[Footnote 238-3: See the exhaustive table in _Euler_, Mémoires de l'Académie de Berlin 1756, in _Süssmilch_, Göttl. Ordnung, I, § 160. Bridge has constructed the following formula: Log. A = Log. P + n x Log.(1+(m-b)/mb). Here P stands for the actually existing population, 1/m = the ratio between the annual mortality and the number of the living, 1/b, the ratio of the number of annual births to the number of the living, n the number of years, A, the population at the end of three years, the quantity sought for.] [Footnote 238-4: _Tucker_, Progress of the United States, 89, ff. 98. Here deduction is already made of immigrants and their posterity, who after subtracting the loss by emigration back to the old country, amounted to over 1,000,000. It probably amounted to more yet. If, as _Wappäus_ does (Bevölekerungsstatistik, 1859, I, 93, 122 ff.), we calculate the rate of increase per annum, we have an average during the first decade of 2.89, during the second of 2.83, the third of 2.74, the fourth of 2.52, the seventh of 2.39, the eighth (1860-70) of probably 2.25 per cent. On the still greater ratio of increase in earlier times, see _Price_, Observations on reversionary Payments, 1769, 4 ed. 1783, I, 282 seq., I, 260. It was nothing unheard of to see an old man with a living posterity of 100. (_Franklin_, Observations concerning the Increase of Mankind, and the Peopling of New Countries, 1751.) It is said that in the region about Contendas, in Brazil, there were on from 70 to 80 births a mortality of from 3 to 4 per annum (how long?), and an unfortunate birth (_unglücklichen_) was scarcely ever heard of. Mothers 20 years of age had from 8 to 10 children; and one woman in the fifties had a posterity of 204 living persons. (_Spix und Martius_, Reise III, 525).] [Footnote 238-5: Immense increase of the Israelites in Egypt. (Genesis 46, 27; Numbers, 1.)] SECTION CCXXXIX. LIMITS TO THE INCREASE OF POPULATION.