In higher mathematics it is possible to start from the idea of the Finite and reach the conception of the Infinite; or to reverse the process, and from the Infinite to deduce the Finite. Thus in the familiar puzzle of the subdivision of the parts of a straight line by halving the remainder, there will be a crowding and a coalescing of the points of division towards one end of the line, the points of division getting infinitely nearer, but the steps will never meet. Here in the centre of a straight line--a limited straight line--we are confronted with the problem of Infinity. Again, from a series of finite numbers we can gain the notion of an infinite series. Take two series which have a correspondence with one another. If for every element of the one we can choose an element of the other, and of the other there is an element for the one, when at any point we cut off its progress to infinity, this happens:-- One series, if summed up, will give a larger numerical result than the other, and therefore can be said to be greater than the second. Let us call the first series A, and the second B. Let us now imagine the two series, though starting at a definite point, are never cut at the further end. Then to all infinity series B is without certain numbers which series A possesses, _and as an infinite series_ is smaller than series A. But, on the other hand, when neither series is cut, series B retains its correspondence with series A. Thus we attain a definition of an infinite series. It is such that the part, while being less than the whole, has yet a complete correspondence with the whole. The whole is greater than the part, but take away the part from the whole and that which remains corresponds to it _in infinity_, because the test of summing the series (which would give a contrary result) involves limitation, and thus cannot be applied. Subtraction can take place in Infinity without loss.