should be acquired, though not, as before said, by excluding all other ideas and operations till this perfection is attained. Get complete possession of addition, with full knowledge of numbers, if possible; without it, if necessary.

Subtraction.-Addition and subtraction are inverse operations. The one implies the other, and in primary operations the two should go together, with the emphasis on addition. Subtraction in actual operations with objects would seem logically to precede addition. If we wish to get a definite idea of a 14-unit quantity, and separate it into two known parts of 8 units and 6 units each, it seems that logically the 6 unit-quantity is taken away from the whole, and both the minor quantities are recognised as parts of the whole before the final process of constructing the whole from the parts is completed. There is no need, therefore, of making a complete separation between these two operations. On the contrary, they should be taught as correlative operations, with addition slightly prominent first for reasons already set forth.

From what has been shown as to the logical and psychological relation between addition and subtraction, it appears that subtraction is the operation of finding the part of a given quantity which remains after a given part of the quantity has been taken away. As in addition, so in subtraction, all the quantities with which we are working-minuend, subtrahend, remainder-must have the same unit of measurement. Further, as in addition we are working from and within a vague whole by means of its given parts, so in subtraction we are working from a defined whole, through a defined

part, in order to make the vaguely conceived "remainder" perfectly definite.

Remainder or Difference.-From the nature of subtraction as related to addition, there seems to be no strong reason for the "important distinction" that should be noted between "taking" one number out of another and finding the difference between two numbers. We can not take away a given portion of a given quantity (to find the remainder) without conceiving this given portion as part of the whole; we can not get a definite idea of the "difference" between two measured quantities without conceiving the less as a part of the greater. If $5 is given as a part that has been taken from $9, we primarily count from 5 to 9 to find the remainder. If $5 and $9 are given as two quantities, we have to count from 5 to 9 to determine the difference. We have to conceive the $5 as a part of the $9.

If the preliminary work of parting and wholing to develop good ideas of number and numerical processeshas been rationally done, there will be but little difficulty in the actual operation in formal subtraction. The following points with respect to the long-time mystic. operations of "borrowing and carrying" may be noticed :

1. The operation involved in, e. g., 7538, may and should be made perfectly clear by counters. The ten-unit in its relation to the unit has been made clear through many constructive acts. The mental process here, then, is indicated simply by

If the pupil has acquired facility in the addition combinations, the operation of adding 10 and 5 and taking 8 from the sum (getting 7) is probably as easy—may become as easily automatic-as taking 8 from the 10 and adding 5 to the difference (getting 7). But the meaning and identity of both processes can be made perfectly clear. The pupil may find it at first a little easier to take 8 from the "borrowed" 10 and add 5 to the remainder (2), than to add 5 to the borrowed 10 and take 8 from the sum 15. But, in any case, these analytic acts are to lead to the clear comprehension of the process, and especially to its automatic use. There should be, of course, large practice in finding the differences of pairs of tens, as well as in finding their sums.

2. The second method of explaining the "borrowing and carrying" in subtraction-that of adding equal quantities to minuend and subtrahend-may be made equally clear. That the difference between two quantities remains the same when each has received equal increments, the pupil will discover for himself by "doing" such operations. In 75-38 we add one ten-unit-i. e., ten ones to the 5 ones, and subtract 8, as in the first case considered; i. e., 15-8, or 10-8 +5; we then increase by 1-ten the 3 tens in the subtrahend, getting 4 tens, which we take from the 7 tens. This process is not a direct solution of the problem, but it is one that can be made quite intelligible. There appears to be but little difference in psychological complexity between the two methods. In both methods 8 is to be taken from 15i. e., we have 10 + 5-8. In the method of borrowing from the tens, we have to bear in mind, when we come to the subtraction of the tens, that the actual number

of tens to be dealt with is one less than the number of written tens. In the case of equal additions, we have to bear in mind that the actual number of tens to be dealt with is one more than the number of written tens.

3. Probably the best way to treat subtraction is the method based on the fact that the sum of the remainder and subtrahend is equal to the minuend. If we wish, for example, to find the difference between $15 and $8, we make up the 8 to 15, i. e., count from 8 up to 15, noting the new count of 7, which is the "difference between 8 and 15. To find the difference between 45 and 38 is to find what number added to 38 will

45

make 45: 38. The 8 units of the subtrahend can not

7

be made up to the 5 units of the minuend; we make it up, therefore, to 15 by adding 7 units, and put down 7 as a supposed part of the remainder. As this addition of 7 to 8 makes 15, we have 1 ten to carry to the 3 tens of the subtrahend, making it 4 tens, which requires no tens to make it up to the 4 tens of the minuend; the remainder is therefore 7. Proceed similarly with 75 — 38, etc.

873478

564693

308785

Take an example with larger numbers. From 873478 take 564693-that is, find what number added to the latter will give a result equal to the former. Write the subtrahend under the minuend, as in the margin, so that the figures of the same decimal order shall be in the same column. To 3, the right-hand figure of the subtrahend, 5 must be added to make up 8, the right-hand figure of the minuend; this is the right-hand figure of the remainder. We add 8 to 9, making the 9 up to 17

(ten-unit), and putting down 8 as the second figure of the remainder. We carry the 1 (hundred) from the 17 (ten) to the 6 hundred in the subtrahend, making it 7 (hundred); this 7 (hundred) is made up to 14 (hundred) by adding 7 (hundred), which is set down in the third place of the remainder; carrying 1 from the 14 to the 4 (thousand) in the subtrahend, we have 5 (thousand), which is made up to 13 (thousand) in the minuend by adding 8 (thousand), and 8 is set down in the thousands' place in the remainder. Similarly, carrying 1 from the made-up 13 to the next figure, 6, of the minuend, we have 7, which requires nothing to make it up to 7, and a zero is therefore set down in the 10-thousands' place in the remainder; finally, 5 requires 3 to make it up to 8, and so 3 is set down as the last figure of the remainder. Using italics to denote the numbers to be set down in figures as the remainder, the statement of the mental process will be: 3 and five, eight; 9 and eight, seventeen; 7 and seven, fourteen; 5 and eight, thirteen; 7 and naught, seven; 5 and three, eight. After some practice the minuend-sums need not be pronounced, and we shall have simply 3 and five, 9 and eight, etc.

This method is usually adopted in making change, and may be used with great facility in making calculations involving both additions and sub- $19128 tractions. Thus, suppose a merchant, having $19128 in bank, cheques out the sums $2714, $996, $3952, $166, $7516, how much has he remaining in bank? The several subtrahends are arranged in columns under the minuend, just as in addition. Add the subtrahends and

2714

996

3952

166

7516

$3784