d'alembert. 271 



preceding table and destroys the colmnn v^, v^, ^3> ••• Then he 

 assumes that the remaining column will shew the correct mortality 

 for the number N—n at starting, where n is the total number who 

 died of small-pox, that is n — v^-\- v^-\-v^+ ... 



M 



Thus if we start w4th the number M of infants ^r^ ?<,. would 



N-n ' 



die on this assumption in the r^^ year. 



There is a certain superficial plausibility in the method, but it 



is easy to see that it is unsound, for it takes too unfavourable a view 



of human life after the eradication of small-pox. For let 



u^ + ^^2 + • • • ^^r = ^r > 



then we know from the observations that at the end of r years 

 there are N —U^— V^ survivors of the original N \ of these w.^^ die 

 in the next year from all diseases excluding small-pox. Thus 

 excluding small-pox 



^r+i 



N-U,- V/ 



is the ratio of those who die in the year to those who are aged 

 r years at the beginning of the year. And this ratio will be the 

 ratio which ought to hold in the new tables of mortality. The 

 method of the savant Geometre gives instead of this ratio the 

 greater ratio 



'•+1 



N- U.- 



n 



488. Thus we see where the savant Geometre was wrong, and 

 the nature of the error. The pages in D'Alembert are 88 — 92 ; 

 but it will require some attention to extricate the false principle 

 really used from the account which D'Alembert gives, which is also 

 obscured by a figure of a curve. In D'Alembert's account regard 

 is paid to the circumstance that Inoculation is fatal to some on 

 whom it is performed ; but this is only a matter of detail : the 

 essential principle involved is that which we have here exhibited. 



489. The next publication of D'Alembert on the subject of 

 Probabilities appears to consist of some remarks in his Melanges 



