OF NUMBER THREE.-Select any two numbers you please, and you will find that either one of the two, or their amount when added together, or their difference, is always 3, or a number divisible by 3. Thus, if the numbers are 3 and 8, the first number is 3; let the numbers be 1 and 2, their sum is 3; let them be 4 and 7, the difference is 3. Again, 15 and 22, the first number is divisible by 3; 17 and 26, their difference is divisible by 3, etc.

OF NUMBER FIVE.—If you multiply 5 by itself, and the quotient again by itself, and the second quotient by itself, the last figure of each quotient will always be 5. Thus, 5 × 5 = 25; 25 × 25 = 625; 125 × 125 = 15,625, etc. proceed in the same manner with the figure 6, will constantly be 6.

Again, if you the last figure

To divide any number by 5, or any multiplicand of that number, by means of simple addition :-To divide by 5, double the number given, and mark off the last figure, which will represent tenths. Thus, to divide 261 by 5.:-261 +261 = 522, or 5 22-10ths. Again, to divide the same number by 25, you must take four times the number to be divided, and mark off the last two figures, which will be hundredths, thus, 261 + 261 +261 +261 = 1044, or 10 44-100ths. OF NUMBER NINE.-The following remarkable properties of the number 9 are not generally known :-Thus, 9 × 1 = 9; 9×2= 18, 1 + 8 = 9; 9 × 3 = 27, 2 + 7 = 9, 9 × 4 = 36, 3+6=9: 9 × 545, 4+ 5 9;9 × 6

9 × 7 63, 6+ 3 9; 9 x 872, 7 + 2 8+1= 9.

54, 549; 9; 9 × 981,

It will be seen by the above that-1. The component figures of the product made by the multiplication of every digit into the number 9, when added together, make NINE. 2. The order of these component figures is reversed, after the said number has been multiplied by 5. 3. The component figures of the amount of the multipliers (viz. 45), when added together, make NINE. 4. The amount of the several products, or multiples of 9 (viz. 405), when divided by 9, gives, for a quotient, 45; that is, 4+5= = NINE.

It is also observable, that the number of changes that may be rung on nine bells is 362,880; which figures, added together, make 27; that is, 2 + 7 = NINE.

And the quotient of 362,880, divided by 9, will be 40,320; that is, 4+0 + 3 + 2 + 0 = NINE.

To add a figure to any given number, which shall render it divisible by Nine :-Add the figures together in your mind, which compose the number named; and the figure which must be added to the sum produced, in order to render it divisible by 9, is the one required. Thus, suppose the given number to be 7521 :

Add those together, and 15 will be produced; now 15 requires 3 to render it divisible by 9; and that number 3, being added to 7521, causes the same divisibility:

7521 3

9)7524(836

This exercise may be diversified by your specifying, before the sum is named, the particular place where the figure shall

be inserted, to make the number divisible by 9; for it is exactly the same thing whether the figure be put at the head of the number, or between any two of its digits.

To multiply by Nine by Simple Subtraction.-Supposing you wish to multiply 67583 by 9, add a cipher to the end of the sum, then place the sum to be divided underneath the amount, and subtract it from the same; the quotient will be the product of 67583 multiplied by 9; thus:

TO DISCOVER A SQUARE NUMBER.

A square number is a number produced by the multiplica. tion of any number into itself; thus, 4 multiplied by 4 is equal to 16, and 16 is consequently a square number, 4 being the square root from which it springs. The extraction of the square root of any number takes some time; and after all your labor you may perhaps find that the number is not a square number. To save this trouble, it is worth knowing that every square number ends either with a 1, 4, 5, 6, or 9, o with two ciphers, preceded by one of these numbers.

Another property of a square number is, that if it be divided by 4, the remainder, if any, will be 1-thus, the square of 5 is 25, and 25 divided by 4 leaves a remainder of 1; and again, 16, being a square number, can be divided by 4 without leav ing a remainder.

A MAGIC SQUARE.

The following arrangement of figures, from 1 to 36, in the form of a square, will amount to the same sum if the numbers are cast up perpendicularly, horizontally, or from

TO WEIGH FROM ONE TO FORTY POUNDS WITH FOUR WEIGHTS.

To weigh any weight from 1 to 40 pounds by the use of four weights only, the weights must be respectively 1, 3, 9, and 27 pounds each. Thus, 2 pounds may be weighed by placing 3 pounds in one scale and I in the other; 5 pounds, by placing 1 and 3 in one scale and 9 pounds in the opposite, and so on.

THE CANCELLED FIGURE GUESSED. To tell the figure a person has struck out of the sum of two given numbers :-Arbitrarily command those numbers only that are divisible by 9; such, for instance, as 36, 63, 81, 117, 126, 162, 261, 360, 315, and 432.

Then let a person choose any two of these numbers; and, after adding them together in his mind, strike out from the sum any one of the figures he pleases.

After he has so done, desire him to tell you the sum of the remaining figures; and it follows, that the number which you are obliged to add to this amount, in order to make it 9 or 18, is the one he struck out. Thus

Suppose he chooses the numbers 162 and 261, making altogether 423, and that he strikes out the centre figure, the two other figures will, added together, make 7, which, to make 9. requires 2, the number struck out.

THE DICE GUESSED UNSEEN.

A pair of dice being thrown, to find the number of points on each die without seeing them :-Tell the person who cast the dice to double the number of points upon one of them, and add 5 to it; then to multiply the sum produced by 5, and to add to the product the number of points upon the other die. This being done, desire him to tell you the amount, and, having thrown out 25, the remainder will be a number consisting of two figures, the first of which, to the left, is the number of points on the first die, and the second figure, to the right, the number on the other. Thus

Suppose the number of points on the first die which comes up to be 2, and that of the other 3; then, if to 4, the double of the points of the first, there be added 5, and the sum

produced, 9, be multiplied by 5, the product will be 45; to which, if 3, the number of points on the other die, be added 48 will be produced, from which, if 25 be subtracted, 23 will remain; the first figure of which is 2, the number of points on the first die, and the second figure 3, the number on the other.

THE SOVEREIGN AND THE SAGE.

A sovereign being desirous to confer a liberal reward on one of his courtiers, who had performed some very important service, desired him to ask whatever he thought proper, assuring him it should be granted. The courtier, 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. The value of the reward was immense; for it will be found that the sixty-fourth term of the double progression divided by 1, 2, 4, 8, 16, 32, etc., is 9,223,372,036,854,775,808. But the sum of all the terms of a double progression, beginning with I, may be obtained by doubling the last term, and subtracting from it 1. The number of the grains of wheat, therefore, in the present case, will be 18,446,744,073,709,551,615. Now, if a pint contain 9216 grains of wheat, a gallon will contain 73,728; and, as eight gallons make one bushel, if we divide the above result by eight times 73,728 we shall have 31,274,997,411,295 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, would exceed all the riches, perhaps, on the globe.

THE HORSE-DEALER'S BARGAIN.

A gentleman taking a fancy to a horse, which a horsedealer 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 cent for the first nail, two for the second, four for the third, and so on to the twentyfourth. The gentleman, thinking he should have a good bargain, accepted the offer; how mistaken he was the result will show.

By calculating as before, the twenty-fourth term of the progression, 1, 2, 4, 8, etc., will be found to be 8,388,608, equal to the number of cents the purchaser gave for the horse; the price, therefore, amounted to $83,886.08.

THE DINNER PARTY.

A club of seven persons agreed to dine together every day successively, so long as they could sit down to table differ ently arranged. How many dinners would be necessary for that purpose? It may be easily found, by the rules of simple progression, that the club must dine together 5,040 times before they would exhaust all the arrangements possible, which would require above thirteen years.

THE BASKET AND STONES.

If a hundred stones be placed in a straight line, at the distance of a yard from each other, the first being at the same dis.

tance from a basket, how many yards must the person walk who ngages to pick them up, one by one, and put them into the basket? It is evident that, to pick up the first stone and put it into a basket, the person must walk two yards; for the second, he must walk four; for the third, six; and so on, increasing by two to the hundredth.

The number of yards, therefore, which the person must walk will be equal to the sum of the progression, 2, 4, 6, etc., the last term of which is 200 (22). But the sum of the progression is equal to 202, the sum of the two extremes, multiplied by 50, or half the number of terms; that is to say, 10, 100 yards, which makes more than 5 miles.

THE CAN OF ALE.

How to divide 8 gallons of ale contained in an 8-gallon can into two equal parts, having only two empty bottles to effect it with, one containing 5 gallons and the other 3. First, fill the 5-gallon bottle, and you will have 3 remaining in the 8-gallon can; fill the 3-gallon bottle from the 5-gallon, which will then contain only 2 gallons; pour the 3 gallons in the 3-gallon bottle into that which holds 8 gallons, which will then contain 6 gallons; then pour the 2 gallons out of the 5-gallon bottle into the 3-gallon bottle, and filling the 5-gallon bottle from the 8-gallon can, which at present contains 6 gallons, you will have I gallon in the can, 5 in the largest bottle, and 2 in the smallest; by filling up the 3-gallon can from the 5-gallon, you leave 4 gallons, or one-half the 8 gallons, in the largest bottle; and, lastly, pouring the contents of the 3-gallon bottle into the 8-gallon can, which contains I gallon, you have the second half, or 4 gallons.

COUNTING A BILLION.

What is a billion? The reply is very simple-a million times a million. This is quickly written, and more quickly still pronounced. But no man is able to count it. You can count 160 or 170 a minute; but let us even suppose that you go as far as 200 in a minute, then an hour will produce 12,000; a day, 288,000; and a year, or 365 days (for every four years you may rest from counting, during leap year), 105,120,000. Let us suppose, now, that Adam, at the beginning of his existence, had begun to count, had continued to do so, and was counting still, he would not, even now, according to the usually supposed age of our globe, have counted nearly enough. For, to count a billion, he would require 9512 years,

On inspecting these amounts it will be seen that the last figures run thus-9, 8, 7, 6, 5, 4, 3, 2, I. Again, if we refer to the sums produced by the multiplication of 73 by 3, 6, 9 12, and 15, it will be found that by reading the two figures to the left of each amount backwards, it will give 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

CURIOUS CALCULATION.

If a pin were dropped into the hold of the Great Eastern steamship, and on that day week a second pin, and on that day week four pins, and so on, doubling the number of pins each week for a year, there would, by the end of the 52 weeks, be deposited no fewer than 4,503,599,627,370,495 pins. lowing 200 to the ounce, the weight of the whole would be 628,292,358 tons; and to carry them all would require 27,924 ships as large as the Great Eastern, which is calculated to hold ` 22,500 tons.

ARITHMETICAL PROGRESSION.

Al

If you could buy a hundred ships, giving a farthing for the first, a halfpenny for the second, a penny for the third, twopence for the fourth, and so on to the last, doubling the sum each time, the whole amount paid would be £557,750,707,053,344,041,463,074,442 18s. 73d.-a sum which in words runs thus: 557 quadrillions, 750,707 trillions, 53,344 billions, 41,643 millions, 74 thousand, 442 pounds, eighteen shillings and sevenpence three farthings. This amount in sovereigns would weigh 3,557,083,590,327,499,123,418 tons.

HE following tables show at a glance the price of any number of articles at any given rate: The first column of figures shows the number of articles, and the second, the total; e. g., 87 articles at cent each, amount to 21 cents; 46 articles at 56 cents each, give for a total $25.87. If the price be one not given in the tables, the sum is easily ascertained by adding together two or more of those which are given, thus: 23 articles at 19 cents, 23 at 10 cents would be 2.30, at 5 cents 1.15, at 2 cents 46 cents, at 11 cents, and at 5; adding these totals, 2.30+ 1.15 +46 + 46 +11+5= $4.541, the amount sought for. 52

These tables, too, will operate in other directions, and show the number of articles, when the total amount and the price per capita are given; also when the number of articles and the total amount are given, the price per capita is easily ascertained.

When fractions of a cent are involved, it is often confusing to an inexpert accountant to readily compute the amount. A reference to these tables will quickly give the required result, thereby saving that which is so important to the business man-time.

The convenience of these tables cannot be overrated. Not only do they save time, but are also conducive to perfect accuracy, a result which is not always certainly obtained in a hurried calculation.