always manages to keep up with the times and exploit the latest mystery.
At one time it was psychic force, then Roentgen or X-rays; lately it has
been attributed to the mysterious effects of radium!
This illustration is more properly a delusion; there is no illusion
about it.
CURIOUS ARITHMETICAL PROBLEMS
THE CHESS-BOARD PROBLEM
An Arabian author, Al Sephadi, relates the following curious anecdote:
A mathematician named Sessa, the son of Dahar, the subject of an Indian
Prince, having invented the game of chess, his sovereign was highly
pleased with the invention, and wishing to confer on him some reward
worthy of his magnificence, desired him to ask whatever he thought
proper, assuring him that it should be granted. The mathematician,
however, only asked for a grain of wheat for the first square of the
chess-board, two for the second, four for the third, and so on to the
last, or sixty-fourth. The prince at first was almost incensed at this
demand, conceiving that it was ill-suited to his liberality. By the
advice of his courtiers, however, he ordered his vizier to comply with
Sessa's request, but the minister was much astonished when, having
caused the quantity of wheat necessary to fulfil the prince's order to
be calculated, he found that all the grain in the royal granaries, and
even all that in those of his subjects and in all Asia, would not be
sufficient.
He therefore informed the prince, who sent for the mathematician, and
candidly acknowledged that he was not rich enough to be able to comply
with his demand, the ingenuity of which astonished him still more than
the game he had invented.
It will be found by calculation that the sixty-fourth term of the double
progression, beginning with unity, is
9,223,372,036,854,775,808,
and the sum of all the terms of this double progression, beginning with
unity, may be obtained by doubling the last term and subtracting the
first from the sum. The number, therefore, of the grains of wheat
required to satisfy Sessa's demand will be
18,446,744,073,709,551,615.
Now, if a pint contains 9,216 grains of wheat, a gallon will contain
73,728, and a bushel (8 gallons) will contain 589,784. Dividing the
number of grains by this quantity, we get 31,274,997,412,295 for the
number of bushels necessary to discharge the promise of the Indian
prince. And if we suppose that one acre of land is capable of producing
in one year, thirty bushels of wheat, it would require 1,042,499,913,743
acres, which is more than eight times the entire surface of the globe;
for the diameter of the earth being taken at 7,930 miles, its whole
surface, including land and water, will amount to very little more than
126,437,889,177 square acres.
If the price of a bushel of wheat be estimated at one dollar, the value
of the above quantity probably exceeds that of all the riches on the
earth.
THE NAIL PROBLEM
A gentleman took a fancy to a horse, and the dealer, to induce him to
buy, offered the animal for the value of the twenty-fourth nail in his
shoe, reckoning one cent for the first nail, two for the second, four
for the third, and so on. The gentleman, thinking the price very low,
accepted the offer. What was the price of the horse?
On calculating, it will be found that the twenty-fourth term of the
progression 1, 2, 4, 8, 16, etc., is 8,388,608, or $83,886.08, a sum
which is more than any horse, even the best Arabian, was ever sold for.
Had the price of the horse been fixed at the value of all the nails, the
sum would have been double the above price less the first term, or
$167,772.15.
A QUESTION OF POPULATION
The following note on the result of unrestrained propagation for one
hundred generations is taken from "Familiar Lectures on Scientific
Subjects," by Sir John F. W. Herschel:
For the benefit of those who discuss the subjects of population, war,
pestilence, famine, etc., it may be as well to mention that the number
of human beings living at the end of the hundredth generation,
commencing from a single pair, doubling at each generation (say in
thirty years), and allowing for each man, woman, and child, an average
space of four feet in height and one foot square, would form a vertical
column, having for its base the whole surface of the earth and sea
spread out into a plane, and for its height 3,674 times the sun's
distance from the earth! The number of human strata thus piled, one on
the other, would amount to 460,790,000,000,000.
In this connection the following facts in regard to the present
population of the globe may be of interest:
The present population of the entire globe is estimated by the best
statisticians at between fourteen and fifteen hundred millions of
persons. This number would easily find standing-room on one half of Long
Island, in the State of New York. If this entire population were to be
brought to the United States, we could easily give every man, woman, and
child, one acre and a half each, or a nice little farm of seven acres
and a half to every family, consisting of a man, his wife, and three
children.
This question has also an important bearing on the preservation of
animals which, in limited numbers, are harmless and even desirable. In
Australia, where the restraints on increase are slight, the rabbit soon
becomes not only a nuisance but a menace, and in this country the
migratory thrush or robin, as it is generally called, has been so
protected in some localities that it threatens to destroy the small
fruit industry.
HOW TO BECOME A MILLIONAIRE
Many plans have been suggested for getting rich quickly, and some of
these are so plausible and alluring that multitudes have been induced to
invest in them the savings which had been accumulated by hard labor and
severe economy. It is needless to say that, except in the case of a few
stool-pigeons, who were allowed to make large profits so that their
success might deceive others and lead them into the net, all these
projects have led to disaster or ruin. It is a curious fact, however,
that some of those who invested in such "get-rich-quickly" schemes were
probably fully aware of their fraudulent character and went into the
speculation with their eyes open in the hope that _they_ might be
allowed to become the stool-pigeons, and in this way come out of the
enterprise with a large balance on the right side. No regret can be felt
when a bird of this kind gets plucked.
But by the following simple method every one may become his own promoter
and in a short time accumulate a respectable fortune. It would seem that
almost any one could save one cent for the first day of the month, two
cents for the second, four for the third, and so on. Now if you will do
this for thirty days we will guarantee you the possession of quite a
nice little fortune. See how easy it is to become a millionaire on
paper, and by the way, it is only on paper that such schemes ever
succeed.
If, however, you should have any doubt in regard to your ability to lay
aside the required amount each day, perhaps you can induce some
prosperous and avaricious employer to accept the following tempting
proposition:
Offer to work for him for a year, provided he pays you one cent for the
first week, two cents for the second, four for the third, and so on to
the end of the term. Surely your services would increase in value in a
corresponding ratio, and many business men would gladly accept your
terms. We ourselves have had such a proposition accepted over and over
again; the only difficulty was that when we insisted upon security for
the last instalment of our wages, our would-be employers could never
come to time. And we would strongly urge upon our readers that if they
ever make such a bargain, they get full security for the last payment
for they will find that when it becomes due there will not be money
enough in the whole world to satisfy the claim.
The entire amount of all the money in circulation among all the nations
of the world (not the _wealth_) is estimated at somewhat less than
$15,000,000,000, and the last payment would amount to fifteen hundred
times that immense sum.
The French have a proverb that "it is the first step that costs" (_c'est
le premier pas qui coute_) but in this case it is the last step that
costs and it costs with a vengeance.
While on this subject let me suggest to my readers to figure up the
amount of which they will be possessed if they will begin at fifteen
years of age and save ten cents per week for sixty years, depositing the
money in a savings bank as often as it reaches the amount required for a
deposit, and adding the interest every six months. Most persons will be
surprised at the result.
THE ACTUAL COST AND PRESENT VALUE OF THE FIRST FOLIO SHAKESPEARE
Seven years after the death of Shakespeare, his collected works were
published in a large folio volume, now known as "The First Folio
Shakespeare." This was in the year 1623. The price at which the volume
was originally sold was one pound, but perhaps we ought to take into
consideration the fact that at that time money had a value, or
purchasing power, at least eight times that which it has at present;
Halliwell-Phillips estimates it at from twelve to twenty times its
present value. For this circumstance, however, full allowance may be
made by multiplying the ultimate result by the proper number.
This folio is regarded as the most valuable printed book in the English
language--the last copy that was offered for sale in good condition
having brought the record price of nearly $9,000, so that it is safe to
assume that a perfect copy, in the condition in which it left the
publisher's hands, would readily command $10,000, and the question now
arises: What would be the comparative value of the present price,
$10,000, and of the original price (one pound) placed at interest and
compounded every year since 1623?
Over and over again I have heard it said that the purchasers of the
"First Folio" had made a splendid investment and the same remark is
frequently used in reference to the purchase of books in general,
irrespective of the present intellectual use that may be made of them.
Let us make the comparison.
Money placed at compound interest at six per cent, a little more than
doubles itself in twelve years. At the present time and for a few years
back, six per cent is a high rate, but it is a very low rate for the
average. During a large part of the time money brought eight, ten, and
twelve per cent per annum, and even within the half century just past it
brought seven per cent during a large portion of the time. Now, between
1623 and 1899, there are 23 periods, of 12 years each, and at double
progression the twenty-third term, beginning with unity, would be
8,388,608. This, therefore, would be the amount, in pounds, which the
volume had cost up to 1899. In dollars it would be $40,794,878.88. An
article which costs forty millions of dollars, and sells for ten
thousand dollars, cannot be called a very good financial investment.
From a literary or intellectual standpoint, however, the subject
presents an entirely different aspect.
Some time ago I asked one of the foremost Shakespearian scholars in the
world if he had a copy of the "First Folio." His reply was that he
could not afford it; that it would not be wise for him to lose $400 to
$500 per year for the mere sake of ownership, when for a very slight
expenditure for time and railway fare he could consult any one of
half-a-dozen copies whenever he required to do so.
ARITHMETICAL PUZZLES
A good-sized volume might be filled with the various arithmetical
puzzles which have been propounded. They range from a method of
discovering the number which any one may think of to a solution of the
"famous" question: "How old is Ann?" Of the following cases one may be
considered a "catch" question, while the other is an interesting
problem.
A country woman, carrying eggs to a garrison where she had three guards
to pass, sold at the first, half the number she had and half an egg
more; at the second, the half of what remained and half an egg more; at
the third the half of the remainder and half an egg more; when she
arrived at the market-place she had three dozen still to sell. How was
this possible without breaking any of the eggs?
At first view, this problem seems impossible, for how can half an egg be
sold without breaking any? But by taking the greater half of an odd
number we take the exact half and half an egg more. If she had 295 eggs
before she came to the first guard, she would there sell 148, leaving
her 147. At the next she sold 74, leaving her 73. At the next she sold
37, leaving her three dozen.
The second problem is as follows: After the Romans had captured Jotopat,
Josephus and forty other Jews sought shelter in a cave, but the refugees
were so frightened that, with the exception of Josephus himself and one
other, they resolved to kill themselves rather than fall into the hands
of their enemies. Failing to dissuade them from this horrid purpose,
Josephus used his authority as their chief to insist that they put each
other to death in an orderly manner. They were therefore arranged round
a circle, and every third man was killed until but two men remained, the
understanding being that they were to commit suicide. By placing himself
and the other man in the 31st and 16th places, they were the last that
were left, and in this way they escaped death.
ARCHIMEDES AND HIS FULCRUM
Next to that of Euclid, the name of Archimedes is probably that which is
the best known of all the mathematicians and mechanics of antiquity, and
this is in great part due to the two famous sayings which have been
attributed to him, one being "Eureka"--"I have found it," uttered when
he discovered the method now universally in use for finding the specific
gravity of bodies, and the other being the equally famous dictum which
he is said to have addressed to Hiero, King of Sicily,--"Give me a
fulcrum and I will raise the earth from its place."
That Archimedes, provided he had been immortal, could have carried out
his promise, is mathematically certain, but it occurred to Ozanam to
calculate the length of time which it would take him to move the earth
only one inch, supposing his machine constructed and mathematically
perfect; that is to say, without friction, without gravity, and in
complete equilibrium, and the following is the result:
For this purpose we shall suppose that the matter of which the earth
is composed weighs 300 pounds per cubic foot, this being about the
ascertained average. If the diameter of the earth be 7,930 miles,
the whole globe will be found to contain 261,107,411,765 cubic
miles, which make 1,423,499,120,882,544,640,000 cubic yards, or
38,434,476,263,828,705,280,000 cubic feet, and allowing 300 pounds to
each cubic foot, we shall have 11,530,342,879,148,611,584,000,000 for
the weight of the earth in pounds.
Now, we know, by the laws of mechanics, that, whatever be the
construction of a machine, the space passed over by the weight, is to
that passed over by the moving power, in the reciprocal ratio of the
latter to the former. It is known also, that a man can act with an
effort equal only to about 30 pounds for eight or ten hours, without
intermission, and with a velocity of about 10,000 feet per hour. If then
we suppose the machine of Archimedes to be put in motion by means of a
crank, and that the force continually applied to it is equal to 30
pounds, then with the velocity of 10,000 feet per hour, to raise the
earth one inch the moving power must pass over the space of
384,344,762,638,287,052,800,000 inches; and if this space be divided by
10,000 feet or 120,000 inches, we shall have for a quotient
3,202,873,021,985,725,440, which will be the number of hours required
for this motion. But as a year contains 8,766 hours, a century will
contain 876,600; and if we divide the above number of hours by the
latter, the quotient, 3,653,745,176,803, will be the number of
centuries during which it would be necessary to make the crank of the
machine continually turn in order to move the earth only one inch. We
have omitted the fraction of a century as being of little consequence in
a calculation of this kind. The machine is also supposed to be
constantly in action, but if it should be worked only eight hours each
day, the time required would be three times as long.
So that while it is true that Archimedes could move the world, the space
through which he could have moved it, during his whole life, from
infancy to old age, is so small that even if multiplied two hundred
million times it could not be measured by even the most delicate of our
modern measuring instruments.
There is a modern saying which has become almost as famous amongst
English-speaking peoples as is that of Archimedes to the world at large.
It is that which Bulwer Lytton puts into the mouth of Richelieu, in his
well-known play of that name:
"Beneath the rule of men entirely great
THE PEN IS MIGHTIER THAN THE SWORD."
About thirty years ago it occurred to the writer that these two
epigrammatic sayings--that of Archimedes and that of Bulwer Lytton,
might be symbolized in an allegorical drawing which would forcibly
express the ideas which they contain, and the question immediately
arose--Where will Archimedes get his fulcrum and what can he use as a
lever?
And the mental answer was: Let the pen be the lever and the printing
press the fulcrum, while the sword, used for the same purpose but
resting on glory, or in other words, having no substantial fulcrum,
breaks in the attempt. The little engraving which, with a new motto,
forms a fitting tail-piece to this volume, was the outcome.
It is true that the pen is mighty, and in the hands of philosophers and
diplomats it accomplishes much, but it is only when resting on the
printing press that it is provided with that fulcrum which enables it to
raise the world by diffusing knowledge, inculcating morality, and
providing pleasure and culture for humanity at large.
When assigned to such a task the sword breaks, and well it may. But we
have a well-grounded hope that through the influence of the pen and the
printing press there will soon come an era of universal
[Illustration: Peace on Earth and Good Will Toward Men.]
INDEX
Absurdities in perpetual motion, 42
Accuracy of modern methods of squaring the circle, 17
Adams, perpetual motion, 71
Ahaz, dial of, 133
Air, liquid, 65
Alkahest, or universal solvent, 104
Altar of Apollo, 30
Angelo, Michael, finely engraved seal, 136
Angle, Trisection of, 33
Apollo, Altar of, 30
Approximations to ratio of diameter to circumference of circle, 17
De Morgan's Illustration of, 18
New Illustration of, 19
Archimedean screw, 49
Archimedes, area of circle, 13
Ratio of circumference to diameter, 14
Archimedes and his fulcrum, 171
Arithmetic of the ancients, 15
Arithmetical problems, 163
Chess-board problem, 163
Nail problem, 164
A question of population, 165
How to become a millionaire, 166
Cost of first folio Shakespeare, 168
Arithmetical puzzles, 170
Archimedes and his fulcrum, 171
Army Medical Museum, 142
Ball, Prof. W. W. R., 39, 129, 133, 134
Balloons for conveying letters, 147
Balls--proportion of weight to diameter, 32
Bean, jumping, 128
Bells kept ringing for eight years, 41
Bible in walnut shell, 136
Bible, written at rate of 22 to square inch, 141
Boat-race without oars, 129
Bolognian phosphorus, 102
Boots--lifting oneself by straps of, 128
Boyle and palingenesy, 107
Bramwell, Sir Frederick, 38
Brick, to look through, 151
Buckle and geometrical lines, 119
"Budget of Paradoxes," De Morgan, 6, 18, 118
Carbon bisulphide for perpetual motion, 67
Capillary attraction, 53
Carpenter, Edward--fourth dimension, 122
Catherine II, 118
"Century of Inventions," 74
Chess-board problem, 163
Child lifting two horses, 131
Perpetual motion by a, 64
Circle, squaring the, 9
Supposed reward for squaring the, 9
Resolution of Royal Academy of Sciences on, 10
What the problem is, 12
Approximation to, by Archimedes, 14
Jews, ratio accepted by, 13
Egyptians, ratio accepted by, 14
Symbol for ratio introduced by Euler, 14
Graphical approximations, 22
Circumference of circle, to find, when diameter is given, 22
Clock that requires no winding, 38
Columbia College seal, 140
Column of De Luc, 40
Compass, watch used as a, 134
Congreve, Sir William, 53
Cube, duplication of, 38
Crystallization seen by microscope, 108
Mistaken for palingenesy, 100
Dancer--microphotographs, 144
Dangerous, fascination of the, 1
Declaration of Independence, 145
De Luc's column, 40
De Morgan--Legend of Michael Scott, 6
Ignorance _v._ learning, 8
Illustration of accuracy of modern attempts to square the circle, 18
"Budget of Paradoxes," 6, 18
Trisection of angle, 34, 118
On powder of sympathy, 112
Anecdote of Diderot, 118
Dial of Ahaz, 133
Diderot, anecdote of, 118
Digby, Sir Kenelm, and palingenesy, 109
Sir Kenelm and powder of sympathy, 111
Dircks, 56, 71, 75
Discoveries, valuable, not due to perpetual-motion-mongers, 36
Duplication of the cube, 30
Elixir of life, 95
Engineering, insect, 130
Euler, 14, 118
Fallacies in perpetual motion, 65
Falstaff and the philosopher's stone, 97
Faraday's discovery, 93
Farrants, Prest. Royal Mic. Soc, 140
Figure, a, enlarged by cutting, 126
First folio Shakespeare, cost of, 168
Fixation of mercury, 92
Follies of Science, The Seven, 2
D'Israeli's list, 2
An inappropriate term, 3
Fourth dimension--conception of, 117
Flatland, 120
Kant and Gauss, 121
Spiritualists, 121
Edward Carpenter on, 122
Possibility of a new sense, 123
Frauds in perpetual motion, 69
Freezing of mercury, 93
Froment, micrographs, 139
Gases, liquefaction of, 93
Geiser's clock, 71
Geometrical quadrature impossible, 21
Gibberish, origin of word, 96
God, demonstration of existence of, 118
Hammer made of solid mercury, 93
Hand, to look through, 156
Heat and cold, illusions, 150
Hesse, Landgrave of, 77
Hindoos, ratio accepted by, 16
Holmes, O. W., and powder of sympathy, 111
Homer's Iliad in nutshell, 136
Honecourt, Wilars de, 42
Horses lifted by child, 131
Hydrofluoric acid, 104
Hydrostatic paradox, 46
Iliad of Homer in nutshell, 136
Impossible, fascination of the, 1
Insect engineering, 130
Irradiation, 152
Jews, ratio accepted by the, 13
Keeley gold cure, 97
Keeley motor, 69
Kircher and palingenesy, 106
Lacomme, on squaring circle, 27
Lamps, ever-burning, 100
Library, Congressional, in hand-bag, 145
Light from electric earth-currents, 103
Lines, geometrical, 119
Lines, direction of, deceptive, 154
Length of, deceptive, 153
Liquid air, 65
Lodge, Sir Oliver, on conservation of energy, 5
Longitude, relation of squaring the circle to, 10
McArthur, on arithmetic of ancients, 15
Machin, 16
Magnetism for perpetual motion, 61
Man lifting himself, 128
Mathematicians--how they go to heaven, 8
Mercury, fixation of, 92
Freezing of, 93
Metals. See _Transmutation_.
Metius, Peter, 16
Micrography, or minute writing, 136
Homer in a nutshell, 136
Michael Angelo's seal, 136
Ten Commandments, 136
Bible in a nutshell, 136
Earliest micrographic engraving, 139
Micrographic copy of seal of Columbia College, 139
Peters' machine, 141
Lord's Prayer written at rate of 22 Bibles to square inch, 141
Webb's fine writing, 142
Calculation in regard to, 143
Microphotographs by Dancer, 144
Pigeon-post in Franco-Prussian War, 146
Millionaire, to become a, 166
Miracle--dial of Ahaz, 133
Morgan. See _De Morgan_.
Morton, President Henry, 66
Motion, perpetual. See _Perpetual motion_.
Muir, Prof. On Archimedes, 14
Musitanus, Carolus, 96
Nail problem, 164
Nicomedean line, 29
Orffyreus--his real name, 77
His fraudulent machine, 77
Overbalancing wheels, 43
Paint, luminous, 102
Palingenesy, 106
Patent office U. S. and perpetual motion, 42
Pen mightier than the sword, 173
Perpetual lamps, 100