Jan. 2, 1879] 



NATURE 



209 



the remainder '/ - i does belong to the leading period, each 

 period v.ill contain an even number of digits, and the first half 

 and second half of each period will be complementary. Thus, 

 for q — 11 there are nine periods : 6136 9863, 6273 9726, 

 64109589, &c., and in each the two halves are complementary. 

 If there is but one period corresponding to q, of coiu-se the 

 remainder ? - i must belong to this period, so that in this case 

 the two halves are always complementary. Returning to the 

 period of ^, we see that it is such that if we multiply it by 4 

 we obtain the same period, only beginning with the last digit, 

 that if we multiply it by 16 we obtain the same period beginning 

 with tlie last d^t but one, and so on. Thus, from knowing 

 that the last figure of the period is t, and that the last remainder 

 is 4, we can obtain the period ; for 4 X I = 4 so that the last 

 figure but one must be 4 , the last two figures must therefore be 

 41, multiply this by 4 we have 164, so 3iat the pre\-ious digit 

 must be 6, and so on. This process^ amounts to multiplying the 

 I by 4, multiplying the 4 by 4, giving 6 and I over, multipl)ing 

 the 6 by 4 and adding the i, giving 25, i.e. 5 with 2 over, and 

 so on, imtil the period is completed. 



In general, in converting — into a circulating decimal, if >6 be 



the last digit of the period, and r the last remainder 10 r - I = kq, 

 so that the last remainder = -z^^kq -^ i) and ^ = 9, 3, 7 or i 

 according as q ends in i, 3, 7, or 9. This is, in fact, the 

 property mentioned by Mr. Chartres and Mr. Toy ; the class of 

 relations to which it belongs, and the reason for Qieir existence, 

 is evident from what has been said above. 



The most direct manner in which the foregoing principles can 

 be applied to abbreviate the labour of division does not consist 

 in multiplying the digits by the remainder from the end but 

 from the beginning. For example, in finding the decimal equi- 

 valent to ^ the first four digits are '0588 and the remainder is 

 4 ; therefore ^V = '05881^. multiplying by 4, we have ^V = 

 •2352^4 whence iV = "058823521^ ; we could then find the 

 next four digits by multiplying the four digits last found by 4 

 and reducing the fi-action J# = 3lf, so that the next mul- 

 tiplication would be a multiplication of the whole period already 

 found by 13 ; but as in this case the remainder does not recur 

 after eight digits (if it did recur after eight digits the remainder 

 would be iV not pf ), it must consist of sixteen digits, and the 

 next eight are the complements of the first eight, and are there- 

 fore 941 17647. 



The principle of the method is to continue the division till 

 a relatively small remainder occurs and then to multiply the 

 figures already found by this remainder, and so on continually 

 till all the figures are obtained. This is the method that has 

 been generally employed in finding the reciprocals of large 

 numbers when the whole period was required. There are 

 several points to be attended to in order that the process may 

 be simplified as much as possible, but these I pass over. The 

 greatest saving of labour afforded by the principle is when a 5 

 or 2 occurs as remainder early in the division, as then we 

 obtain all the remaining digits as fast as the hand can write 

 them by di\-ision by 2 or 5 in the respective cases, without the 

 occurrence of any fractions. Thus, for example, 



^\ = -01639344262295 

 0819672131 147540983606557377049180327868852459 ; 



if we perform the division till we come to the quotient digit 

 5 we then have a 5 remainder, and all the other digits are 

 obtained by halving the figiures from the commencement, 

 viz., 1639. . . . The quotient can also be completed rapidly by 

 division whenever a remainder occurs that is a submultiple of 

 one that has previously occurred. Thus in the case of ^V, the 

 remainder after the first 6 is 24 and after the first 8 is 12, so 

 that the figures that follow the 8, viz., 1967 . . ., are obtained at 

 once by halving those that follow the 6, viz., 3934. . . . 



In the Messenger of Mathematics for April, 1878, I published 

 the following note : — 



"Write down a 5, divide it by 2 giving 2 with I over, divide 

 12 by 2 giving 6, divide 6 by 2 giving 3, divide 3 by 2 giving I 

 with I over, divide ii by 2 giving 5 with I over, divide 15 by 2 

 gi\-iiig 7 with I over, and so on tUl the figures repeat. We thus 

 obtain the figures 5263 15 7894736842 1, and these with a cipher 

 prefixed are the period of ^, viz. — 



,V = '052631578947368421. 

 " If we start with 50 and halve in the same manner, prefixing 

 two ciphers, we obtain the period of -j^, ntz. — 



T^ = '60502512562814070351758793969849246231155775894 

 4723618090452261306532663316582914572864321608040201. 



" Similarly, if we start with 500 and hah'e as before, we 

 obtain, after prefixing three ciphers, — 



TsW = '6005002501250625312656328164082041020510 . . . , 



and, generally, the process gives the reciprocal of i followed by 

 any number of 9's. 



"If we start with 20, 200, 2000, &c., and divide continually 

 by 5 instead of by 2, prefixing one, two, three, &c., ciphers, we 

 obtain the periqds of the reciprocals of 49, 499, 4999, . , . For 

 example, — 



■i^ - •620408163265306122448979591836734693877551 

 -^ ~ '6020040080160320641282565 130260521042084 .... 



"The process is very expeditious, the figures of the periods 

 being obtained as fast as the hand can write them." 



The results stated in this note were obtained as follows : the 

 object was to find the diNdsors for which the first remainder was 

 5, so that the halving should begin from the first significant 

 figure; these numbers are seen at once to be 19, 199, 1999 . . . 

 Similarly the first remainder is 2 for the divisors 49, 499, 4999 . . . 

 It should be mentioned that these are particular cases of Mr. 

 Suffield's method of synthetic division. 



If q be prime and there be only one period corresponding 

 to q (as is the case for q = 7, 17, 19, 23, 29, 47, 49, 59, 61, 97, 



&c.), the f - I fractions-, -, . . . ^—^ — have all the same 

 9 ^ q 



period, viz., the ^ - l digits that form the period of— are such, 



q 



that if we multiply them by 2, 3, 4, 5 ... y - i, we always 

 reproduce these same digits in the same cyclical order, but 

 beginning at a different place. The case of the period of 7, 

 viz., 142857, which is such that, multiplying it by 2, we have 

 285714, by 3 we have 428571, &c., is well known, and is often 

 given as a puzzle ; but the general result is a very remarkable 

 one, e.^. , it is remarkable that it should be possible to write 

 down 96 digifc:, such that their first 96 multiples should consist 

 of the same digits in the same cyclical order. In the forgoing 

 remarks I have confined myself entirely to the statement of the 

 principles connected with the results referred to in Nattthe, and 

 to those which arise directly from them. 



f J. W. L. GUIISHER 



SCIENTIFIC SERIALS 



American Journal of Scieiue and Arts, December, 1878. — In 

 the opening paper Gen. Warren considers that the Minnesota 

 Valley and the Mississippi Valley above the Ohio have been, as 

 a rule, formed since the deposition of the glacial drift, which 

 exists in the banks of the river, and that the Winnipeg basin 

 drained out southward along it ; also, that the loess deposits 

 extending up to the neighbourhood of Savannah are later than 

 the last glacial drift, &c. The hjrpothesis of southern elevation 

 and northern depression (probably reversed sometimes and re- 

 peated) is relied on to explain the effects. — Prof. Dana, con- 

 tinuing his valuable paper on some points in lithology, contends 

 for basing distinction in kinds of rocks on difference in chemical 

 and mineral constitution as regards chief constituents, and offers 

 a cla=sification in eight divisions. — The principle that when the 

 entropy of any isolated material system has reached a maximum 

 the system is in a state of equilibrium, is developed by Mr. Gibbs 

 as a foundation for the general theory of thermodynamic equili- 

 brium. — Mr. McGee distinguishes crania of the mound-builders 

 of the Mississippi Valley from those of modem Indians by a 

 greater development of the posterior molars. — ^An interesting 

 paper by the Rev. C. Hovey, on discoveries in western cares, 

 describes, inter alia, the remarkable acoustic properties of Echo 

 River passage-way (in the Mammoth Cave), where a strong 

 vocal impulse is prolonged with sustained rigour for fifteen 

 seconds or more ; also a locality discovered last April in the 

 Wyandot Cave, in which "pits, miry banks, huge rocks, are 

 overhung by galleries of creamy stalactites, vermicular tubes 

 intertwined, frozen cataract^, and all, in short, that nature could 

 do in her wildest and most fantastic mood." There is a row of 

 musical stalactites, very broad and thin, on which a chord can be 

 struck, or a melody played by a skilful hand. — Prof. Harrington 

 analyses the Chinese official almanac, issued annually in Decern- 



