626 PHILOSOPHICAL TRANSACTIONS. [aNNO 1755. 



shows that, at the present rate of births and burials, it must be 135 years before 

 England can be fully peopled. 



If we suppose, as Sir William Petty does, that the burials are to the births as 

 9 to 10, that is 1 to 1.11 1, which is something less than that of Dr. Derham's 

 proportion, and that 1 dies in 40 in a year; if we substitute these numbers in 

 the formula, we shall then find the time of doubling to be 250 years. For then 



.. ■•! , log 2 0,3010300 , . , 



itwill be^ = ,^^^— — -■--^— — ^= — j^^-- = 250; which shows 



how far Sir William was mistaken in his method of calculation, when he made 

 the time to be 36o years. 



After the same manner, the number of years being given, it will be easy to 

 find the proportional increase. Suppose after 45 years. For then we should 

 have 45 x log. (/ + i — 1) — 45 X log. / = log. p; which will give 45 X 

 O.OOiaoOQ = log./), and therefore p = 1.1443, from which if n be equal to 

 5,467,860, we have pn = 6,256,872. So that it appears if there was 5,467,860 

 people in England at the year 17 10, when the above-mentioned survey was made, 

 there is now 6,250,000; if none were to be deducted on account of our wars, 

 and emigrations to our colonies since that time. 



From what has been found above, that {I -\- b — \y = ply, it is evident, that 

 the ratio of the increase in any number of years may be determined, without the 

 number of people being known, or their proportion to the annual increase ; and 

 also that any one of the quantities /, b, y, p, may be found, the others being 

 known. But if the ratio of the number of people to the annual increase be 

 known; and consequently the proportion, of the number in any one year, to 

 the number next year known, we shall then have a very simple equation. For 

 if we suppose the number of people in any one year, to be to that number with 

 the increase added in the next year, as 1 to r, we shall then have nr' = vp, or 

 r' = /;. And, in like manner, if the proportion of the number of people to 

 their increase, in a given cycle of years, had only been known, and that cycle 



y y 



be c, we should then have nic = np, or n = p. From which formula it would 



be easy to calculate the numbers of mankind, in all ages through the world, if 

 we suppose them to arise from a given number, and the rate of increase known, 

 in any period of years. And this may sometimes be of use to discover the 

 number in any age, that might be possible to reason on, and to find out the truth 

 of any hypothesis. 



XLVI. An Attempt to Explain Tivo Roman Inscriptions, cut on tivo Altars, 

 which were dug up some time since at Bath. By John tVard, LL.D- and 

 V. P. R. S. p. 285. 



These two inscriptions were found near the same time and place, with that 



