that a country is over-populated, we mean that the productiveness of industry in that country is not so great as it would be if the population had not grown so big: we thus admit the idea that there may be too many people. Malthus, on the contrary, was so far infected with the prevalent opinions of his age, that the idea of there being too many people was quite strange to him. If there are too many people the checks to the growth of population cannot have been as strong as it is desirable they should have been-they must have been inefficient. But Malthus denied the possibility, and even the conceivability of the checks to population being inefficient :

'It has been said by some,' he says, that the natural checks to population will always be sufficient to keep it within bounds, without resorting to any other aids; and one ingenious writer has remarked that I have not deduced a single original fact from real observation to prove the inefficiency of the checks which already prevail. These remarks are correctly true, and are truisms exactly of the same kind as the assertion that man cannot live without food. For undoubtedly as long as this continues to be a law of his nature, what are here called the natural checks cannot possibly fail of being effectual.'1

And in a note to the first sentence of this passage, he adds:

'I should like much to know what description of facts this gentleman had in view when he made this observation. If I could have found one of the kind which seems here to be alluded to, it would indeed have been truly original.' 2

1 Appendix to 3d ed. p. 9; 8th ed. p. 488.

2 It may perhaps be remarked that the belief that the checks cannot be inefficient, and so that over-population is impossible, is scarcely consistent with the passages quoted above, p. 136, though the produce of the earth might be increasing every year, population would be increasing much faster; and the redundancy must necessarily be repressed,' and 'the period when the number of men surpass their means of subsistence has long since arrived.' Malthus saw this himself, and altered these passages to though the produce of the earth would be increasing every year, population would have the power of increasing much faster, and this superior power must necessarily be checked,' and 'the period when the number of men surpasses their means of easy subsistence has long since arrived,' 8th ed. pp. 263 and 266. These alterations, together with the substitution of 'the argument of the principle of population,' in the 2d ed. p. 353, for 'the argument of an overcharged population,' in the 1st ed. p. 142, show that it was only by inadvertence that Malthus occasionally seems to admit that over-population is possible.

The question of population with Malthus was not, as it is with us, a question of density of population and productiveness of industry, but a question about the comparative rapidity of the increase of population and of the increase of the annual produce of food. He did not think that the checks upon the growth of population were made necessary by the population having approached or exceeded some economic limit, but simply by the impossibility of increasing the annual produce of food as fast as an 'unchecked' population would increase. His reason for believing it impossible to increase the production of food as fast as the unchecked population was that 'population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio.'1

If this were true, the constant necessity of checks would be proved at once. A quantity increasing like terms in geometrical progression, however small originally, and however small the common ratio by which it is multiplied, must, if given time enough, overtake a quantity which is increasing like terms in arithmetical progression, however large originally, and however large the common difference. To put the same thing into commercial language, the smallest sum accumulating at the smallest rate of compound interest must eventually grow bigger than the largest sum accumulating at the highest rate of simple interest. So, if population

1 1st ed. p. 14. 'It may safely be pronounced, therefore, that population, when unchecked, goes on doubling itself every twenty-five years, or increases in a geometrical ratio' (2d ed. p. 5; 8th ed. p. 4). It may be fairly pronounced, therefore, that, considering the present average state of the earth, the means of subsistence, under circumstances the most favourable to human industry, could not possibly be made to increase faster than in an arithmetical ratio' (2d ed. p. 7; 8th ed. p. 6).

2 Quantities are said to be in geometrical progression when each is equal to the product of the preceding and some constant factor. The constant factor is called the common ratio of the series, or more shortly, the ratio. Thus the following series are in geometrical progression :

3 Quantities are said to be in arithmetical progression when they increase

or decrease by a common difference. Thus the following series are in arith

increased geometrically and subsistence only arithmetically, the increase of population would eventually be checked by want of food, even if there had at first been an enormous surplus annual produce of food. But as a matter of fact there never is any appreciable surplus produce of food in an average year, and so population and subsistence must be supposed, so to speak, to start from the same line. In this case the necessity of checks becomes immediately obvious. The annual addition to the population when unchecked' would be greater every year, but the annual addition to the food could never exceed what it was in the first year.

Now Malthus was, of course, quite right in saying that an increasing population, if the checks on its increase do not alter in force, increases in a geometrical ratio. But he was completely wrong in saying that subsistence 'increases,' or can be increased, only in an arithmetical ratio. His attempt to prove this proposition is extremely feeble:

'Let us now,' he says, 'take any spot of earth, this Island, for instance, and see in what ratio the subsistence it affords can be supposed to increase. We will begin with it under its present state of cultivation.

'If I allow that by the best possible policy, by breaking up more land, and by great encouragements to agriculture, the produce of this Island may be doubled in the first twenty-five years, I think it will be allowing as much as any person can well demand.

'In the next twenty-five years it is impossible to suppose that the produce could be quadrupled.1 It would be contrary to all our knowledge of the qualities of land. The very utmost we can conceive is that the increase in the second twenty-five years might equal the present produce. Let us, then, take this for our rule, though certainly far beyond the truth; and allow that, by great exertion, the whole produce of the Island might be increased every twenty-five years by a quantity of subsistence equal to what it at present produces. The most enthusiastic speculator cannot suppose a greater increase than this. In a few centuries it would make every acre of land in the Island like a garden.

'Yet this ratio of increase is evidently arithmetical.

1 He means 'again doubled.' The original produce is 'quadrupled,' but the quadrupling takes place in the whole fifty years, not in the second twenty-five.

'It may be fairly said, therefore, that the means of subsistence increase in an arithmetical ratio.' 1

He seems to have overlooked the fact that to increase in a geometrical ratio is not necessarily the same thing as doubling every twenty-five years. It was no doubt impossible that the subsistence annually produced in Great Britain could be doubled every twenty-five years for an indefinite period. It was improbable that it could be increased every twenty-five years by an amount equal to the amount produced in 1798. But this does not prove that it could not increase in a geometrical ratio, or that it could only increase in an arithmetical ratio. If the amount produced increased only 1000000 per annum, or if it doubled itself every fifty thousand years, it would be increasing in geometrical progression. Malthus prided himself on relying upon experience, but in this case experience was entirely against him. He admits-indeed, he bases his whole work on the fact, that in the North American colonies the population had increased for a long period in a geometrical ratio.2 This population must have been fed, and consequently the annual produce of food must also have increased in a geometrical ratio. By the time he got to his sixth chapter, Malthus seems to have had some inkling of this objection to his argument, and he endeavours to answer it in a note:

'In instances of this kind,' he says, 'the powers of the earth appear to be fully equal to answer all the demands for food that can be made upon it by man. But we should be led into an error, if we were thence to suppose that population and food ever really increase in the same ratio.'

It is certainly difficult to see how we could be led into an error by supposing what is an admitted fact. However,

'The one,' Malthus continues, 'is still a geometrical and the other an arithmetical ratio; that is, one increases by multiplication and the other by addition.'

But if the population and food increased pari passu, it is

1 Essay, 1st ed. pp. 21-23.

2 Ibid., 1st ed. p. 20; cp. Appendix to 3d ed. p. 12, note (in 8th ed. p. 491, note), quoted below, p. 143.

impossible that the one could have increased in a geometrical and the other in an arithmetical ratio; so Malthus, instead of attempting to prove or explain directly his extraordinary proposition, resorts to his favourite device, and takes refuge in a simile:

'Where there are few people and a great quantity of fertile land, the power of the earth to afford a yearly increase of food may be compared to a great reservoir of water supplied by a moderate stream. The faster population increases, the more help will be got to draw off the water, and consequently an increasing quantity will be taken every year. But the sooner, undoubtedly, will the reservoir be exhausted, and the streams only remain. When acre has been added to acre till all the fertile land is occupied, the yearly increase of food will depend upon the amelioration of the land already in possession; and even this moderate stream will be gradually diminishing. But population, could it be supplied with food, would go on with unexhausted vigour, and the increase of one period would furnish the power of a greater increase the next, and this without any limit.'1

It is doubtless true that if more water runs out of a reservoir than runs in, the reservoir will in time be exhausted, but this does not prevent the outflow from being increased in geometrical ratio until the reservoir is empty; and if it did, that would not disprove Malthus's own fact—that the annual supply of subsistence had doubled every twenty-five years in New Jersey.

In 1803 Malthus bowed to the inevitable, and abandoned the attempt to show, in spite of his own facts, that subsistence never increases in a geometrical ratio. The note just quoted did not appear in its place in the second edition, and only its last three sentences were preserved and introduced into the discussion of 'the rate according to which the productions of the earth may be supposed to increase's in Book I., Chapter i. In that discussion Malthus treads far more gingerly than he did in the first edition. He does not assert that subsistence never has increased in geometrical ratio, and practically admits that it has done so 'sometimes in new

1 Essay, 1st ed. p. 106, note.

2 See p. 338.

2d ed. p. 5. Though he had struck out the reservoir,' Malthus continued to talk of the 'stream,' an oversight which has a curious effect. Eventually he substituted the word 'fund' (8th ed. p. 4).