1878.] Mr Glaisher, On circulating decimals. 195 



The Cambridge University Library contains a copy of tlie ' First 

 centenary, &c.,' 1818, the 'Table of circles,' 1823, and the 

 ' Tabular series,' 1823, the two last being separate. There is 

 bound up with the ' First centenary' the following letter : " Septem- 

 ber 16th, 1831. Mrs Catherine Goodwyn presents to the Library 

 of the University of Cambridge a complete set of the works of her 

 late father, Henry Goodwyn, Esq., of Blackheath, Kent. Royal 

 Hill, Greenwich." At the end of the * Tabular series' which is 

 attached to the ' First centenary' of 1818 is a tract of six leaves 

 (one a folding sheet), entitled " Introductory remarks respecting 

 the imperial gallon and diagonal table," followed by a tract of 

 three leaves (one a folding sheet), entitled "Introduction to a 

 synoptical table of English and French lineal measures" (dated 

 1821, December 13), and three folding sheets relating to weights 

 and measures and simple interest, which were also published by 

 Mr Goodwyn \ 



I have entered thus fully into the description of Mr Goodwyn's 

 works, as they are almost entirely unknown, and the most impor- 

 tant of them are anonj^raous. Reference is made to some of them 

 in De Morgan's article on Tahles in the Penny and English 

 Cyclopaedias, and there is a short account in the British Association 

 report on mathematical tables (1873) pp. 31 — 33 : but I know of 

 but one other place* in which any of them are alluded to. 



§ 6. It will be at once evident from the description that has 

 been given of Mr Goodwyn's tables how admirably they illustrate 

 the theory of circulating decimals as stated in §§ 2 and 3. The 

 most valuable is the 'table of circles' of 1823 which gives all the 

 periods of the numbers prime to 10 up to 1024. No one who has 

 considered the subject of circulating decimals can fail to find this 

 table one of great interest: it affords copious examples of the 

 rules relating to the periods of circulating decimals, and, besides 

 this, is a really important contribution to arithmetic. 



The arrangement adopted by Mr Goodwyn, by which digits and 

 their complements are printed the one under the other is excellent; 

 and the somewhat capricious grouping of the periods which it 

 occasions is no disadvantage. 



§ 7. I have had counted, from Mr Goodwyn's ' table of circles,' 

 the number of digits in a period, and the number of periods 

 corresponding to each number; and the results are given in the 



1 The University Library contains duplicates of all these works except the 

 synoptical table and the three folding sheets ; but there is no copy of the edition of 

 the 'First centenary' of 1816. 



^ Mathematical questions from the 'Educational Times,' vol. ix. (1868), p. 92, 

 where Mr C. W. Merrifield refers to Mr Farey's property. 



