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of the first period, b over the figure to the left of the second period, &c. till all the figures are brought down, as in this example:

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123,456.123,456.123,456.123,456.578,371.123,875.

or, instead of in, b, t, q, qu, &c. put 1, 2, 3, 4, 5, &c. to represent millions once, twice, thrice, &c. repeated; and read thus, one hundred and twenty-three thousand four hundred and fifty-six quintillions ; one hundred and twenty-three thousand four hundred and fifty-six quadrillions; one hundred and twenty-three thousand four hundred and fifty-six trillions; one hundred and twenty-three thousand four hundred and fifty-six billions; five hundred and seventy-eight thousand three hundred and seventy-one millions; one hundred and twentythree thousand eight hundred and seventy-five.

Examples.

1. Write down in words at length the following num

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2. Write down in proper figures the following numbers:

Eighty-nine. Seven hundred and fifty. Five thousand and one. Ten thousand and eighty-seven. Twenty thousand and five.

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Six hundred and eighty-five thousand, three hundred and sixty.

One million five hundred thousand, and one.

Twenty-seven million, three hundred and sixty five thousand.

Three hundred and eighty-five million, seven hundred and forty-eight thousand, three hundred, and five. Eleven thousand, eleven hundred, and eleven. Fifty million, fifty thousand, fifty hundred, and fifty.

SIMPLE ADDITION.

Definition.-Simple Addition is a rule by which seve ral numbers of one denomination are collected together into one sum.

RULE.

Place the numbers under each other, viz. units under `units, tens under tens, &c.; add up the figures in the row of units, and carry as many units to the next row as there are tens contained in the sum: proceed thus till the whole is finished.

For the proof.-Divide the numbers to be added into two parts, then add up each part by itself, and collect these sums together for the whole.

Note 1. If equal numbers be added to equal numbers, the whole will be equal.

2. If several numbers are to be added together, they will amount to the same sum, when placed regularly one under another, whichever line or row of figures stands uppermost.

3. Dr. Wallis, in his Arithmetic, gives the following rule to prove a simple addition sum. Add the figures in the uppermost row together, reject the nines contained in their sum, and set the excess directly even with the figures in the row. Do the same with each row, and set all the excesses of nine together in a line, and find their sum; then, if the excess of nines in this sum (found as before) be equal to the excess of nines in the total sum, the work is right.

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(4.) Add 1473, 40734, 371049, 40057, 3471473, 5734, $7492, and 4718375, together.

(5.) Collect 371434, 278949375, 67149, 3457143, 714934, 9000987, and 5734747, into one sum.

(6.) Add 5714329, 4718714, 34983714, 671493, 74987149, 6777894987, and 19, together.

(7.) Add 571493, 40007, 6493497, 4718349, 3714934, 4934938, 174934, and 147319, together.

(8.) Suppose the distance from London to Biggleswade be 45 miles, thence to Peterborough 36, thence to Lincoln 51, and thence to Hull 41 miles; how many miles are Peterborough, Lincoln, and Hull, from London?

SIMPLE SUBTRACTION.

Definition.-Simple subtraction teaches to deduct, or subtract, a less number from a greater of the same denomination, whereby the remainder or difference is found.

RULE.

Place the less number under the greater, so that units may stand under units, tens under tens, &c. Begin at the unit's place, and subtract each figure in the lower line from the figure above it; if the lower figure be greater than the upper, add ten to the upper figure, from which subtract the lower; set down the remainder, and carry one to the next lower figure.

For the proof.-Add the remainder and less number together, and the sum will be the greater. Or, subtract the remainder from the greater number, and the difference will be the less.

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CLASS II.

(9.) From the Creation to the Flood was 1656 years; thence to the building of Solomon's Temple 1336 years; thence to Mahomet, who lived 622 years after Christ, 1630 years. In what year of the world was Christ then born, and how many years is it since the creation?

(10.) Sir Isaac Newton was born in the year 1642, and died in 1727, how old was he at the time of his decease, and how many years is it since he died?

(11.) A gentleman has two sons, the age of the elder added to his make 126 years, and the age of the younger son is equal to the difference between the age of the father and the elder son. Now, if the father be 80 years of age, how old are each of his sons?

(12.) Three boys, A, B, and C, won together 97 marbles at play; now, if the number of marbles B won be added to the number C won, they will make 60; and, if the number A won be added to the number C won, they will make 62. How many marbles did each boy win separately?

SIMPLE MULTIPLICATION.

Definition 1. Simple multiplication is a rule by which we increase the greater of two given numbers, of the same denomination, as often as there are units in the less; being a compendious method of performing addition.

2. The number to be multiplied is called the multiplicand; the number you multiply by is called the multiplier; and the number produced by multiplication is called the product. These numbers are sometimes called factors, because they are to constitute a factum or product.

The Multiplication Table.

1 2 3 4 5 6 7 8 9 10 11 12

2 4 6 8 10 12 14 16 18 20 22 24 3 6 91215182124 27 30 33 36 4 812 16 2024 2832 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 612182430364248 54 60 66 72 714212835 42 4956 63 70 77 84 816243240485664 72 80 88 96 918273645546372 81 90 99 108 1020304050 60 70 80

1122334455 66 7788

90 100 110 120

99 110 121 132

1224364860 72 8496 108 120 132 144

Proposition 1. To multiply by a single figure, or any number not exceeding 12.

Rule. Begin at the unit's place of the multiplicand, and multiply each figure in it by the multiplier, writing down the whole of such products as are less than 10; but, for such as exceed 10, or a number of tens, write down the excess, and carry an unit, for each 10, to the next product.

Prop. 2. When the multiplier is the product of two, or more, numbers in the table.

Rule. Multiply the multiplicand by one of the component parts, and that product by the other, &c. for the whole product.

Prop. 3. When the multiplier consists of several figures. Rule. The multiplicand must be multiplied by each figure separately, (beginning with the right-hand figure of the multiplier,) and the first figure of every product must stand exactly under the figure you multiply by. Add these products together for the whole product.

Or, begin with the left-hand figure of the multiplier, and multiply every figure in the multiplicand by it; then multiply in a similar manner by the next figure, &c., taking care to place every succeeding product one figure farther out towards the right-hand.

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