VI

differences which we now proceed to develop. The absence of the incommensurables from the series of fractions leaves an absence of endpoints to certain classes. Thus, consider the incommensurable 2 . In the series of real numbers this stands between all the numbers whose squares are less than 2 , and all the numbers whose squares are greater than 2 . But keeping to the series of fractions alone and not thinking of the incommensurables, so that we cannot bring in 2 , there is no fraction which has the property of dividing off the series into two parts in this way, i.e. so that all the members on one side have their squares less than 2 , and on the other side greater than 2 . Hence in the series of fractions

there is a quasi-gap where $\sqrt{2}$ ought to come. This presence of quasi-gaps in the series of fractions may seem a small matter; but any mathematician, who happens to read this, knows that the possible absence of limits

or maxima to a class of numbers, which yet does not spread over the whole series of numbers, is no small evil. It is to avoid this difficulty that recourse is had to the incommensurables, so as to obtain a complete series with no gaps.

There is another even more fundamental difference between the two series. We can rearrange the fractions in a series like that of the integers, that is, with a first term, and such that each term has an immediate successor and (except the first term) an immediate predecessor. We can show how this can be done. Let every term in the series of fractions and integers be written in the fractional form by writing $\frac{1}{1}$ for $1$, $\frac{2}{1}$ for $2$, and so on for all the integers, excluding $0$. Also for the moment we will reckon fractions which are equal in value but not reduced to their lowest terms as distinct; so that, for example, until further notice $\frac{2}{3}$, $\frac{4}{6}$, $\frac{6}{9}$, $\frac{8}{12}$, etc., are all reckoned as distinct. Now group the fractions into classes by adding together the numerator and denominator of each term. For the sake of brevity call this sum of the numerator and denominator of a fraction its index. Thus $7$ is

the index of $\frac{4}{3}$, and also of $\frac{3}{4}$, and of $\frac{2}{5}$. Let the fractions in each class be all fractions which have some specified index, which may therefore also be called the class index. Now arrange these classes in the order of magnitude of their indices. The first class has the index $2$, and its only member is $\frac{1}{1}$; the second class has the index $3$, and its members are $\frac{1}{2}$ and $\frac{2}{1}$; the third class has the index $4$, and its members are $\frac{1}{3}$, $\frac{2}{2}$, $\frac{3}{1}$; the fourth class has the index $5$, and its members are $\frac{1}{4}$, $\frac{2}{3}$, $\frac{3}{2}$, $\frac{4}{1}$; and so on. It is easy to see that the number of members (still including fractions not in their lowest terms) belonging to any class is one less than its index. Also the members of any one class can be arranged in order by taking the first member to be the fraction with numerator $1$, the second member to have the numerator $2$, and so on, up to $(n-1)$ where $n$ is the index. Thus for the class of index $n$, the members appear in the order

$\frac{1}{n-1},\frac{2}{n-2},\frac{3}{n-3},\dots ,\frac{n-1}{1}.$ The members of the first four classes have in fact been mentioned in this order. Thus the whole set of fractions have now been arranged in an order like that of the integers. It runs thus

$\frac{1}{1},\frac{1}{2},\frac{2}{1},\frac{1}{3},\left[\frac{2}{2}\right],\frac{3}{1},\frac{1}{4},\frac{2}{3},\frac{3}{2},\frac{4}{1},\dots ,$

$\frac{n-1}{1},\frac{1}{n-1},\frac{2}{n-2},\frac{3}{n-3},\dots ,\frac{n-1}{1},\frac{1}{n},$

and so on.