Part 24

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.

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.

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.

DECEPTIVE VISION.

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:--

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.

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.

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:--

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.

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.

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.