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 quant.i.ty, 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 quant.i.ty 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 s.p.a.ce 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 mult.i.tudes have been induced to invest in them the savings which had been acc.u.mulated 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 acc.u.mulate 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 circ.u.mstance, 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 a.s.sume 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 pa.s.s, 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