the successive results in their proper places, will be as follows:

174392

6435

871960

523176

697568

1046352

1122212520

In this manner, every example, whatever quantity of figures it may be composed of, will stand.

If any of the figures in the number to be multiplied, which is called the multiplicand, should be an 0, its product into any number whatsoever, is = 0; because 0 times any number whatever, always indicates that the number is not there; the place will, therefore, receive only that number which may be carried over from the preceding multiplication, and if none be carried, only an 0.

If an 0 occur among the numbers by which the multiplication is to be performed, or the multiplier, the whole row of figures to be multiplied by it producing a result = 0, the place where the first number would stand will only be marked by an 0, and the multiplication by the next following number is begun in the same row, immediately after, thus placing each result in its proper place.

The following example will explain both the above cases, where the effect of the two 0's, in the multiplier is shown by the removal towards the left of the two latter rows of figures.

§ 32. It will be proper to exercise the scholar in a variety of examples, until he has become accustomed to the operation, and is able to make any multiplication without error: the younger the scholar may be, the easier the examples must be in the beginning, and must gradually increase in difficulty, by the combination of different cases, and larger numbers. Still, in this it is to be observed: that when the beginner has performed examples gradually with the whole series of the nine simple numbers, it will be proper to show him only, what is the effect of a compound multiplier, as a repetition of the similar operation of one number only, and the addition of the different partial products into one whole; and not to follow servilely the augmentation by one number, (or place of figures,) that he may not, as often happens, consider that he has every time a new difficulty to overcome, but must himself come to the observation, that multiplication by a number of places of figures is a mere repetition of the operation he knows, requiring nothing but a little more attention, and more accuracy in the placing of the figures.

§ 33. DIVISON, is an operation the opposite of Multiplication, as has already been stated; its problem is therefore to find how many times a given number is contained in another given number, which is thus considered as a product of the first and the quantity sought.

The table of products, or multiplication table, given above, may therefore be here applied inversely; a ready and habitual knowledge of its results is therefore also constantly applied in this rule, by the comparison of its results with the quantities presenting themselves in an example.

While all the preceding operations have begun at the unit, this on the contrary must begin by the highest number, or order of symbols; for the greater

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 3, 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

7835921

= 9794901

8

72

7835921

63

56

75

72

39

32

72

71

01

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