mal point. First let the ordinary division be performed:

3-421384)5-879 2314(1-718

3.421 384

2-457 8474

2.394 9688

62 87860

34 21384

28 664760

27 371072

It is plain that to get the last figure of the quotient we needed only the first two figures of the third remainder; so that, if a vertical line were drawn immediately to the right of these two figures, we retain to the left quite enough to determine all the figures of the quotient, unless, indeed, the subtractions, etc., that affect the column of figures to the right of this line may affect the numbers to the left of the line. It will be noticed that to the left of the line (in this case) in the dividend is equal to the number of figures in our result. Let us now try to construct that part of the work which lies to the left of the line. The first step will stand thus:

3.421384)5-879(1

3.421

2.458

and this means that we consider only the first four figures of the divisor (there being nothing to carry from the multiplication of the fifth). But the remainder differs by 1 in the last place, which is due to the "carrying" in the subtraction to the right of the line in the original

division; however, let this difference be overlooked for the present. The next division, in the figures to the left of the line, is concerned with only three figures of the divisor, there being nothing to carry from the multiplication of 1 by 7; we shall now have the work

thus:

3.421)5.879(1.7

3.421

2.458

2.394

64

Here the 1 of the divisor was marked out after the first division. It will be noticed that the remainder is here 64, while the corresponding number in the complete division is 62. Referring to the original work we see that, so far as the figures to the left of the line are concerned, we have to do only with the first two figures of the divisor. We may therefore strike out the third figure, and our work will stand thus:

3-421)5-879(1.71

3.421

2.458

2.394

64

34

30

Here the remainder is 30, while the corresponding remainder in the complete work is 28. Now strike out, for the same reason as before given, the 4 of the divisor; we are then in doubt whether the last figure

should be 9 or 8, as, taking 9, we see that 9 × 3 = 27, which, with the 3 to carry from the multiplication of 4 by 9, makes 30, but we might suspect the 9 to be too great. We see now that if we wish to have four figures, we should start with a divisor of four figures. For the same reasons as given in the case of multiplication, we should also adopt a similar rule for carrying; and, further, if we wish our answer to consist of four figures, we are more likely to be strictly correct to that place if we start with a divisor of five figures, which means that we retain an additional column of figures of the original division. The work would then stand thus:

3-42138)5-8792(1.718

3.4214

2:4578

2.3949

629

342

287

273

We have thus to regard the following:

(1) Find the number of figures that are to be in the

answer.

(2) Start the division with a divisor consisting of a number of digits one more than that number, these digits to be the first digits of the given divisor in order.

(3) After each subtraction, instead of placing a figure (from the dividend) to the right of the remainder, cut off one figure from the right of the divisor.

(4) In multiplying, have regard always to what may be carried from the neglected digit to the right, regard

ing such a number as 48 as giving 5 to carry, such a number as 32 as giving 3.

Manifestly, as in the case of multiplication, there is need of practice to give one confidence, and to educate one's judgment in the matter of deciding what number should be carried in such doubtful cases as may arise.

CHAPTER XV.

PERCENTAGE AND ITS APPLICATIONS.

Percentage. In some text-books on arithmetic percentage is treated as if it were a special process involving certain distinctive principles and therefore entitled to rank as a separate department. In these books, accordingly, percentage has its definitions, its "cases," and its rules and formulas. This elaborate treatment seems to be a mistake on both the theoretical and the practical side: on the theoretical side, because it asserts or assumes a new phase in the development of number; on the practical side, because it substitutes a system of mechanical rules for the intelligent application of a few simple principles with which the student is perfectly familiar. In the growth of number as measurement percentage presents nothing new. It has to do with the ideas and processes of ratio with which fractions are more or less explicitly concerned, and its problems afford excellent practice for enlarging and defining these ideas, and securing greater facility in using them. But the mere fact that, in this new topic with its cases and its rules, a quantity is measured off into a hundred parts instead of into any other possible number of parts, appears to be no valid reason for constituting percentage a new process marking a new phase in the evolution of num