Take a straw, and having bent the thicker end of it in a sharp angle, as in figure subjoined, put this hooked end into the bottle, so that the bent part may rest against its side; you may then take the other end, and lift up the bottle by it, without breaking the straw, and this will be the more readily accomplished as the angular part of the straw approachesnearer to that which comes out of the bottle. It is necessary, in order to succeed in this feat, to be particularly careful in choosing a stout straw, which is neither broken nor bruised; if it have been previously bent or damaged, it is unfit for the purpose of performing this trick, as it will be too weak in the part so bent, or damaged, to support the bottle.
THE DANCING PEA.
If you stick through a pea, or small ball of pith, two pins[8]at right angles, and defend the points with pieces of sealing wax, it may be kept in equilibrio at a short distance from the end of a straight tube by means of a current of breath from the mouth, which imparts a rotary motion to the pea.
THE TOPER'S TRIPOD.
Place three tobacco pipes in the position shown in the engraving, the mouth of the bowls downwards, and the lower end of the stems upon the stem just by the bowls. This tripod, if carefully put together, will support considerably more than a pot of ale.
OBLIQUITY OF MOTION.
Cut a piece of pasteboard into the following shape, and describe on it a spiral line; cut this out with a penknife, and then suspend it on a large skewer or pin, as seen in the engraving. If the whole be now placed on a warm stove, or over the flame of a candle or lamp, it will revolve with considerable velocity. The card, after being cut into the spiral, may be made to represent a snake or dragon, and when in motion will produce a very pleasing effect.
THE BRIDGE OF KNIVES.
Place three glasses,A A A, in the form of a triangle, and arrange three knives upon them, as shown in the figure, the blade of No. 1 over that of No. 2, and that over No. 3, which rests on No. 1. The bridge so made will be self-supported.
SAND IN THE HOUR-GLASS.
It is a remarkable fact, that the flow of sand in the hour-glass is perfectly equable, whatever may be the quantity in the glass; that is, the sand runs no faster when the upper half of the glass is quite full than when it is nearly empty. It would, however, be natural enough to conclude that, when full of sand, it would be more swiftly urged through the aperture, than when the glass was only a quarter full, and near the close of the hour.
The fact of the even flow of sand may be proved by a very simple experiment. Provide some silver sand, dry it over or before the fire, and pass it through a tolerably fine sieve. Then take a tube, of any length or diameter, closed at one end, in which make a small hole, say the eighth of an inch; stop this with a peg, and fill up the tube with the sifted sand. Hold the tube steadily, or fix it to a wall, or frame, at any height from a table; remove the peg, and permit the sand to flow in any measure for any given time, and note the quantity. Then, let the tube be emptied, and only half or a quarter filled with the sand; measure again, for a like time, and the same quantity of sand will flow: even if you press the sand in the tube with a ruler or stick, the flow of the sand through the hole will not be increased.
The above is explained by the fact, that when the sand is poured into the tube, it fills it with a succession of conical heaps, and that all the weight which the bottom of the tube sustains is only that of the heap whichfirstfalls upon it; as the succeeding heaps do not press downwards, but only against the sides or walls of the tube.
RESISTANCE OF SAND.
From the above experiment it may be concluded, that it is extremely difficult to thrust sand out of a tube by means of a fitting plug or piston; and this, upon trial, is found to be the case. Fit a piston to a tube (exactly like a boy's pop-gun,) pour some sand in, and try with the utmost strength of the arm to push out the sand. It will be found impossible to do this: rather than the sand should be shot out, the tube will burst at the sides.
The science of Hydraulics comprehends the laws which regulate non-elastic fluids in motion, and especially water, &c.
Water can only be set in motion by two causes—the pressure of the atmosphere, or its own gravity. The principal law concerning fluids is, that they always preserve their own level. Hence water can be distributed over a town from any reservoir that is higher than the houses to be supplied; and the same principle will enable us to form fountains in a garden, or other place. Should any of our young friends wish to form a fountain, or jet-d'-eau, they may, by bringing a pipe fromT, a water-tank, which should be at the upper part of the house, convey the water down to the garden. Then by leading it through the earth, underneath the path or grass plot, and turning it to a perpendicular position, the water will spring out, and rise nearly as high as the level of that in the tank. The part of the pipe atBshould have a turnkey, so that the water may be let on or shut off at pleasure.
THE PUMP.
The action of the common pump is as follows: When the handleAis raised, the piston-rodBdescends, andbrings the piston-valve, called the sucker, or bucket, to another valve,C, which is fixed, and opens inwards towards the piston. When the handle is drawn down, the piston is raised, and, as it is air-tight, a vacuum is produced between the two valves; the air in the valve of the pump, betwixt the lower valve and the water, then forces open the lower valve, and rushes through to fill up this vacuum; and the air in the pump being less dense than the external atmosphere, the water is forced a short way up the barrel. When the piston again descends to the lower valve, the air between them is again forced out by forcing open the upper valve; and when the piston is raised, a vacuum is again produced, and the air below the lower valve rushes up, and the water in consequence is again raised a little further. This operation continues until the water rises above the lower valve; at every stroke afterwards, the water passes through the valve of the descending piston, and is raised by it, on its ascent, until it issues out of the spout.
THE HYDRAULIC DANCER.
Make a little figure of cork, in the shape of a dancing mountebank, sailor, &c. In this figure place a small hollow cone, made of thin leaf brass. Whenthis figure is placed upon any jet-d'-eau, such as that of the fountain recommended to be constructed, it will be suspended on the top of the water, and perform a great variety of amusing motions. If a hollow ball of very thin copper, about an inch in diameter, be placed on a similar cone, it will remain suspended, turning round and spreading the water all about it.
THE SYPHON.
The syphon is a bent tube, having one leg shorter than the other. It acts by the pressure of the atmosphere being removed from the surface of a fluid, which makes it to rise above its common level atB. In order to make a syphon act, it is necessary first to fill both legs quite full of the fluid; and then the shorter leg must be placed in the vessel to be emptied. Immediately upon withdrawing the finger from the longer leg, the liquor will flow. Any young person may form a syphon by a small piece of leaden pipe, bent into the form above.
THE WATER SNAIL, OR ARCHIMEDIAN SCREW
may easily be constructed. Purchase a yard of small leaden pipe, and twist it round a pole, as in the following figure,A; place a handle at its upper end,B, and let its lower end rest in the water. Between the last turn of the pipe and the orifice place a paddle-wheel,C. Now, should the water be that of a running stream, the force of the stream will turn the pipe, and the water will rise in it till it empties itself into the trough atD. Should the water have no motion, the turning of the handle atBwill elevate the water from the lower to the higher level.
THE BOTTLE EJECTMENT.
Fill a small white glass bottle, with a very narrow neck, full of wine; place it in a glass vase, which must previously have sufficient water in it to rise above the mouth of the bottle. Immediately you will perceive the wine rise, in the form of a little column, toward the surface of the water, and the water will, in the mean time, begin to take the place of the wine at the bottom of the bottle. The cause of this is, that the water is heavier than the wine, which it displaces, and forces it to rise toward the surface.
THE MAGIC OF HYDROSTATICS WITH THE ANCIENTS.
The principles ofHydrostaticswere available in the work of magical deception. The marvelous fountain which Pliny describes in the island of Andros as discharging wine for seven days, and water during the rest of the year,—the spring of oil which broke out in Rome to welcome the return of Augustus from the Sicilian war,—the three empty urns which filled themselves with wine at the annual feast of Bacchus in the city of Elis,—the glass tomb of Belus, which was full of oil, and which, when once emptied by Xerxes, could not again be filled,—the weeping statues, and the perpetual lamps of the ancients,—were all the obvious effect of the equilibrium and pressure of fluids.
TO EMPTY A GLASS UNDER WATER.
Fill a wine-glass with water, place over its mouth a card, so as to prevent the water from escaping, and put the glass, mouth downwards, into a basin of water. Next, remove the card, and raise the glass partly above the surface, but keep its mouth below the surface, so that the glass still remains completely filled with water. Then insert one end of a quill or reed in the water below the mouth of the glass, and blow gently at the other end, when air will ascend in bubbles to the highest part of the glass, and expel the water from it; and, if you continue to blowthroughthe quill, all the water will be emptied from the glass, which will be filled with air.
Acoustics is the science relating to sound and hearing. Sound is heard when any shock or impulse is given to the air, or to any other body which is in contact directly or indirectly with the ear.
DIFFERENCE BETWEEN SOUND AND NOISE.
Noises are made by the cracks of whips, the beating of hammers, the creak of a file or saw, or the hubbub of a multitude. But when a bell is struck, the bow of a violin drawn across the strings, or the wetted finger turned round a musical glass, we have what are properly called sounds.
SOUNDS, HOW PROPAGATED.
Sounds are propagated on all bodies much after the manner that waves are in water, with a velocity of 1,142 feet in a second. Sounds in liquids and in solids are more rapid than in air. Two stones rubbed together may be heard in water at half a mile; solid bodies convey sounds to great distances, and pipes may be made to convey the voice over every part of the house.
VISIBLE VIBRATION.
Provide a glass goblet about two thirds filled with colored water, draw a fiddle bow against its edge, and the surface of the water will exhibit a pleasing figure, composed of fans, four, six, or eight in number, dependent on the dimensions of the vessel, but chiefly on the pitch of the note produced.
Or, nearly fill a glass with water, draw the bow strongly against its edge, the water will be elevated and depressed; and when the vibration has ceased, and the surface of the water has become tranquil, these elevations will be exhibited in the form of a curved line, passing round the interior surface of the glass, and above the surface of the water. If the action of the bow be strong, the water will be sprinkled on the inside of the glass, above the liquid surface, and this sprinkling will show the curved linevery perfectly, as in the engraving. The water should be carefully poured, so that the glass above the liquid be preserved dry; the portion of the glass between the edge and curved line will then be seen partially sprinkled; but, between the level of the water and the curved line, it will have become wholly wetted, thereby indicating the height to which the fluid has been thrown.
TRANSMITTED VIBRATION.
Provide a long, flat glass ruler or rod, as in the engraving, and cement it with mastic to the edge of a drinking glass, fixed into a wooden stand; support the other end of the rod very lightly on a piece of cork, and strew its upper surface with sand; set the glass in vibration by a bow, at a point opposite where the rod meets it, and the motions will be communicated to the rod without any change in their direction. If the apparatus be inverted, and sand be strewed on the under side of the rod, the figures will be seen to correspond with those produced on the upper surface.
DOUBLE VIBRATION.
Provide two disks of metal or glass, precisely of the same dimensions, and a glass or metal rod; cement the two disks at their centers to the ends of the rod, as in the engraving, and strew their upper surfaces with sand. Cause one of the disks, viz., the upper one, to vibrate by a bow, and its vibration will be exactly imitated by the lower disk, and the sand strewed over both will arrange itself in precisely the same forms on both disks.
CHAMPAGNE AND SOUND.
Pour sparkling champagne into a glass, until it is half full, when the glass will lose its power of ringing by a stroke upon its edge, and will emit only a disagreeable and puffy sound. Nor will a glass ring while the wine is brisk, and filled with air-bubbles; but as the effervescence subsides, the sound will become clearer and clearer, and when the air-bubbles have entirely disappeared, the glass willring as usual. If a crumb of bread be thrown into the champagne, and effervescence be reproduced, the glass will again cease to ring. The same experiment will also succeed with soda water, ginger wine, or any other effervescing liquid.
MUSIC OF THE SNAIL.
Place a garden snail upon a pane of glass, and in drawing itself along, it will frequently produce sounds similar to those of musical glasses.
THE TUNING-FORK A FLUTE PLAYER.
Take a common tuning-fork, and on one of its branches fasten with sealing-wax a circular piece of card, of the size of a small wafer, or sufficient nearly to cover the aperture of a pipe, as the sliding of the upper end of a flute with the mouth stopped: it may be tuned in unison with the loaded tuning-fork (a C fork), by means of the moveable stopper or card, or the fork may be loaded till the unison is perfect. Then set the fork in vibration by a blow on the unloaded branch, and hold the card closely over the mouth of the pipe, as in the engraving, when a note of surprising clearness and strength will be heard. Indeed, a flute may be made to "speak" perfectly well, by holding close to the opening a vibrating tuning-fork, while the fingering proper to the note of the fork is at the same time performed.
MUSICAL BOTTLES.
Provide two glass bottles, and tune them by pouring water into them, so that each corresponds to the sound of a different tuning-fork. Then apply both tuning-forks to the mouth of each bottle alternately, when that sound only will be heard, in each case, which is reciprocated by the unisonant bottle; or, in other words, by that bottle which contains a column of air, susceptible of vibrating in unison with the fork.
THEORY OF WHISPERING.
Apartments of a circular or elliptical form are bestcalculatedfor the exhibition of this phenomenon. If a person stand near the wall, with his face turned to it, and whispera few words, they may be more distinctly heard at nearly the opposite side of the apartment, than if the listener was situated near to the speaker.
THEORY OF THE VOICE.
Provide a species of whistle, common as a child's toy or a sportsman's call, in the form of a hollow cylinder,aboutthree fourths of an inch in diameter, closed at both ends by flat circular plates, with holes in their centers. Hold this toy between the teeth and lips; blow through it, and you may produce sounds varying in pitch with the force with which you blow. If the air be cautiously graduated, all the sounds within the compass of a double octave may be produced from it; and, if great precaution be taken in the management of the wind, tones even yet graver may be brought out. This simple instrument, or toy, has indeed the greatest resemblance to the larynx, which is the organ of voice.
TO TUNE A GUITAR WITHOUT THE ASSISTANCE OF THE EAR.
Make one string to sound, and its vibrations will, with much force, be transferred to the next string: this transference may be seen, by placing a saddle of paper (like an inverted Λ) upon the string, at first in a state of rest. When this stringhearsthe other, the saddle will be shaken, or fall off; when both strings are in harmony, the paper will be very little, or not at all shaken.
PROGRESS OF SOUND.
When a bow is drawn across the strings of a violin, the impulses produced may be rendered evident by fixing a small steel bead upon the bow; when looked at by light, or in sunshine, the bead will seem to form a series of dots during the passage of the bow.
TO MAKE AN ÆOLIAN HARP.
This instrument consists of a long narrow box of very thin pine, about six inches deep, with a circle in the middle of the upper side, of an inch and half in diameter, in which are to be drilled small holes. On this side seven, ten, or more strings of very fine catgut are stretched over bridges at each end like the bridge of a fiddle, and screwed up or relaxed with screw pins. The strings must all be tuned toone and the same note,[9]and the instrument should be placed in a window partly open, in which the width is exactly equal to the length of the harp, with the sash just raised to give the air admission. When the air blows upon these strings with different degrees of force it will excite different tones of sound. Sometimes the blast brings out all the tones in full concert, and sometimes it sinks them to the softest murmurs.
A colossal imitation of the instrument just described was invented at Milan in 1786, by the Abbate Gattoni. He stretched seven strong iron wires, tuned to the notes of the gamut, from the top of a tower sixty feet high, to the house of a Signor Moscate, who was interested in the success of the experiment, and this apparatus, called the "giant's harp," in blowing weather yielded lengthened peals of harmonious music. In a storm this music was sometimes heard at the distance of several miles.
THE INVISIBLE GIRL.
The facility with which the voice circulates through tubes was known to the ancients, and no doubt has afforded the priests of all religions means of deception to the ignorant and credulous. But of late days the light of science dispels all such wicked deceptions. A very clever machine was produced at Paris several years ago, and afterwards exhibited in New York and other cities in the United States, under the name of the "Invisible Girl," since the apparatus was so constructed that the voice of a female at a distance was heard as if it originated from a hollow globe, not more than a foot in diameter. It consisted of a wooden frame something like a tent bedstead, formedby four pillarsa a a a, connected by upper cross railsb b, and similar rails below, while it terminated above in four bent wiresc c, proceeding at right angles of the frame, and meeting in a central point. The hollow copper balld, with four trumpets,t t, crossing from it at right angles, hung in the center of the frame, being connected with the wires alone by four narrow ribbonsr r. The questions were proposed close to the open mouth of one of these trumpets, and the reply was returned from the same orifice. The means used in the deception were as follows: a pipe or tube was attached to one of the hollow pillars, and carried into another apartment, in which a female was placed; and this tube having been carried up the leg or pillar of the instrument to the cross-rails, had apertures exactly opposite two of the trumpet mouths; so that what was spoken was immediately answered through a very simple mode of communication.
THE MAGIC OF ACOUSTICS.
The science ofAcousticsfurnished the ancient sorcerers with some of their best deceptions. The imitation of thunder in their subterranean temples could not fail to indicate the presence of a supernatural agent. The golden virgins whose ravishing voices resounded through the temple of Delphos,—the stone from the river Pactolus, whose trumpet notes scared the robber from the treasure which it guarded,—the speaking head which uttered its oracular responses at Lesbos,—and the vocal statue of Memnon, which began at the break of day to accost the rising sun,—were all deceptions derived from science, and from a diligent observation of thephenomenaof nature.
Take a long piece of wood, such as the handle of a hair broom, and placing a watch at one end, apply your ear to the other, and the tickings will be distinctly heard.
TO SHOW THAT SOUND DEPENDS ON VIBRATION.
Touch a bell when it is sounding, and the noise ceases; the same may be done to a musical string with the same results. Hold a musical pitchfork to the lips, when it is made to sound, and a quivering motion will be felt from its vibrations. These experiments show that sound is produced by the quick motions and vibrations of different bodies.
OR, CURIOUS PROBLEMS IN ARITHMETIC.
As the principal object of this volume is to enable the young reader to learn something in his sports, and to understand what he is doing, we shall, before proceeding to the curious tricks and feats connected with the science of numbers, present him with some arithmetical aphorisms, upon which most of the following examples are founded.
APHORISMS OF NUMBER.
1. If two even numbers be added together, or subtracted from each other, their sum or difference will be an even number.
2. If two uneven numbers be added or subtracted, their sum or difference will be an even number.
3. The sum or difference of an even and an uneven number added or subtracted, will be an uneven number.
4. The product of two even numbers will be an even number, and the product of two uneven numbers will be an uneven number.
5. The product of an even and uneven number will be an even number.
6. If two different numbers be divisible by any one number,their sum and their difference will also be divisible by that number.
7. If several different numbers, divided by 3, be added or multiplied together, their sum and their product will also be divisible by 3.
8. If two numbers, divisible by 9, be added together, the sum of the figures in the amount will be either 9, or a number divisible by 9.
9. If any number be multiplied by 9, or by any other number divisible by 9, the amount of the figures of the product will be either 9, or a number divisible by 9.
10. In every arithmetical progression, if the first and last term be each multiplied by the number of terms, and the sum of the two products be divided by 2, the quotient will be the sum of the series.
11. In every geometric progression, if any two terms be multiplied together, their product will be equal to that term, which answers to the sum of these two indices. Thus, in the series—
123452481632
If the third and fourth terms 8 and 16 be multiplied together, the product 128 will be the seventh term of the series. In like manner, if the fifth term be multiplied into itself, the product will be the tenth term, and if that sum be multiplied into itself, the product will be the twentieth term. Therefore, to find the last, or any other term of a geometric series, it is not necessary to continue the series beyond a few of the first terms.
Previous to the numerical recreations, we shall here describe certain mechanical methods of performing arithmetical calculations, such as are not only in themselves entertaining, but will be found more or less useful to the young reader.
PALPABLE ARITHMETIC.
The blind mathematician, Dr. Saunderson, adopted a very ingenious device for performing arithmetical operations by the sense of touch.
Small cubes of wood were provided, and in one face of each, nine holes were pierced, thus:
123ooo456ooo789ooo
These holes represented the nine digits, as in the figure, and to denote any figure, a small peg was inserted into the hole corresponding to it. If the number consisted of several figures, more cubes were used, one for each. A cipher was represented by a peg of different shape from that of the others, and inserted in the central hole.
To perform any arithmetical process, a square board was provided, divided by ridges into recesses of the same width as the cubes, and by this the cubes were retained in the required horizontal andperpendicularlines. Suppose it was necessary to add together the numbers 763, 124, 859, the cubes and pegs would be arranged thus:
THE ABACUS.
This instrument is used for teaching numeration, and the first principles of arithmetic.
Upon a frame are placed wires, parallel to one another, and at equal distances. Ten small balls are strung upon each wire, being placed as in the margin. The right wire denotes units, the next tens, and so on, the 7th wire being the place of millions. In using the abacus, all the balls are first ranged at one end, and a number of them are then moved to the other end of each wire, to correspond to the figures required. The example given in the margin is 15,781, the height of Mount Blanc.
NAPIER'S RODS.
The object of this contrivance is to render arithmetical multiplication more easy, and to secure its correctness; it was much used by astronomers before the invention of logarithms.
To appreciate the merits of this invention, we must consider the process of multiplication as usually performed. Suppose we had to multiply 8,679 by 8:
8,6798———69,432
8,6798———69,432
We first multiply 9 by 8 = 72, and putting down 2 as the first figure in the product, carry the 7 to add to the next product of 7 by 8 = 56; this gives us 63, the 3 being put down as the second figure; 6 is carried to add to the product of 6 by 8, and so on.
A blunder may be made in each part of this process; for 1st, we might reckon 8 times 9 as some other number than 72; 2d, after multiplying the 7 by the 8, we might add to the resulting 56 some other figure than the 7, which we carried; 3d, we may add 56 to 7 inaccurately, making some other sum of it than the right one, 63. Errors in a long multiplication problem are usually made in one of these three ways, and to prevent such errors, Lord Napier[10]introduced this useful contrivance. Thin strips of card, wood, or bone, 9 times as long as they are broad, are each divided into 9 equal squares, a figure is printed or written on the top square, and in each of the squares underneath is the product of multiplying that figure by 2, 3, 4, &c., up to 9.
To use these in multiplication, select the strips, the top figures of which make the number to be multiplied. For example:
To multiply 8,679 by 8, look at the eighth line of squares from the top, and on that line will be found the product of each of the integers 8, 6, 7, 9, when multiplied by 8. We have then to write down the 2 as the first figure of the product,add 7 and 6 together = 13; write 3 as the next figure, carry 1 to add to the sum of 8 and 5, and so on.
The reason for dividing the figures in each square by a diagonal line, and for placing the left-hand figure higher than the right is, that the eye may be thus assisted in adding the carried figure of one slip to the unit of the next.
To provide for the occurrence of more than one of the same figures in the multiplicand, there should be several slips or rods for each of the digits.
In practice the rods are placed on a flat piece of wood, with two ridges at right angles, by which they are preserved in a proper position.
This instrument can be made useful in "divisions," by making by means of it a table of the product of the divisor, multiplied by each of the numbers 1 to 9.
THE ARITHMETICAL BOOMERANG.
The boomerang is an instrument of peculiar form, used by the natives of New South Wales, for the purpose of killing wild fowl and other small animals. If projected forwards, it at first proceeds in a straight line, but afterwards rises in the air, and after performing sundry peculiar gyrations, returns in the direction of the place where it was thrown.
The term is applied to those arithmetical processes by which you can divine a number thought of by another. You throw forwards the number by means of addition and multiplication, and then, by means of subtraction and division, you bring it back to the original starting point, making it proceed in a track so circuitous as to evade the superficial notice of the tyro.
TO FIND A NUMBER THOUGHT OF.
First Method.
This is an arithmetical trick which, to those who are unacquainted with it, seems very surprising; but, when explained, it is very simple. For instance, ask a person tothinkof any number under 10. When he says he has done so, desire him to treble that number. Then ask him whether the sum of the number he has thought of (now multiplied by 3) be odd or even; if odd, tell him to add 1 to make the sum even. He is next to halve the sum, and then treble that half. Again ask whether the amount be odd or even.If odd, add 1 (as before) to make it even, and then halve it. Now ask how many nines are contained in the remainder. The secret is, to bear in mind whether the first sum be odd or even; if odd, retain 1 in the memory; if odd a second time, retain 2 more (making in all 3 to be retained in the memory;) to which add 4 for every nine contained in the remainder.
For example, No. 7 is odd the first and also the second time; and the remainder (17) contains one nine; so that 1, added to 2, make 3, and 3, added to 4, make 7, the number thought of. No. 1 is odd the first time (retain 1), and even the second (of which no notice is taken), but the remainder is not equal to nine. No. 2 is even the first and odd the second time (retain 2), but the remainder contains no nine. No. 3 is odd the first and the second time, still there is no nine in the remainder. No. 4 is even both times, and contains one nine. No. 5 is odd the first time and the remainder contains one nine. No. 6 is odd the second time, and contains one nine in the remainder. No. 8 is even both times, and the remainder contains two nines. No notice need be taken of any overplus of a remainder, after being divided by nine.
The following are illustrations of the result with each number:
Second Method.
EXAMPLE.
Let a person think of a number, say61. Let him multiply it by 3182. Add 1193. Multiply by 3574. Add to this the number thought of63
Let him inform you what is the number produced; it will always end with 3. Strike off the 3, and inform him that he thought of 6.
Third Method.
EXAMPLE.
Suppose the number thought of to be61. Let him double it122. Add 4163. Multiply by 6804. Add 12925. Multiply by 10920
Let him inform you what is the number produced. You must in every case subtract 320; the remainder is, in this example, 600; strike off the two ciphers, and announce 6 as the number thought of.
Fourth Method.
Desire a person to think of a number, say 6. He must then proceed—
EXAMPLE.
1. To multiply this number by itself362. So take 1 from the number thought of53. To multiply this by itself254. To tell you the difference between this product and the former11You must then add 1 to it12And halve this number6
Which will be the number thought of.
Fifth Method.
Desire a person to think of a number, say 6. He must then proceed as follows:
EXAMPLE.
1. Add 1 to it72. Multiply by 3213. Add 1 again224. Add the number thought of28
Let him tell you the figures produced (28):
5. You then subtract 4 from it246. And divide by 46
Which you can say is the number thought of.
Sixth Method.
EXAMPLE.
Suppose the number thought of61. Let him double it122. Desire him to add to this any number you tell him, say 4163. To halve it8
You can then tell him that if he will subtract from this the number he thought of, the remainder will be, in the case supposed, 2.
Note.—The remainder is always half of the number you tell him to add.
TO DISCOVER TWO OR MORE NUMBERS THAT A PERSON HAS THOUGHT OF.
1st Case.—Where each of the numbers is less than 10. Suppose the numbers thought of were 2, 3, 5.
EXAMPLE.
1. Desire him to double the first number making42. To add 1 to it53. To multiply by 5254. To add the second number28
There being a third number, repeat this process—
5. To double it566. To add 1 to it577. To multiply by 52858. To add the third number290
And to proceed in the same manner for as many numbers as were thought of. Let him tell you the last sum produced (in this case 290). Then, if there were two numbers thought of, you must subtract 5; if three, 55; if four, 555. You must here subtract 55, leaving a remainder of 235, which are the numbers thought of, 2, 3 and 5.
2d Case.—Where one or more of the numbers are 10, or more than 10, and where there is anoddnumber of numbers thought of.
Suppose he fixes upon five numbers, viz. 4, 6, 9, 15, 16.
He must add together the numbers as follows, and tell you the various sums: