January 1, 1890.] 



KNOWLEDGE. 



47 



This method sufficps not only for Primes, but also for 

 all numbers ending in the digits 1, 3, 7, or 9. But some- 

 one may say, How do you get the niultiyliers after you have 

 obtained your last figure of the period ? Nothing is easier. 

 If your number X ends in 9 take the tens and add one. 

 Thus, for Jg our multiplier is 8 (since 7 + 1 = H), and we 



get the period of yV= " 012658227H4K1. 



For numbers ending in miity take the tens from the 

 given number. Thus for jL take 4 from 41 = 37, which 

 is our multiplier, and 4^ = ' 02439. 



For numbers ending in 3 multiply by 7, and for numbers 

 ending in 7 multiply by 3, and in either case take the tens' 

 digit from the given number. Thus, for ^\j- we have 13 - 9 

 = 4, and the period= '076923, and the multiplier for 17 



is 12. Thus, j-V = - 0588235294117647. 



This remarkable rule is intimately connected with the 

 new method of divisidn, an elementary explanation of 

 which I now give. 

 Let .Y=;'/,' ; say 2091 = 177,'. 



3.V=3/'/,'=10.Vi-f»i ; say 6273 = 517;=10 x 627 + 3. 

 Let A'=(3/'-l)/10 say 5 = (3x 17-1)/10 in this case. 

 Then 3.VA'=3P7,'A'=iai7iA +)(,.V. 

 And consequently l'l!^ — M^ - »]A ; 



Say 177/j = 627-3x5 = 612 = 10.1/j + »„. 

 Similarly 7 '7.'„ —^[„— n„ A ; 



Say 177^,= 61 -2 x 5 = 51 = 10.1/;j + «;,. 

 Also l>I!.^=M.^-)i^X; say 177;., = 5 - 1 x 5 = O=10J/j+ /(^. 

 Now compare the process witli the method described. 

 627-3 = 177.'= 10.1/, + «i 

 15 

 61-2 =17//, = 10.1/„ + »., 

 10 



5-1 =l7/.'.,^l0.1/.5 + ».5 

 5 



1) =1777, 



Thus 2091 -=-17 = 123. 



Proceeding to the general case we get 



P7?„, = .17,,. - »„, A= 10.17,,, + 1 -I- »,„ + J. 

 Again we have — .1/j — «,.V=10.1/'.i-|-»,,. 

 Thus— .17i- 10.17,,= »i A -(-»., . . ". (T). 

 Again— .17„-«.,A= 10,17., -h)(,, . . . (2). 

 Thus— 10i7j='l00.173 + io«, + 10».oA. 

 Therefore .1/, = lOO.l/., -f- 10»., -|- 10«o.V + ii^X + n,(l & 2) . 

 .-. 3.V= 1000.1/3-1- lOO/;., + IOOmo.V + 10 /(,.Y 4- 10/;., + 11^. 

 But ,V., = »,..Y. ' 



Thus— 3.V4(10 )> 1) {100«.5 + lOii, + II ^). 

 That is— .V=/' (10h3-|-10»j + »|). " 

 And generally — .V=7' (lOrt,';,' + 'lo"i'i„'_i . . . -|-,). 

 The above is a proof for any divisor ending in 7. The 

 process is exactly the same for divisors ending in 1, 8, or 

 9, iind the only difference is in the value of .\. 

 (I'll hr nuitiiilied.) 



GROWTH AND DECAY OF MIND. 



By the I, ATI'. 1!. \. Pkiictok, 

 fContiininl jraiii jki,/,' 33. / 



PI', lilLVPS no more remarkable instance could be 

 cited of the the ])osscssion of power in old age 

 with the want of elasticity I'cfcrred to in the 

 lii.st paper— than the remarkable papers on the 

 universe, written by Sir W. Herschel in the years 

 IK17 mid IHlS, that is, in his seventy-ninth and eiglitieth 

 years. We find the veteran astronomer proceeding in 

 the path which, nu)re than forty years before, he had 

 marked out for himself ; but the very steadiness and 



strength of purpose with which he pursues it indicates 

 the degree to which his mind had lost its wonted 

 elasticity. In 17H4 and 1785 he was traversing a por- 

 tion of the same road. But then he was in the prime 

 of liis powers, and accordingly we recognise a versatility 

 which enabled iiim to test and reject the methods of 

 research which presented themselves to his mind. It was 

 in those years that he invented his famous method of star- 

 gauging, which our text-books of astronomy preposterously 

 adopt as if it were an established and recognised method 

 of scieiititic research. But Herschel himself, after trying 

 it, and satisfying himself that it was unsound in principle, 

 abandoned it altogether. In 1817 he adopted a method of 

 research equally requiring to be tested, and, in my con- 

 viction, equally incapable of standing the test; but he 

 now worked upon the plan he had devised, without sub- 

 jecting it to any test. Nay. results which only a few years 

 before he would certainly have rejected — for he did then 

 actually reject results which were open to the same objec- 

 tion — passed muster in 1817 and 1818, and are recorded 

 in his papers of those dates without comment. We may 

 recognise another illustration of the loss of elasticity with 

 advancing years, in the obstinacy — one may even say the 

 perversity — with which Sir Isaac Newton, in the latter 

 years of his life, adhered to opinions on certain points 

 where, as has since been shown, he was unquestionably 

 wrong, and where, had he possessed his former mental 

 versatility, he must have perceived as much. Compare 

 this with bis conduct in earlier years, when for nineteen 

 years he allowed his theory of gravitation to rest in abey- 

 ance — though he had fully recognised its surpassing 

 importance — simply because certain minute details were 

 not satisfactorily accounted for. Many other instances 

 might be cited, were it worth while, to show how the 

 mind commonly changes when approaching an advanced 

 age, in a manner corresponding to that bodily change — 

 that stifi'ness and want of elasticity, without any marked 

 loss of power, which comes on with advancing years. 

 That old age does not necessarily involve any loss of power 

 for routine work, has been clearly shown in the lives of 

 many eminent men of our own era. 



It is well pointed out by Dr. Beard, in the lecture to 

 which I have already referred, that "we must not expect 

 to find at one age the mental qualifications due to another 

 age— we nmst not look for experience and caution in youth, 

 or for suppleness and versatility in age. ^^'c ought also to 

 apportion to the various ages of a man the kind of work 

 most suitable to them. Positions which require mainly 

 enthusiasm and original work should be filled by the young 

 and middle-iiged ; positions that require mamly experience 

 and routine work, should be tilled by those in mature and 

 advanced life, or (as in (derkships) by the young who iiave 

 not yet reached the golden decade. The enormous stupidity ,_ 

 and backwardness, and red-tapeism of all departments of 

 governments everywhere, are partly due to the fact that 

 they are too much controlled by age. The conservatism 

 and inferiority of colleges are similarly explained. Some 

 of those who control the policy of colleges— presidents and 

 trustees— should be young and midiUe-aged. .lomnalism. 

 on flic other hand, has suffered from relative excess of 

 youth and entliusiasm." 



Before passing from the lecture of l>r. Heard, I shall 

 venture to quot^e the remarks which he makes on the 

 evidence sometimes afforded of approaching mental decay 

 by a decline jn moral sensitiveness. " Moral decline in 

 oid age," he says " means— • Take care ; for the brain is 

 giving way.' It is very frequently accompanied or pre- 

 ceded by "sleeplessness. Decline of the moral faculties, 

 like the" decline of other functions, may be relieved, 



