V

the fundamental laws of algebra:–- Cf. Note A, noteA.60

x + y = y + x, \tag{\quad\ensuremath{(1)}} \ (x + y) + z = x + (y + z), \tag \ x × y = y × x, \tag}{\quad\ensuremath{(3)}} \ (x × y) × z = x × (y × z), \tag}} \ x × (y + z) = (x × y) + (x × z). \tag*{\quad\ensuremath{(5)}

Here (1) and (2) are called the commutative and associative laws for addition, (3) and (4)

are the commutative and associative laws for multiplication, and (5) is the distributive law relating addition and multiplication. For example, without symbols, (1) becomes: If a second number be added to any given number the result is the same as if the first given number had been added to the second number.

This example shows that, by the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain.

It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle–-they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

One very important property for symbolism to possess is that it should be concise, so as to be visible at one glance of the eye and to be rapidly written. Now we cannot place symbols more concisely together than by placing them in immediate juxtaposition. In a good symbolism therefore, the juxtaposition of important

symbols should have an important meaning. This is one of the merits of the Arabic notation for numbers; by means of ten symbols, $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and by simple juxtaposition it symbolizes any number whatever. Again in algebra, when we have two variable numbers $x$ and $y$, we have to make a choice as to what shall be denoted by their juxtaposition $xy$. Now the two most important ideas on hand are those of addition and multiplication. Mathematicians have chosen to make their symbolism more concise by defining $xy$ to stand for $x\times y$. Thus the laws (3), (4), and (5) above are in general written, $xy=yx,(xy)z=x(yz),x(y+z)=xy+xz,$ thus securing a great gain in conciseness. The same rule of symbolism is applied to the juxtaposition of a definite number and a variable: we write $3x$ for $3\times x$, and $30x$ for $30\times x$.

It is evident that in substituting definite numbers for the variables some care must be taken to restore the $\times$, so as not to conflict with the Arabic notation. Thus when we substitute $2$ for $x$ and $3$ for $y$ in $xy$, we must write $2\times 3$ for $xy$, and not $23$ which means $20+3$.

It is interesting to note how important for the development of science a modest-looking symbol may be. It may stand for the emphatic presentation of an idea, often a very

subtle idea, and by its existence make it easy to exhibit the relation of this idea to all the complex trains of ideas in which it occurs. For example, take the most modest of all symbols, namely, $0$, which stands for the number

zero. The Roman notation for numbers had no symbol for zero, and probably most mathematicians of the ancient world would have been horribly puzzled by the idea of the number zero. For, after all, it is a very