion as you are able to make this perfectly, you will succeed in drawing gracefully. I must presume that very many of my readers will have no difficulty in copying the few natural objects suggested below. Practice upon your slate or board the figure (67) until you can do it easily. Then, for the purposes of "sport," proceed as follows. You wish to produce a droll "bit" of landscape. Take any simple view, such as submitted in Fig. 68.

In this you will readily discover, as I said above, the curved, graceful lines of beauty--in the clouds, the outline of the distant hills, the foliage, the meandering stream. Let me advise you to practice this lesson somewhat perseveringly; apart from the new source of amusement which it is the object of these papers to open up, a beautiful lesson could be impressed upon the mind. Nothing is more calculated to refine the mind, to ennoble the thoughts, than to withdraw one's self from the artificial world, and to gaze upon the fresh face of nature. And if we are able to do this intelligently--in other words, if, having learned the alphabet, we are able to peruse as it were the book of nature--the delight and the advantage is proportionably increased. Now turn to the example shown in Fig. 69. What do we see? The lines of beauty have given place to others less pleasing to the eye, and (except as a source of merriment) less acceptable to the mind. Fig. 69 is a comic landscape; how it has become so must be clearly apparent.

In Fig. 70, the same process is carried out, and the result is similar.

I hope that I shall not be understood to mean, that in order to be graceful everything must be round, or that everything round is graceful, or that every square object is ungraceful; or, again, that by making any curved line into a square or straight one the end we propose is to be obtained. Doubtless many round things are ungraceful, as many others composed entirely of straight lines at various angles to each other are exceedingly graceful. But what is meant is this: that natural objects, in which, left to themselves, the curved line predominates, are made odd and comic-looking when drawn upon the square.

In Fig. 71, not only the lines of the shepherd's form are curved lines, and, therefore, conducive in a degree to its general pleasing character, but the _attitude_ is formed upon a curved line. This will be perceived clearer by reference to the next figure, (72,) in which, without using a single straight line in the parts of the form, the oddity is attained by making the whole attitude stiff and angular.

The student will find no difficulty in multiplying examples for himself: those given will suffice as hints. We must now proceed to show how comic designs may be made and applied to the slides of magic lanterns. The proper course of procedure is as follows:

Procure a piece of clear common window glass, without specks or scratches; let this be made perfectly clean. Prepare your design, which should be made the exact size you intend it to be painted upon the glass; color it; and when quite dry, place it beneath your slide of glass, to which it might be fastened at the corners by means of a little gum or varnish. Now commence to paint upon the glass an exact fac simile of the design, which, of course, you see clearly enough through the glass.

Common camel's hair brushes will do; those made of sable are, however, much better; but the first will suffice for ordinary purposes.

The colors necessary are what are called silica colors, and are procurable of most artist's color makers.

It will be necessary to let your first colors dry before putting on your shades; and it is desirable not to work in too hot a room, as the nature of the varnish with which you work is to dry very rapidly.

Bear in mind too, that upon glass you cannot wash in a tint. Broad surfaces, such as skies, must be stippled in, as in painting upon ivory.

In originating this paper on "The Magic of Art," the author did not propose to himself to give a complete treatise, but simply to point out, by some very easy processes, at source of amusement and instruction, available to almost every intelligent reader. It is hoped that, in this subject, he has not entirely failed, and that all will find some entertainment from,

"Art in Sport."

FOOTNOTES:

[Footnote 14: This is done easily enough, but the following directions may not be needless for some. Draw a straight line for a base of any length. If you wish to form a equilateral triangle, _i. e._ one of which the three sides are equal, divide this base line by two, and at the point of division set up an upright line; then from each end of the base line slant against the central upright line one the length of the base. These, of course, will meet at the top, and the triangle is formed. Any other triangle may be formed in a similar manner, the length of the sides being at the choice of the artist.]

SECRET WRITING.

The art of communicating secret information by means of writing, which is intended to be illegible except by the person for whom it is destined, is very ancient. The ancients sometimes shaved the head of a slave, and wrote upon the skin with some indelible coloring matter, and then sent him, after his hair had been grown again, to the place of his destination. This is not, however, properly secret writing, but only a concealment of writing. Another kind, which corresponds better with the name, is the following, used by the ancients. They took a small stick, and wound around it bark or papyrus, upon which they wrote. The bark was then unrolled and sent to the correspondent, who was furnished with a stick of the same size. He wound the bark again round this, and thus was enabled to read what had been written.

This mode of concealment is evidently very imperfect. Cryptography properly consists in writing with signs, which are legible only to him for whom the writing is intended, or who has a key or explanation of the signs. The most simple method is to choose for every letter of the alphabet some sign, or only another letter. But this sort of cryptography (chiffre) is also easy to be deciphered without a key. Hence many illusions are used. No separation is made between the words, or signs of no meaning are inserted between those of real meaning. Various keys are also used according to rules before agreed upon. By this means the deciphering of the writing becomes difficult for a third person not initiated, but it is also extremely troublesome to the correspondents themselves, and a slight mistake often makes it illegible even to them.