# Peano paragraphing

Every mathematician has heard of the Peano axioms of arithmetic. Here is a lesser known contribution of Giuseppe Peano: the “Peano paragraphing method”. This is a numbering system for sections/subsections/etc in books where the different items are identified by a decimal number (e.g., 9.132), where the integral part is the chapter number, and the decimal part is arranged in increasing order within each chapter. So for instance 9.301 is a subsubsection lying between 9.3 and 9.31.

I had noticed this system in Titchmarsh’s book “The theory of functions”, from 1932, without understanding it (it is not explained, nor attributed to Peano). Then I saw it again just recently as I was looking up a reference in Whittaker and Watson’s “A course of modern analysis” from 1927, where the explanation and attribution are given in a remark at the beginning. This greatly clarified my previous perplexity in navigating the book of Titchmarsh, which I had found extremely confusing; for instance in Chapter 9, we have

9.1, 9.11 up to 9.15, 9.2, 9.3, 9.31, 9.32, 9.4, 9.41 to 9.45, 9.5, 9.51 to 9.55, 9.6, 9.61, 9.62, 9.621 to 9.623, 9.7…

Looking into other classical books, I can see this system in Watson’s treatise on Bessel functions, but it is not used in either Hardy and Wright’s “Introduction to the theory of numbers”, nor in Titchmarsh’s “The theory of the Riemann zeta function”. It is also absent from Zygmund’s “Trigonometric series” (which, on the other hand, uses a continuous numbering scheme X.Y (Chapter.Item) both for equations, theorems, etc), and from Hardy and Rogosinsky’s “Fourier series”.

Note finally that it seems rather euphemistic to say that this is “lesser known”: neither Google nor Wikipedia seem to be able to give a reference or explanation!