296 THE WONDERFUL CENTURY. CHAP. xvm. 



years since, to show that the rarity of second attacks 

 may, in all probability, be fully explained by the doc- 

 trine of chances. But I had not statistics sufficient to 

 prove this. Professor Vogt, however, having the sta- 

 tistical tables of all Europe at his command, is able to 

 show, not only that the calculus of probabilities itself 

 explains the rarity of a second attack of small-pox, but 

 that second attacks occur more frequently than they 

 should do on the doctrine of chances alone, indicating 

 that, instead of there being any immunity, there is really 

 a somewhat increased susceptibility to a second attack! * 



1 Brief statement of the argument : 



The chances of a person having small-pox a second time mny be 

 roughly estimated thus : Suppose the average annual death-rate by 

 small-pox to be 500 per million, and the average duration of life 

 forty years. Then the proportion of the population that die of 

 small-pox will be 500 X 40 = 20,000 per million. If the proportion 

 of deaths to cases is one to five, there will be 100,000 cases of small- 

 pox per million during the life of that million, so that one-tenth of 

 the whole population will have small-pox once during their lives. 



Now, according to the law of probabilities alone, the chances of a 

 person having small-pox twice will be the square of this fraction, or 

 one hundredth : so that on the average only one person in 100 

 would have small-pox twice if it were a matter of pure chance, and if 

 nothing interfered with that chance. But there are interferences 

 which modify the result. (1) Those that die of the first attack can- 

 not possibly have it a second time. (2) It is most frequent in the very 

 young, so that the chances of having it later in life are not equal. 

 (3) It is an especially epidemic disease, only occurring at considerable 

 intervals, which reduces the chances of infection to those who have 

 had it once. (4) It is probable that most persons are only liable to 

 infection at certain periods of life, having passed which without 

 infection may never take the disease. It seems probable, therefore, 

 that these several conditions would greatly diminish the chances in 

 the case of any person who had once had small-pox, so that perhaps, 

 under the actual state of things, chance alone would only lead to one 

 person in two hundred having the disease a second time. 



The above is only an illustration of the principle. Professor Vogt 



