This proof is not a perfectly sure test of accuracy. It does not point out an error of 9, or of a multiple of 9, in the product. Thus, if 0 has been written for 9 or 9 for 0, or if a partial product has been set down in the wrong place, or if one or more noughts have been inserted or omitted in any of the products, or if two figures have been interchanged, or if one figure set down is as much too great as another is too small, casting out the nines will fail to detect the error, for the remainder from dividing by 9 will not be affected. Still the proof is interesting, as throwing light upon the decimal system of notation.

The Multiplication Table.-The sure groundwork for this table is, of course, facile mastery of the addition and subtraction tables. Though scraps of it given from time to time-as the 2's and 3's in 6-are perhaps of no great value as contributing to the making and mastering of the entire table, yet some complete parts of the table-as, for example, two times, three times, ten times may be kept in view, and may be expertly handled quite early in the course. It has been said that the table is a grand effort of the special memory for symbols and their combinations, and that the labour can not be extenuated in any way. The labour is, indeed, heavy enough, but it is believed that it may be somewhat lightened. The table, as the key to arithmetic, must be learned, and it must be learned perfectly-i. e., so that any pair of factors instantly suggests the product; there must be no halting memory summoning attention and

judgment to its aid. It is therefore worth while to "extenuate" the labour of learning it, if this can possibly be done. To this end some suggestions are made which are believed to be rational, while they have certainly stood the test of experience.

means.

1. The Meaning of the Table.-Pupils rightly taught know how to construct the table; they know what it The symbol memory, like every other kind of memory, is always aided where intelligence is at work. In former times, not so long past, the table used to be said or sung-rattled off in some familiar tune—without a glimmer of what it all meant; but under rational instruction the children know several important things about it, and the teacher should use these things in lessening the labour of complete mastery.

2. Memory aided by Intelligence.-(1) The pupils have learned how to construct any part of the table, two times, three times, etc.

(2) They know exactly what such construction means, for they have acquired a fair idea of times-of number as denoting repetition of a measuring unit. They know, therefore, the meaning of every product: 6 oranges at 5 cents apiece, 6 yards of calico at 9 cents a yard, etc.

(3) They can derive the product of any pair of factors from the product of the immediately preceding pair. Knowing that 6 yards of cloth at 8 cents a yard cost 48 cents, they know that 7 yards cost 8 cents more. Similarly they quickly learn that if 10 oranges cost 50 cents, 9 oranges will cost five cents less, and 8 oranges one ten less, etc. Thus they will have various ways of constructing, and recovering when momentarily forgotten, the product of any pair of digits.

3. The Commutation of Factors.-In learning the table the relation of the factors must be kept in view. This greatly reduces the labour. There ought to be little difficulty in this if a fair idea of the relation between parts and times has been brought out. At 3 cents apiece, 5 oranges cost 3 cents X 5; this is seen to be identical with 5 cents X 3. Each of these expresses measured quantity, a sum of money; the thought "oranges disappears from this conception. The table is often taught without reference to this principle, and so the labour of learning it is at least one half greater than it ought to be. In our boyhood we learned 9 × 6 = 54, without a suspicion that 6 × 954. Let us see to it that the present things be made better than the former.

4. Memory further aided.-Associations. In this connection the following suggestions are worthy of attention:

(1) The thing to be kept in view is that, so far as possible, associations are to be formed directly between a product and one or both of the factors which produce it.

(2) Ten times is already learned in addition—in the counting of the tens. The pupil knows how to "multiply" any number by ten by simply affixing a zero to the number. The association of product and factors is direct; the product is the multiplicand with the zero of the 10 affixed. Ten times, then, is well in hand.

(3) Eleven times is almost equally easy. The product in each case, for the first nine digits, is directly associated with the digit; the digit is simply repeated11, 22, 33, etc. Eleven tens is known from ten elevens, and the other two products (11 × 11, 12 × 11) must be built up from this.

(4) Nine times can be formed and remembered in a similar way. The pupil will note: (a) That a product is made up of tens and units. (6) That in 9 times (up to 10 X 9) the number of tens is always one less than the number multiplied. (c) That in every product the sum of the digits is 9; and thus, having written down the tens directly from the multiplicand, he can at once write the units. He should be led to notice also that (b) holds good as to the law of tens up to 10 X 9, after which the number of tens is two less than the multiplicand up to and including 20 X 11, after which the number of tens is three less, etc. He should note, too, in his formed table, how the tens increase by one and the units decrease by one.

This

may seem somewhat complex, but it works well. We have known a boy of six years to construct and learn 9 times up to 9 times 10 in fifteen minutes.

(5) Probably two times has been completely learned before a formal attack is made upon the table as a whole. There has been much practice in counting by 2's-backward and forward-and by 3's, etc. There seems to be no way of making a mnemonic association between a product and its factors; but addition by two is an easy operation, and two times is quickly learned.

(6) In twelve times (assuming two times) the memory can be aided by association. The product of any multiplicand may be obtained by taking 12, the multiplicand, as so many tens, and doubling it for the units; twelve times 3 three tens and six (twice 3) units. For 5 and up to 9, doubling the unit gives more than 10, but the additions are easy. 12 times 5 five tens and ten Or, consider the products

units (twice five)

=

60.

thus the products of 14 are 12, 24, 36, 48; those of 59 are each one ten more than the multiplicand, and the units increase by 2's-i. e., 0, 2, 4, 6, 8; the products of 5, 6, 7, 8, 9 are therefore 60, 72, 84, 96, 108.

(7) Some assistance from association may be had in learning 5 times by observing: (a) that the units are alternately 5 and 0, 5 for the odd multiplicands, 0 for the even ones; (b) if the multiplicand is even, the tens are half of it; if odd, the tens are half the next lower number: 8 X 54 tens and 0 units; 9 X 54 tens and 5 units, etc. More advanced students will take pleasure in extending the multiplication table according to these laws, as well as in accounting for the laws. For example in 9 times, why are the tens one less than the multiplicand up to 10 X 9, then two tens less up to 20 × 9? etc. In 8 times why are the tens one less than the multiplicand up to 5 × 8, two less from 6 × 8 to 10 X 8? etc.

Division.-Division is, we have seen, the operation of finding either of two factors when their product and the other factor are given. After what has been said in Chapter V upon the nature of division and its relation to multiplication and fractions, little further need be added, especially as most of the text-books explain clearly enough the actual arithmetical work. A few points, however, may be briefly noticed: (1) If, in the method of teaching, the idea of number as measurement has been kept steadily in view, the nature of division as the inverse of multiplication will be fully understood. (2) Knowing the relation of the factors in multiplication, the pupil will, with but little difficulty,