Part 25
The early mathematicians, in their attempts to solve this problem, generally proceeded on the following plan. If we draw a square exterior to a circle, that is, touching the square in four points, each side of the square being equal to the diameter of the circle, we can soon convince ourselves that the boundary of the square will be greater than the circumference of the circle, and the area of the former greater than that of the latter. But if the square be drawn within the circle, so that only the four corners touch it, then it is equally evident that the circle is larger, both in boundary and area, than the square. By this proceeding, we arrive at the conclusion that a circle is _smaller_ than a square _external_ to it, and _larger_ than one _internal_ to it. Let us next suppose that we draw a regular pentagon, that is, a figure of five equal sides, exterior to the circle, and touching it on five points; then it is evident that as the circle is wholly contained within the pentagon, it must be smaller than that which contains it. But if the pentagon be described within the circle, touching it at the five angular points, then of course the circle is larger than the pentagon which it contains.
Now, in geometry, my young readers must bear in mind, the exact periphery or circumference, and the exact area of any figure bounded by straight lines, may be determined with rigorous accuracy; and if we draw two polygons--say of one hundred sides, one within and one without the circle--we can ascertain the exact area of those polygons, and affirm that the area of a circle is greater than a certain amount, and less than another certain amount. These two amounts, if the number of the sides of the polygon be so large as we here suppose, may be so very nearly alike, that either one will give the area of the circle with great closeness.
By some such means as these Archimedes found that if the diameter of a circle be called 7, then the circumference will be nearly 22; and that if the square of the diameter be 14, then the area of the circle will be equal to about 11; but this computation was slightly in error, and gave to the area of the circle too great a measure by about one three-thousandth part of the whole. At a later period, however, a European mathematician, named Metius, discovered a method which makes an extraordinary approach to accuracy, and is at the same time easily remembered. He found that if the diameter be considered equal to 113, then the circumference would equal 355; or if we multiply the square of the radius by 355, and divide it by 113, the area will be given. Now this method is so very nearly correct, that the area of a circle one foot in diameter is given within the fifty-thousandth part of a square inch.
Other mathematicians have carried the approximation still further. Ludolph Van Ceulen worked it out to 36 places of figures, showing that if the diameter be 1, the circumference will be
3.14,159,265,358,979,323,846,264,338,327,950,288.
or that if the last figure be 8, the result will be a little below the truth, and if 9, a little above it.
Since this, Mr. Sharp, an English mathematician, carried the approximation to 72 places of figures; Mr. John Machin to 100 figures, and eclipsed all others. M. de Lagny worked it out to 128 places of figures, and of the degree of _nearness_ to which this computation brings the proportion, Montucla says, "If we suppose a circle, the _diameter_ of which is a thousand million times greater than the distance between the sun and the earth, the error in the proportion of the circumference would be a thousand million times less than the thickness of a hair."
But after all, none of these computations are quite correct; they all deviate from the truth, and bring us to the conclusion that there are no numbers or collection of numbers which will give the exact ratio of the circumference, or of the area of a circle to its diameter. We offer this explanation on the subject to our young friends that they may not be puzzled by the question; and that should they be asked to square the circle, or hear any one assert that he can do so, they may be able to show that they are "awake" to the question, and know how to explain it.
FOOTNOTES:
[Footnote 13: _I. e._, at the fourth square from the right angle.]
PRACTICAL PARADOXES AND PUZZLES.
A puzzle is not solved, impatient sirs, By peeping at its answer, in a trice-- When Gordius, the plow-boy, king of Phrygia, Tied up his implements of husbandry In the far-famed knot, rash Alexander Did not undo, by cutting it in twain.
Paradoxes and Puzzles, although by many persons looked upon as mere trifles, have, in numerous instances, cost their inventors considerable time, and exhibit a great degree of ingenuity. We can readily imagine that some of the complicated puzzles in the ensuing pages may have been originally constructed by captives, to pass away the hours of a long and dreary imprisonment; thus does the misery of a few frequently conduce to the amusement of many. We look upon a Paradox as a sort of superior riddle, and a tolerable Puzzle, in our opinion, takes precedence of a first-rate rebus. There is often considerable thought, calculation, patience, and management, required to solve some of these strange enigmas; and we have, ere now, followed the mazes of a Puzzle so ardently, as to be entirely absorbed in devising means to extricate ourself from its bewildering difficulties; and felt almost as much pleasure in eventually achieving victory over it, as we have in conquering an adversary at some superior game of skill. It is "in good sooth, a right dainty and pleasant pastime," to watch the stray wanderings of another person attempting to elucidate a Paradox, or perform a Puzzle, with which one is previously acquainted. It is laughable to see him elated with hope at the apparent speedy end of his troubles, when you know that, at that moment, he is actually farther from his object than he was when he began; and it is no less amusing to watch his increasing despair, as he conceives himself to be getting more and more involved, when you are well aware that he is within a single turn of a happy termination of his toils; but what a mirthful moment is that, when there being only two ways to turn, the one right and the other wrong, as is usually the case, he takes the latter, and becomes more than ever
"Pozed, puzzled, and perplexed."
Puzzles are by no means of modern origin; the Sphynx puzzled the brains of some of the heroes of antiquity, and even Alexander the Great, as it is written, made several essays to untie the knot with which Gordius, the Phrygian king, who had been raised from the plow to the throne, tied up his implements of husbandry in the temple, in so intricate a manner, that universal monarchy was promised to the man who could undo it: after having been repeatedly baffled, he, at length, drew his sword, considering that he was entitled to the fulfillment of the promise, by cutting the Gordian knot.
1. THE CHINESE CROSS.
Have six pieces of wood, bone, or metal, made of the same length as No. 6, in the above figures, and each piece of the same size as No. 7. It is required to construct a cross, with six arms, from these pieces, and in such a manner that it shall not be displaced when thrown upon the floor.
The shaded parts of each figure represent the parts that are cut _out_ of the wood, and each piece marked _a_ is supposed to be facing the reader, while the pieces marked _b_ are the _right_ side of each piece turned over _towards_ the left, so as to face the reader. No. 7 represents the end of each piece of wood, &c., and is given to show the dimensions.
2. THE PARALLELOGRAM.
A parallelogram, as in the illustration, fig. 1, may be cut into two pieces, so that by shifting the position of the pieces, two other figures may be formed, as shown by figs. 2 and 3.
3. THE DIVIDED GARDEN.
A person let his house to several inmates, who occupied different floors, and having a garden attached to the house, he was desirous of dividing it among them. There were ten trees in the garden, and he was desirous of dividing it so that each of the five inmates should have an equal share of garden and two trees. How did he do it?
4. THE ENDLESS STRING.
Now, sir, your coat is off! And see-- Your right-hand pocketed! So let it be: While o'er your arm An endless string-- Some three yards round-- Hangs like a sling. Take the string off-- But, just for fun, It must be done Keeping your right-hand in its place, And not a smile must stir your face. Until you find this puzzle out, No coat shall wrap your back about.
5. CHINESE MAZE. THE WILLOW-PATTERN PLATE.
Ye fair ones who, in continent or isle, Long for delights which love alone can bring; Whilst ruby lips display affection's smile, Haste through the maze, and reach the "wedding ring" The sweet Koong-see, whose spirit hovers near, Shall watch thee wand'ring through the doubtful way; And when thou showest aught of hope or fear, Shall whisper to thee, as thy footsteps stray!
6. THE VERTICAL LINE PUZZLE.
Draw six vertical lines, as below, and, by adding five other lines to them, let the whole form nine.
7. THE THREE RABBITS.
Draw three rabbits, so that each shall appear to have two ears, while, in fact, they have only three ears between them.
8. THE ACCOMMODATING SQUARE.
Make eight squares of card, then divide four of them from corner to corner, so that you will now have twelve pieces. Form a square with them.
9. THE CIRCLE PUZZLE.
Draw a circle upon a piece of paper, and thrust a pin through it without crossing the circle, or thrusting it downward through the center.
10. THE CARDBOARD PUZZLE.
Take a piece of cardboard or leather, of the shape and measurement indicated by the diagram, cut it in such a manner that you yourself may pass through it, still keeping it in one piece.
11. THE BUTTON PUZZLE.
In the center of a piece of leather make two parallel cuts with a penknife, and just below a small hole of the same width; then pass a piece of string under the slit and through the hole, as in the figure, and tie two buttons much larger than the hole to the ends of the string. The puzzle is, to get the string out again without taking off the buttons.
12. THE QUARTO PUZZLE.
Divide this figure into four equal parts, each of the same figure.
13. THE PUZZLE OF FOURTEEN.
Cut out fourteen pieces of paper, card, or wood, of the same size and shape as those shown in the diagram, and then form an oblong with them.
14. THE SQUARE AND CIRCLE PUZZLE.
Get a piece of cardboard, the size and shape of the diagram, and punch in it twelve circles or holes in the position shown. The puzzle is, to cut the cardboard into four pieces of equal size, each piece to be of the same shape, and to contain three circles, without cutting into any of them.
15. THE SCALE AND RING PUZZLE.
Provide a thin piece of wood of about two inches and a half square; make a round hole at each corner, sufficiently large to admit three or four times the thickness of the cord you will afterwards use, and in the middle of the board make four smaller round holes in the form of a square, and about half an inch between each. Then take four pieces of thin silken cord, each about six inches long, pass one through each of the four corner holes, tying a knot underneath at the end, or affixing a little ball or bead to prevent its drawing through; take another cord, which, when doubled, will be about seven inches long, and pass the two ends through the middle holes _a a_, from the front to the back of the board, (one cord through each hole,) and again from back to front through the other holes _b b_; tie the six ends together in a knot, so as to form a small scale, and proportioning the length of the cords, so that when you hold the scale suspended, the middle cord, besides passing through the four center holes, will admit of being drawn up into a loop of about half an inch from the surface of the scale; provide a ring of metal or bone, of about three quarters of an inch in diameter, and place it on the scale, bringing the loop through its middle; then, drawing the loop a little through the scale toward you, pass it, double as it is, through the hole at the corner A, over the knot underneath, and draw it back; then pass it in the same way through the hole at corner B, over the knot, and draw it back; then, drawing up the loop a little more, pass it over the knot at top, and afterward through the holes C and D in succession, like the others, and the ring will be fixed.
16. THE HEART PUZZLE.
Cut a piece of thin wood the shape indicated by the diagram, and having perforated it as above, draw a piece of string, with a smaller heart attached at the end, through No. 1, pass it behind, and bring it through 2 before, and through 3, and so on to 6, when a loop must be made so as to enclose that part of the string which runs from 2 to 3. The puzzle is to remove the string from the large heart altogether, without unfastening the loop.
Care should be taken to avoid twisting or entangling the string. The length of the string should be proportioned to the size of the heart; if you make the heart two inches and a half high, the string when doubled should be about nine inches long.
17. THE CROSS PUZZLE.
Cut three pieces of paper to the shape of No. 1, one to the shape of No. 2, and one to that of No. 3. Let them be of proportional sizes. Then place the pieces together so as to form a cross.
18. THE YANKEE SQUARE.
Cut as many pieces of each figure in cardboard as they have numbers marked on each; then form the pride of the American army.
19. THE CARD PUZZLE.
One of the best puzzles hitherto made is represented in the annexed cut. A is a piece of card; _b b_ a narrow slip divided from its bottom edge, the whole breadth of the card, except just sufficient to hold it on at each side; _c c_ is another small slip of card with two large square ends, _e e_; _d_ is a bit of tobacco-pipe, through which _c c_ is passed, and which is kept on by the two ends _e e_. The puzzle consists in getting the pipe off without breaking it or injuring any other part of the puzzle. This, which appears to be impossible, is done in the most simple manner. On a moment's consideration it will appear plainly that there must be as much difficulty in getting the pipe in its present situation, as there can be in taking it away. The way to put the puzzle together is as follows: The slip _c c e e_ is cut out of a piece of card in the shape delineated in Fig. 3. The card in the first figure must then be gently bent at A, so as to allow of the slip at the bottom of it being also bent sufficiently to pass double through the pipe, as in Fig. 2. The detached slip with the square ends (Fig. 3) is then to be passed half way through the loop _f_ at the bottom of the pipe; it is next to be doubled in the center at _a_, and pulled through the pipe, double; by means of the loop of the slip to the card. Upon unbending the card the puzzle will be complete, and appear as represented in Fig. 1.
20. THREE SQUARE PUZZLE.
Cut seventeen slips of cardboard of equal lengths, and place them on a table to form six squares, as in the diagram. It is now required to take away five of the pieces, yet to leave but three perfect squares.
21. THE CYLINDER PUZZLE.
Cut a piece of cardboard about four inches long, of the shape of the diagram, and make three holes in it, as represented. The puzzle is, to make one piece of wood to pass through, and also exactly to fill, each of the three holes.
22. PUZZLE OF THE FOUR TENANTS.
I have a square plot of ground, in one quarter of which I have built a house, which I have let to four tenants. I tell them that if they can divide the remaining ground into four equal plots, alike in shape, and each containing one of the four apple trees I have planted, they shall have it without any increase of rent. How may they succeed?
23. THE PUZZLE WALL.
Suppose there was a pond, around which four poor men built their houses, thus:
Suppose four evil-disposed rich men afterwards built houses around the poor people, thus:
and wished to have all the water of the pond to themselves. How could they build a high wall, so as to shut out the poor people from the pond?
24. THE NUNS.
Twenty-four nuns were arranged in a convent by night, by a sister, to count nine each way, as in the diagram. Four of them went out for a walk by moonlight. How were the remainder placed in the square so as still to count nine each way? The four who went out returned, bringing with them four friends; how were they all placed still to count nine each way, and thus to deceive the sister, as to whether there were 20, 24, 28, or 82, in the square?
25. THE HORSE SHOE PUZZLE.
Cut a piece of apple or turnip into the shape of a horse shoe, stick six pins in it for nails, and then, by two cuts, divide it into six parts, each to contain one pin.
26. THE CARD SQUARE.
With eight pieces of card or paper, of the shape of Fig. _a_, four of Fig. _b_, and four of Fig. _c_, and of proportionate sizes, form a perfect square.
27. THE DOG PUZZLE.
The dogs are, by placing two lines upon them, to be suddenly aroused to life and made to run. Query, How and where should these lines be placed, and what should be the forms of them?
28. PUZZLE OF THE TWO FATHERS.
Two fathers have each a square of land. One father divides his so as to reserve to himself one fourth; thus--
The other father divides his so as to reserve to himself one fourth in the form of a triangle; thus--
They each have four sons, and each divides the remainder among his sons in such a way that each son will share equally with his brother, and in similar shape. How were the two farms divided?
29. THE TRIANGLE PUZZLE.
Cut twenty triangles out of ten square pieces of wood; mix them together, and request a person to make an exact square with them.
30. CUTTING OUT A CROSS.
How can be cut out of a single piece of paper, and with one cut of the scissors, a perfect cross, and all the other forms as shown in the cuts?
31. ANOTHER CROSS PUZZLE.
With three pieces of cardboard of the shape and size of No. 1, and one each of No. 2 and 3, to form a cross.
32. THE FOUNTAIN PUZZLE.
A is a wall, B C D three houses, and E F G three fountains or canals.
It is required to bring the water from E to D, from G to B, and from F to C, without one crossing the other, or passing outside of the wall A.
33. THE PUZZLE OF THE STARS.
Friends one and all, I pray you show How you _nine stars_ would so bestow, _Ten_ rows to form--in each row _three_-- Tell me, ye wits, how this can be?
34. THE COUNTER PUZZLE.
Place eight counters or coins, as in the diagram; it is then required to lay them in four couples, removing only one at a time, and in each removal passing the one in the hand over _two_ on the table.
35. THE JAPAN SQUARE PUZZLE.
Cut out ten pieces of card or wood of the same sizes and shapes as in the diagram, and then form a square with them.
36. THE CABINET MAKER'S PUZZLE.
A cabinet-maker has a circular piece of veneering with which he has to veneer the tops of two oval stools; but it so happens that the area of the stools, exclusive of the hand-holes in the center, and that of the circular piece, are the same. How must he cut his stuff so as to be exactly sufficient for his purpose?
37. THE STRING AND BALLS PUZZLE.
Get an oblong strip of wood or ivory, and bore three holes in it, as shown in the cut. Then take a piece of twine, passing the two ends through the holes at the extremities, fastening them with a knot, and thread upon it two beads or rings, as depicted above. The puzzle is to get both beads on the same side, without removing the string from the holes, or untying the knots.
38. THE DOUBLE-HEADED PUZZLE.
Cut a circular piece of wood as in the cut No. 1, and four others, like No. 2. The puzzle consists in getting them all into the cross-shaped slit, until they look like Fig. 3.
39. ARITHMETICAL PUZZLE.
The sum of four figures in value will be. Above seven thousand nine hundred and three; But when they are halved, you'll find very fair The sum will be nothing, in truth I declare.
40. GRAMMATICAL PUZZLE.
Let the rich, great, and noble, banquet in the festal halls, And pass the hours away, as the most thoughtless revel; Then seek the poor man's dreary home, whose very dingy walls Proclaim full well to all how low his rank and level.
Take away one letter from a word in the above stanza, and substitute another, leaving the word so metamorphosed still a word of the English language; and, by that change, totally after the syntactical construction of the whole sentence, changing the moods and tenses of verbs, turning verbs into nouns, nouns into adjectives, and adjectives into adverbs, &c., and so make the entire stanza bear quite a different meaning from that which it has as it stands above.
41. THE TREE PUZZLE.
Plant an orchard of twenty-one trees, so that there shall be nine straight rows, with five trees in each row, the _outline_ a regular geometrical figure, and the trees all at unequal distances from each other.
42. AN EPITAPH ON ELLINOR BACHELLOR, AN OLD PYE WOMAN.
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43. A CURIOUS LETTER.