six hundred and seventy-four; and nine hundred and eight millions and six.

Ans. Nine hundred and twenty-one millions two hundred and fourteen thousand six hundred.

7. Required the sum of thirty-seven; six hundred and fifty-five; three thousand and seventy; twenty millions ninetyfour thousand and fifty-seven; seventy-four thousand and fortyseven; six thousand and ninety-nine; three hundred and twenty-eight; fifty-seven thousand five hundred and six; and six hundred and six millions five hundred thousand seven hundred.

Ans. Six hundred and twenty-six millions seven hundred and thirty-six thousand four hundred and ninety-nine.

8. What is the sum of fifty-eight; three thousand and four; eighty-three; ten thousand four hundred and ninety; three hundred and fourteen; six millions six hundred and four thousand and seventy-five; two hundred and twenty-four; eleven thousand and fifty; and five billions one hundred and four thousand and seventy-two.

Ans. Five billions six millions seven hundred and thirtythree thousand three hundred and seventy.

Let the following examples be performed and proved, as above:

74. The unit figure of the sum of any column, being of the order of the column, is written underneath; and as the other figures of this sum express (48) a number of tens with regard to that figure, they are carried, as that number, to the next column on the left; the figures of which, (36) in the same sense, are also tens. Therefore, when the sum of a column equals or exceeds 100, the number to be carried will be 10 or more: when the sum equals or exceeds 200, the number to be carried will be 20 or more: when the sum equals or exceeds 300, the number to be carried will be 30 or more, &c. On reaching 100, the scholar may make a dash at the side of his work, and proceed with the surplus, as at first.

Addition may be considered the principal, and, indeed, the most laborious operation of the Counting-house. The pupil, therefore, should, by all means, endeavour to render himself as skilful as possible in this most important operation. The old adage" practice makes perfect"-is certainly applicable here, if anywhere; and there can be little doubt that one of the very best methods to adopt is to repeat the same operation many times, writing the result each time, by which the student will not only clearly perceive his liability to err, but also the progress he is hourly making towards perfection. He should also frequently repeat each operation by taking the figures in an inverted or contrary order.

The answers to these questions are omitted, because the insertion of them might induce carelessness in the scholar, and because the teacher, having once seen them, will, by a mere glance of the eye, determine whether the work offered for in- spection is correct or not; and the scholar should not present his work till he has found the same result by two consecutive additions performed in a contrary order.

75. THE word Subtraction is derived from the Latin words sub, out or away, and trahere, to draw, to take; and, therefore, signifies to draw out or take away.

76. There can be no difference between two equals. Also, if the whole of a thing be taken away, there can be no remain

Subtraction, therefore, as an arithmetical operation, is the method of finding the difference between two unequal numbers, by arranging their orders so as to diminish the greater by as many units as are contained in the less. The remaining part of the greater number is, consequently, called remainder or difference.

77. The greater number, then, is composed of the remainder and the less number; consequently, to prove subtraction, we add the remainder and less number together, the sum of which equals the greater, when the work is right.

78. If we take either of two numbers from their sum, the remainder will be the other. Wherefore, when the work is right, if we subtract the remainder from the greater number, we shall obtain the less.

79. Of two unequal numbers, let 19 be the greater and 10 the less. From 19, to subtract 10, we take away the unit 1, which is ten, and the remainder is 9. If to this remainder 9 we add the less number 10, we have the greater number 19: and this is done by replacing the unit on the left of 9. Again: if from 19 we take the remainder 9, we have the less number 10.

Hence, we see that to subtract 10 from any of the numbers 11, 12, 13, &c., 19, is merely to take away the unit in the place of tens. Also, to add 10 to any of the digits 1, 2, 3, &c., 9, is merely to place a unit on the left of it.

80. As the sign minus represents a simple subtraction, so the sign represents a continual subtraction, which divides the number from which we subtract, into equal parts. In both cases-that is, when either of these signs is placed between two numbers-it is always the number on the right of the sign which is to be subtracted from that on the left. Thus, 9—3—6. Here, we say: 3 from 9, six; and, inversely, (78,) 6 from 9,

9

three. Again: 93 or 3

9 3 3, &c., as often as we can.

6-33, and 3 3

= 0. =

3 Thus, 9 6; Where, having made three subtractions and arrived at 0, we say that 9 contains 3 just 3 times; that is, when 9 is divided into equal parts of 3 units each, the three subtractions show there are 3 of those parts; and this number 3 is thence called the quotient, (quoties, Lat., how often?)

81. The frequent repetition or conning of the following Table will greatly facilitate the progress of the learner. In