who was well acquainted with the science of numbers, only requested that the monarch would give him a quantity of wheat equal to that which would arise from one grain doubled sixty-three times successively. What was the value of the reward? It will be found by calculation that the sixty-fourth term of the double progression 1:2:4:8:16: 32: &c. is 9223372036854775808. But the sum of all the terms of a double progression, beginning with unity, may be obtained by doubling the last term, and subtracting from it unity. The number of the grains of wheat, therefore, in the present case will be 184467440 73709551615. Now, if a standard pint contains 9216 grains of wheat, a gallon will contain 73728: and, as eight gallons make one bushel, if we divide the above result by eight times 73728, we shall have 31274997 411295, for the number of the bushels of wheat equal to the above number of grains, a quantity greater than what the whole surface of the earth could produce in several years, and which, in value, exceeds all the riches perhaps on the globe of the earth. Another problem of the same kind is proposed in the following manner: A gentleman, taking a fancy to a horse, which a horse-dealer wished to dispose of at as high a price as he could, the latter, to induce the gentleman to become a purchaser, offered to let him have the horse for the value of the twenty-fourth nail in his shoes, reckoning one farthing for the first nail, two for the second, four for the third, and so on to the twenty-fourth. The gentleman, thinking he should have a good bargain, accepted the offer: what was the price of the horse? By calculating as before, the twenty-fourth term of the progression 1:2:4:8; &c. will be found to be 8388608, equal to the number of farthings the purchaser gave for the horse. The price therefore, amounted to £8738. 2s. 8d. which is more than any Arabian horse, even of the noblest breed, was ever sold for. We shall conclude this article with some physicomathematical observations on the prodigious fecundity and progressive multiplication of animals and vegetables, which would take place if the powers of nature were not continually meeting with obstacles. 1st. It is not astonishing, that the race of Abraham, after sojourning 260 years in Egypt, should have formed a nation capable of giving uneasiness to the sovereigns of that country. We are told, in the sacred writings, that Jacob settled in Egypt with seventy persons: now, if we suppose, that among these seventy persons, there were twenty too far advanced in life, or too young to have children; that of the remaining fifty, twenty-five were males and as many females, forming twenty-five married couples, and that each couple in the space of twenty-five years, produced, one with another, eight children, which will not appear incredible in a country celebrated for the fertility of its inhabitants; we shall find that, at the end of twenty-five years, the above seventy persons may have increased to two hundred and seventy; from which, if we deduct those who died, there will, perhaps, be no exaggeration in making them amount to two hundred and ten. The race of Jacob, therefore, after sojourning twenty-five years in Egypt, may have been tripled. In like manner, these two hundred and ten persons, after twenty-five years more, may have increased to six hundred and thirty; and so on in triple geometrical progression: hence it follows, that, at the end of two hundred and twenty-five years, the population may have amounted to 1377810 persons, among whom there might easily be five or six hundred thousand adults fit to bear arms. 2nd. If we suppose that the race of Adam, making a proper deduction for those who died, may have been doubled every twenty years, which certainly is not inconsistent with the powers of nature, the number of men at the end of five centuries, may have amounted to 1048576. Now, as Adam lived about 900 years, may have seen, therefore, when in the prime of life, a posterity of 1048576 persons. he 3rd. How great would be the multiplication of many ánimals, did not the difficulty of finding food, the continual war which they carry on against each other, or the numbers of them consumed by man, set bounds for their propagation! It might easily be proved, that the breed of a sow, which brings forth six young, two males and four females, if we suppose that each female produces every year after six young, four of them females and two males, would in twelve years amount to 33554230. Several other animals, such as rabbits and cats, which go with young only for a few weeks, would multiply with still greater rapidity; in half a century the whole earth would not be sufficient to supply them with food, nor even to contain them. If all the ova of a herring were fecundated, a very few years would be sufficient to make its posterity fillthe whole ocean: for every oviparous fish contains thousands of ova, which it deposits in spawning time. Let us suppose, that the number of ova amounts only to 2000, and that these produce as many fish, half males and half females; in the second year there would be more than 200000; in the third, more than 200000000; and in the eighth year, the number would exceed that expressed by 2 followed by twenty-four cyphers. As the earth contains scarcely so many cubic inches, the ocean, if it covered the whole globe, would not be sufficient to contain all these fish, the produce of one herring in eight years! 4. Many vegetable productions, if all their seeds were put into the earth, would, in a few years cover the whole surface of the globe The hyosciamus, which, of all the known plants produces, perhaps, the greatest number of seeds, would, for this purpose require no more than four years. According to some experiments, it has been found that one stem of the hyosciamus produces sometimes more than 50000 seeds: now, if we admit the number to be only 10000, at the fourth crop it would amount to a 1 followed by sixteen cyphers. But, as the whole surface of the earth contains no more than 5507634452576256 square feet, if we allow to each plant only one square foot, it will be seen that the whole surface of the earth would not be sufficient for the plants produced from one hyosciamus at the end of the fourth year! EXERCISES IN THE SINGLE AND COMPOUND (26.) We shall confine ourselves to a small number of examples in this rule, which we briefly explained in the introduction. SINGLE RULE OF THREE DIRECT. EXAMPLE I. If 40 pioneers can dig a trench 268 yards long, in a certain time, how many yards can 60 pioneers dig in the same time? 40 60 268 x 402. (23.) EXAMPLE II. If a ship with a fresh breeze, sails 200 leagues in three days, how long time will she require to sail 2000 leagues, every other circumstance being the same? 200 2000 :: 3 : x = 30. EXAMPLE III. If 52 yards 2 feet and 5 inches of mason work cost £168. 98. 4d. what will be the expence of 77 yards, 2 feet 8 inches of the like kind of work, at the same rate? To render the solution of this problem easier, the quantity of each piece of work must be reduced to inches, by multiplying the yards by 3 and 12; and, for the same reason, the price of the work must be reduced to pence. We shall then have the following propor Inches, Inches. Peuce. 1901 2804 :: 40432: x. (23.) SINGLE RULE OF THREE INVERSE. EXAMPLE I. If 30 men can perform a certain piece of work in 25 days, how many men will be requisite to perform the same work in 10 days? It is here evident that, as the work is to be done in a shorter time, it will require more men. Consequently, the proportion must be expressed in this manner: Days. Days. Men. 10: 25: 30 : x = 75. EXAMPLE II. A vessel has provisions for 15 days, but being obliged by certain circumstances, to continue at sea for 20 days, to what quantity must the daily ration of each man be reduced, to make the provisions last during that time? If the quantity of provisions consumed daily be represented by unity, it is evident that the reduced quantity must be as much below 1 as 15 are less than 20. We shall therefore have 20: 15 1 : x = 2. COMPOUND RULE OF THREE. EXAMPLE I. If 30 men perform 132 yards of work in 18 days, how much will 54 men perform in 28 days? Thirty men, working 18 days, will perform the same work as 18 times 30 men 540, in one day; and, in like manner, 54 men, in 28 days, will perform the same work as 54 times 28 men 1512, in one day. We have, therefore, the following proportion: 540 132: 1512; x. |