pArte 1
9 Long-fel-low 15 7
It will be observed that there is always a difference of 8 between the numbers of the columns, so that it is necessary to recollect only one of them. Perhaps some of our readers who wish to be adepts in this game, would prefer recollecting the above table if put in this form:
2-3 1-2 3 1-2-3 1-3 2 none -------------------------------- 1 2 3 4 8 13 15
where the upper line denotes the cases in which the "larger half" was taken, and the lower line the numbers of the left hand column above given.
_Another Method._
The person having chosen any number from one to fifteen, he is to add twenty-one to that number, and triple the amount. Then,
1st. He is to take half of that triple, and triple that half.
2nd. To take the half of the last triple, and triple that half.
3rd. To take the half of the last triple.
4th. To take the half of the last half.
In this operation there are four distinct cases or stages where the half is to be taken. The three first are denoted by one of the eight following Latin words, each word being composed of three syllables, and the syllables containing the letter i corresponding in numerical order with the cases where the half cannot be taken without a fraction; consequently, in those cases the person who makes the deduction is to add one to the number to be divided. The fourth case shows which of the two numbers corresponding to each word has been chosen. For if the fourth half can be taken without adding one, the number chosen is in the first, or left-hand column; but if not, it is in the second column to the right.
The words. The numbers denoted. Mi-ser-is 8 0 Ob-tin-git 1 9 Ni-mi-um 2 10 No-tar-i 3 11 In-fer-nos 4 12 Or-di-nes 13 5 Ti-mi-di 6 14 Te-ne-ant 15 7
_Example._--Suppose the number chosen to be nine, to which is to be added one, making ten, and which last, being tripled, gives thirty. Then:
1st case. The half of the triple is 15 which tripled, makes 45 2nd case. The half of that triple, 1 being added to make an even number, is 23 and that tripled, makes 69 3rd case. The half of the last triple, 1 being added, is 35 4th case. The half of the last half, 1 being again added, is 18
Here we see, that in the second and third case, one had to be added, and, looking at the table, we find that the only corresponding word having an i in its second and third syllables is _Ob-tin-git_, which represents the figures one and nine. Then, as one had to be added in the fourth case, we know by the rule, that the figure in the second column, 9, is the one required. Observe, that if no addition be required at any of the four stages, the number thought of will be fifteen; and if one addition only be required at the fourth stage, the number will be seven.
WHO WEARS THE RING?
This is an elegant application of the principles involved in discovering a number fixed upon. The number of persons participating in the game should not exceed nine. One of them puts a ring on one of his fingers, and it is your object to discover--1st. The wearer of the ring. 2d. The hand. 3d. The finger. 4th. The joint.
The company being seated in order the persons must be numbered 1, 2, 3, &c.; the thumb must be termed the first finger, the fore finger being the second; the joint nearest the extremity must be called the first joint; the right hand is one, and the left hand two.
These preliminaries having been arranged, leave the room in order that the ring may be placed unobserved by you. We will suppose that the third person has the ring on the right hand, third finger, and first joint; your object is to discover the figures 3131.
Desire one of the company to perform secretly the following arithmetical operations:
1. Double the number of the person who has the ring; in the case supposed, this will produce 6 2. Add 5 11 3. Multiply by 5 55 4. Add 10 65 5. Add the number denoting the hand 66 6. Multiply by 10 660 7. Add the number of the finger 663 8. Multiply by 10 6630 9. Add the number of the joint 6631 10. Add 35 6666
He must apprise you of the figures now produced, 6666; you will then in all cases subtract from it 3535; in the present instance there will remain 3131, denoting the person No. 3, the hand No. 1, the finger No. 3, and the joint No. 1.
PROBABILITIES.[12]
When we look around us at results happening daily, of the causes of which we are ignorant, we are led to regard them as isolated incidents subject to no law or rule; but could we see and understand the secret workings and connection existing between cause and effect, we might frequently discover that all works by rule. As it is, we may readily mark the boundaries, within which events must happen in very many instances; and do much to estimate their probability. We speak of _Chance_, as something without plan or design, but taking in a large range, our calculations will approximate closely to the truth. When we throw a copper into the air, the chances of "heads or tails," as the boys say, are equal, and though one or the other may occur most frequently for a few throws, in a large number, say a thousand, the results will be about equally divided. In this case the sides of the coin must be equal in weight, else it will be like the grumbler's bread and butter:
"I never had a piece of bread,
## Particularly good and wide,
But fell upon the sanded floor, And always on the buttered side."
Had he put on less butter, perhaps the sides would have been more equal in weight, and the probability of the buttered side being uppermost would have been increased. Disturbing causes unknown to us, may often shape the result; but in the absence of these, we may pretty accurately estimate our chances.
We see accidents from fire and flood, happening at times and points least expected; but the insurer has learned by observation to estimate probabilities, and by taking a wide range of country and a period of years, he does a comparatively safe business. Death takes the young and the old; but the life insurer has conned the bills of mortality and studied the ages of those who have died, until he can estimate at once the probability of duration of life, and determine what he can afford to pay for an annuity contingent on life, or engage for a present sum, or an annual sum paid for life, to pay the heirs at the death of the insured. In one instance his estimate may fall short, and in another exceed, but the average will be about right.
So, too, the man who deals in lotteries and games of chance, knows the data and calculates carefully the probabilities, and though "luck" may sometimes be against him, his estimates of probabilities are based on mathematical principles, and he is secure in being ultimately the gaining party.
How these chances are calculated, depends on the data in each case, and it is not within the range of our present plan to attempt more than giving a general idea of the subject; and this with any one of ordinary prudence, will be sufficient to prevent all intermeddling with lotteries and every other species of gambling. The probabilities are always against the casual operator, even if all be conducted fairly; what then must they be when fraud and dishonesty are superadded? It is downright swindling!
In lottery schemes generally, fifteen per cent. is reserved as profit, but this is a small part of what may be secured; yet even this amounts to a great deal. If a man were to draw a prize nominally of $100,000, fifteen thousand would be deducted at once, and he would be entitled to only $85,000. It is true that in his good fortune he would not probably regard the abatement, but that does not change the principle.
VARIATIONS.
It is obvious that if we have a number of single things arranged in any order, we may change the arrangement into a variety of forms, and in doing so, we may take all together, or we may take only part at once. For instance, we may arrange the six vowels, a e, i, o, u, y, in a great number of ways, as a e i o u y, a i e o u y, e a i o u y, &c., &c.; or we may form them into groups, as ae, io, uy, ai, eu, oy, &c.; or, we may take three, four, five, or, as above, all at a time; and it is reasonable to suppose that the number of possible changes may, in all cases, be calculated.
When all are taken together, the operation is called _Permutation_; but if a part only be taken, it is called either a _Variation_ or a _Combination_; a e, i o, u y, are distinct combinations, and are also considered one of the variations of two of which those six letters are susceptible; e a, o i, y u, are three other variations, but they are the same combinations; for a change of order will constitute a new variation but not a new combination; hence the number of variations will always exceed the number of combinations.
The doctrine of variations and combinations forms the basis of many forms of lotteries, and of other calculations used in practical life.
COMBINATIONS AND PERMUTATIONS.
"Combinations" are the different ways in which a certain number of things can be selected out of a larger number, when taken 1 at a time, 2 at a time, or any other number each time, but without regard to the order in which the selected numbers can be arranged among themselves. The latter is the province of "Permutation," which refers to the different ways in which a number can be selected out of one that is larger, and, _in addition to this_, to the different ways of _grouping_ these selected numbers.
Thus 4 things can be taken 2 at a time in 6 different ways; for instance, the letters a, b, c, d, can be taken 2 at a time thus, a and b, a and c, a and d, b and c, b and d, c and d; if we regard the _order_ of the selected letters we shall find that these 4 letters are capable of 12 different permutations, as ab, ba, ac, ca, ad, da, bc, cb, bd, db, cd, dc.
If we selected 3 letters at a time we could make 4 different selections, and 24 different changes of grouping.
The rule to compute the number of these different ways is very simple, but sometimes involves a multitude of figures.
To determine the number of permutations, commence with unity, and multiply by the successive terms of the natural series 1, 2, 3, &c., until the highest multiplier shall express the number of individual things. The last product will indicate the number of possible changes.
_Example 1._ How many changes can be made in the arrangement of 5 grains of corn, all of different colors, laid in a row?
_Solution._ 1 × 2 × 3 × 4 × 5 = 120, _Ans._
This may seem improbable, the number being so great, but if there were but a single grain more, the possible changes would be 720; and another would extend the limit to 5040; and so onward in a constantly increasing ratio. The reason, however, will be obvious on a little scrutiny. If there were but one thing, as _a_, it would admit of but one position; but if two, as _a b_, it would admit of two positions, _ab_, _ba_. If three things, as _a b c_, then they will admit of 1 × 2 × 3 = 6 changes, for the last two will admit of two variations, as _a b c_, _a c b_, and each of the three may successively be placed first, and two changes made to each of the others, so that 3 × 2 = 6, the number of possible changes. In the same way we may show that if there be four individual things, each one will be first in each of the six changes which the other three will undergo, and consequently, there will be 24 changes in all. In this way we might show that when there are 5 individual things, there will be 5 times as many changes as when there were but 4; and when 6, there will be 6 times as many changes as when there are only 5; and so on _ad infinitum_, according to the same law.
_Example 2._ In how many ways may a family of 10 persons seat themselves differently at dinner? _Ans._ 3,628,800.
When we consider that this would require a period of 9935-55/487 years, the mind is lost in astonishment. The story of the man who bought a horse at a farthing for the first nail in his shoe, a penny for the second, &c., is thrown into the shade; and we incline to doubt whether there is not some mistake; and yet on just such chances as one to all these, do gamblers constantly risk their money!
_Example 3._ I have written the letters contained in the word N I M R O D on 6 cards; being one letter on each, and having thrown them confusedly into a hat, I am offered $10 to draw the cards successively, so as to spell the name correctly. What is my chance of success worth? _Ans._ 1-7/18 cents.
_Example 4._ In order to form a lottery scheme, I have put into the wheel as many cards as I can put 4 letters of the word Charleston on, without having the same letters in the same order upon any two cards. I offer $100 to him who draws the card having on it the first four letters of the said word in their natural order (Char). What is the chance of drawing a prize worth?
There are 10 letters in the word, and the combination is of the 4th class; and, according to the mode of determining combinations with repetitions, we find the whole number of combinations of the 4th class which the word admits of is 210. Then he has one chance in 210 of drawing the letters Char, in _some_ order. The number of permutations of four individual things is 1 × 2 × 3 × 4 = 24, and 210 × 24 = 5040 his chance of drawing them in the right order, and $100 divided by 5040 gives _Ans._ 1 52/63 cents.
Suppose that the numbers from 1 to 78, inclusive, be placed upon 78 cards, and the cards placed in a wheel by which they are thoroughly mixed; and then 13 cards be successively drawn out, by a person who has no means of choosing, and the numbers on them registered. Suppose also that tickets have been issued, containing each three of the 78 numbers, but no two having _all_ the same numbers, and that he who holds the ticket having on it the first three drawn numbers in their regular order, shall be entitled to $100,000; what would the probability of drawing such a ticket be worth?
_Ans._ 21 5183/5858 cents.
_Note._--It is usual also, to give smaller prizes to the holders of tickets having the numbers in any order, or having any two or one of the drawn numbers. Lotteries may be arranged on a great diversity of plans, and in each the probability of drawing prizes will vary.
A speaks the truth 3 times in 4; B 4 times in 5, and C 6 times in 7. What is the probability of an event which A and B assert, and C denies? _Ans._ 140/143
Suppose a coin be thrown up, having two faces; what is the probability that the obverse (heads) side will fall upward, and what the reverse?
Here there are only two possible cases, and one favors each of the contingencies the probability of each will be 1/(1 + 1) = 1/2; there being no reason why one side should fall uppermost rather than the other.
What would be the probability of either side presenting upwards twice in two throws?
Here we have 4 possible cases, viz.:
Obverse and reverse; Obverse both times; Reverse and obverse; Reverse both times.
Of the 4 possibilities there is only one which favors the turning up of the obverse twice in succession, and the same is true of the reverse, hence the probability of either is only 1/4.
In like manner we might show that the probability of the obverse presenting upwards three times in succession will be 1/8, or 1/2 × 1/2 × 1/2; the general principle being to multiply successively together the independent probabilities of an event for the fraction expressing the chance of all the events happening.
THE VISITORS TO THE CRYSTAL PALACE.
In a family consisting of 8 young people, it was agreed that 3 at a time should visit the Crystal Palace, and that the visit should be repeated each day as long as a different trio could be selected. In how many days were the possible combinations of 3 out of 8 completed?
We must multiply 8 × 7 × 6, and also 3 × 2 × 1, and divide the product of the former, 336, by the product of the latter, 6; the result is 56, the number of visits, a different three going each time. So much gratified were they with the results of their agreement, that they wished to be allowed another series of visits, to be continued as many days as they could group 3 together in different order when starting. If Paterfamilias had granted such permission he would have had to wait 56 multiplied by 3 × 2 × 1, or 336 days, before this "new series" of visits would have come to a _finis_.
HOW MANY CHANGES CAN BE GIVEN TO 7 NOTES OF A PIANO?
That is to say, in how many ways can 7 keys be struck in succession, so that there shall be some difference in the order of the notes each time?
The result of multiplying
7 × 6 × 5 × 4 × 3 × 2 × 1
is 5,040, the number of changes.
THE ARITHMETICAL TRIANGLE.
This name has been given to a contrivance said to have originated with the famous Pascal, or to have been perfected by him.
1 2 1 3 3 1 4 6 4 1 5 10 10 5 1 6 15 20 15 6 1 7 21 35 35 21 7 1 8 28 56 70 56 28 8 1 &c. &c.
This peculiar series of numbers is thus formed: Write down the numbers 1, 2, 3, &c., as far as you please, in a vertical row. On the right hand of 2 place 1, add them together, and place 3 under the 1; then 3 added to 3 = 6, which place under the 3; 4 and 6 are 10, which place under the 6, and so on as far as you wish. This is the second vertical row, and the third is formed from the second in a similar way.
This triangle has the property of informing us, without the trouble of calculation, how many combinations can be made, taking any number at a time out of a larger number.
Suppose the question were that just given; how many selections can be made of 3 at a time out of 8? On the horizontal row commencing with 8, look for the third number; this is 56, which is the answer.
HOW MANY DIFFERENT DEALS CAN BE MADE WITH 13 CARDS OUT OF 52?
To discover this we must make a continued multiplication of 52 × 51 × 50 × 49 × 48 × 47 × 46 × 45 × 44 × 43 × 42 × 41 × 40, being 13 terms for the 13 cards, also a continued multiplication of 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, and having found the two products, we must divide one by the other, and the quotient is the number of different deals out of 52 cards. This "sum," that looks so formidable with natural figures, is a very short one by logarithms.
THE THREE GRACES.
Three articles, or three names inscribed on cards, having been distributed between three persons, you are to tell which article or card each person has.
Designate the three persons in your own mind, as 1st, 2d, and 3d, and the three articles, A, E, I. Provide 24 counters, and give 1 to the first person, 2 to the 2d, 3 to the 3d. Place the remaining 18 on the table. Request that the three persons will distribute among themselves the three articles, and that, having done so, the person who has the one which you have secretly denoted by A, will take as many counters as he may have already; the holder of E must take twice as many as he may have; and the holder of I must take four times as many. Then leave the room, in order that the distribution of articles and of counters may be made unobserved by you. We will suppose that the three articles are three cards, on which are the words Clara, Rosa, Emily, which you will yourself secretly denote by the letters A, E, I. Suppose also that in the division the first person has Emily (I), the second has Clara (A), and the third has Rosa (E), then the 1st will take four times as many counters as he has (1), and will therefore take 4; the 2d will take as many as he has (2), and will therefore take 2; the 3d will take 6, being twice as many as he has (3). On the table will be left 6 counters. The distribution having been made, you will return and observe the number of counters on the table, from which you can find who is the holder of each card by the following method.
It is plain that if the cards held by the 1st and 2d can be told, that held by the 3d will be known. It will be found that only six numbers can remain, viz. 1, 2, 3, 5, 6, 7; never 4, and never more than 7. Now the 6 combinations of a, e, and i, here given, represent the articles held by the 1st and 2d persons.
1 2 3 4 5 6 7 ae ea ai -- ei ia ie
In the case supposed, 6 counters being on the table, the combination _ia_ indicates that the first person has the card you have called I (Emily), the 2d has A (Clara), so that the 3d has E (Rosa).
In order to recollect the combinations of A, E, and I, it will be best to keep in memory some 7 words which form a sentence, and which contain these vowels in the order just given.
Our young friends can amuse themselves in forming a sentence for themselves, but as examples we supply three.
1 2 3 4 5 6 7 _ae_ _ea_ _ai_ -- _ei_ _ia_ _ie_ James easy admires now reigning with a bride. Anger, fear, pain may be hid with a smile. Graceful Emma, charming she reigns in all circles.
Or, if they prefer Latin, they can use the pentameter made up by the inventor of this beautiful pastime:
1 2 3 5 6 7 Salve certa animæ semita vita quies.
ANOTHER METHOD.
The performer must mentally distinguish the articles by the letters A, B, C, and the persons as 1st, 2d, and 3d. The persons having made their choice, give 12 counters to the 1st, 24 to the 2d, and 36 to the 3d. Then request the 1st person to add together the half of the counters of the person who has chosen A, the 3d of the person who has chosen B, and the 4th of those of the person who has chosen C, and then ask the sum, which must be either 23, 24, 25, 27, 28, or 29, as in the following table:
First. Second. Third. 12 24 36 A B C 23 A C B 24 B A C 25 C A B 27 B C A 28 C B A 29
This table shows that if the sum be 25, for example, the 1st person must have chosen B, the 2d A, and the 3d C; or if it be 28, the 1st must have chosen B, the 2d C, and the 3d A.
ANOTHER METHOD.
Three things having been divided between three persons, you are to determine the holder of each.
Call the persons in your own mind 1st, 2d, 3d.
Give to the 1st a card on which you have written the number 12; to the 2d the number 24; to the 3d 36.
The three things you must denote as A, E, I.
To simplify it you may have three cards with a name upon each, of which the initial letters are A, E, I, as Anna, Emma, Isabel.
Request your friends to divide between them the three articles, and then to add together certain parts of the numbers on their cards, as follows:
Whoever has A must supply one half of the number on his card;
Whoever has E must supply one third;
Whoever has I must supply one fourth;
This half, third and fourth having been added together, the sum must be announced to you on your return; and from this number you can tell who has A, who has E, and who has I.
If the No. is the 1st has the 2d has the 3d has 23 A E I 24 A I E 25 E A I 27 I A E 28 E I A 29 I E A
The sum which will be given to you can be one of six only. There are only six ways in which the articles can be divided, and there is a definite number for each of them.
The number 26 can never occur, and to recollect the six which do occur, and which you perceive are consecutive, you need take note only of what the 1st and 2d persons have.
23 24 25 26 27 28 29 ae ai ea -- ia ei ie
If you make up a line of good (or bad) English, having the vowels in the order here given, you will find it will aid you in their recollection. We give one as a specimen:
ae ai ea -- ia ei ie Brave dashing sea, like a giant revives itself.
THE FORTUNATE NINTH.
A sharp youth, fresh from school, having gone to visit a good-natured uncle, the latter placed on a table fifteen fine oranges and fifteen apples, and desired his young friend to take half. He, not liking the apples, was about to take the fifteen oranges; but this monopoly of the best fruit being objected to, the old gentleman told him to range all the fruit in a circle, and to take every ninth. The clever fellow ranged them in such a way as that, by taking away every ninth, all the apples were left on the table, and all the oranges were transferred to his capacious pockets. How did he arrange them?
He placed them as in the margin, A representing apples, and O oranges; and it will be found that, by commencing at the four apples, and going round and round the circle, taking away every ninth, all the oranges will be removed, and all the apples will remain.
[Illustration]
If we let the vowels a e i o u denote the figures 1 2 3 4 5,
the arrangement of the figures 4, 5, 2, 1, &c., can be easily recollected by the following line:
Our earth's final fate--enigma ever dark 45 21 3 1 1 2 2 3 1 2 2 1 or, Our dear Richard's tale begins at the sea. 45 21 3 1 1 2 2 3 1 2 21
Our young friends may find amusement in forming lines for themselves as much superior to these as possible.
THE TEN TENS.
Take ten pieces of card, and upon each write any ten words; there is no restriction as to the initial letter of nine of the words, but the last word on each card must commence with certain letters which you must in your own mind associate with the numbers 1 to 10, so that by knowing the initial letter of the last word on each card, you can determine its number.
Here are ten cards, (call these the _Selecting Cards_,) which we give by way of example, though our readers will perhaps prefer having words of their own selection.
Jane. Ellen. George. James. Newton. Mary. Fanny. William. Clement. Davy. Matilda. Caroline. Frederick. Edward. Morse. Sarah. Isabel. Robert. Ralph. Fulton. Rosa. Flora. Edmund. Francis. Franklin. Elizabeth. Laura. John. Edwin. Arago. Harriet. Maria. Alfred. Walter. Spurzheim. Ann. Frances. Albert. Charles. Laplace. Emily. Edith. Henry. Samuel. Steers. =E=mma. =D=orothea. =I=saac. =T=heodore. =H=erschel.
Sister. Rose. Friendship. Putnam. Clay. Brother. Violet. Happiness. Lafayette. Webster. Uncle. Lupin. Industry. Steuben. Calhoun. Aunt. Daisy. Ambition. Scott. Benton. Grandmother. Tulip. Energy. Taylor. Jefferson. Grandfather. Peony. Fidelity. Green. Adams. Nephew. Hyacinth. Affection. Harrison. Madison. Niece. Pink. Hope. Hamilton. Jackson. Cousin. Snowdrop. Justice. Wayne. Monroe. =F=ather. =L=ily. =O=rder. =W=ashington. =N=apoleon.
For these the key words are, "Edith Flown," so that the letters
E D I T H F L O W N Stand for 1 2 3 4 5 6 7 8 9 10
For the success of the game, the key words and the numbers denoted by their letters, must be carefully concealed.
Take ten other cards, which call the "_grouped cards_," and upon one write down the first word from each of the selecting cards, being careful to write them in the same order. Let another card contain all the words which are second from the top, and so on till all the words have been grouped together. As an example, we give the 1st and 4th grouped cards.
1st. 4th. Jane. Sarah. Ellen. Isabel. George. Robert. James. Ralph. Newton. Fulton. Sister. Aunt. Rose. Daisy. Friendship. Ambition. Putnam. Scott. Clay. Benton.
The object of the game is to guess which of the words from any of the _selecting cards_ any person may have fixed upon.
Let any one choose a card out of the _selecting cards_, and after he has fixed upon a word, give it back to you; when receiving it, carefully note the last word upon it, which will give you, by the aid of the key word, the number of the card; this you must keep secret, and you then give him all the _grouped cards_, and request him to show you the cards which contain the words he fixed upon.
You can then announce the word; for the number of the word from the top on the grouped card is the same as the number of the selecting card, from which he made his choice.
Suppose he made his choice from the card which has Theodore for its last word--this is No. 4; when he shows you the grouped card, which he says contains the selected word, you will know that Ralph, the fourth from the top, is the name he fixed upon.
DIVIDING THE BEER.
During the siege of Sebastopol, when the troops were on "short allowance," a can of eight pints of porter was ordered to be equally divided between two messes; but having only a five pint can, and one which held three pints, it was found impossible to make this division, till one of the clever sappers suggested the following method; and, to understand it, we will put down the contents of each of the three cans at each stage of the process; commencing with
8-pt. 5-pt. 3-pt. The 8-pint can full, and the others empty, 8 0 0 1. Filled the 5-pint can 3 5 0 2. Filled the 3-pint can from the 5-pint 3 2 3 3. Pour the contents of the 3-pint into the 8-pint 6 2 0 4. Transfer the 2 pints from the 5-pint to the 3-pint 6 0 2 5. Filled the 5-pint from the 8-pint 1 5 2 6. Fill up the 3-pint from the 5-pint 1 4 3 7. Poured the 3 pints into the 8-pint; completing the feat 4 4 0
This was a dexterous expedient of the worthy sapper, the only objections to it being the time the thirty men had to wait, and the resulting flat condition of the beer.
THE DIFFICULT CASE OF WINE.
A gentleman had a bottle containing 12 pints of wine, 6 of which he was desirous of giving to a friend; but he had nothing to measure it, except two other bottles, one of 7 pints, and the other of 5. How did he contrive to put 6 pints into the 7-pint bottle?
12-pt. 7-pt. 6-pt. Before he commenced, the contents of the bottles were 12 0 0 1. He filled the 5-pint 7 0 5 2. Emptied the 5-pint into the 7-pint 7 5 0 3. Filled again the 5-pint from the 12-pint 2 5 5 4. Filled up the 7-pint from the 5 2 7 3 5. Emptied the 7-pint into the 12-pint 9 0 3 6. Poured the 3 pints from the 5 into the 7 9 3 0 7. Filled the 5-pint from the 12-pint 4 3 5 8. Filled up the 7-pint from the 5-pint 4 7 1 9. Emptied the 7-pint into the 12-pint 11 0 1 10. Poured 1 pint from the 5-pint into the 7-pint 11 1 0 11. Filled the 5-pint from the 12-pint 6 1 5 12. Poured the contents of the 5-pint into the 7-pint 6 6 0
ANOTHER DECIMATION OF FRUIT.
On the next visit of the youth to his uncle, the latter produced thirty apples and ten oranges, and offered him the favorite oranges, if his nephew could arrange them in an oval, so that by taking every twelfth the apples should remain. But this he could not accomplish, and the old gentleman, being well versed in the "Recreations in Science," proceeded to arrange them thus:
[Illustration]
The places which the oranges here occupy can be easily remembered, being Nos. 7, 8, 11, 12, 21, 22, 24, 34, 36, 37.
THE WINE AND THE TABLES.
A certain hotel-keeper was dexterous in contrivances to produce a large appearance with small means. In the dining-room were three tables, between which he could divide 21 bottles, of which 7 only were full, 7 half full, and 7 apparently just emptied, and in such a manner that each table had the same number of bottles, and the same quantity of wine. He did this in two ways:
Table. Full Hf. full. Empty. Table. Full Hf. full. Empty. 1 2 3 2 | 1 3 1 3 2 2 3 2 | 2 3 1 3 3 3 1 3 | 3 1 5 1
He also performed a similar exploit with 24 bottles, 8 full, 8 half-full, and 8 empty:
Table. Full Hf. full. Empty. Table. Full Hf. full. Empty. 1 3 2 3 | 1 2 4 2 2 3 2 2 | 2 2 4 2 3 2 4 2 | 3 4 0 4
Also with 27 bottles, 9 full, 9 half-full, and 9 empty:
Table. Full Hf. full. Empty. Table. Full Hf. full. Empty. 1 2 5 2 | 1 1 7 1 } 2 3 3 3 | 2 4 1 4 } 3 4 1 4 | 3 4 1 4 }
THE THREE TRAVELERS.
Three men met at a caravansary or inn, in Persia; and two of them brought their provision along with them, according to the custom of the country; but the third not having provided any, proposed to the others that they should eat together, and he would pay the value of his proportion. This being agreed to, A produced 5 loaves, and B 3 loaves, all of which the travelers ate together, and C paid 8 pieces of money as the value of his share, with which the others were satisfied, but quarreled about the division of it. Upon this the matter was referred to the judge, who decided impartially. What was his decision?
At first sight it would seem that the money should be divided according to the bread furnished; but we must consider that, as the 3 ate 8 loaves, each one ate 2⅔ loaves of the bread he furnished. This from 5 would leave 2⅓ loaves furnished the stranger by A; and 3 - 2⅔ = ⅓ furnished by B, hence 2⅓ to ⅓ = 7 to 1, is the ratio in which the money is to be divided. If you imagine A and B to furnish, and C to consume all, then the division will be according to amounts furnished.
WHICH COUNTER HAS BEEN THOUGHT OF OUT OF SIXTEEN?
Take sixteen pieces of card, and number them 1 to 16. Arrange them in two rows, as at A B.
A B C B D M E B F N G B H 1 9 1 9 2 2 2 9 4 2 2 9 6 2 10 3 10 4 4 6 10 8 6 1 10 5 3 11 5 11 6 6 1 11 3 1 4 11 8 4 12 7 12 8 8 5 12 7 5 3 12 7 5 13 13 1 13 4 13 6 14 14 3 14 8 14 7 15 15 5 15 3 15 8 16 16 7 16 7 16
Desire a person to think of one of the numbers, and to tell you in which row it is. Suppose he fixes on 6; he will tell you that the row A contains the number he thought of.
Take up the row A, and arrange the numbers on each side of the row B, as shown at C D, so that the first number of the row A may be the first of the row C, the second of A be the first of D, the third of A be the second of C, and so on.
Ask in which of the rows, C or D, is the number thought of: in the case supposed it is in D.
Take up the rows C D, and put one underneath the other, as at M, taking care that the half-row in which is the number thought of, shall be above the other.
Divide it again into two rows, as at E F, on each side of B, in the same way as before. Ask again in which row it is: it is now in E.
Place one row under the other, as at N, and divide again into two rows, which will now be as G H.
You will be informed that the number is in row H, and you may then announce it to be the top number of that row.
The number thought of will always be _at the top of one of the rows after three transpositions_. If there were 32 counters it would be at the top after four transpositions.
MAGIC SQUARES.
The name "Magic Square" is given to a square divided into several smaller squares, in which numbers are placed in such a manner that every column of numbers, whether vertical, horizontal, or from corner to corner, shall amount to the same sum.
They are divided into three principal classes: 1st, Those which have an odd number of squares in each band; 2d, Those which have an even number of squares in each band, this even number being divisible exactly by 4; 3d, Where the even number of squares in each band cannot be divided by 4 without a fraction.
ODD MAGIC SQUARES.
Squares of this kind are formed thus. Imagine an exterior line of squares above the magic square you wish to form, and another exterior line on the right hand of it These two imaginary lines are shown in the figure.
Then attend to the two following rules:
1st. In placing the numbers in the squares we must go in an ascending oblique direction from left to right; any number which, by pursuing this direction, would fall into the exterior line, must be carried along that line of squares, whether vertical or horizontal, to the last square. Thus, 1 having been placed in the center of the top line, (see the first table on p. 228,) 2 would fall into the exterior square above the fourth vertical line; it must be therefore carried down to the lowest square of that line; then, ascending obliquely 3 falls into the square, but four falls out of it, to the end of a horizontal line, and it must be carried along that line to the extreme left, and there placed. Resuming our oblique ascension to the right, we place 5, where the reader sees it, and would place 6 in the middle of the top band, but finding it occupied by 1, we look for direction to the 2d Rule, which prescribes that, when in ascending obliquely, we come to a square already occupied, we must place the number, which according to the first rule should go into that occupied square, directly under the last number placed. Thus, in ascending with 4, 5, 6, the 6 must be placed directly under the 5, because the square next to 5 in an oblique direction is "engaged."
[Illustration:
........................................... . . . . . . . . . . . . . . . . 18 . 25 . 2 . 9 . . . . . . . . . +----------------------------------+....... | | | | | | . | | | | | | . | 17 | 24 | 1 | 8 | 15 | 17 . | | | | | | . |------+------+------+------+------+....... | | | | | | . | | | | | | . | 23 | 5 | 7 | 14 | 16 | 23 . | | | | | | . |------+------+------+------+------+....... | | | | | | . | | | | | | . | 4 | 6 | 13 | 20 | 22 | 4 . | | | | | | . |------+------+------+------+------+....... | | | | | | . | | | | | | . | 10 | 12 | 19 | 21 | 3 | 10 . | | | | | | . |------+------+------+------+------+....... | | | | | | . | | | | | | . | 11 | 18 | 25 | 2 | 9 | . | | | | | | . +------+------+------+------+------+....... ]
[Illustration:
+------+------+------+------+------+------+------+ | | | | | | | | | | | | | | | | | 30 | 39 | 48 | 1 | 10 | 19 | 28 | | | | | | | | | |------+------+------+------+------+------+------| | | | | | | | | | | | | | | | | | 38 | 47 | 7 | 9 | 18 | 27 | 29 | | | | | | | | | |------+------+------+------+------+------+------| | | | | | | | | | | | | | | | | | 46 | 6 | 8 | 17 | 26 | 35 | 37 | | | | | | | | | |------+------+------+------+------+------+------| | | | | | | | | | | | | | | | | | 5 | 14 | 16 | 25 | 34 | 36 | 45 | | | | | | | | | |------+------+------+------+------+------+------| | | | | | | | | | | | | | | | | | 13 | 15 | 24 | 33 | 42 | 44 | 4 | | | | | | | | | |------+------+------+------+------+------+------| | | | | | | | | | | | | | | | | | 21 | 23 | 32 | 41 | 43 | 3 | 12 | | | | | | | | | |------+------+------+------+------+------+------| | | | | | | | | | | | | | | | | | 22 | 31 | 40 | 49 | 2 | 11 | 20 | | | | | | | | | +------+------+------+------+------+------+------+ ]
Magic squares of this class, however large in the number of compartments, can be easily filled up by attending to these two rules.
We give opposite, a seven-placed square.
There are various other kinds of magic squares; but explanations of them would be too lengthy for our work.
The invention of these contrivances has been traced back to the early ages of science, and talismanic properties were attributed to them. Modern philosophers have amused themselves in bringing them to perfection, and none has contributed so much as "the model of practical wisdom," Dr. Franklin.
THE SQUARE OF GOTHAM.
The wise men of Gotham, famous for their eccentric blunders, once undertook the management of a school; they arranged their establishment in the form of a square divided into 9 rooms. The playground occupied the center, and 24 scholars the rooms around it, 3 being in each. In spite of the strictness of discipline, it was suspected that the boys were in the habit of playing truant, and it was determined to set a strict watch. To assure themselves that all the boys were on the premises, they visited the rooms, and found three in each, or 9 in each row. Four boys then went out, and the wise men soon after visited the rooms, and finding 9 in each row, thought all was right. The four boys then came back, accompanied by four strangers; and the Gothamites, on their third round, finding still 9 in each row, entertained no suspicion of what had taken place. Then 4 more "chums" were admitted; but the clever men, on examining the establishment a fourth time, still found 9 in each row, and so came to an opinion that their previous suspicions had been unfounded. How was all this possible?
The following figures represent the contents of each room at the four different visits; the first, at the commencement of the watch; the second, when four had gone out; the third, when these 4, accompanied by another 4, had returned; and the fourth, when 4 more had joined them.
I. II. III. IV. ___________ ___________ ___________ ___________ | | | | | | | | | | | | | | | | | 3 | 3 | 3 | | 4 | 1 | 4 | | 2 | 5 | 2 | | 1 | 7 | 1 | |___|___|___| | __|___|___| | __|___|___| | __|___|___| | | | | | | | | | | | | | | | | | 3 | | 3 | | 1 | | 1 | | 5 | | 5 | | 7 | | 7 | |___|___|___| | __|___|___| | __|___|___| | __|___|___| | | | | | | | | | | | | | | | | | 3 | 2 | 3 | | 4 | 1 | 4 | | 2 | 5 | 2 | | 1 | 7 | 1 | |___|___|___| | __|___|___| | __|___|___| | __|___|___|
On each change the boys arranged themselves in the rooms in such a manner that, when the corner rooms were counted as a part of two rows, each entire row of three rooms contained the same number of boys. The illusion of the wise men was due to their mistake in counting each corner room twice.
THE MATHEMATICAL BLACKSMITH.
A blacksmith had a stone weighing 40 lbs. A mason coming into the shop, hammer in hand, struck it and broke it into four pieces. "There," says the smith, "you have ruined my weight." "No," says the mason, "I have made it better, for whereas you could before weigh but 40 lbs. with it, now you can weigh every pound from 1 to 40." Required size of the pieces?
_Ans._ 1, 3, 9, 27; for in any geometrical series proceeding in a triple ratio, each term is 1 more than twice the sum of all the preceding, and the above series might proceed to any extent. In using the weights, they must be put in one or both scales as may be necessary: as to weigh 2, put 1 in one scale, and 3 in the other.
CURIOUS PROPERTIES OF SOME FIGURES.
Select any two numbers you please, and you will find that one of the two, their amount when added together, or their difference, is always 3, or a number divisible by 3.
Thus, if the numbers are 3 and 8, the first number is 3; let the numbers be 1 and 2, their sum is 3; let them be 4 and 7, the difference is 3. Again, 15 and 22, the first number is divisible by 3: 17 and 26, their difference is divisible by 3, &c.
All the odd numbers above 3, that can only be divided by 1, can be divided by 6, by the addition or subtraction of a unit. For instance, 13 can only be divided by 1; but after deducting 1, the remainder can be divided by 6; for example, 5 + 1 = 6 ; 7 - 1 = 6; 17 + 1 = 18; 19 - 1 = 18; 25 - 1 = 24, and so on.
If you multiply 5 by itself, and the quotient again by itself, and the second quotient by itself, the last figure of each quotient will always be 5. Thus 5 × 5 = 25; 25 × 25 = 125; 125 × 125 = 625, &c. Again, if you proceed in the same manner with the figure 6, the last figure will constantly be 6; thus, 6 × 6 = 36; 36 × 36 = 216; 216 × 216 = 1,296, and so on.
To multiply by 2 is the same as to multiply by 10 and divide by 5.
Any number of figures you may wish to multiply by 5, will give the same result if divided by 2--a much quicker operation than the former; but you must remember to annex a cipher to the answer where there is no remainder, and where there is a remainder, annex a 5 to the answer. Thus, multiply 464 by 5, the answer will be 2320; divide the same number by 2, and you have 232, and as there is no remainder you add a cipher. Now, take 357, and multiply by 5--the answer is 1785. On dividing 357 by 2, there is 178, and a remainder; you therefore place 5 at the right of the line, and the result is again 1785.
There is something more curious in the properties of the number 9. Any number multiplied by 9 produces a sum of figures which, added together, continually makes 9. For example, all the first multiples of 9, as 18, 27, 36, 45, 54, 63, 72, 81, sum up 9 each. Each of them multiplied by any number whatever produces a similar result; as 8 times 81 are 648, these added together make 18, 1 and 8 are 9. Multiply 648 by itself, the product is 419,904--the sum of these digits is 27, 2 and 7 are 9. The rule is invariable. Take any number whatever and multiply it by 9; or any multiple of 9, and the sum will consist of figures which, added together, continually number 9. As 17 × 18 = 306, 6 and 3 are 9; 117 × 27 = 3,159, the figures sum up 18, 8 and 1 are 9; 4591 × 72 = 330,552, the figures sum up 18, 8 and 1 are 9. Again, 87,363 × 54 = 4,717,422; added together the product is 27, or 2 and 7 are 9, and so always. If any row of two or more figures be reversed and subtracted from itself, the figures composing the remainder, will, when added horizontally, be a multiple of nine:
42 886 326 24 688 1623 -- --- ---- 18 - 9 × 2. 198 - 9 × 2. 1638 - 9 × 2.
If a multiplicand be formed of the digits in their regular order, omitting the 8, a multiplier may be found by a rule, which will give a product, each figure of which shall be the same. Thus if 12345679 be given, and it be required to find a multiplier which shall give the product all in 2, that multiplier will be 18: if in 3, the multiplier will be 27: if all 4, it will be 36--and so forth.
12345679 12345679 12345679 18 27 36 -------- -------- ------- 98765432 86419753 74074074 12345679 24691358 37037037 --------- --------- --------- 222222222 333333333 444444444
The rule by which the multiplier is discovered (but which we do not attempt to explain) is this: Multiply the last figure (the 9) of the multiplicand by the figure of which you wish the product to be composed, and that number will be the required multiplier. Thus, when it was required to have the product composed of 2, the 2 multiplied by 9 gives 18, the multiplier: 3 multiplied by 9 gives 27, the multiplier to give the product in 3; &c.
If a figure, with a number of ciphers attached to it, be divided by 9, the quotient will be composed of one figure only, namely, the first figure of the dividend, as--
9)600,000 9)40,000 ---------- --------- 66,666-6 4,444-4
{ 9)549 If any sum of figures can be divided by 9 as, {------ { 61
the amount of these figures, when added together, can be divided by 9:--thus, 5, 4, 9, added together, make 18, which is divisible by 9. If the sum 549 is multiplied by any figure, the product can also be divided by 9, as--
} { 3 549 } { 2 6 } { 9 ------ } And the amount of the figures of { 4 9)3294 } the product can also be divided by { ---- ----- } 9, thus, { 2)18 366 } { --- } { 9
To multiply by 9, add a cipher, and deduct the sum that is to be multiplied: thus,
43,260 } { 4,326 4,326 } Produces the same result as { 9 ----- } { ---- 38,934 } { 38,934
In the same manner, to multiply by 99, add two ciphers; by 999, three ciphers, &c. These properties of the figure 9 will enable the young arithmetician to perform an amusing trick, quite sufficient to excite the wonder of the uninitiated.
Any series of numbers that can be divided by 9, as 365, 472,821,754, &c., being shown, a person may be requested to multiply secretly either of these series by any figure he pleases, to strike out one number of the quotient, and to let you know the figures which remain, in any order he likes; you will then, by the assistance of the knowledge of the above properties of 9, easily declare the number which has been erased. Thus, suppose 365,472 are the numbers chosen, and the multiplier is six; if then, 8 is stuck out, the numbers returned to you will be
} 2 } 1 } 9 365472 } 2 6 } 3 ------ } 2 219232 } -- } 19
The amount of these numbers is 19; but 19, divided by 9, leaves a remainder of 1; you, therefore, want 8 to complete another 9: 8, then, is the number erased.
The component figures of the product made by the multiplication of every digit into the number 9, when added together, make NINE.
The order of these component figures is reversed after the said number has been multiplied by 5.
The component figures of the amount of the multipliers (viz. 45,) when added together, make NINE.
The amount by the several products, or multiples of 9 (viz. 405,) when divided by 9, gives for a quotient, 45; that is, 4 + 5 = NINE.
The amount of the first product (viz. 9,) when added to the other product, whose respective component figures make 9, is 81; which is the square of NINE.
The said number 81, when added to the above mentioned amount of the several products, or multiples of 9 (viz. 405) makes 486, which, if divided by 9, gives for a quotient 54: that is, 5 + 4 = NINE.
It is also observable, that the number of changes that may be rung on nine bells is 362,880; which figures, added together, make 27; that is, 2 + 7 = NINE.
And the quotient of 362,880, divided by 9, will be 40,320; that is 4 + 0 + 3 + 2 + 0 = NINE.
If number 37 be multiplied by any of the progressive numbers arising from the multiplication of 3 with any of the units, the figures in the quotient will be similar, and the result may be known beforehand by merely inspecting the progressive numbers, thus, 3, 6, 9, 12, 15, 18, 21, 24, 27, &c., are the progressive numbers formed by 3 multiplied by the units 1 to 9; and the result of the multiplication of any of these numbers with 37 may be seen in the following examples:--37 × 3 = 111; 37 × 6 = 222; 37 × 12 = 444; 37 × 24 = 888; by which it appears that the numbers of which the quotient is formed are the same as the units by which number 3 was multiplied to obtain the respective progressive numbers. Thus--3 multiplied by 2 is equal to 6, and 37 multiplied by 9 is equal to 222; so, again, 4 multiplied by 3 produces 12, and 37 multiplied by 12 is equal to 444, and so on.
THE INDUSTRIOUS FROG.
There was a well 30 feet deep, and at the bottom a frog anxious to get out. He got up 3 feet per day, but regularly fell back 2 feet at night. Required the number of days necessary to enable him to get out?
The frog appears to have cleared one foot per day, and at the end of 27 days, he would be 27 feet up, or within 3 feet of the top, and the next day he would get out. He would therefore be 28 days getting out.
THE COUNCIL OF TEN.
Ten cards or counters, numbered from one to ten, or the first ten playing cards of any suit disposed in a circular form may be employed with great convenience for performing this feat. The accompanying figure shows the cards thus arranged, number one, or the ace, designated by A, and the ten by K.
3 C 2 B D 4 1 A E 5 10 K F 6 9 I G 7 H 8
Having placed the cards in the above order, desire a bystander to think of a card or number, and when he has done so, to touch any other card or number. Request him then to add to the number of the card touched the number of the cards employed, which in this case is ten. Then desire him to count the sum in an order contrary to that of the natural numbers, beginning at the card he touched, and assigning it the number of the card he thought of. By counting in this manner, he will end at the number or card he thought of, and consequently you will immediately know it.
Thus, for example, suppose the person had thought of 3 C, and touched 6 F; then, if 10 be added to 6, the sum will be 16; and if that number be counted from F, the number touched, towards E D B C A, and so on, in the retrograde order, counting F three, the number thought of, E five, D six, and so round to sixteen, that number will terminate at C, showing that the person thought of 3, the number which corresponds to C.
A greater or less number of cards or counters may be employed at pleasure; but in every instance the whole number of cards must be added to the number of the card touched.
THE TWO TRAVELERS.
Two travelers trudged along the road together, Talking, as Yankees do, about the weather; When, lo! beside their path the foremost spies Three casks, and loud exclaims "A prize, a prize!" One large, two small, but all of various size. This way and that they gazed, and all around, Each wondering if an owner might be found: But not a soul was there--the coast was clear, So to the barrels they at once drew near, And both agree whatever may be there In friendly partnership they'll fairly share. Two they find empty, but the other full, And straightway from his pocket one doth pull A large clasp-knife. A heavy stone lay handy, And thus in time they found their prize was brandy. 'Tis tasted and approved: their lips they smack, And each pronounces 'tis the famed Cognac. "Won't we have many a jolly night, my boy! May no ill luck our present hopes destroy!" 'Twas fortunate one knew the mathematics, And had a smattering of hydrostatics; Then measured he the casks, and said, "I see This is eight gallons, those are five and three." The question then was how they might divide The brandy, so that each should be supplied With just four gallons, neither less nor more. With eight, and five, and three they puzzle sore, Filled up the five--filled up the three, in vain; At length a happy thought came o'er the brain Of one: 'twas done, and each went home content, And their good dames declared 'twas excellent. With those three casks they made division true; I found the puzzle out, say, friend, can you?
The five-gallon barrel was filled first, and from that the three-gallon barrel, thus leaving two gallons in the five-gallon barrel; the three-gallon barrel was then emptied into the eight-gallon barrel, and the two gallons poured from the five-gallon barrel into the empty three-gallon barrel; the five-gallon barrel was then filled, and one gallon poured into the three-gallon barrel, therefore leaving four gallons in the five-gallon barrel, one gallon in the eight-gallon barrel, and three gallons in the three-gallon barrel, which was then emptied into the eight-gallon barrel. Thus each person had four gallons of brandy in the eight and five-gallon barrels respectively.
ARITHMETICAL PUZZLE.
If from 6 you take 9, and from 9 you take 10; and if 50 from 40 be taken, there will just half a dozen remain.
ANSWER.
From SIX From IX From XL Take IX Take X Take L --- -- -- S I X Remains.
THE MONEY GAME.
A person having in one hand a piece of gold, and in the other a piece of silver, you may tell in which hand he has the gold, and in which the silver, by the following method: Some value, represented by an even number, such as 8, must be assigned to the gold; and a value represented by an odd number, such as three, must be assigned to the silver; after which, desire the person to multiply the number in the right hand by any even number whatever, such as 2, and that in the left by an odd number, as 3; then bid him add together the two products, and if the whole sum be odd, the gold will be in the right hand, and the silver in the left; if the sum be even, the contrary will be the case.
To conceal the artifice better, it will be sufficient to ask whether the sum of the two products can be halved without a remainder; for in that case the total will be even, and in the contrary case odd.
It may be readily seen, that the pieces, instead of being in the two hands of the same person, may be supposed to be in the hands of two persons, one of whom has the even number, or piece of gold, and the other the odd number, or piece of silver. The same operations may then be performed in regard to these two persons, as are performed in regard to the two hands of the same person, calling the one privately the right, and the other the left.
THE PHILOSOPHER'S PUPILS.
To find a number of which the half, fourth, and seventh added to three shall be equal to itself.
This was a favorite problem among the ancient Grecian arithmeticians, who stated the question in the following manner: "Tell us, illustrious Pythagoras, how many pupils frequent thy school?" "One half," replied the philosopher, "study mathematics, one fourth natural philosophy, one seventh observe silence, and there are three females besides."
The answer is, 28: 14 + 7 + 4 + 3 = 28.
TO DISCOVER A SQUARE NUMBER.
A square number is a number produced by the multiplication of any number into itself; thus, 4 multiplied by 4 is equal to 16, and 16 is consequently a square number, 4 being the square root from which it springs. The extraction of the square root of any number takes some time; and after all your labor you may perhaps find that the number is not a square number. To save this trouble, it is worth knowing that every square number ends either with a 1, 4, 5, 6, or 9, or with two cyphers, preceded by one of these numbers.
Another property of a square number is, that if it be divided by 4, the remainder, if any, will be 1--thus, the square of 5 is 25, and 25 divided by 4 leaves a remainder of 1; and again, 16, being a square number, can be divided by 4 without leaving a remainder.
THE SHEEP-FOLD.
A farmer had a pen made of 50 hurdles, capable of holding 100 sheep only; supposing he wanted to make it sufficiently large to hold double that number, how many additional hurdles would he have occasion for?
_Answer._--Two. There were 24 hurdles on each side of the pen; a hurdle at the top, and another at the bottom; so that, by moving one of the sides a little back, and placing an additional hurdle at the top and bottom, the size of the pen would be exactly doubled.
COUNTRYWOMAN AND EGGS.
A countrywoman carried eggs to a garrison, where she had three guards to pass, sold to the first guard half the number she had, and half an egg more; to the second, the half of what remained, and half an egg besides; and to the third guard she sold the half of the remainder, and half another egg. When she arrived at the market-place, she had three dozen still to sell; how was this possible, without breaking any of the eggs? It would seem at the first view that this is impossible, for how can half an egg be sold without breaking any of the eggs? The possibility of this seeming impossibility will be evident, when it is considered, that by taking the greater half of an odd number, we take the exact half + 1/2. When the countrywoman passed the first guard, she had 295 eggs; by selling to that guard 148, which is the half + 1/2, she had 147 remaining; to the second guard she disposed of 74, which is the major half of 147; and, of of course, after selling 37 out of 74 to the last guard, she had still three dozen remaining.
HOW TO RUB OUT TWENTY CHALKS AT FIVE TIMES, RUBBING OUT EVERY TIME AN ODD ONE.
To do this trick, you must make twenty chalks, or long strokes, upon a board, as in the margin:
1 ---- 2 ---- 3 ---- 4 ---- 5 ---- 6 ---- 7 ---- 8 ---- 9 ---- 10 ---- 11 ---- 12 ---- 13 ---- 14 ---- 15 ---- 16 ---- 17 ---- 18 ---- 19 ---- 20 ----
Then begin and count backwards, as 20, 19, 18, 17, rub out these four; then proceed saying, 16, 15, 14, 13, rub out these four; and begin again, 12, 11, 10, 9, and rub out these; and proceed again, 8, 7, 6, 5, then rub out these; and lastly say, 4, 3, 2, 1, when these four are rubbed out. The whole twenty are rubbed out at five times, and every time an odd one, that is, 17th, 13th, 9th, 5th, and 1st.
This is a trick which, if once seen, may be easily retained; and the puzzle at first is, it not occurring immediately to the mind to begin to rub them out backwards. It is as simple as any thing possibly can be.
THE IMPOSSIBLE TRIANGLE.
The longest side of a triangle is 100 rods; and each of the other sides 50. Required the value of the grass at $5 per acre.
This is a catch question, as a triangle cannot be formed unless any two of the lines are longer than the third.
ODD OR EVEN.
Every odd number multiplied by an odd number produces an odd number; every odd number multiplied by an even number produces an even number; and every even number multiplied by an even number also produces an even number. So, again, an even number added to an even number, and an odd number added to an odd number, produce an even number; while an odd and even number added together produce an odd number.
If any one holds an odd number of counters in one hand, and an even number in the other, it is not difficult to discover in which hand the odd or even number is. Desire the party to multiply the number in the right hand by an even number, and that in the left hand by an odd number, then to add the two sums together, and tell you the last figure of the product; if it is even, the odd number will be in the right hand; and if odd, in the left hand; thus, supposing there are 5 counters in the right hand, and 4 in the left hand, multiply 5 by 2, and 4 by 3, thus: 5 × 2 = 10, 4 × 3 = 12, and then adding 10 to 12, you have 10 + 12 = 22, the last figure of which, 2, is even, and the odd number will consequently be in the right hand.
THE FIGURES, UP TO 100, ARRANGED SO AS TO MAKE 505 IN EACH COLUMN, WHEN COUNTED IN TEN COLUMNS PERPENDICULARLY, AND THE SAME WHEN COUNTED IN TEN FILES HORIZONTALLY.
+-----+----+----+----+----+----+----+----+----+----+ | 10 | 92 | 93 | 7 | 5 | 96 | 4 | 98 | 99 | 1 | | 11 | 19 | 18 | 84 | 85 | 86 | 87 | 13 | 12 | 90 | | 71 | 29 | 28 | 77 | 76 | 75 | 24 | 23 | 22 | 80 | | 70 | 62 | 63 | 37 | 36 | 35 | 34 | 68 | 69 | 31 | | 41 | 52 | 53 | 44 | 46 | 45 | 47 | 58 | 59 | 60 | | 51 | 42 | 43 | 54 | 56 | 55 | 57 | 48 | 49 | 50 | | 40 | 32 | 33 | 67 | 65 | 66 | 64 | 38 | 39 | 61 | | 30 | 79 | 78 | 27 | 26 | 25 | 74 | 73 | 72 | 21 | | 81 | 89 | 88 | 14 | 15 | 16 | 17 | 83 | 82 | 20 | | 100 | 9 | 8 | 94 | 95 | 6 | 97 | 3 | 2 | 91 | +-----+----+----+----+----+----+----+----+----+----+
[Sidenote: Each of these files, when added up, makes 505.]
Each of these ten columns, when added up, makes 505.
THE OLD WOMAN AND HER EGGS.
At a time when eggs were scarce, an old woman who possessed some remarkably good-laying hens, wishing to oblige her neighbors, sent her daughter round with a basket of eggs to three of them; at the first house, which was the squire's, she left half the number of eggs she had and half a one over; at the second she left half of what remained and half an egg over; and at the third she again left half of the remainder, and half a one over; she returned with one egg in her basket, not having broken any. Required--the number she set out with. _Ans._ 15 eggs.
THE MATHEMATICAL FORTUNE TELLER.
Procure six cards, and having ruled them the same as the following diagrams, write in the figures neatly and legibly.
It is required to tell the number thought by any person, the numbers being contained in the cards, and such numbers not to exceed 60. How is this done?
+----+----+----+----+----+----+ +----+----+----+----+----+----+ | 3 | 5 | 7 | 9 | 11 | 1 | | 5 | 6 | 7 | 13 | 12 | 4 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 13 | 15 | 17 | 19 | 21 | 23 | | 14 | 15 | 20 | 21 |22 | 23 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 25 | 27 | 29 | 31 | 33 | 35 | | 28 | 29 | 30 | 31 | 36 | 37 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 37 | 39 | 41 | 43 | 45 | 47 | | 52 | 38 | 39 | 44 | 45 | 46 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 49 | 51 | 53 | 55 | 57 | 59 | | 47 | 53 | 54 | 55 | 60 | 13 | +----+----+----+----+----+----+ +----+----+----+----+----+----+
+----+----+----+----+----+----+ +----+----+----+----+----+----+ | 9 | 10 | 11 | 12 | 13 | 8 | | 3 | 6 | 7 | 10 | 11 | 2 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 14 | 15 | 24 | 25 | 26 | 27 | | 14 | 15 | 18 | 19 | 22 | 23 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 28 | 29 | 30 | 31 | 40 | 41 | | 26 | 27 | 30 | 31 | 34 | 35 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 42 | 43 | 44 | 45 | 46 | 47 | | 38 | 39 | 42 | 43 | 46 | 47 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 56 | 57 | 58 | 59 | 60 | 13 | | 50 | 51 | 54 | 55 | 58 | 59 | +----+----+----+----+----+----+ +----+----+----+----+----+----+
+----+----+----+----+----+----+ +----+----+----+----+----+----+ | 17 | 18 | 19 | 20 | 21 | 16 | | 33 | 34 | 35 | 36 | 37 | 32 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 22 | 23 | 24 | 25 | 26 | 27 | | 38 | 39 | 40 | 41 | 42 | 43 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 28 | 29 | 30 | 31 | 48 | 49 | | 44 | 45 | 46 | 47 | 48 | 49 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 50 | 51 | 52 | 53 | 54 | 55 | | 50 | 51 | 52 | 53 | 54 | 55 | +----+----+----+----+----+----+ +----+----+----+----+----+----+ | 56 | 57 | 58 | 59 | 30 | 60 | | 56 | 57 | 58 | 59 | 60 | 41 | +----+----+----+----+----+----+ +----+----+----+----+----+----+
Request the person to give you the cards containing the number, and then add the right hand upper corner figures together, which will give the correct answer. For example: suppose 10 is the number thought of, the cards with 2 and 8 in the corners will be given, which makes the answer 10, and so on with the others.
THE DICE GUESSED UNSEEN.
A pair of dice being thrown, to find the number of points on each die without seeing them. Tell the person who cast the dice to double the number of points upon one of them, and add 5 to it; then to multiply the sum produced by 5, and to add to the product the number of points upon the other die. This being done, desire him to tell you the amount, and, having thrown out 25, the remainder will be a number consisting of two figures, the first of which, to the left, is the number of points on the first die, and the second figure, to the right, the number on the other. Thus:
Suppose the number of points of the first die which comes up to be 2, and that of the other 3; then, if to four, the double of the points of the first, there be added 5, and the sum produced, 9, be multiplied by 5, the product will be 45; to which, if 3, the number of points on the other die, be added, 48 will be produced, from which, if 25 be subtracted, 23 will remain; the first figure of which is 2, the number of points on the first die, and the second figure 3, the number on the other.
THE SOVEREIGN AND THE SAGE.
A sovereign being desirous to confer a liberal reward on one of his courtiers, who had performed some very important service, desired him to ask whatever he thought proper, assuring him it should be granted. The courtier, who was well acquainted with the science of numbers, only requested that the monarch would give him a quantity of wheat equal to that which would arise from one grain doubled sixty-three times successively. The value of the reward was immense; for it will be found by calculation that the sixty-fourth term of the double progression divided by 1, 2, 4, 8, 16, 32, &c., is 9223372036854775808. But the sum of all the terms of a double progression, beginning with 1, may be obtained by doubling the last term, and subtracting from it 1. The number of the grains of wheat, therefore, in the present case, will be 18446744073709551615. Now, if a pint contain 9216 grains of wheat, a gallon will contain 73728; and, as eight gallons make one bushel, if we divide the above result by eight times 73728 we shall have 31274997411295 for the number of the bushels of wheat equal to the above number of grains, a quantity greater than what the whole surface of the earth could produce in several years, and which in value would exceed all the riches, perhaps, on the globe.
THE KNOWING SHEPHERD.
A shepherd was going to market with some sheep, when he met a man who said to him, "Good morning, friend, with your score." "No," said the shepherd, "I have not a score; but if I had as many more, half as many more, and two sheep and a half, I should have just a score." How many sheep had he?
He had 7 sheep: as many more 7; half as many more, 3½; and 2½; making in all 20.
THE CERTAIN GAME.
Two persons agree to take, alternately, numbers less than a given number, for example, 11, and to add them together till one of them has reached a certain sum, such as 100. By what means can one of them infallibly attain to that number before the other?
The whole artifice in this consists in immediately making choice of the numbers 1, 12, 23, 34, and so on, or of a series which continually increases by 11, up to 100. Let us suppose that the first person, who knows the game, makes choice of 1; it is evident that his adversary, as he must count less than 11, can at most reach 11, by adding 10 to it. The first will then take 1, which will make 12; and whatever number the second may add the first will certainly win, provided he continually add the number which forms the complement of that of his adversary to 11; that is to say, if the latter take 8, he must take 3; if 9, he must take 2; and so on. By following this method he will infallibly attain to 89, and it will then be impossible for the second to prevent him from getting first to 100; for whatever number the second takes he can attain only to 99; after which the first may say--"and 1 makes 100." If the second take 1 after 89, it would make 90, and his adversary would finish by saying--"and 10 make 100." Between two persons who are equally acquainted with the game, he who begins must necessarily win.
THE ASTONISHED FARMER.
A and B took each 30 pigs to market, A sold his at 3 for a dollar, B at 2 for a dollar, and together they received $25. A afterwards took 60 alone, which he sold _as before_, at 5 for $2, and received, but $24; what became of the other dollar?
This is rather a catch question, the insinuation that the first lot were sold at the rate of five for $2, being only true in part. They commence selling at that rate, but after making ten sales, A's pigs are exhausted, and they have received $20: B still has 10 which he sells at "2 for a dollar" and of course receives $5; whereas had he sold them at the rate of 5 for $2, he would have received but $4. Hence the difficulty is easily settled.
MAGICAL CENTURY.
If the number 11 be multiplied by any one of the nine digits, the two figures of the product will always be alike, as appears in the following example:--
11 11 11 11 11 11 11 11 11 1 2 3 4 5 6 7 8 9 -- -- -- -- -- -- -- -- -- 11 22 33 44 55 66 77 88 99 -- -- -- -- -- -- -- -- --
Now, if another person and yourself have fifty counters a-piece, and agree never to stake more than ten at a time, you may tell him that if he permit you to stake first, you always complete the even century before him.
In order to succeed, you must first stake 1, and remembering the order of the above series, constantly add to what he stakes as many as will make one more than the numbers 11, 22, 33, &c., of which it is composed, till you come to 89, after which your opponent cannot possibly reach the even century himself, or prevent you from reaching it.
If your opponent has no knowledge of numbers, you may stake any other number first, under 10, provided you subsequently take care to secure one of the last terms, 56, 67, 78, &c.; or you may even let him stake first, if you take care afterward to secure one of these numbers.
This exercise may be performed with other numbers; but, in order to succeed, you must divide the number to be attained by a number which is a unit greater than what you can stake each time, and the remainder will then be the number you must first stake. Suppose, for example, the number to be attained be 52 (making use of a pack of cards instead of counters), and that you are never to add more than 6; then, dividing 52 by 7, the remainder, which is 3, will be the number which you must first stake; and whatever your opponent stakes, you must add as much to it as will make it equal to 7, the number by which you divided, and so in continuation.
THE UNLUCKY HATTER.
A blackleg passing through a town in Ohio, bought a hat for $8 and gave in payment a $50 bill. The hatter called on a merchant near by, who changed the note for him, and the blackleg having received his $42 change went his way. The next day the merchant discovered the note to be a counterfeit, and called upon the hatter, who was compelled forthwith to borrow $50 of another friend to redeem it with; but on turning to search for the blackleg he had left town, so that the note was useless on the hatter's hands. The question is, what did he lose--was it $50 besides the hat, or was it $50 including the hat?
This question is generally given with names and circumstances as a real transaction, and if the company knows such persons so much the better, as it serves to withdraw attention from the question; and in almost every case the first impression is, that the hatter lost $50 besides the hat, though it is evident he was paid for the hat, and had he kept the $8 he needed only to have borrowed $42 additional to redeem the note.
THE BASKET OF NUTS.
A person remarked that when he counted over his basket of nuts, two by two, three by three, four by four, five by five, or six by six, there was one remaining; but when he counted them by sevens, there was no remainder. How many had he?
The least common multiple of 2, 3, 4, 5, and 6 being 60, it is evident, that if 61 were divisible by 7, it would answer the conditions of the question. This not being the case, however, let 60 × 2 + 1, 60 × 3 + 1, 60 × 4 + 1, &c., be tried successively, and it will be found that 301 = 60 × 5 + 1, is divisible by 7; and consequently this number answers the conditions of the question. If to this we add 420, the least common multiple of 2, 3, 4, 5, 6 and 7, the sum 721 will be another answer; and by adding perpetually 420, we may find as many answers as we please.
THE UNITED DIGITS.
Arrange the figures 1 to 9 in such order that, by adding them together, they amount to 100.
15 36 47 -- 98 2 --- 100
DECEMBER AND MAY.
An old man married a young woman; their united ages amounted to C. The man's age multiplied by 4 and divided by 9, gives the woman's age. What were their respective ages?
ANSWER.--The man's age, 60 years 12 weeks; the woman's age, 30 years 40 weeks.
THE TWO DROVERS.
Two drovers, A and B, meeting on the road, began discoursing about the number of sheep they each had. Says B to A, "Pray give me one of your sheep and I will have as many as you." "Nay," replied A, "but give me one of your sheep and I will have as many again as you." Required to know the number of sheep they each had?
A had seven and B had five sheep.
THE BASKET AND STONES.
If a hundred stones be placed in a straight line, at the distance of a yard from each other, the first being at the same distance from a basket, how many yards must the person walk who engages to pick them up, one by one, and put them into the basket? It is evident that, to pick up the first stone, and put it into the basket, the person must walk two yards; for the second, he must walk four; for the third, six: and so on increasing by two, to the hundredth.
The number of yards, therefore, which the person must walk will be equal to the sum of the progression, 2, 4, 6, &c., the last term of which is 200 (22). But the sum of the progression is equal to 202, the sum of the two extremes, multiplied by 50, or half the number of terms: that is to say, 10,100 yards, which makes more than 5½ miles.
THE FAMOUS FORTY-FIVE.
How can number 45 be divided into four such parts that, if to the first part you add 2, from the second part you subtract 2, the third part you multiply by 2, and the fourth part you divide by 2, the sum of the addition, the remainder of the subtraction, the product of the multiplication, and the quotient of the division be all equal?
The 1st is 8; to which add 2, the sum is 10 The 2nd is 12; subtract 2, the remainder is 10 The 3rd is 5; multiplied by 2, the product is 10 The 4th is 20; divided by 2, the quotient is 10 -- 45
Required to subtract 45 from 45, and leave 45 as a remainder?
Solution.--9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 -------------------------------------- 8 + 6 + 4 + 1 + 9 + 7 + 5 + 3 + 2 = 45
SUBTRACTION.
From 1 mile subtract 7 furlongs, 39 rods, 5 yards, 1 foot, 5 inches.
miles, furlongs, rods, yards, feet, inches. From 1 0 0 0 0 0 Take 0 7 39 5 1 5 -------------------------------------- 0 0 0 0 0 1
In this problem, instead of borrowing 1 foot, we borrow ½ a foot = 6 inches, from which we take 5 inches, and 1 remains; we then carry ½ to 1, and borrowing ½ a yard = 1½ feet, we have 1½ from 1½ = 0, and afterwards proceed as usual.
THE EXPUNGED FIGURE.
In the first place desire a person to write down secretly, in a line, any number of figures he may choose, and add them together as units; having done this, tell him to subtract that sum from the line of figures originally set down; then desire him to strike out any figure he pleases, and add the remaining figures in the line together as units, (as in the first instance,) and inform you of the result, when you will tell him the figure he has struck out.
76542-24 24 ------ 76518
Suppose, for example, the figures put down are 76542; these, added together, as units, make a total of 24: deduct 24 from the first line, and 76518 remain; if 5, the center figure be struck out, the total will be 22. If 8, the first figure be struck out, 19 will be the total.
In order to ascertain which figure has been struck out, you make a mental sum one multiple of 9 higher than the total given. If 22 be given as the total, then 3 times 9 are 27, and 22 from 27 show that 5 was struck out. If 19 be given, that sum deducted from 27 shows 8.
Should the total be equal multiplies of 9, as 18, 27, 36, then 9 has been expunged.
With very little practice any person may perform this with rapidity, it is therefore needless to give any further examples. The only way in which a person can fail in solving this riddle is, when either the number 9 or a cipher is struck out, as it then becomes impossible to tell which of the two it is, the sum of the figure in the line being an even number of nines in both cases.
THE MYSTERIOUS ADDITION.
It is required to name the quotient of five or three lines of figures--each line consisting of five or more figures--only seeing the first line before the other lines are even put down. Any person may write down the first line of figures for you. How do you find the quotient?
EXAMPLE.--When the first line of figures is set down, subtract 2 from the last right-hand figure, and place it before the first figure of the line, and that is the quotient for five lines. For example, suppose the figures given are 86,214, the quotient will be 286,212. You may allow any person to put down the two first and the fourth lines, but you must always set down the third and fifth lines, and in doing so, always make up 9 with the line above, as in the following example:
Therefore in the annexed diagram you will see that you have made 9 in the third and fifth lines with the lines above them. If the person desire to put down the figures should set down a 1 or 0 for the last figure, you must say we will have another figure, and another, and so on until he sets down something above 1 or 2.
86,214 42,680 57,319 62,854 37,145 ------ Qt. 268,212
67,856 47,218 52,781 ------ Qt. 167,855
In solving the puzzle with three lines, you subtract 1 from the last figure, and place it before the first figure, and make up the third line yourself to 9. For example: 67,856 is given, and the quotient will be 167,855, as shown in the above diagram.
TO TELL AT WHAT HOUR A PERSON INTENDS TO RISE.
Let the person set the hand of the dial of a watch at any hour he pleases, and tell you what hour that is; and to the number of that hour you add in your mind 12; then tell him to count privately the number of that amount upon the dial, beginning with the next hour to that on which he proposes to rise, and counting backwards, first reckoning the number of the hour at which he has placed the hand. For example:
Suppose the hour at which he intends to rise be 8, and that he has placed the hand at 5; you will add 12 to 5, and tell him to count 17 on the dial, first reckoning 5, the hour at which the index stands, and counting backwards from the hour at which he intends to rise; and the number 17 will necessarily end at 8, which shows that to be the hour he chose.
TO FIND THE DIFFERENCE BETWEEN TWO NUMBERS, THE GREATEST OF WHICH IS UNKNOWN.
Take as many nines as there are figures in the smallest number, and subtract that sum from the number of nines. Let another person add the difference to the largest number, and talking away the first figure of the amount add it to the last figure, and that sum will be the difference of the two numbers.
For example: John, who is 22, tells Thomas, who is older, that he can discover the difference of their ages; he therefore privately deducts 22 from 99 (his age consisting of two figures, he of course takes two nines); the difference, which is 77, he tells Thomas to add to his age, and to take away the first figure from the amount, and add it to the last figure and that will be the difference of their ages; thus,
The difference between John's age and 99 is 77 To which Thomas adding his age 35 --- The sum is 112 Then by taking away the first figure 1, and adding it to the figure 2, the sum is 13 Which add to John's age 22 --- Gives the age of Thomas 35
THE REMAINDER.
A very pleasing way to arrive at an arithmetical sum, without the use of either slate or pencil, is to ask a person to think of a figure, then to double it, then add a certain figure to it, now halve the whole sum, and finally to subtract from that the figure first thought of. You are then to tell the thinker what is the remainder.
The key to this lock of figures is, that HALF of whatever sum you request to be added during the working of the sum is THE REMAINDER. In the example given, five is the half of ten, the number requested to be added. Any amount may be added, but the operation is simplified by giving only even numbers, as they will divide without fractions.
_Example._
Think of 7 Double it 14 Add 10 to it 10 --- Halve it 2)24 --- Which will leave 12 Subtract the number thought of 7 --- THE REMAINDER will be 5
A PERSON HAVING AN EQUAL NUMBER OF COUNTERS, OR PIECES OF MONEY, IN EACH HAND, TO FIND HOW MANY HE HAS ALTOGETHER.
Request the person to convey any number, as 4, for example, from the one hand to the other, and then ask how many times the less number is contained in the greater. Let us suppose that he says the one is the triple of the other; and, in this case, multiply 4, the number of the counters conveyed, by 3, and add to the product the same number, which will make 16. Lastly, take 1 from 3, and if 16 be divided by the remainder 2, the quotient will be the number contained in each hand, and consequently the whole number is 16.
This curious problem deserves another example. Let us again suppose that 4 counters are passed from one hand to the other, and the less number is contained in the greater 2⅓ times. In this case, we must, as before, multiply 4 by 2⅓, which will give 9⅓; to which, if 4 be added, we shall have 13⅓, or 40/3; if 1, then, be taken from 2⅓, the remainder will be 1⅓, or 4/3, by which, if 40/3 be divided, the quotient 10 will be the number of counters in each hand.
THE THREE JEALOUS HUSBANDS.
Three jealous husbands, A, B, and C, with their wives, being ready to pass by night over a river, find at the water side a boat which can carry but two at a time, and for want of a waterman they are compelled to row themselves over the river at several times. The question is how those six persons shall pass, two at a time, so that none of the three wives may be found in the company of one or two men, unless her husband be present?
This may be effected in two or three ways; the following may be as good as any: Let A and wife go over--let A return--let B's and C's wives go over--A's wife returns--B and C go over--B and wife return, A and B go over--C's wife returns, and A's and B's wives go over--then C comes back for his wife. Simple as this question may appear, it is found in the works of Alcuin, who flourished a thousand years ago, hundreds of years before the art of printing was invented.
THE FALSE SCALES.
A cheese being put into one of the scales of a false balance, was found to weigh 16 lbs., and when put into the other only 9 lbs. What is the true weight?
The true weight is a mean proportional between the two false ones, and is found by extracting the square root of their product. Thus 16 × 9 = 144; and square root 144 = 12 lbs., the weight required.
THE APPLE WOMAN.
A poor woman, carrying a basket of apples, was met by three boys, the first of whom bought half of what she had, and then gave her back 10; the second boy bought a third of what remained, and gave her back 2; and the third bought half of what she had now left, and returned her 1; after which she found she had 12 apples remaining. What number had she at first?
From the 12 remaining, deduct 1, and 11 is the number she sold the last boy, which was half she had; her number at that time, therefore, was 22. From 22 deduct two, and the remaining 20 was 2/3 of her prior stock, which was therefore 30. From 30 deduct 10, and the remainder 20 is half her original stock; consequently she had at first 40 apples.
THE GRACES AND MUSES.
The three Graces, carrying each an equal number of oranges, were met by the nine Muses, who asked for some of them; and each Grace having given to each Muse the same number, it was then found that they had all equal shares. How many had the Graces at first?
The least number that will answer this question is twelve; for if we suppose that each Grace gave one to each Muse, the latter would each have three, and there would remain three for each Grace. (Any multiple of 12 will answer the conditions of the question.)
THE JESUITICAL TEACHER.
A teacher, having fifteen young ladies under her care, wished them to take a walk each day of the week. They were to walk in five divisions of three ladies each, but no two ladies were to be allowed to walk together twice during the week. How could they be arranged to suit the above conditions?
SUN. MON. TUES. WEDN. THURS. FRID. SAT. a b c|a d g|a k n|a e l|a h o|a f p|a i m d e f|b e h|b l o|b f m|b i p|b d n|b g k g h i|c m p|c f i|c g n|c d k|c h l|c e o k l m|f k o|d h m|d i o|e m n|e i k|d l p n o p|i l n|e g p|h k p|f g l|g m o|h f n
QUAINT QUESTIONS.
What is the difference between twenty four quart bottles, and four and twenty quart bottles?
_Ans._--56 quarts difference.
What three figures, multiplied by 4, will make precisely 5?
_Ans._--1¼, or 1·25.
What is the difference between six dozen dozen, and half-a-dozen dozen?
_Ans._--792: Six dozen dozen being 864, and half-a-dozen dozen, 72.
Place three sixes together, so as to make seven.
_Ans._--6⁶/₆.
Add one to nine and make it twenty.
_Ans._ IX--cross the _1_, it makes XX.
Place four fives so as to make six and a half. _Ans._--5⁵/₅ ·5.
A room with eight corners had a cat in each corner, seven cats before each cat, and a cat on every cat's tail. What was the total number of cats?
_Ans._--Eight cats.
Prove that seven is the half of twelve. _Ans._--Place the Roman figures on a piece of paper, and draw a line through the middle of it, the upper will be VII.
THE FOX, GOOSE AND CORN.
A countryman having a Fox, a Goose, and a peck of Corn, came to a river, where it so happened that he could carry but one over at a time. Now as no two were to be left together that might destroy each other, he was at his wit's end, for says he "Though the corn can't eat the goose, nor the goose eat the fox; yet the fox can eat the goose, and the goose eat the corn." How shall he carry them over, that they shall not destroy each other?
Let him first take over the Goose, leaving the Fox and Corn; then let him take over the Fox and bring the Goose back; then take over the Corn; and lastly take over the Goose again.
MULTIPLYING MONEY BY MONEY.
Amongst the various questions that are given for the purpose of puzzling the unwary arithmetician, the multiplication of money by money is one of the most curious: take for instance the following problems:
Multiply £99 19_s._ 11-3/4_d._ by £99 19_s._ 11-3/4_d._ Multiply £11 11_s._ 11_d._ by £11 11_s._ 11_d._
To the uninitiated they usually appear easy of solution but the various modes of working them out, and the different results obtained, prove that there is something absurd and wrong in the questions themselves. Some reduce all to farthings, and after multiplying one term by the other, return the product into pounds, shillings, and pence. Others convert them into decimals; whilst some work the problem in the style of duodecimals.
Having sufficiently puzzled the tyros, the querist remarks: "The problem itself is absurd, it is incapable of solution; for what is the nature of the product of pounds, shillings, and pence multiplied by pounds, shillings and pence? We know that a yard multiplied by a yard is a square yard, but who can tell what is a penny multiplied by a penny, or a penny by a pound?"
Now all this is quite correct, provided the question is limited, as above to the product of pounds, shillings, and pence, into pounds, shillings, and pence; but suppose the problem were put in this form--If a capital of £1 produces by compound interest, in a certain time, £99 19_s._ 11¾_d._, how much would be produced by a capital of £99 19_s._ 11¾_d.?_ It is evident that, to answer this, we must multiply £99 19_s._ 11¾_d._ by £99 19_s._ 11¾_d._: these are in fact the second and, third terms of an ordinary "rule of three;" and though one of the terms is a "concrete" quantity of pounds, shillings, and pence, the other must be regarded as an "abstract" mathematical quantity, being 99 and a fraction, of which the number of farthings in a pound is the denominator 960, and the number of farthings in the third term is the numerator, 959; or, instead of this, the shillings and pence might be converted into decimals of a pound, or into aliquot parts. The product of multiplying £99 19_s._ 11¾_d._ by 99-959/960 is £9,999 15_s._ 10 1/3840_d._; the quickest way of doing this, is to multiply by 100, and to subtract from the product the 960th part of the multiplicand.
In the other question proposed, the product of £11 11_s._ 11_d._ into £11 11_s._ 11_d._, or 11 143/240, is £134 9_s._ 3 49/240_d._
Number and value are distinct abstract ideas, and cannot, without committing a logical absurdity, be confused. To multiply is to repeat a certain number of _times_, and it is obviously impossible to bring _value_ into the question. Value is arbitrary; number is fixed. Put it in this way, and the absurdity is evident: One pound is equivalent to 20 shillings, or 240 pence, or 960 farthings. In value there is no difference whatever; but what an enormous difference between multiplying by 1, 20, 240, or 960!
THE UNFAIR DIVISION.
A gentleman rented a farm, and contracted to give to his landlord 2/5 of the produce; but prior to the time of dividing the corn, the tenant used 45 bushels. When the general division was made, it was proposed to give to the landlord 18 bushels from the heap, in lieu of his share of the 45 bushels which the tenant had used, and then to begin and divide the remainder as though none had been used. Would this method have been correct?
The landlord would lose 7⅕ bushels by such an arrangement, as the rent would entitle him to ⅖ of the 18. The tenant should give him 18 bushels from his own share after the division is completed, otherwise the landlord would receive but 2/7 of the first 63 bushels.
A POPULAR FALLACY.
It is often suggested from the pulpit and elsewhere, that enough persons have lived and died in the world to cover its whole surface with bodies; and even two or three strata deep. Is this probable?
Say the earth has existed 6000 years, the population always having been 800,000,000, and the average life of man 30 years; this being the utmost that could be claimed. Allow then the State of Virginia to contain 70,000 square miles, and each grave to occupy a space of 6 feet by 2; the territory of the State would contain 162,624,000,000; while the mighty army of the dead would number only 160,000,000,000; leaving 2,624,000,000 graves yet unoccupied. How wide of truth then is the position often set forth so positively!
FOOTNOTES:
[Footnote 10: Ancestor of the fighting and writing Napiers of our day.]
[Footnote 11: When an _exact half_ cannot be taken without a fraction, he must take the _larger half_--you must tell him this before he commences. Here it is the _larger half_.]
[Footnote 12: From Parkes' Philosophy of Arithmetic, a capital work published by Moss & Bro. Philadelphia.]
TRICKS IN GEOMETRY.
"Let young beginners come and try Their hands at our geometry."
The word Geometry is derived from the Greek, and signifies the art of measuring land. The invention of it is ascribed by some to the Chaldeans and Babylonians, by others to the Egyptians, who were obliged to determine the boundaries of their fields after the inundation of the Nile, by geometrical measurements. According to Cassiodorus, the Egyptians either derived the art from the Babylonians, or invented it after it was known to them. Thales, a Phœnician, who died 548 years B.C., and Pythagoras of Samos, who flourished about 520 B.C., introduced it from Egypt into Greece. In elementary geometry, Euclid of Alexandria, as everybody knows, is particularly distinguished. Archimedes measured the sphere, and after him other philosophers prosecuted the science with the utmost assiduity. In Italy, where the sciences first revived after the dark ages, several mathematicians were distinguished in the 16th century. The French, and after them the Germans, followed; while in England, Hook, Newton, and others, carried the science to the highest pitch of usefulness, and through its aid made the most prodigious discoveries. It is not, however, our province to enter into a long disquisition on the subject, but simply to set before the young reader some of the more curious properties of the science, that he may be excited to study it for himself; and we will promise him that should he devote his mind to its study, he will be amply repaid for any amount of labor he may bestow upon it.
GEOMETRICAL DEFINITIONS.
In geometry a _point_ is said to have neither breadth, length, nor thickness. A _line_ is the distance between two points; parallel lines always keep at the same distance from each other. A _right_ line is what is commonly called a straight line. A _curve_ is a line which continually changes its direction. An _angle_ is the inclination or opening of two lines meeting in a point. A _figure_ is a bounded space, and is either a superficies or a solid. A _triangle_ is a figure with three sides and three angles. A _square_ has four equal sides, and four right angles. A _circle_ is a plane figure bounded by a curved line running into itself. Its diameter is a straight line drawn from one extremity of its circumference to the other, and its center is equally distant from every part of the circumference. A _solid_ is any body which has length, breadth, and thickness; and a sphere is a solid, terminated by a convex surface, every part of which is at an equal distance from a point within, called its center.
THE FIVE GEOMETRICAL SOLIDS.
[Illustration: _Fig. 1._]
[Illustration: _Fig. 2._]
[Illustration: _Fig. 3._]
[Illustration: _Fig. 4._]
The following figures will show how the five geometrical solids may be cut out of a piece of cardboard. Where the lines are drawn the board is to be partly cut through with a penknife, so as to render the angles of the models as sharp and as straight as possible. The edges which require joining are to be fastened together with a slip of thin paper and gum dissolved in just sufficient water to bring it to the consistence of treacle. Fig. 1 will form a tetrahedron, a figure with four sides, each shaped like an equilateral triangle. Fig. 2 forms a cube or hexahedron. Fig. 3 an octohedron, with eight triangular sides. Fig. 4, a dodecahedron, with twelve sides shaped like pentagons, with five equal sides. Fig. 5, an isocahedron, with twenty sides, formed of equilateral triangles.
[Illustration: _Fig. 5._]
HOW TO MAKE FIVE SQUARES INTO A LARGE ONE WITHOUT ANY WASTE OF STUFF.
Suppose you have five squares of cloth, or anything else, as in Fig. 7; find the center of one side of four of these squares, and cut them from that point to the opposite corner, then place the perfect square in the centre, and the other pieces round, as seen in Fig. 8.
[Illustration: _Fig. 7._]
[Illustration: _Fig. 8._]
DECEPTIVE VISION.
[Illustration]
The following sleight shows how easily the eye may be deceived. Take a piece of pasteboard, an inch and a half in width, and five inches in length, and divide it by inked lines into thirty squares, then cut it from corner to corner, so as to form two triangles. After this cut off the top of these triangles at C and D,[13] and arrange the pieces in this manner:--
[Illustration]
On counting the squares in the first figure, there appear to be thirty, but the other arrangement of the same card seems to contain thirty-two. It does so, however, only in appearance, but it is only a very correct eye that can detect the imperfection.
THE CARPENTER PUZZLED.
[Illustration]
A carpenter having a piece of mahogany of a triangular form, (see Fig.) wished to know how he could make it up to the best advantage. His first idea was to make an oblong square table of it, but he found that if he did so the waste of the wood would be very great. After consideration he discovered that the most economical method of using the wood would be to form it into an oval. To make this oval contain as much wood as possible, he proceeded in the following manner: Let B G D be the triangular piece of wood; take G H one half of the base, and divide the triangle by drawing a line from H to B. Take G H in the compasses, and set it off on one of the sides from G to E, draw the line E F, and the point I will be the center of the oval; draw K L parallel to E F, and at the same distance from the center as the base G. The points A and C are found by dividing the line from E to K and drawing A C, or by drawing the dotted lines D A and G C through the center at I. These points being found, the oval must be completed by the eye of the draughtsman.
THE BRICKLAYER PUZZLED.
[Illustration: _Fig. 1._]
A bricklayer had to construct a wall, whose length in the direction A B C was twenty-four feet. The one half of this wall, namely from B to C, had to be built over a piece of rising ground, so that the base of this part of the wall would necessarily be more than twelve feet. In making out his account he charged more for this half of the wall than for that which was built on level ground from A to B. A geometrician assured him that the square contents of both portions of the wall were exactly alike; which may be proved in the following manner:--
[Illustration: _Fig. 2._]
[Illustration: _Fig. 3._]
[Illustration: _Fig. 4._]
Cut two pieces of cardboard, in the form shown in Figs. 2 and 3, to represent the two parts of the wall; lay the piece representing the straight wall on the curved piece, and it will be found that the angles which project at A and B will exactly fill up the spaces at E and F. The piece of board representing the straight wall may thus be found to be exactly sufficient to form a piece equal to that representing the curved wall. You may then lay the curved piece upon the straight one, and reversing the experiment prove that the curved piece is capable of forming a rectangular piece equal to the other.
TRIANGULAR PROBLEM.
Take four square pieces of pasteboard of the same dimensions, and divide them diagonally, that is, by drawing a line from two opposite angles, as in the figures, into eight triangles. Paint seven of these triangles with the prismatic colors, red, orange, yellow, green, blue, indigo, violet, and let the eighth be white. To find how many chequers or regular four-sided figures, different either in form or color, may be made out of these eight triangles.
[Illustration]
First, by combining two of these triangles there may be formed, either the triangular square A, or the inclined square B, called a Rhomb. Secondly, by combining four of the triangles the large square C may be formed, or the long square D, called a parallelogram. Now the first two squares, consisting of two parts out of eight, may each of them by the eighth rank of the triangle be taken twenty-eight different ways, which makes fifty-six. And the last two squares, consisting of four parts, may each be taken by the same rank of the triangle seventy times, which makes 140.
TO FORM A SQUARE.
Take a piece of card of the shape and size or proportions of the subjoined, and cut it into three parts, and with these three form a perfect square.
[Illustration]
To do this, cut it in the direction of the dotted lines, and it will then be easy to lay down the pieces to form a perfect square.
SQUARING THE CIRCLE.
"Squaring the circle," as it is called, is the puzzle of puzzles, and there are many persons who fancy this can be accomplished, as there are also many who believe that they can discover "perpetual motion."
The meaning of this phrase _squaring_ is scientifically expressed by the term finding the quadrature of the circle; that is, the act of producing a square equal to a given circle; and many persons but slightly acquainted with mathematics have puzzled their brains to effect this object. The Cardinal de Cusa rolled a cylinder over a plane, till the point which was first in contact with the plane touched it again; and then, by a train of reasoning very unmathematical, he endeavored to determine the length of the line thus described. Oliver de Serras worked a circle, and also a triangle equal to an equilateral triangle, inscribed within the circle, and imagined that the former was exactly equal to two of the latter, forgetting that the double of this triangle is equal to the hexagon inscribed within the circle, and therefore smaller than the circle itself. A Frenchman challenged the world, and deposited 10,000 livres as a stake, that he could accomplish the feat. He reduced the problem to the mechanical process of dividing a circle into four quarters, and then turning these with their angles outwards, so as to form a square, which he asserted to be equal to the circle; this however was soon proved to be ridiculous.
Some persons have taken a piece of pasteboard, and cutting it out into a circular form, and by cutting that circular disc into pieces of a square form and definite dimensions, and fitting the same turned pieces one into the other, have come _near_ to a notion of the superficial area of a circle. But this kind of demonstration is purely mechanical, and is neither geometrical nor scientific, and, is in fact, no demonstration according to mathematics. For if we take the pieces of card, however exactly they may appear to be formed, and examine them with a microscope, we shall soon find that none of them are geometrically true, nor of the same length or breadth, and therefore the conclusion arrived at is a false one.
[Illustration]
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.
[Illustration]
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.
[Illustration]
FOOTNOTES:
[Footnote 13: _I. e._, at the fourth square from the right angle.]
PRACTICAL PARADOXES AND PUZZLES.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration:
| | | | | | | | | | | | | | | | | | ]
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.
[Illustration:
+-----------+ | 3 Inches. | | | |6 | | | |I | |n | |c | |h | |e | |s | |. | | | +-----------+ ]
11. THE BUTTON PUZZLE.
[Illustration]
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.
[Illustration:
|\ /| | \ / | | \ / | | \ / | | \ / | | \ / | | \/ | | | | | | | +--------------+ ]
Divide this figure into four equal parts, each of the same figure.
13. THE PUZZLE OF FOURTEEN.
[Illustration]
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.
[Illustration:
+---------------+ |o o o| | | | | |o oo o| | oo | | | | | |o o o| +---------------+ ]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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:
[Illustration]
Suppose four evil-disposed rich men afterwards built houses around the poor people, thus:
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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--
[Illustration:
+----------------+ | | | | | | +------+ | | | | | | | | | | +------+---------+ ]
The other father divides his so as to reserve to himself one fourth in the form of a triangle; thus--
[Illustration:
+--------+ | | | | | /\ | | / \ | | / \ | |/ \| +--------+ ]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
38. THE DOUBLE-HEADED PUZZLE.
[Illustration]
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.
Bene A. Thin Thed Ustt HEMO. Uld yo L.D.C. RUSTO! Fnel L.B. Ach El Lor. Lat. ELY, Wa. S. shove N. W. How--Ass! kill'd I. N. T. H. Ear T. Sofp, I, Escu Star. D. San D T Art. San D K. N E. W. E Ver----Yus E.--Oft He ove N W. Hens He 'Dli V'DL. on geno Ug H S hem A.D.E. he R. la Stp. Uf----fap Uf. F. B Y he. R hu S. Ban D. M. Uch pra is 'D. No. Wheres Hedot HL. i. e. Tom. A kead I.R.T.P. Yein hop Esthathe R. C. RUSTWI, L L B. Era is '----D!
43. A CURIOUS LETTER.
Friends Sir, friends, stand your disposition; I bearing a man the world is whilst the contempt, ridicule. are ambitious.
44. A PUZZLING INSCRIPTION.
P R S V R Y P R F C T M N V R K P T H S P R C P T S T N.
The two lines above were affixed to the communion table of a small church in Wales, and continued to puzzle the learned congregation for several centuries, but at length the inscription was deciphered. What was it?
45. THE PUZZLING RINGS.
This perplexing invention is of great antiquity, and was treated on by Cardan, the mathematician, at the beginning of the sixteenth century. It consists of a flat piece of thin metal or bone, with ten holes in it; in each hole a wire is loosely fixed, beaten out into a head at one end, to prevent its slipping through, and the other fastened to a ring, also loose. Each wire has been passed through the ring of the next wire, previously to its own ring being fastened on; and through the whole of the rings runs a wire loop or bow, which also contains, within its oblong space, all the wires to which the rings are fastened; the whole presenting so complicated an appearance, as to make the releasing the rings from the bow appear an impossibility. The construction of it would be found rather troublesome to the amateur, but it may be purchased at most of the toy shops very lightly and elegantly made. It also exists in various parts of the country, forged in iron, perhaps by some ingenious village mechanic, and aptly named "The Tiring Irons." The following instructions will show the principle on which the puzzle is constructed, and will prove a key to its solution.
Take the loop in your left hand, holding it at the end, B, and consider the rings as being numbered 1st to 10th. The 1st will be the extreme ring to the right, and the 10th the nearest your left hand.
[Illustration]
It will be seen that the difficulty arises from each ring passing round the wire of its right-hand neighbor. The extreme ring at the right hand, of course, being unconnected with any other wire than its own, may at any time be drawn off the end of the bow at A, raised up, dropped through the bow, and finally released. After you have done this, try to pass the second ring in the same way, and you will not succeed, as it is obstructed by the wire of the first ring; but if you bring the first ring on again, by reversing the process by which you took it off, viz., by putting it up through the bow, and on to the end of it, you will then find that by taking the first and second rings together, they will both draw off, lift up, and drop through the bow. Having done this, try to pass the third ring off, and you will not be able; because it is fastened on one side to its own wire, which is within the bow, and on the other side to the second ring, which is without the bow. Therefore, leaving the third ring for the present, try the fourth ring, which is now at the end all but one, and both of the wires which affect it being within the bow, you will draw it off without obstruction; and, in doing this, you will have to slip the third ring off, which will not drop through for the reasons before given; so, having dropped the fourth ring through, you can only slip the third ring on again. You will now comprehend that (with the exception of the first ring) the only ring which can at any time be released is that which happens to be second on the bow, at the right-hand end; because both the wires which affect it being within the bow, there will be no impediment to its dropping through. You have now the first and second rings released, and the fourth also--the third still fixed; to release which we must make it last but one on the bow, and to effect which pass the first and second rings together through the bow, and on to it; then release the first ring again by slipping it off and dropping it through, and the third ring will stand as second on the bow, in its proper position for releasing, by drawing the second and third off together, dropping the third through, and slipping the second on again. Now to release the second, put the first up, through and on the bow; then slip the two together off, raise them up, and drop them through. The sixth will now stand second, consequently in its proper place for releasing; therefore draw it toward the end, A, slip the fifth off, then the sixth, and drop it through; after which replace the fifth, as you cannot release it until it stands in the position of a second ring; in order to effect this you must bring the first and second rings together, through and on to the bow; then in order to get the third on, slip the first off and down through the bow; then bring the third up, through and on to the bow; then bring the first ring up and on again, and, releasing the first and second together, bring the fourth through and on to the bow, replacing the third; then bring the first and second together on, drop the first off and through, then the third the same, replace the first on the bow, take off the first and second together, and the fifth will then stand second, as you desired; draw it toward the end, slip it off and through, replace the fourth, bring the first and second together up and on again, release the first, bring on the third, passing the second ring on to the bow again, replace the first, in order to release the first and second together; then bring the fourth toward the end, slipping it off and through, replace the third, bring the first and second together up and on again, release the first, then the third, replacing the second, bring the first up and on, in order to release the first and second together, which having done, your eighth ring will then stand second, consequently you can release it, slipping the seventh on again. Then to release the seventh, you must begin by putting the first and second up and on together, and going through the movements in the same succession as before, until you find you have only the tenth and ninth on the bow; then slip the tenth off and through the bow, and replace the ninth. This dropping of the tenth ring is the first effectual movement toward getting the rings off, as all the changes you have gone through were only to enable you to get at the tenth ring. You will then find that you have only the ninth left on the bow, and you must not be discouraged on learning, that in order to get that ring off, all the others to the right hand must be put on again, beginning by putting the first and second together, and working as before, until you find that the ninth stands as second on the bow, at which time you can release it. You will then have only the eighth left on the bow; you must again put on all the rings to the right hand, beginning by putting up the first and second together, till you find the eighth standing as second on the bow, or in its proper position for releasing; and so you proceed until you find all the rings finally released. As you commence your operations with all the rings ready fixed on the bow, you will release the tenth ring in one hundred and seventy moves; but as you then have only the ninth on, and as it is necessary to bring on again all the rings up to the ninth, in order to release the ninth, and which requires fifteen moves more, you will, consequently, release the ninth ring in two hundred and fifty-six moves; and, for your encouragement, your labor will diminish, by one half, with each following ring which is finally released. The eighth comes off in one hundred and twenty-eight moves, the seventh in sixty-four moves, and so on, until you arrive at the second and first rings, which come off together, making six hundred and eighty-one moves, which are necessary to take off all the rings. With the experience you will by this time have acquired, it is only necessary to say, that to replace the rings, you begin by putting up the first and second together, and follow precisely the same system as before.
46. MOVING THE KNIGHT OVER ALL THE SQUARES ALTERNATELY.
[Illustration: EULER'S METHOD.]
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."
[Illustration]
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.
[Illustration]
48. A MAZE OR LABYRINTH.
[Illustration]
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.
[Illustration]
[Illustration]
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.
[Illustration]
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.
[Illustration:
Fig 1 Fig 2 Fig 3 ]
2. ANSWER TO THE "PARALLELOGRAM."
[Illustration]
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.
[Illustration]
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.
[Illustration:
|\ | | |\ | +----- | \ | | | \ | | | \ | | | \ | +----- | \ | | | \ | | | \| | | \| +----- ]
7. THE THREE RABBITS, ANSWER.
[Illustration]
8. THE ACCOMMODATING SQUARE.
[Illustration]
9. ANSWER TO THE CIRCLE PUZZLE.
Thrust it upwards from the other side.
10. ANSWER TO THE CUT CARD PUZZLE.
[Illustration]
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.
[Illustration]
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.
[Illustration]
14. ANSWER TO THE SQUARE AND CIRCLE PUZZLE.
[Illustration:
+---------------+ |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.
[Illustration]
18. ANSWER 10 THE YANKEE SQUARE.
[Illustration]
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.
[Illustration: _Fig. 2_]
[Illustration: _Fig. 3_]
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.
[Illustration]
22. ANSWER TO THE FOUR TENANTS.
[Illustration]
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.
[Illustration]
24. ANSWER TO THE NUN'S PUZZLE.
[Illustration]
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.
[Illustration]
27. THE DOGS PUZZLE ANSWERED SEE DOTTED LINES.
[Illustration]
28. PUZZLE OF THE TWO FATHERS.
The first father divided the land in this way:
[Illustration]
The second father divided the land in the following manner:
[Illustration]
The different colors represent the several sons' portions.
29. THE TRIANGLE PUZZLE.
[Illustration]
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.
[Illustration]
31. ANSWER TO ANOTHER CROSS PUZZLE.
[Illustration]
32. ANSWER TO THE FOUNTAIN PUZZLE.
[Illustration]
33. ANSWER TO THE STAR PUZZLE.
Good-tempered friends! here _nine_ stars see: _Ten_ rows there are, in each row _three!_
[Illustration]
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.
[Illustration]
36. ANSWER TO THE CABINET MAKERS' PUZZLE.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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:
[Illustration]
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.
[Illustration]
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.
[Illustration]
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration: 25]
[Illustration: 26]
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.
[Illustration: 27]
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?
[Illustration: 28]
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.
[Illustration: 29]
[Illustration: 30]
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.
[Illustration: 31]
[Illustration: 32]
[Illustration: 33]
Fig. 33 is the application of the preceding.
[Illustration: 34]
[Illustration: 35]
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.
[Illustration: 36]
[Illustration: 37]
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.
[Illustration: 38]
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.
[Illustration: 39]
[Illustration: 40]
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.
[Illustration: 41]
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.
[Illustration: 42]
[Illustration: 43]
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.)
[Illustration: 44]
[Illustration: 45]
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.
[Illustration: 46]
[Illustration: 47]
As a pendant to the comical landscape given No. 27 I give you the annexed (Fig. 48).
[Illustration: 48]
[Illustration: 49]
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.
[Illustration: 50]
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.
[Illustration: 51]
[Illustration: 52]
[Illustration: 53]
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.
[Illustration: 54]
[Illustration: 55]
[Illustration: 56]
[Illustration: 57]
[Illustration: 58]
[Illustration: 59]
[Illustration: 60]
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.
[Illustration: 61]
[Illustration: 62]
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.
[Illustration: 63]
[Illustration: 64]
[Illustration: 65]
[Illustration: 66]
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.
[Illustration: 67]
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.
[Illustration: 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.
[Illustration: 69]
In Fig. 70, the same process is carried out, and the result is similar.
[Illustration: 70]
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.
[Illustration: 71]
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.
[Illustration: 72]
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.
[Illustration]
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.
Another mode of communicating intelligence secretly, viz. to agree upon some printed book, and mark the words out, is also troublesome, and not at all safe. The method of concealing the words which are to convey the information intended in matter of a very different character, in a long letter which the correspondent is enabled to read by applying a paper to it, with holes corresponding to the places of the significant words, is attended with many disadvantages: the paper may be lost, the repetition of certain words may lead to a discovery, and the difficulty of connecting the important with the unimportant matter, so as to give to the whole the appearance of an ordinary letter, is considerable.
There are many kinds of sympathetic inks. They are so called because the writings or drawings made by them are illegible, till by the action of some chemical agents, such as light, heat, acids, or other substances are brought in contact with them, when they appear. A weak sulphate of iron will be invisible in writing till washed over with a weak solution of prussiate of potass, which turns it of a beautiful blue. If we write with the nitro-muriate of gold, and afterwards brush the letters over with dilute muriate of tin, the writing will appear of a beautiful purple. If we write with a diluted solution of muriate of copper, and when dry present it to the fire, it will be of a yellow color.
Chemistry was also in great request for secret writing, and various substances were found to afford a fluid which would leave no mark behind the pen, until some chemical agent were applied. For example, if a letter be written with a pen dipped in the juice of lemon, the words will be invisible until the paper is held before the fire. This is caused by the
## action of the heat. Again, if a solution of nitrate of iron be the fluid
used, the writing cannot be seen until it is dipped into a solution of galls, or even into tea, which will act upon the iron, and become ink. It was found that if a plain sheet of paper were sent, and intercepted, the very fact of its being plain rendered it suspicious, and every means were used to render visible any writing that might be on it. A letter was therefore written with ordinary ink, on indifferent subjects, and between the lines the required information was added in some sympathetic ink. But writing with these or other sympathetic inks is unsafe, because the agents employed to render them visible are too generally known. Hence, the chiffre indéchiffrable, as it is called, has come very much into use, because it is easily applied, difficult to be deciphered, and the key may be preserved in the memory and easily changed. It consists of a table in which the letters of the alphabet, or any other signs agreed upon, are arranged as follow:
z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d c f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z a b c d e f g h i k l m n o p q r s t u v w x y z
Any word is now taken for a key. The word _Paris_, for example. This is a short word, and for the sake of secresy it would be well to choose for the key some one or more words less striking. Suppose we wish to write in this cypher with this key the phrase, "We lost a battle," we must write _Paris_ over the phrase, repeating it as often as is necessary, thus:
Pa risP a risPar We lost a battle.
We now take cypher for _w_, the letter which we find in the square opposite _w_ in the left margin column, and under _p_ on the top, which is _m_. Instead of _e_ we take the letter opposite _e_, and under _a_, which is _f_; for _l_, the letter opposite _c_, and under _z_, and so on.
Proceeding thus, we should obtain the following series of letters:
mf cxli b tkmimw
The person who receives the epistle writes the key over the letters
P a r i s P a r i s P a r m f c x l i b t k m i m w
He now goes down in the perpendicular line, at the top of which is _p_, until he meets _m_, opposite to which, in the left marginal column, he finds _w_. Next, going in the line of _a_ down to _f_ he finds, on the left, _e_. In the same way _r_ gives _l_, _i_ gives _o_, and so on. Or you may reverse the process; begin with _p_, in the left marginal column, and look along horizontally till you find _m_, over which, in the top line, you will find _w_. It is easily seen that the same letter is not always designated by the same cypher; thus _e_ and _a_ occur twice in the phrase selected, and they are designated respectively by the cyphers _f_ and _w_, _b_ and _k_. Thus the possibility of finding out the secret writing is almost impossible.
The key may be changed from time to time, and a different key may be used with each correspondent. The utmost accuracy is necessary, because one character accidentally omitted changes the whole cypher. The best way of determining the key word is to arrange that any word which occurs at a certain distance from the beginning or end shall be the key word--the tenth from the beginning, for example. The key word will thus change every time, and any combination of letters will make it. This will make it impossible to be guessed.
The easiest method of working this square is to cut a piece of thin wood like a carpenter's square, and by applying it to the alphabet the letter is at once seen in the angle. For example, supposing such a square to be applied so that one side is on the letter _p_ at the top, and the other on the letter _w_ at the left hand, the letter _m_ will be in the angle, so that the trouble of following the lines with the eye will be avoided.
Here is another specimen of secret writing.
A LOCK FOR MR. HOBBS TO PICK.
T:2 21rt:(,)t:2 s21(,)t:2 st1rr6 s,6(,) 1r2 86p : 2rs wr3t 76 : 1-9 93v3-2(,) T : 1t : 1-9 w:38: t5-29 t:23r : 1r ? 4-6(,) 1-9 7192 t:23r v1r329 .!4r325 s:3-2(:) 3- t:2? 22- :21t:2-s 262 ?16 s22 S6?74!s 3- 5-2 .r1-9 tr3t: 84?75-2(;) 76t 3- t:2 744, 40 744,s t:2r2 !32s 1 ,26 34 r219 t:23r ?6st2r32s (.)
T:2- !2t -4t 013t:!2ss t4-.52 19v1-82 3ts s:1!!4w v15-ts(,)-4r s84002r 91r2 5p4- t:2 71sl2 40 3.-4r1-82 T4 r13s2 1 str58t5r2 40 92sp13r(!) T:45.: 044!s ?16 .3v2 t:2 w4r!9s t4 8:1-82(,) 92s3.- 1-9 pr4v392-82 1r2 t:2r2(.) :4w 7!3-929(,)w:4 1t 921t: 1!4-2 T:23r 922p s3.-30381-86 4w-(!)
HERE IS THE ANSWER.
The letters are represented by the figures and symbols below them. With this key the lock may be opened.
a b c d e f g h i j k l m n o u y 1 7 8 9 2 0 . : 3 ; , ! ? - 4 5 6
The stops enclosed in brackets, are used in their capacity of stops: thus, (,) (;) &c.
The earth, the sea, the starry sky, Are cyphers writ by hand divine, That hand which tuned their harmony, And bade their varied glories shine; In them e'en heathen's eye may see Symbols in one grand Truth combine; But in the book of books there lies A key to read their mysteries.
Then let not faithless tongue advance Its shallow vaunts, nor scoffer dare Upon the base of ignorance To raise a structure of despair! Though fools may give the world to chance Design and Providence are there: How blinded, who at death alone Their deep significancy own!
THE CIRCULAR CYPHER.
To carry on a correspondence without the possibility of the meaning of the letter being detected, in case it should be opened by any other person, has employed the ingenuity of many. No method will be found more effectual for this purpose, or more easy, than the following.
[Illustration: Fig. 1.]
Provide a piece of square card or pasteboard, and draw a circle on it, which circle is to be divided into 27 equal parts, in each of which parts must be written _one_ of the capital letters of the alphabet, and the &, as in the figure. Let the center of this circle be blank. Then draw another circle, also divided into 27 equal parts, in each of which write one of the small letters of the alphabet, and the &. This circle must be cut round, and made exactly to fit the blank space in the center of the large circle, and must run round a pivot or pin. The person with whom you correspond must have a similar dial, and at the beginning of your letter you must put the capital letter, and at the end the small letter, which answer to each other when you have fixed your dial.
Suppose what you wish to communicate is as follows:
_I am so watched I cannot see you as I promised; but I will meet you to-morrow in the park, with the letters, &c._
You begin with the letter _T_, and end with the letter _m_, which shows how you have fixed the dial, and how your correspondent must fix his, that he may decipher your letter. Then, for _I am_, you write _b uf_, and so of the rest, as follows:
_T b uf lh pumwayx b wugghm lyy rhn ul b ikhfblyx vnm b pbee fyym rhn mh fhkkhp bg may iukd pbma may eymmykl, tw._
_m._
_Another Way._
Take two pieces of card, pasteboard, or stiff paper, through which you cut long squares at different distances. One of these you keep yourself, and the other you give to your correspondent. You lay the pasteboard on a paper, and, in the spaces cut out write what you would have understood by him only; then fill the intermediate spaces with any words that will connect the whole together, and make a different sense. When he receives it, he lays his pasteboard over the whole, and those words which are between crotchets [ ] form the intelligence you wish to communicate. For example: suppose you want to express these words,
"_Don't trust Robert: I have found him a villain._"
"[Don't] fail to send my books. I [trust] they will be ready when [Robert] calls on you. [I have] heard that you have [found] your dog. I call [him a villain] who stole him." You may place a pasteboard of this kind three other ways--the bottom at top--the top at bottom, or by turning it over; but in this case you must previously apprize your correspondent, or he may not be able to decipher your meaning.
SECRET CORRESPONDENCE BY MUSIC.
[Illustration: Fig. 2.]
Form a circle like Fig. 2, divided into twenty-six parts, with a letter of the alphabet written in each. The interior of the circle is movable, like that in Fig. 1, and the circumference is to be ruled like music paper. Place in each division a note different in figure or position.
Within the musical lines place the three keys, and on the outer circle the figures to denote time. Then get a ruled paper, and place one of the keys (suppose _ge-re-sol_) against the time 2-4ths, at the beginning of the paper, which will inform your correspondent how to place his circle. You then copy the notes that answer to the letters of the words you intend to write, in the manner expressed above.
[Illustration]
THE MAGIC OF STRENGTH.
"Not two strong men the enormous weight could raise, Such men as live in these degenerate days."--POPE'S HOMER.
The mechanical knowledge of the ancients was principally theoretical; and though they seem to have executed some minor pieces of mechanism which were sufficient to delude the ignorant, yet there is no reason for believing that they have executed any machinery that was capable of exciting much surprise, either by its ingenuity or its magnitude. The properties of the mechanical powers, however, seem to have been successfully employed in performing feats of strength which were beyond the reach even of strong men, and which could not fail to excite the greatest wonder when exhibited by persons of ordinary size.
Firmus, a native of Seleucia, who was executed by the Emperor Aurelian for espousing the cause of Zenobia, was celebrated for his feats of strength. In his account of the life of Firmus, who lived in the third century, Vopiscus informs us that he could suffer iron to be forged upon an anvil placed upon his breast. In doing this, he lay upon his back, and, resting his feet and shoulders against some support, his whole body formed an arch, as we shall afterward more particularly explain. Until the end of the sixteenth century, the exhibition of such feats does not seem to have been common. About the year 1703, a native of Kent, of the name of Joyce, exhibited such feats of strength in London and other parts of England, that he received the name of the second Samson. His own personal strength was very great; but he had also discovered, without the aid of theory, various positions of his body in which men even of common strength could perform very surprising feats. He drew against horses, and raised enormous weights; but as he actually exhibited his powers in ways which evinced the enormous strength of his own muscles, all his feats were ascribed to the same cause. In the course of eight or ten years, however, his methods were discovered, and many individuals of ordinary strength exhibited a number of his principal performances, though in a manner greatly inferior to Joyce.
Some time afterward, John Charles Van Eckeberg, a native of Harzgerode, in Anhault, traveled through Europe under the appellation of Samson, exhibiting very remarkable examples of his strength. This, we believe, is the same person whose feats are particularly described by Dr. Desaguliers. He was a man of the middle size, and of ordinary strength; and as Dr. Desaguliers was convinced that his feats were exhibitions of skill, and not of strength he was desirous of discovering his methods, and, with this view, he went to see him, accompanied with the Marquis of Tullibardine, Dr. Alexander Stuart, Dr. Pringle, and his own mechanical operator. They placed themselves round the German, so as to be able to observe accurately all that he did, and their success was so great, that they were able to perform most of the feats the same evening by themselves, and almost all the rest when they had provided the proper apparatus. Dr. Desaguliers exhibited some of the experiments before the Royal Society, and has given such a distinct explanation of the principles on which they depend, that we shall endeavor to give a popular account of them.
[Illustration: FIG. 1.]
1. The performer sat upon an inclined board A B, placed upon a frame C D E, with his feet abutting against the upright board C. Round his loins was placed a strong girdle FG, to the iron ring of which at G was fastened a rope by means of a hook. The rope passed between his legs through a hole in board C, and several men, or two horses, pulling at the other end of the rope, were unable to draw the performer out of his place. His hands at G seemed to pull against the men, but they were of no advantage to him whatever.
2. Another of the German's feats is shown in Fig. 2. Having fixed the rope above mentioned to a strong post at A, and made it pass through a fixed iron eye at B, to the ring in his girdle, he planted his feet against the post at B, and raised himself from the ground by the rope, as shown in the figure. He then suddenly stretched out his legs and broke the rope, falling back on a feather bed at C, spread out to receive him.
[Illustration: FIG. 2.]
[Illustration: FIG. 3.]
3. In imitation of Firmus, he laid himself down on the ground, as shown in Fig. 3, and when an anvil A was placed upon his breast, a man hammered with all his force the piece of iron B, with a sledge hammer, and sometimes two smiths cut in two with chisels a great cold bar of iron laid upon the anvil. At other times a stone of huge dimensions, half of which is shown at C, was laid upon his belly, and broken with a blow of the great hammer.
[Illustration: FIG. 4.]
4. The performer then placed his shoulders upon one chair, and his heels upon another, as in Fig. 4, forming, with his back-bone, thighs, and legs, an arch springing from its abutments at A and B. One or two men then stood upon his belly, rising up and down while the performer breathed. A stone, one and a half feet long, one foot broad, and half a foot thick, was then laid upon his belly and broken by a sledge hammer, an operation which may be performed with much less danger than when his back touched the ground, as in Fig. 3.
5. His next feat was to lie down on the ground, as in Fig. 5. A man being then placed on his knees, he draws his heels towards his body, and, raising his knees, he lifts up the man gradually, till, having brought his knees perpendicularly under him, as in Fig. 6, he raises his own body up, and, placing his arms around the man's legs, he rises with him, and sets him down on some low table or eminence of the same height as his knees. This feat he sometimes performed with two men in place of one.
[Illustration: FIG. 5.]
[Illustration: FIG. 6.]
[Illustration: FIG. 7.]
6. The last, and apparently the most wonderful, performance of the German, is shown in Fig. 7, where he appears to raise a cannon A, placed upon a scale, the four ropes of the scale being fixed to a rope or chain attached to his girdle, in the manner already described. Previous to the fixing of the ropes, the cannon and scale rest upon two rollers B C, but when all is ready, the two rollers are knocked from beneath the scale, and the cannon is sustained by the strength of his loins.
The German also exhibited his strength in twisting into a screw a flat piece of iron like A, Fig. 8. He first bent the iron into a right angle, as at B, and then wrapping his handkerchief about its broad upper end, he held that end in his left hand, and with his right applied to the other end, twisted about the angular point, as shown at C. Lord Tullibardine succeeded in doing the same thing, and even untwisted one of the irons which the German had twisted.
It would lead into details by no means popular, were I to give a minute explanation of the mechanical principles upon which these feats depend. A few general observations will perhaps be sufficient for ordinary readers. The feats No. 1, 2, and 7, depend entirely on the natural strength of the bones of the pelvis, which form a double arch, which it would require an immense force to break, by any external pressure directed to the center of the arch; and, as the legs and thighs are capable of sustaining four or five thousand pounds when they stand quite upright, the performer has no difficulty in resisting the force of two horses, or of sustaining the weight of a cannon weighing two or three thousand pounds.
[Illustration: FIG. 8.]
The feat of the anvil is certainly a very surprising one. The difficulty, however, really consists in sustaining the anvil; for when this is done, the effect of the hammering is nothing. If the anvil were a thin piece of iron, or even two or three times heavier than the hammer, the performer would be killed by a few blows; but the blows are scarcely felt when the anvil is very heavy, for the more matter the anvil has, the greater is its inertia, and it is the less liable to be struck out of its place; for when it has received by the blow the whole momentum of the hammer, its velocity will be so much less than that of the hammer, as its quantity of matter is greater. When the blow, indeed, is struck, the man feels less of the weight of the anvil than he did before, because, in the reaction of the stone, all the parts of it round about the hammer rise towards the blow. This property is illustrated by the well known experiment of laying a stick with its ends upon two drinking glasses full of water, and striking the stick downwards in the middle with an iron bar. The stick will in this case be broken without breaking the glasses, or spilling the water. But if the stick is struck upwards, as if to throw it up in the air, the glasses will break if the blow be strong, and if the blow is not very quick, the water will be spilled without breaking the glasses.
When the performer supports a man upon his belly, as in Fig. 4, he does it by means of the strong arch formed by his backbone, and the bones of his legs and thighs. If there were room for them, he could bear three or four, or, in their stead, a great stone to be broken with one blow.
A number of feats of real and extraordinary strength were exhibited about a century ago, in London, by Thomas Topham, who was five feet ten inches high, and about thirty-one years of age. He was entirely ignorant of any of the methods for making his strength appear more surprising; and he often performed, by his own natural powers, what he learned had been done by others by artificial means. A distressing example of this occurred in his attempt to imitate the feat of the German Samson by pulling against horses. Ignorant of the method which we have already described, he seated himself on the ground with his feet against two stirrups, and by the weight of his body he succeeded in pulling against a single horse; but in attempting to pull against two horses, he was lifted out of his place, and one of his knees was shattered against the stirrups, so as to deprive him of most of the strength of one of his legs. The following are the feats of real strength which Dr. Desaguliers saw him perform.
1. Having rubbed his fingers with coal-ashes to keep them from slipping, he rolled up a very strong and large pewter plate.
2. Having laid seven or eight short and strong pieces of tobacco-pipe on the first and third fingers, he broke them by the force of his middle finger.
3. He broke the bowl of a strong tobacco-pipe placed between his first and third fingers, by pressing his fingers together sideways.
4. Having thrust such another bowl under his garter, his legs being bent he broke it to pieces by the tendons of his hams, without altering the bending of his leg.
5. He lifted with his teeth, and held in a horizontal position for a considerable time, a table six feet long, with half a hundred weight hanging at the end of it. The feet of the table rested against his knees.
6. Holding in his right hand an iron kitchen poker three feet long and three inches round, he struck upon his bare left arm, between the elbow and the wrist, till he bent the poker nearly to a right angle.
7. Taking a similar poker, and holding the ends of it in his hands, and the middle against the back of his neck, he brought both ends of it together before him, and he then pulled it almost straight again. This last feat was the most difficult, because the muscles which separate the arms horizontally from each other are not so strong as those which bring them together.
8. He broke a rope about two inches in circumference, which was partly wound about a cylinder four inches in diameter, having fastened the other end of it to straps that went over his shoulder.
9. Dr. Desaguliers saw him lift a rolling stone of about 800 pounds' weight with his hands only, standing in a frame above it, and taking hold of a frame fastened to it. Hence Dr. Desaguliers gives the following relative view of the strengths of individuals:
Strength of the weakest men, 125 pounds. Strength of very strong men, 400 Strength of Topham, 800
The weight of Topham was about 200.
One of the most remarkable and inexplicable experiments relative to the strength of the human frame, which we have ourselves seen and admired, is that in which a heavy man is raised with the greatest facility, when he is lifted up the instant that his own lungs, and those of the persons who raise him, are inflated with air. This experiment was, I believe, first shown in England a few years ago by Major H., who saw it performed in a large party at Venice under the direction of an officer of the American navy. As Major H. performed it more than once in my presence, I shall describe, as nearly as possible, the method which he prescribed. The heaviest person in the party lies down upon two chairs, his legs being supported by the one and his back by the other. Four persons, one at each leg and one at each shoulder, then try to raise him, and they find his dead weight to be very great, from the difficulty they experience in supporting him. When he is replaced in the chairs, each of the four persons takes hold of the body as before, and the person to be lifted gives two signals by clapping his hands. At the first signal he himself and the four lifters begin to draw a long and full breath, and when the inhalation is completed, or the lungs filled, the second signal is given, for raising the person from the chairs. To his own surprise and that of his bearers, he rises with the greatest facility, as if he were no heavier than a feather. On several occasions I have observed that when one of the bearers performs his