He could not have chosen a worse example. For, without having made an examination of English text-books on mathematics contemporary with the publication of this essay, the

assumption is a fairly safe one that Taylor's theorem was enunciated and proved wrongly in every one of them. Accordingly, the anxious precision of modern mathematics is necessary for accuracy. In the second place it is necessary for research. It makes for clearness of thought, and thence for boldness of thought and for fertility in trying new combinations of ideas. When the initial statements are vague and slipshod, at every subsequent stage of thought common sense has to step in to limit applications and to explain meanings. Now in creative thought common sense is a bad master. Its sole criterion for judgment is that the new ideas shall look like the old ones. In other words it can only act by suppressing originality.

In working our way towards the precise definition of continuity (as applied to functions) let us consider more closely the statement that there is no relation of "graduality" between numbers. It may be asked, Cannot one number be only slightly greater than another number, or in other words, cannot the difference between the two numbers be small? The whole point is that in the abstract, apart from some arbitrarily assumed application, there is no such thing as a great or a small number. A million miles is a small number of miles for an astronomer investigating the fixed stars, but a million

pounds is a large yearly income. Again, one-quarter is a large fraction of one's income to give away in charity, but is a small fraction of it to retain for private use. Examples can be accumulated indefinitely to show that great or small in any absolute sense have no abstract application to numbers. We can say of two numbers that one is greater or smaller than another, but not without specification of particular circumstances that any one number is great or small. Our task therefore is to define continuity without any mention of a "small" or "gradual" change in value of the function.

In order to do this we will give names to some ideas, which will also be useful when we come to consider limits and the differential calculus.

An "interval" of values of the argument $x$

of a function $f (x)$ is all the values lying between some two values of the argument. For example, the interval between $x = 1$ and $x = 2$ consists of all the values which $x$ can take lying between $1$ and $2$ , i.e. it consists of all the real numbers between $1$ and $2$ . But the bounding numbers of an interval need not be integers. An interval of values of the argument contains a number $a$ , when $a$ is a member of the interval. For example, the interval between $1$ and $2$ contains $\frac{3}{2}$ , $\frac{5}{3}$ , $\frac{7}{4}$ , and so on.

A set of numbers approximates to a number $a$

within a standard $k$ , when the numerical difference between $a$ and every number of the set is less than $k$ . Here $k$ is the "standard of approximation." Thus the set of numbers $3$ , $4$ , $6$ , $8$ , approximates to the number $5$ within the standard $4$ . In this case the standard $4$ is not the smallest which could have been chosen, the set also approximates

to $5$ within any of the standards $3.1$ or $3.01$ or $3.001$ . Again, the numbers, $3.1$ , $3.141$ , $3.1415$ , $3.14159$ approximate to $3.13102$ within the standard $.032$ , and also within the smaller standard $.03103$ .

These two ideas of an interval and of