{"id":58,"date":"2008-06-03T10:50:24","date_gmt":"2008-06-03T09:50:24","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/2008\/06\/03\/peano-paragraphing\/"},"modified":"2024-02-20T13:40:59","modified_gmt":"2024-02-20T11:40:59","slug":"peano-paragraphing","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2008\/06\/03\/peano-paragraphing\/","title":{"rendered":"Peano paragraphing"},"content":{"rendered":"<p>Every mathematician has heard of the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Peano_axioms\">Peano axioms<\/a> of arithmetic.  Here is a lesser known contribution of Giuseppe Peano: the &#8220;Peano paragraphing method&#8221;.  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.<\/p>\n<p>I had noticed this system in Titchmarsh&#8217;s book <a href=\"http:\/\/books.google.com\/books?id=ebnitl7SNLQC\">&#8220;The theory of functions&#8221;<\/a>, 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&#8217;s <a href=\"http:\/\/books.google.com\/books?id=ULVdGZmi9VcC\">&#8220;A course of modern analysis&#8221;<\/a> 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<\/p>\n<blockquote><p> 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&#8230;<\/p><\/blockquote>\n<p>Looking into other classical books, I can see this system in Watson&#8217;s treatise on Bessel functions, but it is not used in either Hardy and Wright&#8217;s &#8220;Introduction to the theory of numbers&#8221;, nor in Titchmarsh&#8217;s &#8220;The theory of the Riemann zeta function&#8221;. It is also absent from Zygmund&#8217;s &#8220;Trigonometric series&#8221; (which, on the other hand, uses a continuous numbering scheme X.Y  (Chapter.Item) both for equations, theorems, etc),  and from Hardy and Rogosinsky&#8217;s &#8220;Fourier series&#8221;.<\/p>\n<p>Note finally that it seems rather euphemistic to say that this is &#8220;lesser known&#8221;: neither Google nor Wikipedia seem to be able to give a reference or explanation!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Every mathematician has heard of the Peano axioms of arithmetic. Here is a lesser known contribution of Giuseppe Peano: the &#8220;Peano paragraphing method&#8221;. 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 &hellip; <a href=\"https:\/\/blogs.ethz.ch\/kowalski\/2008\/06\/03\/peano-paragraphing\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Peano paragraphing<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[362,2],"tags":[],"class_list":["post-58","post","type-post","status-publish","format-standard","hentry","category-language","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/58","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=58"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/58\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=58"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=58"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=58"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}