Dec, 19, 1889] 



NATURE 



16; 



on the South African gold-fields, which include much informa- 

 tion on the present condition of the whole of South Africa as 

 far north as the Zambesi. The observer points out that, while 

 the Delagoa Bay and other lines of communication are much 

 discussed, the fine artery of the perfectly navigable Limpopo is 

 entirely neglected, notwithstanding Captain Chaddock's naviga- 

 tion of it a few years ago. The writer remarks that " this river 

 flows mainly through regions under the influence or protectorate 

 of England ; the Transvaal people on the one side, and those of 

 Matabeleland on the other, would certainly be glad to avail them- 

 selves of this outlet for their produce. As it traverses only a 

 small tract of Portuguese territory about its estuary, I hope and 

 believe that Portugal will not be allowed to treat the Limpopo 

 as she is now attempting to treat the Zambesi. The subject is 

 of such importance that it cannot fail soon to be brought before 

 the British Parliament." Referring to the negotiations at pre- 

 sent going on in connection with the Swaziland question, he 

 observes, in the same spirit : — " The Swazi people must, sooner 

 or later, yield either to the Transvaal or to England, and if to 

 the former, it must be to the entire detriment of British interests. 

 England, as the suzerain power in South Africa, should be the 

 first in the field, both in her own interest and in that of her 

 other colonies and subjects. If she does not assume the pro- 

 tectorate of Swaziland, besides losing the control of a vast and 

 rich mineral district, she will deprive the colony of Natal of all 

 further hope of expansion. If she ignores her responsibility in 

 this matter, and allows the Transvaal Republic to absorb Swazi- 

 land, she will add another to the long list of blunders that 

 threaten to destroy all prospect of consolidating a dominion as 

 large as Canada, and may end disastrously for British interests 

 in South Africa." 



A French traveller has just achieved a feat of great interest. 

 Captain Trivier, equipped by the newspaper Za Giroizde, started 

 some eighteen months ago for the Congo State. He went up 

 the river to Stanley Falls, and thence proceeded to Central 

 Africa and the Lake region, accompanying caravans. He has 

 just arrived at Mozambique. 



Globus reports that during the past summer M. Thoroddsen, 

 the well known student of Iceland, has carried out a journey in 

 the waste region known as Fiskivotn, lying between Hecla and 

 the Vatna Jokul, which has hitherto been unvisited for the most 

 part by any inquirer. To the east and north of Hecla he dis- 

 covered a new obsidian region. Crossing the Tunguaa, he 

 went to the Fiskivotn group of lakes, all true crater lakes. The 

 district between this and the Vatna Jokul has absolutely no 

 plant-life whatever ; it consists of lava-fields, and plains of vol- 

 canic sand. In it he found a lake, Thorisvatn, the second 

 largest in the island. Thence, after a day's journey through an 

 utterly desolate district, he reached the hitherto unknown source 

 of the Tunguaa. To the south of this he discovered, between 

 three ranges of hills, previously unknown, a new and very long 

 lake. 



Mr. Dauvergne has, says the Times of India, completed an 

 adventurous journey in the regions of North- West Cashmere. 

 His course was from Leh northwards to the Kilian Pass, in 

 Kashgaria, and then northwards across the Pamir to the Upper 

 Oxus. He reached Sarhad in safety, and after six days' halt 

 there, crossed the Hindu Kush by the Baroghil Pass, as he did 

 not wishto visit Chitral. He then turned eastwards, and after 

 a trying journey through the snow, crossed the Ishkaman Pass, 

 north of Yasin. Thence he travelled southwards by the 

 Karambar Valley, and eventually reached Gilgit, a short time 

 after Captain Durand had started for Chitral. Mr. Dauvergne 

 reports that the Russian explorer, Captain Grombchevsky, 

 whose attempt to reach Kafiristan was noticed some time 

 ago, was stopped at Kila Panjah on the Oxus, by the Afghan 

 authorities. 



THE ST. PETERSBURG PROBLEM. 

 'T'HIS celebrated problem, which is first mentioned before 

 1708 in a letter from the younger Nicholas Bernoulli to 

 Montmort, has been frequently discussed by Daniel Bernoulli 

 (1730) and other eminent mathematicians. It may be briefly 

 stated as follows : — 



A tosses a coin, and undertakes to pay B a florin if head 

 comes up at the first throw, two florins if it comes up at the 

 second, four florins if it be deferred until the third throw, and so 

 on. What is the value of B's expectation ? 



The chance of head appearing at the 

 1st, 2nd, 3rd, 4th .... wth throw is 

 \, \, \, iV • • • • i"- A promises to pay for head 

 I, 2, 4, 8 . . . . 2" ~ ' florins, hence B's expectation is 



*, h %, I'V . . . • a'-Vz" = \ florin. 



Hence the total value of B's expectation is an infinite series, 

 each term of which is a shilling, or it is infinite. 



This result of the theory of probability is apparently directly 

 opposed to the dictates of common-sense, since it is supposed 

 that no one would give even a large finite sum, such as £'^0, for 

 the prospect above defined. 



Almost all mathematical writers on probability have allowed 

 the force of the objection, which they have endeavoured to evade 

 by various ingenious artifices all more or less unsatisfactory. 



The real difficulty of the problem seems to lie in the exact 

 meaning oi infinite and value of the expectation. 



Since the infinite value of the result is only true if an infinite 

 number of trials are paid for and made, all such considerations 

 as want of time and the bankruptcy of A or B are precluded by 

 the terms of the question. 



The value of B s expectation is frequently confused with how 

 much he can or ought to pay for it ; thus Mr. Whitworth 

 ("Choice and Chance," p. 234) finds that if B have 1024 florins, 

 he may give very little more than 6 florins for the venture. This 

 ingenious, solution seems to have no reference to the original 

 problem, which has been modified by Mr. Whitworth's introduc- 

 tion of the word "advantageously" (p. 232). 



B can pay for his expectation in three ways : (i.) a sum before 

 each toss ; (ii.) a sum before each series of tosses ending with 

 head ; (iii.) a sum for the total result of A's operations. 



Mr. Whitworth apparently assumes the first method of pay- 

 ment, and shows that the larger B's funds are the more he may 

 safely pay for each toss, since he can continue to' play longer. 

 Many mathematicians take the second method of payment. 

 "However large a fee I pay for each of these sets, I shall be 

 sure to make it up in time ' (" Logic of Chance," p. 155). 



It is easy to show in this case also that what may be safely 

 paid before each series increases with the number of series. 



Suppose a very large number of tosses made, about half 

 would come up heads and half tails ; each head would end 

 a series, when a fresh payment must be made by B. Suppose 

 the tosses limited to one series, if B pays one florin he cannot 

 possibly lose, if he pay anything more he may lose by head 

 coming up the first time, and the more he pays the greater will 

 his chance of loss be, since the series of tails must be longer to 

 cover it. But, however large a finite sum he pays, he is not 

 certain to lose, e.g. head may not come up till the hundred and 

 first toss, when he would receive 



2^0" = 1,267650,600228,229401,496 703,205 376 florins. 



If the sets are limited to one hundred, about 



50 heads would probably come up the 1st toss. 



25 



13 



6 



3 

 2 

 I 



2nd 

 3rd 

 4th 

 5th 

 6th 

 7th 



B would 

 receive for 

 each series 

 50 florins. 



Hence for the hundred sets, B would receive about 350 florins, 

 or he could pay without loss seven shillings for each set. 



If N be the number of sets, the total amount received by B 

 will probably not be less than n terms of the series 



/N X 

 t 2I 



N X 2^ 



-f- &c. 



} =«{i}N, 



but n is the number of times which N is successively divisible 

 by 2, or 2" = N, or « = log N/log 2. But the amount x which 

 B can afford to pay per set when multiplied by the number of 

 sets is equal to the amount which he r eceives, or — 



xN=Mi}N, 

 log 2 

 hence x ~ log N/o'6 nearly. 



This formula, though inexact for low, is very convenient for 

 high, values of N. 



N= I X = o N=io8 X = 10 



= 50 = 2*7 =10^ =15 



= 100 = 3'3 = lo'*^ = 20 



= 1000 =5 = 10^^ = 25 



X increases with, though much more slowly than, N, and 



becomes infinite when N does. But to justify a payment of 



