It appears, however, that it is not to the use of mathematical words, but only to the use of those words in their right senses that Mr Sadler objects. The law of inverse variation, or inverse proportion, is as much a part of mathematical science as the law of geometric progression. The only difference in this respect between Mr Malthus and Mr Sadler is, that Mr Malthus knows what is meant by geometric progression, and that Mr Sadler has not the faintest notion of what is meant by inverse variation. Had he understood the proposition which he has enounced with so much pomp, its ludicrous absurdity must at once have flashed on his mind.
Let it be supposed that there is a tract in the back settlements of America, or in New South Wales, equal in size to London, with only a single couple, a man and his wife, living upon it. The population of London, with its immediate suburbs, is now probably about a million and a half. The average fecundity of a marriage in London is, as Mr Sadler tells us 2.35. How many children will the woman in the back settlements bear according to Mr Sadler's theory? The solution of the problem is easy. As the population in this tract in the back settlements is to the population of London, so will be the number of children born from a marriage in London to the number of children born from the marriage of this couple in the back settlements. That is to say—
2 : 1,500,000 :: 2.35 : 1,762,500.
The lady will have 1,762,500 children: a large "efflux of the fountain of life," to borrow Mr Sadler's sonorous rhetoric, as the most philoprogenitive parent could possibly desire.
But let us, instead of putting cases of our own, look at some of those which Mr Sadler has brought forward in support of his theory. The following table, he tells us, exhibits a striking proof of the truth of his main position. It seems to us to prove only that Mr Sadler does not know what inverse proportion means.
Countries Inhabitants on a Children to a
Square Mile, about Marriage
Cape of Good Hope 1 5.48
North America 4 5.22
Russia in Europe 23 4.94
Denmark 73 4.89
Prussia 100 4.70
France 140 4.22
England 160 3.66
Is 1 to 160 as 3.66 to 5.48? If Mr Sadler's principle were just, the number of children produced by a marriage at the Cape would be, not 5.48, but very near 600. Or take America and France. Is 4 to 140 as 4.22 to 5.22? The number of births to a marriage in North America ought, according to this proportion, to be about 150.
Mr Sadler states the law of population in England thus:—
"Where the inhabitants are found to be on the square mile,
From To Counties Number of births to 100 marriages
50 100 2 420
100 150 9 396
150 200 16 390
200 250 4 388
250 300 5 378
300 350 3 353
500 600 2 331
4000 and upwards 1 246