order of magnitude, namely, with a first term, and such that each term has a couple of next-door neighbours, one on either side, with the exception of the first term which has, of course, only one next-door neighbour. Thus, if $m$ be any integer (not zero), there will be always an $m$ th term. A series with a finite

number of terms (say $n$ terms) has the same characteristics as far as next-door neighbours are concerned as an infinite series; it only differs from infinite series in having a last term, namely, the $n$ th.

The important thing to do with a series of numbers–-using for the future "series" in the restricted sense which has just been mentioned–-is to add its successive terms together.

Thus if $u_{1}$ , $u_{2}$ , $u_{3}$ , , $u_{n}$ , are respectively the $1$ st, $2$ nd, $3$ rd, $4$ th, , $n$ th, terms of a series of numbers, we form successively the series $u_{1}$ , $u_{1} + u_{2}$ , $u_{1} + u_{2} + u_{3}$ , $u_{1} + u_{2} + u_{3} + u_{4}$ , and so on; thus the sum of the $1$ st $n$ terms may be written $u_{1} + u_{2} + u_{3} + \dots + u_{n} .$

If the series has only a finite number of

terms, we come at last in this way to the sum of the whole series of terms. But, if the series has an infinite number of terms, this process of successively forming the sums of the terms never terminates; and in this sense there is no such thing as the sum of an infinite series.

But why is it important successively to add the terms of a series in this way? The answer is that we are here symbolizing the fundamental mental process of approximation. This is a process which has significance far

beyond the regions of mathematics. Our limited intellects cannot deal with complicated material all at once, and our method of arrangement is that of approximation. The statesman in framing his speech puts the dominating issues first and lets the details fall naturally into their subordinate places. There is, of course, the converse artistic method of preparing the imagination by the presentation of subordinate or special details, and then gradually rising to a crisis. In either way the process is one of gradual summation of effects; and this is exactly what is done by the successive summation of the terms of a series. Our ordinary method of stating numbers is such a process of gradual summation, at least, in the case of large numbers. Thus $568, 213$ presents itself to the mind as:–- $500, 000 + 60, 000 + 8, 000 + 200 + 10 + 3 .$

In the case of decimal fractions this is so more avowedly. Thus $3.14159$ is:–- $3 + \frac{1}{10} + \frac{4}{100} + \frac{1}{1000} + \frac{5}{10000} + \frac{9}{100000} .$ Also, $3$ and $3 + \frac{1}{10}$ , and $3 + \frac{1}{10} + \frac{4}{100}$ , and $3 + \frac{1}{10} + \frac{4}{100} + \frac{1}{1000}$ , and $3 + \frac{1}{10} + \frac{4}{100} + \frac{1}{1000} + \frac{5}{10000}$ are successive approximations to the complete result $3.14159$ . If we read $568, 213$ backwards from right to left, starting with the $3$ units,

we read it in the artistic way, gradually preparing the mind for the crisis of $500, 000$ .

The ordinary process of numerical multiplication proceeds by means of the summation of a series, Consider the computation $* 6 @ c @ & & & 3 & 4 & 2 & & & 6 & 5 & 8 \cline 4 - 6 \Strut & & 2 & 7 & 3 & 6 & 1 & 7 & 1 & 0 & 2 & 0 & 5 & 2 & & \cline 1 - 6 \Strut 2 & 2 & 5 & 0 & 3 & 6$

Hence the three lines to be added form a series of which the first term is the upper line. This series follows the artistic method of presenting the most important term last, not from any feeling for art, but because of the convenience gained by keeping a firm hold on the units' place, thus enabling us to omit some $0$ 's, formally necessary.