from fourteen and two remains ; six into twenty-one, three times six are eighteen, from twenty-one and three remains; six into thirty-six, six times six are thirty-six, and nothing remains.

You observe we always begin our division from the left, not as in the other rules from the right; therefore to give our remainders their proper value, we place them before the next figure of our dividend. In the above example, the seven represents seven millions; the one remainder by being placed in our mind before the two, represented one million; the next step there was no remainder; the next figure being a cipher, there was nought written in the quotient, to preserve the value of the figures already written on its left; the next figure 1 would not contain the divisor, and for the same reason we write another cipher in the quotient; the one being in the fourth place or that of thousands, we place it before the four, and these two when divided leave a remainder of two, which two being a remainder of a figure representing hundreds, we place before the one; these two when divided leave a remainder of three, which from its place represents tens; we therefore place it before the unit six, which being divided leaves no remainder, but if there had been, it would be the remainder of a unit, and must consequently be less than one.

When you have to divide by ten, one hundred, or one thousand, cut off as many figures on the right of your dividend as there are ciphers in your divisor; the figures so cut off will be the remainder, those on their left will be the quotient.

Work for practice the following sums.-(Sign of division, and of equality =). 1025, when read, is ten divided by two is equal to five.

The divisors of the above sums are twelve, or something less than twelve, with one exception, and for that there were ample instructions given. You were therefore able to divide them in one line; but it often occurs that your divisor consists of three, four, or more figures, and in that case, with your present knowledge of arithmetic you could not divide them in one line. But there is another method equally simple, which is called Long Division, as the former is called Short Division. As this long division requires a different mode of statement, you will observe the following examples; and as you read them, work them carefully on your slate.

Example 1st.-Divide 842786 by 78.

In this example, instead of a line drawn underneath, there are two curved lines drawn, one on the left and one on the right; on the left is written

the divisor, and on the right the quotient, figure by figure, as we proceed with our division. We find the divisor is contained once in the first two figures of the dividend,

78) 842786 ( 10804 78

627

624

386

312

74

which one is written in the quotient, and the divisor seventy-eight, when subtracted from the first two figures of the dividend eighty-four, leaves a remainder of six; to this six I bring down the next figure of the

dividend, two, under which I make a dot, to shew that it is brought down, and when written on the right of the remainder it makes sixty-two, which is still less than the divisor; I write a cipher in the quotient, and I write the next figure of the dividend seven on the right of the 62, making it six hundred and twenty-seven, in which the divisor is contained eight times; I write eight in the quotient, and the product of the divisor 78 multiplied by eight being 624, I subtract it from 627 and I find the remainder is three; on the right of this three I write the next figure of the dividend eight, making it thirty-eight, which being less than the divisor, I write a cipher in the quotient and bring down the next figure of the dividend six, which being written on the right, makes three hundred and eighty-six, in which the divisor is contained four times, which four being written in the quotient and the divisor multiplied by it, and the product three hundred and twelve being subtracted from three hundred and eighty-six, it leaves a remainder of seventy-four, and the last figure of the dividend being brought down, such remainder is less than a unit.

Example 2nd.-Divide 842786 by 978.

In this example the divisor being more than the three first figures of the dividend, in separating the latter for our purpose, we take the four first figures of the dividend, which make 8427, and we find the divisor is contained in this number eight times, and that when multiplied by eight, its product is 7824, which leaves a remainder of 603; to this

978) 842786 (861

7824..

6038

5868

1706

978

728

remainder we bring down the next

figure of the dividend, which make 6038; in this sum

the divisor is contained six times, and we find its product, when multiplied by six, is 5868, which leaves a remainder of 170, on the right of which we write the next figure of the dividend six, which makes it 1706; in this number the divisor is contained once, with a remainder of 728; there being no more figures in the dividend, the work is complete.

In this manner can sums, however large, be divided; the chief dexterity of the calculator consisting in being able to determine at a glance how often the divisor is contained in that portion of the dividend he selects for his operations. This can only be acquired by practice; and the only rule I can lay down for his guidance is: First, to see that his remainder is always less than his divisor, for if greater, or if equal to it, the quotient figure is too small, and must be increased: Secondly, to dot each figure of the dividend as he brings it down, otherwise he may bring down the same figure twice, and may proceed for some time in error.

You will, after working the above examples, do the following sums.

(25.) Divide £76494 between 294 persons.

(26.) A ray of light travels from the sun to the earth in 495 seconds; how many miles does it travel a second, the sun's distance from the earth being 95173000 miles?

(27.) A ship bound for Jamaica set sail from Liverpool

on the 24th of January, 1846, and arrived at that island on the 28th of March; what was her rate of sailing per day and per hour, the distance being 4558 miles?

(28.) The time in which the planet Jupiter revolves in its orbit round the sun is 4330 of our days, which is one year of that planet; how many years of ours, reckoning 365 days for each year, are equal to five years of Jupiter?

(29.) If a chest of oranges, containing 1292 in number, were distributed, one moiety between 19 boys, and the other between 17 girls; how much will fall to the share of each ?

(30.) The circumference of the earth's orbit, or annual path round the sun, is about 596440000 miles; there are 8766 hours in the year; how many miles in an hour, and how many in a minute, are we carried by this motion?

(31.) In how many years would a mail-coach, travelling at the rate of ten miles an hour (without intermission) be going from the earth to the sun, the distance being ninety-five millions of miles, and the length of the year 365 days?

(32.) The quotient and remainder of a sum in simple division being each 23, and the divisor ten more than both, it is required to find the dividend?*

PROOF OF MULTIPLICATION AND DIVISION.

These rules being converse to each other—that is, multiplication being, as we know, adapted for increasing,

*The sum of the quotient and remainder when added make forty-six ;. the divisor being ten more, must be fifty-six, therefore you multiply the divisor, fifty-six, by the quotient, twenty-three, adding the remainder, and the product will be the dividend.