An Introduction to the Theory of Numbers

1. Farey Series. The theory of numbers has always occupied a peculiar position among the purely mathematical sciences. It has the reputation of great difficulty and mystery among many who should be competent to judge; I suppose that there is no mathematical theory of which so many wellqualified mathematicians are so much afraid. At the same time it is unique among mathematical theories in its appeal to the uninstructed imagination and in its fascination for the amateur. It would hardly be possible in any other subject to write books like Landau's Vorlesungen or Dickson's History, six great volumes of overwhelming erudition, better than the football reports for light breakfast table reading. The excursions of amateur mathematicians into mathematics do not usually produce interesting results. I wish to draw your attention for a moment to one very singular exception. Mr. John Farey, Sen., who lived in the Napoleonic era, has a notice of twenty lines in the Dictionary of National Biography, where he is described as a geologist. He received as a boy "a good mathematical training". He was at one time agent to the Duke of Bedford, but afterwards came to London, where he acquired an extensive practice as a consulting surveyor, which led him to travel much about the country and "collect minerals and rocks". His principal work was a geological survey of Derbyshire, undertaken for the Board of Agriculture, but he also wrote papers in the Philosophical Magazine, on geology and on many other subjects, such as

1. 法里数列。数论在纯数学学科中一直占据着独特的地位。在许多有能力评判的人当中，它享有难度大且神秘的名声；我想没有哪一种数学理论会让这么多优秀的数学家如此畏惧。同时，它在数学理论中独一无二，能吸引未经训练的想象力，对业余爱好者也极具吸引力。在其他任何学科中，都几乎不可能写出像兰道的《讲义》或迪克森的《史》这样的书，六大卷博大精深的学识，比早餐桌上轻松阅读的足球报道还要好。业余数学家涉足数学通常不会产生有趣的结果。我想提请你们注意一个非常特殊的例外。老约翰·法里先生生活在拿破仑时代，在《英国传记大词典》中有20行的介绍，在那里他被描述为一位地质学家。他小时候接受过“良好的数学训练”。他曾一度是贝德福德公爵的代理人，但后来来到伦敦，在那里他作为一名咨询测量师业务广泛，这使他在全国各地游历颇多，并“收集矿物和岩石”。他的主要工作是为农业委员会进行的德比郡地质调查，但他也在《哲学杂志》上发表过关于地质以及许多其他学科的论文，比如