V

subtle idea, not at all obvious. A great deal of discussion on the meaning of the zero of quantity will be found in philosophic works. Zero is not, in real truth, more difficult or subtle in idea than the other cardinal numbers. What do we mean by $1$ or by $2$, or by $3$? But we are familiar with the use of these ideas, though we should most of us be puzzled to give a clear analysis of the simpler ideas which go to form them. The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought. Many important services are rendered by the symbol $0$, which stands for the number zero.

The symbol developed in connection with the Arabic notation for numbers of which it is an essential part. For in that notation the

value of a digit depends on the position in which it occurs. Consider, for example, the digit $5$, as occurring in the numbers $25$, $51$, $3512$, $5213$. In the first number $5$ stands for five, in the second number $5$ stands for fifty, in the third number for five hundred, and in the fourth number for five thousand. Now, when we write the number fifty-one in the symbolic form $51$, the digit $1$ pushes the digit $5$ along to the second place (reckoning from right to left) and thus gives it the value fifty. But when we want to symbolize fifty by itself, we can have no digit $1$ to perform this service; we want a digit in the units place to add nothing to the total and yet to push the $5$ along to the second place. This service is performed by $0$, the symbol for zero. It is extremely probable that the men who introduced for this purpose had no definite conception in their minds of the number zero. They simply wanted a mark to symbolize the fact that nothing was contributed by the digit's place in which it occurs. The idea of zero probably took shape gradually from a desire to assimilate the meaning of this mark to that of the marks, $1$, $2$, , $9$, which do represent cardinal numbers. This would not represent the only case in which a subtle idea has been introduced into mathematics by a symbolism which in its origin was dictated by practical convenience.

Thus the first use of $0$ was to make the arable notation possible–-no slight service. We can imagine that when it had been introduced for this purpose, practical men, of the sort who dislike fanciful ideas, deprecated the silly habit of identifying it with a number zero. But they were wrong, as such men always are when they desert their proper function of masticating food which others have prepared. For the next service performed by the symbol $0$ essentially depends upon assigning to it the function of representing the number zero.

This second symbolic use is at first sight so absurdly simple that it is difficult to make a beginner realize its importance. Let us start with a simple example. In II. we mentioned the correlation between two variable numbers $x$ and $y$ represented by the equation $x+y=1$. This can be represented in an indefinite number of ways; for example, $x=1-y$, $y=1-x$, $2x+3y-1=x+2y$, and so on. But the important way of stating it is $x+y-1=0.$ Similarly the important way of writing the equation $x=1$ is $x-1=0$, and of representing the equation $3x-2=2x^2$ is $2x^2-3x+2=0$. The point is that all the symbols which represent variables, e.g. $x$ and $y$, and the symbols

representing some definite number other than zero, such as $1$ or $2$ in the examples above, are written on the left-hand side, so that the whole left-hand side is equated to the number zero. The first man to do this is said to have been Thomas Harriot, born at Oxford

in 1560 and died in 1621. But what is the importance of this simple symbolic procedure? It made possible the growth of the

modern conception of algebraic form.