number of times, which one quantity may be contained in another is necessarily to be taken out, or considered, first, the inferior numbers will then follow in their regular order, and keeping account of the value of any remainder from the preceding operation in its proper rank, as in the following example, which we shall express in the manner that has been shown in § 20, in order to accustom the learner to keep the systematic language of the operation itself, which is always the most preferable method; with this view we shall draw a horizontal line under the dividend, under which we shall place the divisor, and the result, or quotient, will be written on the right hand side of the sign of equality which follows them, thus:

Here we say 3, in 8, is contained twice, and having written the 2, as the first number to the quotient, we must make the product of it by the divisor, write it under the corresponding number of the dividend, and subtract it from it; this product being 6, in this case the subtraction leaves 2, as a remainder. Now, for the sake of easier distinc

tion we place the next number by the side of this remainder, which being 4, gives for the next number to be divided 24. Now 3, is in 24, contained: 8 times; placing the 8 in the quotient, multiplying the 3 by it, the product of 3 times 8, placed under the 24, being also 24, leaves no remainder; placing the next number 2 down, we find, that 3 not being contained in it, we must indicate this by an 0, in the quotient, for the rank or order of the numeric system corresponding, which being done, the next number, 3, is taken down to the right side of the 2, which making 23, we say 3 in 23 will be contained 7 times; writing the 7 in the quotient, multiplying the 3 by it, and subtracting the product 21 from the 23, we obtain the remainder 2; taking down the 1 which gives 21, we say again, 3 in 21, is contained 7 times, and the product 3 times 7 being equal to 21, leaves no remainder; lastly, bringing down the 6, we find 3 in 6 twice, and writing the 2 in the quotient, and subtracting its product by S, from the 6, we obtain the exact quotient 280772.

Division being the opposite of multiplication, we have the means of proving this result, by the multiplication of the quotient by the divisor; the product of which must be equal to the dividend, as is evident from the definitions given of this operation.

Writing then the divisor under the quotient, and performing the multiplication, the product resulting will be equal to the dividend, if the whole operation has been rightly performed.

34. If the divisor is not contained an exact whole number of times in the dividend there will remain at the end of the division, a number smaller than this divisor, which is called the remainder. In order to indicate fully the actual result of the division, this number is yet to be placed at the end of the quotient, with the divisor written under it, and a horizontal

line between them, to indicate that this division should yet be made.

Such numbers as indicate a division which cannot be executed, are called proper fractions, while every division, indicated as above, of a number larger than the divisor, is, in comparison with these, called an improper fraction: and, when considered in this point of view, the number corresponding to the dividend, is called the numerator, and the number corresponding to the divisor is called the denominator; while the quotient, whatever it may be, will always represent the value of the fraction.

This general idea of fractions, the origin of which it is proper to show here, will hereafter be the fundamental idea from which the calculation of this kind of quantities is to be deduced.

The following is an example that will show such a division, and the mode of operating in the case. Being given to divide

In this example: we see that the first number of the highest order being smaller than the divisor, we

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must take it jointly with the next following lower number and say: 8, in 78, is contained 9 times; and the 9 is written as the first number in the quotient; then making the product 8.972, and writing it under the 78, from which it is subtracted, and leaves 6, which being written below the line, and the next lower number 3, written down to it, gives 63 for the next number, to be divided by 8, which being contained 7 times in it, 7 being written in the quotient, the product 7 X 8 56 is written under the 63, the subtraction performed, and the 5 or next following number placed down to the 7 that remains from the subtraction; the operation is thus continued, exactly as in the former example, until when the last number, 1, is set down at the side of the 0, we find that 8 is no longer contained in 1, and therefore write an 0 in the quotient, and having no more numbers in the dividend, we find that I ought yet to be divided by 8, which we write in the quotient, as stated above, like an unexecuted division, or a proper fraction.

When we make the proof of this example, as has been done in the preceding one, we consider the 1 as a remainder, and in the multiplication of the quotient by the divisor, add it to the product; so that we would here say 8 times 0, is 0, and the remainder 1 added, gives 1 for the first number of the product, exactly as in the dividend, and then continue the multiplication through the whole quotient obtained, as in the former example.

35. When the divisor is a number composed of more than one figure, the principles of the operation remain the same; but it becomes necessary to pay attention to the effect of the multiplication of the quotient into the whole number of the divisor; which may render it necessary to take this quotient smaller than might appear from a mere comparison of the first numbers of the divisor and the divi

dend; all the rest of the operation is only an extension of the operations explained in the preceding examples, which have been described in detail, with the express view of giving a full explanation of the first elementary principles. Reasoning with the same details upon the following example, the operation of a division, with a divisor composed of more than one figure, will also be clear.

Here in considering only the first number of the divisor, and comparing it with the two first of the dividend, we would find 7 in 64 contained 9 times; but we must take into consideration the multiples of the numbers which follow the hundreds. The 5 tens, or 50, multiplied by 9 would give 45 tens, or 450, and 7 X 963 would leave only 1, which, considered as hundreds, as must be done in this case, would not allow us to take the 4 hundreds from it. We find, therefore, that the quotient 9, is too large. Taking 8, we find that 7 × 856, leaves 8 as remainder; and if we consider now the 58, as multiplied by 8, we find that the 4, which comes here again as hundreds to be subtracted from volqmazo quiegvidt