Section I.: introductory observations.

“It has been said,” says Mr. Malthus, “that I have written a quarto volume to prove, that population increases in a geometrical, and food in an arithmetical ratio; but this is not quite true. The first of these propositions I considered as proved the moment the American increase was related, and the second proposition as soon as it was enunciated. The chief object of my work was to enquire, what effects those laws, which I considered as established in the first six pages, had produced, and were likely to produce, on society; a subject not very readily exhausted.” This, it must be acknowledged, is meeting his adversary fairly in the open field. His means of defence are displayed before us; and we shall see if Ins” first six pages” form an impenetrable shield.

The argument of Mr. Malthus is founded on the comparison of numerical progressions; and, as mathematical science has always been held as the only one that is demonstrably true, this comparison of numerical progressions has been hastily allowed as infallibly certain, and has obtained for its author all that implicit faith, which has been granted by his disciples to the corollaries which he has drawn.

It may be premised however, that the science of mathematics is a science of certainty, only in as far as it is a science of abstraction;—that when it ceases to be abstract, and becomes what is termed “mixed mathematics,” the numbers or quantities, assume definite designations; and the reasonings thence derived are true, or false, according to the accuracy, or inaccuracy, with which these numbers or quantities are designated.—Two times two, for instance, is four, because four is the term by which we have agreed to denominate the result of this multiplication; but when, in measuring surfaces, it is said that two feet multiplied by two feet will make four feet, there is an error in the syllogism; because the word “feet” in the product, has not the same meaning that it has in the factors:—in the latter it is lineal, and in the former superficial. In application then, mathematical reasoning partakes of all the uncertainty of the branches of knowledge with which it is combined; and hence the numerous paradoxes which, even in this science of demonstration, have puzzled the pupils, and have scarcely been explained by their masters. These observations are not foreign to our purpose, when we have to speak of the ratio of increase in human population.

Mr. Malthus, if he himself understood the subject, has taken it for granted that his comparison of ratios would escape the notice of mathematicians. He asserts that human population, if allowed to expand freely, would increase in a geometrical ratio, in the order of

1, 2, 4, 8, 16, 32, 64, 128, 256, &c.

Now it is obvious that this series can represent no connected chain of the expansion of human life. The quantity represented by 1 (the first term in the series) does not at any moment become 2 (the second term), but there are an indefinite number of terms of different magnitudes, to be interjected between these terms (and so between every two other successive ones) to fill up the links of the chain. This then supposes time: time therefore, the most metaphysical of all metaphysical beings, is an ingredient mixed up with the consideration of the abstract numbers of the progression. This time Mr. Malthus has specified to be 25 years. The series then which denotes the increase of human beings in America, is thus represented.

The philosophy of Mr. Malthus is not the method of induction. He perpetually appeals lo principles which have never been brought into action, and which are opposed to all experience. He speaks of tendencies to human increase, and of powers of population, which “in no state that we have yet known have been left to exert themselves with perfect freedom.” This is exactly in the style of those dreamers, who predict of the future something unlike and opposite to what has ever appeared in the past. They too talk of secret springs, that have never yet displayed their elasticity,—of latent energies which have never been exerted.

Latent signifies concealed, and consequently the latent power of increase in the human species is what we shall never know; but, even granting for a moment that the 3 or 4 censuses which have been taken in America do exhibit something like a duplication in 25 years;—granting too that this increase has arisen solely from propagation, independent of emigration, there certainly exist no data from which to infer the law of the series. We have only 4, or at most 5 terms given us,— some of them extracted at intervals of time by no means regular,—from a series perpetually flowing, and of the ebbs and floods of whose motion we know nothing; and from these the ordinary reader is presented with a picked set of numbers, in geometrical progression, with the ratio of two. From such an increasing series as the human race may be supposed to exhibit, any form of a progression may be taken:— why not that of 1, 4, 9, 16, 25, &c. which increase as the squares of the terms 1, 2, 3, 4, 5, &c.? For aught that Mr. Malthus has discovered this may be the latent law of increase. All that he has demonstrated, even granting his American censuses, as we for the moment have done, is that human beings are capable of increasing their numbers; or, rather that they have been found to do so for a specific time: but the series which would mark the Law of that Increase, he has either been unwilling or unable to develop.

“The rate,” says Mr. Malthus, “according to which the productions of the earth may be supposed to increase, it will not be easy to determine. Of this however we may be perfectly certain, that the ratio of their increase must be totally of a different nature from the ratio of the increase of population.” This passage is much more modest than that which we quoted at the beginning of this Dissertation, where he says, that food increases in an arithmetical ratio, and that he considered this proposition as proved the moment it was enunciated; but, as he proceeds, this modesty vanishes, and he comes to an undoubting conclusion that food can increase only in the series 1, 2, 3, 4, 5, 6, 7, 8, 9, &c. (an arithmetical progression whose ratio is one) and that the period between the terms, or time of increase, is also 25 years.

If the quantity of the food of man be increased, it is obvious that the increase will not be by starts every 25 years; but that it will be increased through many intervening times; and, consequently, even granting that such quantities as 1, 2, 3, 4, &c. were extracted from the flow of increase, at certain periods, the arithmetical progression thus exhibited would be a picked set of numbers, (as we stated respecting population) and might have been any other series rather than that which Mr. Malthus has chosen, for aught that experience has told him on the subject. The successive terms of all increasing series whatever present nothing but additions. The mathematician forms series at his own pleasure, where the additions are regulated by certain laws. It is not so with those of Nature. Whether her series alternately progress and retrograde;—whether they circulate, or decrease, or flow in straight and eternal lines, is beyond the ken of the philosopher. He snatches at intervals, a few links in the immeasurable and ever moving chain of the universe; and dividing these links into such portions as are perceived by the glance of a moment, he cries out in extacy, “I have found it!” This remark however is general. It refers to the boundaries of human knowledge, and is applicable to a Newton as well as to a Malthus:—Our business is with the latter.

Taking the series 1, 2, 3, 4, &c. as that of the increase of human subsistence, or of any thing else, we may, by picking the terms, extract from it any progression we chuse: for instance, in — 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16, &c. the 1st, 2d, 4th, 8th, and 16th terms, form the precise geometrical progression, which Mr. Malthus has chosen to represent the increase of human beings. The progressions themselves then would have signified nothing, had not Mr. Malthus assumed the principle, that an equal period of time, 25 years, was to elapse between the production of every subsequent term of either progression. Thus arranged:

he believes that he has demonstrated that in 8 periods of time of 25 years each, the population if unchecked, would increase to 256 times its present number, while the food would only be 9 times what it now is. Let us endeavour to view the grounds on which these different progressions have been raised.