Consider, for example, the product of four numbers, abcd. We will endeavour to show that this is the same thing as the product acbd. The symbol abcd means that we are to take c groups of d things and then b groups like the aggregate so formed, and then finally a groups of bed things.

Now, by what we have already proved, b groups of cd things come to the same number as c groups of bd things. Consequently, a groups of bed things are the same as a groups of cbd things; that is to say, abcd= acbd.

It will be quite clear that this reasoning will hold no matter how many letters come after d. Suppose, for example, that we have a product of six numbers abcdef. This means that we are to multiply ƒ by e, the result by d, then def by c, and so on.

Now in this case the product def simply takes the place which the number d had before. And b groups of c times def things come to the same number as c groups of b times def things, for this is only the product of three numbers, b, c, and def. Since then this result is the same in whatever order b and c are written, there can be no alteration made by multiplications coming after, that is to say if we have to multiply by ever so many more numbers after multiplying by a. It follows therefore that no matter how many numbers are multiplied together, we may change the places of any two of them which are close together without altering the product.

In the next place let us prove that we may change the places of any two which are not close together. For example, that abcdef is the same thing as accdbf, where b and e have been interchanged. We may do this by first making the e march backwards, changing

places successively with d and c and b, when the product is changed into aebcdf; and then making b. march forwards so as to change places successively with c and d, whereby we have now got e into the place of b.

Lastly, I say that by such interchanges as these we can produce any alteration in the order that we like. Suppose for example that I want to change abcdef into defbea. Here I will first get d to the beginning; I therefore interchange it with a, producing dbcaef. Next, I must get c second; I do this by interchanging it with b, this gives dcbaef. I must now put ƒ third by interchanging it with b, giving defaeb, next put b fourth by interchanging it with a, producing dcfbea. This is the form required. By five such interchanges at most, I can alter the order of six letters in any way I please. It has now been proved that this alteration in the order may be produced by successive interchanges of two letters which are close together. But these interchanges, as we have before shown, do not alter the product; consequently the product of six numbers in any order is equal to the product of the same six numbers in any other order; and it is easy to see how the same process will apply to any number of numbers.

But is not all this a great deal of trouble for the sake of proving what we might have guessed beforehand? It is true we might have guessed beforehand that a product was independent of the order and grouping of its factors; and we might have done good work by developing the consequences of this guess before we were quite sure that it was true. Many beautiful theorems have been guessed and widely used before they were conclusively proved; there are some even now in that state. But at some time or other the

inquiry has to be undertaken, and it always clears up our ideas about the nature of the theorem, besides giving us the right to say that it is true. And this is not all; for in most cases the same mode of proof or of investigation can be applied to other subjects in such a way as to increase our knowledge. This happens with the proof we have just gone through; but at present, as we have only numbers to deal with, we can only go backwards and not forwards in its application. We have been reasoning about multiplication; let us see if the same reasoning can be applied to addition.

What we have proved amounts to this. A certain result has been got out of certain things by taking them in a definite order; and it has been shown that if we can interchange any two consecutive things without altering the result, then we may make any change whatever in the order without altering the result. Let us apply this to counting. The process of counting consists in taking certain things in a definite order, aud applying them to our fingers one by one; the result depends on the last finger, and its name is called the number of the things so counted. We learn then that this result will be independent of the order of counting, provided only that it remains unaltered when we interchange any two consecutive things; that is, provided that two adjacent fingers can be crossed, so that each rests on the object previously under the other, without employing any new fingers or setting free any that are already employed. With this assumption we can prove that the number of any set of things is independent of the order of counting; a statement which, as we have seen, is the foundation of the science of number.

4. The Distributive Law.

There is another law of multiplication which is, if possible, still more important than the two we have already considered. Here is a particular case of it: the number 5 is the sum of 2 and 3, and 4 times 5 is the sum of 4 times 2 and 4 times 3. We can make this visible by an arrangement of dots as follows:

Here we have four rows of five dots each, and each row is divided into two parts, containing respectively two dots and three dots. It is clear that the whole number of dots may be counted in either of two ways; as four rows of five dots, or as four rows of two dots together with four rows of three dots. By our general principle the result is independent of the order of counting, and therefore

4 x 5 = (4 × 2) + (4 × 3);

or, if we put in evidence that 5=2+3,

4 (2+3)= (4 × 2) + (4 × 3).

The process is clearly applicable to any three numbers whatever, and not only to the particular numbers 4, 2, 3. We may construct an oblong containing a rows of b+c dots; and this may be divided by a vertical line into a rows of b dots and a rows of c dots. Counted in one way, the whole number of dots is a(b+c);

counted in another way, it is ab+ac. Hence we must always have

a (b + c) = ab + ac.

This is the first form of the distributive law.

Now the result of multiplication is independent of the order of the factors, and therefore

a (b + c) = (b + c) a,

abba,

ac = ca;

so that our equation may be written in the form

(b + c) a = ba + ca.

This is called the second form of the distributive law. Using the numbers of our previous example, we say that since 5 is the sum of 2 and 3, 5 times 4 is the sum of 2 times 4 and 3 times 4. This form may be arrived at independently and very simply as follows. We know that 2 things and 3 things make 5 things, whatever the things are; let each of these things be a group of 4 things; then 2 fours and 3 fours make 5 fours, or

(2 × 4) + (3 × 4)=5 × 4.

The rule may now be extended. It is clear that our oblong may be divided by vertical lines into more parts than two, and that the same reasoning will apply. This

figure, for example, makes visible the fact that just as 2 and 3 and 4 make 9, so 4 times 2, and 4 times 3, and 4 times 4 make 4 times 9. Or generally