74. How many minutes, are there, from the 13th July, at 43 minutes past 9 in the morning to the 5th Nov., at 19 m. past 3 in Ans. 165,936. 75. How many sec. in 8 yrs. 3 mo. 2 wks. 2 d. 19 h. 43 m. 57 sec., allowing 13 mo. 1 d. 6 h. to the year? Ans. 261,171,837.

the afternoon.

76. How many seconds old are you?

77. In 156 T. 1 cwt. 0 qr. 2 lb. 6 oz. how many oz.?

78. In 19 lb. 13 oz. 7 dr. how many dr.? 79. In 90€. 17. 9d. i qr. how many qrs.?

Ans. 5,592,870.

Ans. 5,079. Ans. 87,253.

80. In 34 lbs. 5 oz. 19 dwts. 10 gr. how many gr.? A. 198,706. 81. In 27 th. 93. 23. 19. 16 gr. how many gr. ?

82. At 12 cts a grain, what must be paid for a gold cup, weighing 1 lb. 3 oz. 13 dwt. 17 gr.?

83. At 6 cts. a pint, what cost 27 bu. 3 pks. 7 qt. 1 pt. of wheat? 84. At 5 cts. a pint, what cost 16 butts, 2 bar. 1 kil. 1 fir. 7 gal. 3 qts. 1 pt. of ale?

85. In 16 yrs. 24d how many sec. allowing 365 d. to the year? Ans. 506,649,600.

86. A merchant sold 84 yds. 3 qrs. 2 na. of broadcloth at $0.50 a nail. In return he received 25 boxes of raisins at $4.00 a box, and the rest in flour at $11.00 a barrel. How much flour did he receive?

87. A farmer sold 6 loads of wood, at $3.00 a load; and 3 cwt. of butter, at 9 cts. a pound; and 70 bushels of wheat, at $2.125 a bushel; and 93 bushels of rye, at 75 cts. a bushel. In return he received ploughs, at $12.00 apiece; 1 yoke of oxen, $60.00; 1 pr. of horses, at 75 dollars apiece; and the rest in money. How much money did he receive? Ans. $20.74. NOTE. The last two questions belong to the rule, commonly called Barter.

OBSERVATIONS ON MULTIPLICATION, FOR ADVANCED PUPILS. § XIX. The order of writing numbers for multiplication, requi. red by the rule, is only adopted for the sake of convenience. We need some certain method, and this is as good as any.

It is manifestly of no consequence which of the figures of the multiplier is first used, since if the partial products are set down correctly, the answer will be the same, whether we begin on the left, on the right, or somewhere in the middle.

In order to multiply with ease, it is necessary that the several products of all the possible combinations of two digits, should be fixed in the memory; exactly as in Addition, it was found necessary to retain the sums of the same combinations in the mind. A little reflection will show, that it is even more necessary to be acquainted with a multiplication table, than with a table for addition. For in adding, we might add numbers unit by unit, as children do; but in multiplying, this process would become very tedious, and liable to

error.

The following table, generally supposed to have been in

vented by PYTHAGORAS, a Greek philosopher, contains all the products which are absolutely necessary to be learned. It is convenient to know the products of numbers still higher.

In order to find the product of two numbers in this table; look for one of the numbers in the left hand column, and for the other in the upper row. Then, in a line with the former, and directly under the latter, you will find the product sought. Thus, opposite 9, and under 6, will be found 54, the product of 9 and 6.

The construction of this table is very simple. It may be entirely formed by Addition. In the upper row, let the nine digits be written down in order. Then, let every figure in this row be added to itself, and the amount placed directly under the figure thus added. By this process, the second row is found. For the third, add each number in the second, to the one above it in the first. For the fourth, add, in the same manner, the third and first; or add the second to itself. For the fifth, add the fourth and first; or the third and second. For the sixth, add the fifth and first; or the fourth and second; or add the third to itself, and so on.

The former of the two modes of proof, given in the last section. is that found in common Arithmetics. The latter embraces several varieties, which the inquisitive student will discover for himself. The natural mode of proof, by reversing the operation, would be to divide the product by one of the factors. The quotient, of course, ought to be the other factor.

Multiplication may also be proved by the casting out of nines, as

To multiply 672 by 9 and 2, then, is the same, in effect, as to multiply it by 11. 692×9, is, of course, even 9s. In multiplying 672 by 2, multiply first its even 9s by 2. This product, is, likewise, even 9s. Then, multiply the 6, (excess,) by 2. If this product be even 9s, the whole is plainly even 9s. If not, the whole will have the same excess, which the product of 6 and 2 has. 6×2=12-9+3. Then, 672×11, ought to have 3 for its excess of 9s. Casting out 9s from 7,392, the product obtained above, 3 remains. Hence, we conclude the work right.

From this explanation, we derive the following rule:

I. REJECT THE 9S FROM THE MULTIPLIER AND MULTIPLICAND.

II. MULTIPLY THE EXCESSES TOGETHER, and REJECT THE 9S FROM

III. REJECT THE 9S FROM THE PRODUCT, AND IF THE EXCESS, THUS found, agreE WITH THE LAST, THE OPERATION MAY BE CONSIDERED AS CORRECTLY PERFORMED.

We may prove Multiplication in the same manner, by casting out 3s, for the reasons mentioned under Addition.

We have seen that Multiplication is an abbreviation of Addition. It may seem singular that addition should be employed to abbreviate multiplication. It is a fact, however, that by means of what are called Logarithms, the nature of which we cannot explain here, tedious multiplications may be performed by the mere addition of two numbers.

We will here insert some useful hints, not found in the ordinary Arithmetics. They may be classed under the head of

ABBREVIATED MULTIPLICATION.

ART. I. It is easier, always, to multiply by small figures, than by large; because we have smaller numbers to carry.

Let us take the following example.

1275

93

Here, we shall find it most convenient, after obtaining a product by 3, to multiply that product by 3 again, instead of multiplying the multiplicand by 9. The result will be the saine in either case-for

3825 11475

3 times 3 are 9.

118575

It will be observed that this operation is the same, in principle, with the rule for composite numbers. Take another example.

8136

65088 195264

Here, after multiplying by 8, we multiply the product thus ob 248 tained, by 3; which is equivalent to multiplying the multiplicand by 24, the remaining part of the multiplier. In this case, we obtain a single product, where the common rule would make two necessary. Hence, WE MAY OFTEN OBTAIN A PARTIAL PRODUCT, FROM ONE ALREADY OBTAINED, BY MULTIPLYING ACCORDING TO 2017728 THE RULE FOR COMPOSITE NUMBERS. When the larger number comes first, we may often find a second partial product by division.

ART. II. We have seen, (§ xv.) that annexing a cypher to any number, multiplies that number by 10. Now 5 is half of ten; and 5 times any number, is half of 10 times the same number. If, then, we annex a cypher to any number, and then divide that number, with the cypher annexed, by 2; we shall obtain a result, equal to 5 times the original number.

Thus, to multiply 724 by 5; annex first a cypher, thus, 7,240. Then, take half of this, and it is 3,620, which is 5 times 724.

To multiply 531 by 5. Annex a cypher, thus, 5,310, result is 2,655, which is 5 times 531.

Divide by 2, and the

Hence, To multiply by 5, ANNEX A CYPHER TO THE MULTIPLICAND, AND DIVIDE BY 2.

ART. III. 15 is 10 and 5. Therefore, if we multiply a number by 10, and then by 5, and add the two products, their sum will be 15 times the number mul. tiplied. But, (9 xv.) annexing a cypher is multiplying by 10; and (11.) half the product by 10, is the product by 5.

Then to multiply 337 by 15.
Multiply first by 10: thus,
Divide by 2; thus,

Add these results: thus,

3370

1685

5055 15 times 337.

Hence, to multiply by 15, ANNEX A CYPHER TO THE MULTIPLICAND, AND

ADD THIS RESULT TO HALF ITSELF.

ART. IV. To multiply by 100, (§ xv.) we annex two cyphers. But 25 is a quarter of a hundred. Therefore, 25 times a number, is a quarter as much as 100 times the same number. Of course, if we multiply by 100, and divide the result by 4, we shall obtain a result equal to 25 times the number multiplied. Thus, multiply 452 by 25.

It is multiplied by 100 by annexing two cyphers; thus, 45,200
Divide this result by 4, and we obtain

11,300 25 times 452. Hence, To multiply by 25, ANNEX TWO CYPHERS TO THE MULTIPLICAND, AND DIVIDE BY 4.

ART. V. 9 is 1 less than 10. Therefore, 9 times a number is once that number less than 10 times the same number. Now annexing a cypher, (§ xv.) multiplies by 10. If, then, after multiplying by 10, we take away once the number, we have 9 times the number left.

Thus, to multiply 87 by 9.

First annex a cypher, which multiplies by 10, and then take away once 87.

870

87

783 9 times 87.

So, because 99 is 1 less than 100, we can multiply by 99, by annexing two cyphers, and subtracting once the multiplicand. And, to multiply by 999, we may annex three cyphers and subtract once the multiplicand, and so on. Hence,

To multiply by 9, or by a number consisting of several 98, ANNEX TO THE MULTIPLICAND AS MANY CYPHERS AS THERE ARE 98 IN THE MULTI. PLIER, AND SUBTRACT ONCE THE MULTIPLICAND.

ART. VI. 98 is two less than 100. We may, therefore, multiply by 98, by annexing two cyphers, and subtracting twice the multiplicand. 97 is three less than 100. We may, therefore, multiply by 97, by annexing two cyphers, and

subtracting three times the multiplicand. 998 is two less than 1,000. We may, therefore, multiply by 998, by annexing three cyphers, and subtracting twice the multiplicand, and so on. Hence,

When the multiplier consists of 9 or 98, with other figures following, ANNEX AS MANY CYPHERS AS THERE ARE FIGURES, AND SUBTRACT AS MANY TIMES THE MULTIPLICAND, AS THE MULTIPLIER FALLS SHORT OF A UNIT OF THE NEXT HIGHER ORDER.

The above abbreviations are the most important of those which may be made in multiplication, without the use of Logarithms. Many others might be given; but the thorough student will be able to derive them from the rules in the preceding sections, combined with the instances noticed here. Such are rules for multiplying by 19, 29, 39, 49, &c.; and also for multiplying by 55, 155, 95, 995, 955, 555, 50, 550, &c. Indeed, the practised scholar in multiplication will find few cases in which he will not be able to introduce # some mode of operation, shorter than that prescribed by the general rule for multiplication, in § xvII.

SUBTRACTION.

MENTAL EXERCISES.

§ XX. 1. 3 boys were skating. One fell down. How many did not fall? 1 from 3 leave how many?

2. 5 chairs are standing together, but 2 have broken backs. How many are whole? 2 from 5 leave how many?

3. 6 tumblers stand on a table, and 4 are full of cider. How many are empty? 4 from 6 leave how many?

4. 8 bottles are on a shelf, and 3 have their noses broken off. How many are whole? 3 from 8 leave how many?

How

5. George put 9 marbles on a table, and 3 rolled off. How many were left? 3 from 9 leave how many? 6. 7 lamps stand on a table, and 4 are burning. many are not burning? 4 from 7 leave how many? 7. A boy had 9 quills. He made pens of 4. How many were left? 4 from 9 leave how many? 8. 8 birds were on a tree. A man shot 4. were left? 4 from 8 leave how many ?

9. 7 penknives are on a card. 5 are open. ny are shut? 5 from 7 leave how many ?

How many

How ma

10. 6 chickens are together. 2 are drinking out of a basin. How many are not drinking? 2 from 6 leave how many?