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 0 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 articles 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, 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 dividend, 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 divi

sor 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.

It may also be remarked that, in striking out two ciphers at the end of the given numbers, they have been, at the same time, both of them divided by 100; for it follows from article 31, that in striking out 1, 2, or 3 ciphers on the right of any number, the number is divided by 10, or 100, or 1000, &c.

48. Division and multiplication mutually prove each other, like subtraction and addition, for according to the definition of division, (36), we ought, by dividing the product by one of the factors, to find the other; and multiplying the divisor by the quotient, we ought to reproduce the dividend (37).

FRACTIONS.

49. DIVISION cannot always be exactly performed, because any number whatever of units taken a certain number of times, does not always compose any other number whatever. Exam

ples of this have already been seen in the table of Pythagoras, which contains only the product of the 9 first numbers multiplied two and two, but does not contain all the numbers between 1 and 81, the first and last numbers in it. The method hitherto given shows then, only how to find the greatest multiple of the divisor, that can be contained in the dividend.

If we divide 239 by 8, according to the rule in article 46,

we have, for the last partial dividend, the number 79, which does not contain 8 exactly, but which, falling between the two numbers, 72 and 80, one of which contains the divisor, 8, nine times, and the other ten, shows us that the last part of the quotient is greater than 9, and less than 10, and consequently, that the whole quotient is between 29 and 30. If we multiply the unit figure of the quotient, 9, by the divisor, 8, and subtract the product from the last partial dividend, 79, the remainder, 7, will evidently be the excess of the dividend, 239, above the product of the factors, 29 and 8. Indeed, having, by the different parts of the operation, subtracted successively from the dividend, 239, the product of each figure of the quotient by the divisor, we have evidently subtracted the product of the whole quotient by the divisor, or 232; and the remainder, 7, less than the divisor, proves, that 232 is the greatest multiple of 8, that can be contained in 239.

50. It must be perceived, after what has been said, that to reproduce any dividend, we must add to the product of the divisor by the quotient, the sum which remains when the division cannot be performed exactly.

51. If we wished to divide into eight equal parts a sum of whatever nature, consisting of 239 units, we could not do it without using parts of units or fractions. Thus, when we have taken from the number 239. the 8 times 29, units contained in it, there will remain 7 units, to be divided into 8 parts; to do this, we may divide each of these units, one after the other, into 8 parts, and then take one part out of each unit, which will give 7 parts to be joined to the 29 whole units, to form the eighth part of 239, or the exact quotient of this number, by 8.

The same reasoning may be applied to every other example of division in which there is a remainder, and in this case the quotient is composed of two parts; one, consisting of whole units, while the other cannot be obtained until the concrete or material units of the remainder have been actually divided into the number of parts denoted by the divisor; without this it can only be indicated by supposing, a unit of the dividend to be divided into as many parts as there are units in the divisor, and so many of these parts, as there are units in the remainder, taken to complete the quotient required.

52. In general, when we have occasion to consider quantities less than unity, we suppose unity divided into a certain number of parts, sufficiently small to be contained a certain number of times in these quantities, or to measure them. In the idea thus formed of their magnitude there are two elements, namely, the number of times the measuring part is contained in unity, and the number of these parts found in the quantities.

A nomenclature has been made for fractions, which answers to this manner of conceiving and representing them.

That which results from the division of unity

into 2 parts is called a moiety or half,

and so on, adding after the two first, the termination th to the number, which denotes how many parts are supposed to be in unity.

Every fraction then is expressed by two numbers; the first, which shows how many parts it is composed of, is called the numerator, and the other, which shows how many of these parts are necessary to form an unit, is called the denominator, because the denomination of the fraction is deduced from it. Five sixths of an unit is a fraction, the numerator of which is five, and the denominator six.

The numerator and the denominator together are called the two terms of the fraction.

Figures are used to shorten the expression of fractions, the