VOL. XLIX.3 PHILOSOPHICAL TRANSACTIONS. 625 



to find in what time the people shall be in any given proportion, to what they are 

 at present. Suppose n to be the unknown number of the people at present, and 

 let the living be to the dead, in one year, as / to ], and the dead to the births 

 as 1 to b, the proportion given to what their number is at present as p to 1 , and 

 the number of years required to be y. It is plain then, that the dead at the end 



of the first year will be - , and the births y, and the whole number of people 



must hen -\- j. In like manner, at the end of the 2d year, the dead will 



be v-^ , and the births — r " ~ " , and the whole number of people 



. I , bn n , Ibn + hbn — nb , n — In — bn J + 6 — 1, ., . , 



must be n + y — y -j ~ 1 ^ = {-^ yn. And so at 



the end of the 3d year the number of people will be ( -^^— )^«. From which 



at length it is evident by induction, that the number of people at the end of the 

 required number of years will be ( ■,-^^) n. . But as the proportion is then to 



beas j& to 1, we shall have ( — j^^-Yn = pn, and thence (/ + b — \y = pi". 

 And because the logarithms of equal quantities must be equal, we shall have 

 y X log. (/ + i - 1) = log.p + 7/ X log. /, and alsoy = _^^-J°liL_-_. 

 And therefore the number of years y is determined by the logarithms of known 

 quantities, when the people shall be in the given proportion of /> to 1 . 



It may be observed that the quantity ( j-^^)*/j may be considered as the or- 

 dinate of the logarithmic curve, whose abscisse is the index y, and that the ordi- 

 nate passing through the beginning of the abscisse, where y = o, must be 

 equal to n. 



If now it be required to know when the people shall be doubled; let us sub- 

 stitute in the above formula, instead of b, I, p, the respective numbers 1.12, 40, 

 2. and it will be w = , — -^f^ — rr ; — —; and then the logarithms being 



•^ log. (40 + 1.12 — 1) — log. 40 o o 



taken we shall have y = ' — = 231; which shows that, according to the 



present state of births and burials, the people could not be doubled in less than 

 231 years. And by the same method it appears, changing the signs o( b — 1, 

 that 230 years ago, in the time of Henry the 8th, the number could not be 

 above -^ of what it is now, that is about 3 millions. 



And so if we were to find, when the number of people in England would be 

 increased to Q millions, which, by what has been said above, is near about the 

 outmost that can be maintained, from the natural produce of the country; we 

 should then have/) = -J- = 1.5, because Q millions is to the present number as 



o* „ J 1 log. 1.5 ,' 0,17609!3 ,„^ ... 



3 to 2, and also y = ^--^-^^--—^— = -^^^^ = 135; which 



VOL. X. 4L 



