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 at D. Should the water have no motion, the turning of the handle at
B will elevate the water from the lower to the higher level.
[Illustration]
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 of _Hydrostatics_ were 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 blow through
the quill, all the water will be emptied from the glass, which will be
filled with air.
TRICKS IN ACOUSTICS.
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.
[Illustration]
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
line very 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.
[Illustration]
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.
[Illustration]
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 will ring 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.
[Illustration]
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 best calculated for the
exhibition of this phenomenon. If a person stand near the wall, with his
face turned to it, and whisper a 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, about three 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 string _hears_ the 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 to one 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.
[Illustration]
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, formed by four pillars _a a a a_, connected by upper cross
rails _b b_, and similar rails below, while it terminated above in four
bent wires _c c_, proceeding at right angles of the frame, and meeting in
a central point. The hollow copper ball _d_, 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 ribbons _r 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 of _Acoustics_ furnished 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 the
phenomena of nature.
TO SHOW HOW SOUND TRAVELS THROUGH A SOLID.
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.
FOOTNOTES:
[Footnote 9: D is a good note for it. The upper string may be tuned to the
upper D, and the two lower to the lower D, and D D. The "harmonics," are
the sounds produced.]
THE MAGIC OF NUMBERS
OR, CURIOUS PROBLEMS IN ARITHMETIC.
[Illustration]
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--
1 2 3 4 5
2 4 8 16 32
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:
1 2 3 o o o
4 5 6 o o o
7 8 9 o o o
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 and perpendicular lines.
Suppose it was necessary to add together the numbers 763, 124, 859, the
cubes and pegs would be arranged thus:
o o o o o o o o ☻
o o o o o o o o o
o o o o o o o o o
☻ o o o ☻ o o o o
o o o o o o ☻ o o
o o o o o o o o o
o o o o o o o o o
o o o o ☻ o o o o
o ☻ o o o o o o ☻
☻ o o o o o o o o o o o
o o o o o o ☻ o o o o ☻
o o o o o o o o o o o o
THE ABACUS.
This instrument is used for teaching numeration, and the first principles
of arithmetic.
[Illustration]
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:
8,679
8
------
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.
[Illustration:
+-----+-----+-----+-----+-----+
| | | | | |
| 1 | 8 | 6 | 7 | 9 |
|-----+-----+-----+-----+-----|
| | | | | |
| 2 | 1/6 | 1/2 | 1/4 | 1/8 |
|-----+-----+-----+-----+-----|
| | | | | |
| 3 | 2/4 | 1/8 | 2/1 | 2/7 |
|-----+-----+-----+-----+-----|
| | | | | |
| 4 | 3/2 | 2/4 | 2/8 | 3/6 |
|-----+-----+-----+-----+-----|
| | | | | |
| 5 | 4/0 | 3/0 | 3/5 | 4/5 |
|-----+-----+-----+-----+-----|
| | | | | |
| 6 | 4/8 | 3/6 | 4/2 | 5/4 |
|-----+-----+-----+-----+-----|
| | | | | |
| 7 | 5/6 | 4/2 | 4/9 | 6/3 |
|-----+-----+-----+-----+-----|
| | | | | |
| 8 | 6/4 | 4/8 | 5/6 | 7/2 |
|-----+-----+-----+-----+-----|
| | | | | |
| 9 | 7/2 | 5/4 | 6/3 | 8/1 |
+-----+-----+-----+-----+-----+
]
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 to _think_ of 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: