The logarithmic scale. Let us consider once more the series of functions \[x,\quad \sqrt{x},\quad \sqrt[3]{x},\ \dots,\quad \sqrt[n]{x},\ \dots,\] which possesses the property that, if \(f(x)\) and \(\phi(x)\) are any two of the functions contained in it, then \(f(x)\) and \(\phi(x)\) both tend to \(\infty\) as \(x \to \infty\), while \(f(x)/\phi(x)\) tends to \(0\) or to \(\infty\) according as \(f(x)\) occurs to the right or the left of \(\phi(x)\) in the series. We can now continue this series by the insertion of new terms to the right of all those already written down. We can begin with \(\log x\), which tends to infinity more slowly than any of the old terms. Then \(\sqrt{\log x}\) tends to \(\infty\) more slowly than \(\log x\), \(\sqrt[3]{\log x}\) than \(\sqrt{\log x}\), and so on. Thus we obtain a series \[x,\quad \sqrt{x},\quad \sqrt[3]{x},\ \dots,\quad \sqrt[n]{x},\ \dots\quad \log x,\quad \sqrt{\log x},\quad \sqrt[3]{\log x},\ \dots\quad \sqrt[n]{\log x},\ \dots\] formed of two simply infinite series arranged one after the other. But this is not all. Consider the function \(\log\log x\), the logarithm of \(\log x\). Since \((\log x)/x^{\alpha} \to 0\), for all positive values of \(\alpha\), it follows on putting \(x = \log y\) that \[(\log\log y)/(\log y)^{\alpha} = (\log x)/x^{\alpha} \to 0.\] Thus \(\log\log y\) tends to \(\infty\) with \(y\), but more slowly than any power of \(\log y\). Hence we may continue our series in the form \[\begin{gathered} x,\quad \sqrt{x},\quad \sqrt[3]{x},\ \dots\qquad \log x,\quad \sqrt{\log x},\quad \sqrt[3]{\log x},\ \dots\\ \log\log x,\quad \sqrt{\log\log x},\ \dots\quad \sqrt[n]{\log\log x},\ \dots;\end{gathered}\] and it will by now be obvious that by introducing the functions \(\log\log\log x\), \(\log\log\log\log x\), … we can prolong the series to any extent we like. By putting \(x = 1/y\) we obtain a similar scale of infinity for functions of \(y\) which tend to \(\infty\) as \(y\) tends to \(0\) by positive values.^{1}

- For fuller information as to ‘scales of infinity’ see the author’s tract ‘Orders of Infinity’,
*Camb. Math. Tracts*, No. 12.↩︎

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