208 



NATURE 



\_yan. 2, 1879 



apparatus was designed by Mr. Whitehead, the inventor of the 

 celebrated torpedo, and is said to have worked satisfactorily. 



A CORRESPONDENT Writes to us that in looking through some 

 of the drawings and prints, &c., of Old London, belonging to 

 Mr. J. E. Gardner, F.S.A., of Park House, he came upon the 

 following interesting handbill : — 



London, 1775 



Proposals 



for a 



Short course of lectures 



on 



Fossils 



by 



Emanuel Mendes da Costa. 



The course will consist of only 

 TWELVE LECTURES. 

 A public Introductory lecture will be given gratis to 

 any one who chuse to come. 

 To begin on Wednesday, "j June, at noon, at the Author's 

 appartments at a shoemaker's opposite ArundelStreet in the Strand 

 and the future Lecture Hours will be determined by the sub- 

 scribers. 



The conditions are 



One Guinea the course. 



To be paid on Subscribing. 



Single lectiu-es at two shillings and sixpence each. 



Subscriptions are taken in at Mr. Elmsley's, Bookseller, 



opposite Southampton St., Strand ; Mr. White, bookseller in 



Fleet-street ; Mr, Humphreys, dealer in shells and other 



curiosities in St. Martin's Lane, near Charing Crois ; and by the 



author at his said appartments. 



N.B. — The Introductory lecture will be repeated on 

 Thursday evening at six o'clock. 



It is proposed to hold an anthropological exhibition at Moscow 

 in the coming summer, together with a general meeting of 

 anthropologists from all parts of the world. 



We have on our table the following works : — "Plistoiy of 

 the Steam Engine," R. H. Thurston, Kc-Sjan Paul and Co. ; 

 " Etudes et Lectures sur 1' Astronomic," Camille Flammarion, 

 G. Villars, Paris ; " Catalogue des Etoiles Doubles et Multiples,'' 

 Camille Flammarion, G. Villars, Paris ; " Sport and Work on 

 the Nepaul Frontier," "Maori," Macmillan and Co. ; "Mathe- 

 matical Problems," J. Wolstenholme, Macmillan and Co. ; 

 " The Fairy-Land of Science," Arabella B. Buckley, E. 

 Stanford; " Das Leben," Philipp Spiller, Gerstmann, Berlin j 

 " Wanderings in Patagonia," Julius Beerbohm, Chatto and 

 Windus ; " Natural History of Victoria," Frederick McCoy> 

 Triibner; "Fourth Annual Report of the Imperial Mint;" 

 "Extra Physics and the Mystery of Creation," Hodder and 

 Stoughton ; " From Kulja Across the Tian Shan to Lob Nor," 

 Col. N. Prejevalsky, Sampson Lov/ and Co.; "The Heart 

 of Africa," Dr. Georg Schweinfurth, Sampson Low and 

 Co. ; " The Philosophy of Science, Experience, and Reve- 

 lation," John Coutts, F. Pitman ; "The Native Flowers and 

 Ferns of the United States," Parts 13, 14, 15, 16, Thomas 

 Mechan, L. Prangola, Boston; " The Principles of Light 

 and Colour," Edwin B. Babbitt, Triibner and Co. 



The additions to the Zoological Society's Gardens during the 

 past week include a White-whiskered Paradoxure [Paradoxurus 

 leucomystax) from East India, presented by Mr. W. G. Wilson ; 

 a Common Barn Owl {Sirix Jlammea), British, presented by 

 Mr. W. Davies ; a Common Coot {Fullca atra), British, pre- 

 sented by Mr. F. H. O'Donoghue ; two Philantomba Antelopes 

 ( Cephalophus tnax7uelli) from West Africa, two Egj-ptian Jerboas 

 {Dipus ccgyptius') from Egypt, purchased ; three River Jack 

 Vipers {Vipei-a rhinoceros) from West Africa, deposited. 



CIRCULATING DECIMALS 



nPHE properties of circulating decimals mentioned by Mr. R. 

 ■*■ Chartres and by Mr. E. P. Toy in Nature (vol. xviii. 

 pp. 291, 541) are particular cases of very general laws relating 

 to the periods of circulating decimals of which, as they are not 

 stated with any approach to completeness in any work on arith- 

 metic with which I am acquainted, it may be worth while to 

 give a brief explanation. 



Consider the process of converting a vulgar fraction into a 

 circulating decimal ; take for example -gV. The work ij — 



39) roD (-625641 



78 



220 

 195 



250 

 234 



i6o 

 156 



40 

 39 



10 



which may be more concisely and better arranged thus : — 

 39) I (-6 



ID 2 

 22 5 



25 6 

 16 4 



4 i 



10, 22, 25, 16, 4 being, the remainders and the corresponding 

 quotient figures being written at the side. From this it is clear 

 that— 



■gl^ = -625641, \% = -256410, II = -564102, II = 641025, 



\% - •410256, -iTi = -102564, 



and the numbers i, 10, 22, 25, '16, 4 form a cycle such that if 

 we divide any one of them by 39 we obtain the others as re- 

 mainders in this order, and all the fractions give rise to the same 

 period, though the beginning is made in each case at a different 

 place in the period. 



The following are three other divisions arranged in the same 

 manner : — 



39) 2 (-6 

 20 5 



5 I 

 II 2 

 32 8 



82 35 8 i\ i 



The four divisions thus give the values of the periods of the 

 fractions ^V. ^V. uV. in7> • • • If, i->:-, of all the proper fractions 

 in their lowest terms, having 39 as denominator. In this case, 

 therefore, there are four distinct periods, or, say, four periods 

 each containing six figures ; one of these, viz., that to which -^-^ 

 belongs, may be called the leading period. 



In general if q be any number prime to 10, and if all the 

 proper fractions in their loM'est terms having q for denominator 

 be converted into decimals there will be f periods each contaiu- 

 a digits, and a and/will be connected by the relation af— (t>{q)> 

 where p(q) denotes the number of numbers less than q and prime 

 to it. If 17 be a prime, ^((7) = q - i. 



It is to be observed that if we divide r and q - r respectively 

 by q the digits of the periods will in the two cases be comple- 

 mentary, i.e., the sum of each corresponding pair will be 9. Thus 

 in tlie case of 39 



^V = -625641" ^ = -651282 



U = -974358 f^ = -948717 



and 9 -f o = 9, 7 -f 2 = 9, &c. Also, the sum of each pair ot 

 corresponding remainders is q ; e.g., in the divisions for ^V and 



11, -fs^ and f J, the sum of each pair of corresponding remainders 

 is 39. 



If, as in the case of 39, the remainder q - i does not belong 

 to the leading period, the periods may be arranged in pairs, the 

 periods in each pair being complementary to one another. If 



