The dividend and divisor are separated by a line, and another line is drawn under the divisor, to mark the place of the quotient. This being done, we take on the left of the dividend the part 16, capable of containing the divisor, 3, and dividing it by this number, we get 5 for the first figure of the quotient on the left; then taking the product of the divisor by the number just found, and subtracting it from 16, the partial dividend, we write, underneath, the remainder, 1, by the side of which we bring down the 5 tens of the dividend. Considering the number, as it now stands, a second partial dividend, we divide it also by the divisor, 3, and obtain 5 for the second figure of the quotient; we then take the product of this number by the divisor, and subtracting it from the partial dividend, get 0 for the remainder. We then bring down the last figure of the dividend, 6, and divide this third partial dividend by the divisor, 3, and get 2 for the last figure of the quotient.

41. It is manifest that, if we find a partial dividend which can. not contain the divisor, it must be because the quotient has no units of the order of that dividend, and that those which it contains arise from the products of the divisor by the units of the lower orders in the quotient; it is necessary, therefore, whenever this is the case, to put a 0 in the quotient, to occupy the place of the order of units that is wanting.

For instance, let 1535 be divided by 5.

The division of the 15 hundreds of the dividend, by the divi sor, leaving no remainder, the 3 tens, which form the second partial dividend, do not contain the divisor. Hence it appears, that the quotient ought to have no tens; consequently this place must be filled with a cipher, in order to give to the first figure of the quotient the value it ought to have, compared with the others;

then bringing down the last figure of the dividend, we form a third partial dividend, which, divided by 5, gives 7 for the units of the quotient, the whole of which is now 307.

42. The considerations, presented in article 40, apply equally to the case, in which the divisor consists of any number of figures.

If, for instance, it were required to divide 57981 by 251, it would easily be seen, that the quotient can have no figures of a higher order than hundreds, because, if it had thousands, the dividend would contain hundreds of thousands, which is not the case; further, the number of hundreds should be such, that, multiplied by 251, the product would be 579, or the multiple of 251 next less than 579; this restriction is necessary on account of the reserved numbers which may have been furnished by the multiplication of the other figures of the quotient by the divisor. The number, which answers to this condition, is 2; but 2 hundreds, multiplied by 251, give 502 hundreds, and the divisor contains 579; the difference, 77 hundreds, arises then from the reserved numbers resulting from the multiplication of the units and tens of the quotient, by the divisor.

If we now subtract the partial product, 502 hundreds, or 50200, from the total product, 57981, the remainder, 7781, will contain the products of the units and tens of the quotient by the divisor, and the operation will be reduced to finding a number, which, multiplied by 251, will give for a product 7781.

Thus, when the first figure of the quotient shall have been determined, it must be multiplied by the divisor; the product being subtracted from the whole dividend, a new dividend will be the result, which must be operated upon like the preceding; and so on, till the whole dividend is exhausted.

It is always necessary, for obtaining the first figure of the quotient, to separate, on the left of the dividend, so many figures, as, considered as simple units, will contain the divisor, and admit of this partial division.

43. Disposing of the operation as before, the calculation, just explained, is performed in the following order;

The 3 first figures, on the left of the dividend, are taken to form the partial dividend; they are divided by the divisor, and the number 2, thence resulting, is written in the quotient; the divisor is then multiplied by this number, and the product, 502, is written under the partial dividend, 579. Subtraction being performed, the 8 tens of the dividend are brought down to the side of the remainder, 77; this new partial dividend is then divided by the divisor, and 3 is obtained for the second figure of the quotient; the divisor is multiplied by this, the product subtracted from the corresponding partial dividend, and to the remainder, 25, is brought down the last figure of the dividend, 1; this last partial dividend, 251, being equal to the divisor, gives 1 for the units of the quotient.

44. When the divisor contains many figures, some difficulty may be found in ascertaining how many times it is contained in the partial dividends. The following example is designed to show how it may be known.

It is necessary at first to take four figures on the left of the dividend, to form a number which will contain the divisor; and then it cannot be immediately perceived how many times 485 is contained in 4234. To aid us in this inquiry, we shall observe, that this divisor is between 400 and 500; and if it were exactly one or the other of these numbers, the question would be reduced to finding how many times 4 hundred or 5 hundred is contained in the 42 hundreds of the number 4234, or, which amounts to the same thing, how many times 4 or 5 is contained in 42. For the first of these numbers we get 10, and for the second 8; the quotient must now be sought between these two. We see at first that we cannot employ 10, because this would imply, that the order of units in the dividend above hundreds contained the divisor, which is not the case. It only remains, then, to try which of the two numbers 9 or 8, used as the multiplier of 485, gives a product that can be subtracted from 4234, and 8 is found to be the one. Subtracting from the partial dividend the product of the divisor multiplied by 8, we get, for the remainder, 354; bringing down then the O tens in the dividend, we form a second partial dividend, on which we operate as on the preceding; and so with the others.

45. The recapitulation of the preceding article gives us this rule, To divide one number by another, place the divisor on the right of the dividend, separate them by a line, and draw another line under the divisor, to make the place for the quotient. Take, on the left of the dividend, as many figures as are necessary to contain the divisor; find how many times the number expressed by the first figure of the divisor, is contained in that, represented by the first figure or two first figures of the partial dividend; multiply this quotient, which is only an approximation, by the divisor, and, if the product is greater than the partial dividend, take units from the quotient continually, till it will give a product that can be subtracted from the partial dividend; subtract this product, and if the remainder be greater than the divisor, it will be a proof that the quotient has been too much diminished; and, consequently, it must be increased. By the side of the remainder bring down the next figure of the dividend, and find, as before, how many times this partial dividend contains the divisor; continue thus, until all the figures of the given

dividend are brought down. When a partial dividend occurs, which does not contain the divisor, it is necessary, before bringing down another figure of the dividend, to put a cipher in the quotient.

46. The operations required in division may be made to occupy a less space, by performing mentally the subtraction of the products given by the divisor and each figure of the quotient, as is exhibited in the following example;

After having found that the first partial dividend contains 4 times the divisor, 39, we multiply at first the 9 units by 4, which gives 36; and, in order to subtract this product from the partial dividend, we add to the 5 units in the dividend 4 tens, making their sum 45, from which taking 36, 9 remains. We then reserve 4 tens to join them, in the mind, to 12, the product of the quotient by the tens in the divisor, making the sum 16; in taking this sum from 17, we take away the 4 tens, with which we had augmented the units of the dividend, in order to perform the preceding subtraction. We then operate in the same manner on the second partial dividend, 195, saying; 9 times 5 make 45, taken from 45, nought remains; then 5 times 3 make 15, and 4 tens, reserved, make 19, taken from 19, nought remains.

We see sufficiently by this in what manner we are to perform any other example, however complicated.

47. Division is also abbreviated when the dividend and divisor are terminated by ciphers, because we can strike out, from the end of each, as many ciphers as are contained in the one that has the least number.

If, for instance, 84000 were to be divided by 400, these numbers may be reduced to 840 and 4, and the quotient would not be altered; for we should only have to change the name of the units, since, instead of 84000, or 840 hundreds, and 400, or 4 hundreds, we should have 840 units and 4 units, and the quotient of the numbers 840 and 4 is always the same, whatever may be the denomination of their units.