will serve to guide us to the real nature of Political Economy, and of the evidence of which alone it is susceptible, and on which it must rest, before it can be rendered a clear, solid, and permanent science.

There are two circumstances essentially and indispensably necessary to the acquisition of knowledge, and even the continuance of the human race; if both, or either of these circumstances were annihilated, or were constituted differently from what they are, we could know nothing, and mankind would speedily become extinct. One is the permanence or stability of the appearances and operations of nature: the other is, that fundamental law of the human mind, on which rests the association of our ideas.

Let us imagine, for an instant, that the permanence of the appearances and operations of nature were destroyed; that the food which when first eaten pleased our palates, and supplied a wholesome and nutritious aliment, the next time we used it, was bitter and disagreeable, and afforded no nourishment, or absolutely proved poisonous; let us extend this supposition to every other thing surrounding us, which we observed, or did, or used, on which we acted, or which acted upon us;-and whence could our knowledge be derived, or how could we continue to exist? So far as regards our knowledge-the point to which at present we must exclusively direct our attention the experience or observation of this moment would be contradicted by the experience or observation of the next: and it is too evident to require illustration or proof, that in such a state of things, we could anticipate nothing-we could know nothing, we could believe nothing, but what would deceive us.

The other circumstance, not being so obvious and direct, may not appear at first sight so absolutely necessary to the acquisition of knowledge, or an indispensable and essential instrument even to the obtaining of its simplest rudiments. But let us suppose, that the fundamental law of the mind, by which our ideas are associated, were annihilated: that the course of nature in her appearances and operations continued, as it is, permanent and stable; -that the sun continued to rise, and set, and give heat, and fertility, and health; that the earth yielded its pro

duce to human labour and skill; and that that produce continued to afford an agreeable and wholesome nutriment to man: all these things remain exactly in every respect as they were. The constitution of the human mind alone undergoes an essential change; all things that surround us— indeed, all that we see, and do, or by which we are acted upon, remain as formerly; our senses perform their functions as usual; but the association of our ideas is destroyed. What would be the result? as we are constituted, the sun and the idea of warinth are so indissolubly connected in our mind, that the appearance of the one, immediately, without an effort of the mind, or process of reasoning, calls up the expectation of the other; and on this expectation, we act and calculate. Suppose our ideas no longer to be associated, that every impression in our mind was single and insulated: the sun, though it warmed us the first time we felt its rays play upon us, would raise no expectation of future warmth. In short, if our ideas were not associated, we could have no knowledge of any kind; for if we attend to what passes in our own minds, we shall be convinced that knowledge is nothing else but the association of ideas, by whatever means this association takes place, whether from what we are taught and accustomed to do, or from our own observation and experience. If our ideas were no longer subject to the law of association, we could no longer be taught anything: habits could no longer be formed: and nature would in vain exhibit a permanence and stability in her appearances and operations.

But this very law of association on which depends the whole fabric of human intellect, happiness, and even existence, is itself the source of our prejudices, errors, and misery. No ap pearance manifests itself, no operation or event takes place in the three departments of the universe in which we have an opportunity of seeing the regular order of nature displayed, viz. the phenomena of inanimate matter, the phenomena of the lower animals, and the phenomena exhibited by the human race, which is not surrounded by a variety of circumstances. It may be that the phenomena depend on one alone of all these circumstances; or on several, or possibly on the whole

of them; and it may be, that, from some simple circumstance making a strong impression on our senses or feelings at the time the phenomena were witnessed, they become associated in our minds with it, though in no respect its cause. Hence, error in our thoughts, and mistakes in our conduct, arising from the very law of associa tion on which human intellect and happiness essentially depend. But the phenomena of the order of nature, aided by this law of association, correct the error, and remove the mistake which the latter has occasioned. We observe and experiment again and again at every time, some circumstances preceding, attending, or following the phenomenon, change, and some remain unaltered:-if the one which we at first connected with it, as its cause, disappears while the phenomenon continues, or continues while the phenomenon disappears,-in either case, the association in our minds between them is destroyed, and a new association between those circumstances that uniformly precede, attend, and follow the phenomenon, and the phenomenon itself, is formed. Hence the utility, or rather the absolute necessity, of repeated observations and experiments, if we wish to avoid error or wrong associations, and to attain truth, or an association of ideas in our minds, exactly throughout similar to the regular order displayed in the three departments of the universe, already particularized.

The order of nature, therefore, being permanent and stable, and the association of ideas being a fundamental law of the human intellect, which is the source at once of all our errors and all our knowledge, it becomes a question of infinite importance, how we should proceed, in order to render this law as little injurious, and as highly beneficial, as possible.

There are two grand and paramount objects to which we must direct our attention and researches, if we wish to attain the truth, and to render it, when attained, useful and valuable: we must, in the first place, find out what the general laws of nature are, and, in the next place, learn to apply them with propriety and effect to the extension of our knowledge and regulation of our conduct. By a law of nature is meant a statement of some general fact with respect to the order of na

ture-a fact which has been found to hold uniformly in our past experience, and on the continuance of which, in future, the constitution of our mind, as exhibited in the association of our ideas, determines us confidently to rely.

But it is evident that the general fact, or permanent principle, on which nature proceeds, cannot be determined, unless after a great number and variety of observations and experiments, so as to enable us to separate those circumstances that are accidental from those that are necessary: by necessary, all that we can really mean or understand, except in the case of mathematics-amounts to this, that with them, the result takes place ;— without them, it does not. This separation is indispensable in order to destroy erroneous associations, and to establish those that in all respects correspond with the general laws of na

ture.

The first object, therefore, is to attend to what is passing around and within us; the next, to separate accidental from necessary circumstances. It must be obvious, that those general facts will be ascertained with the least trouble, and in the shortest time, which are attended with the smallest number and variety of circumstances; as the circumstances increase in either or both these respects, the difficulty of separating the accidental from the necessary proportionally increases, and we are the more exposed to error and prejudice in our opinions, and to hurtful mistakes in our conduct. The general law, which we call gravitation, is one of the simplest and most obvious in nature: the circumstances which seem to suspend or modify it are few, and may be easily ascertained, accounted and allowed for. On the other hand, the law which nature follows in proportioning the births of the sexes, and in regulating the duration of human life, appears, even after long and close attention to the facts from which it must be drawn, so varying and contradictory, that we are disposed to regard it as beyond the limit of human knowledge, or as having no real existence. And yet how wonderfully shall we find the balance between the sexes preserved in the case of a numerous society, and in a long list of persons of the same age, and placed in the same circumstances! the mean du

ration of life, too, is found to vary within very narrow limits. It is a just remark, that how accidental soever circumstances, and how much soever they may be placed, when individually considered, beyond the reach of our calculations, experience shews that they are, somehow or other, mutually adjusted, so as to produce a certain degree of uniformity in the result; and this uniformity is the more complete, the greater is the number of circumstances combined.

This separating of those circumstances which uniformly precede a result, from those which are accidental and inoperative, as well as from those that prevent the result from taking place, or alter and modify it, is, in fact, the induction which Bacon recommends; and where the mind is not powerfully warped by prejudice, and the necessary observations and experiments are made with care and attention, is a natural consequence of that law of association, to which we have already alluded.

There is only one branch of knowledge which does not require induction or the association of ideas for its attainment, though it may rest on these-this is Mathematics. There has been much controversy on the nature of mathematical evidence; by most it is represented as something abstract, and entirely independent of experiment, or even of the senses; or, to use the expression of M. Prevost, in his Philosophical Essays, Mathematics is a science of pure reasoning. Others, on the contrary, and particularly Dr Beddoes, maintain that mathematical truths, like all other truths, must be drawn entirely and exclusively from observation and experiment; and that so they ought to be taught and communicated. This is an important and interesting topic; but it would lead us far beyond our limits, as well as our special subject, to enter on it here: a few remarks, however, may be made.

The demonstration of all the theorems in the elements of plane geometry, in which different spaces are compared together, when traced back to its first principles, terminates in the fourth proposition of Euclid's first Book; and this rests entirely on a supposed application of the one triangle to the other. Indeed, according to D'Alembert, we might go farther; for this author, who certainly is a

competent judge, and cannot be suspected of a wish to bring down Mathematics to the level of an experimental science-expressly states, that the fundamental principles of Geometry may be reduced to two: the measurement of angles by circular arches, and the principle of superposition. Afterwards, however, he maintains, and indeed proves, that the measurement of angles by circular arches, is, itself, dependent on the principle of superposition. On this latter principle, therefore, according to D'Alembert, the whole structure of Geometry rests. The attempt of this author, and, long prior to him, of Barrow, to rescue Mathematics from the character of being an experimental science, we cannot think happy or successful. The superposition, it is contended, not being actual-not the applying of one figure to another, to judge by the eyes if there is really a difference, as a workman applies his foot-measure to a line to measure it ;-but an imaginary or ideal superposition, consisting in supposing one figure placed on the other

the evidence is addressed to the understanding alone, and cannot fairly be characterized as nothing but an ultimate appeal to external observation.

But, if the whole structure of Geometry is grounded on the principle of superposition, will not the basis of this structure be more stable and permanent, if that superposition is actually performed, than if it is only supposed or imagined to be so?

Mr Stewart, who coincides with the opinion of D'Alembert, that the whole structure of Geometry rests on this principle; repels the inference that it is a mechanical science. Alluding to the fourth proposition of the first book, he says, that the reasoning employed rests solely on hypotheses and definitions; and therefore possesses the peculiar characteristic which distin guishes mathematical evidence from that of all the other sciences. In the case of this proposition, the hypotheses are, that the sides of two triangles are equal, each to each, and that the angles included between the respectively equal sides, are also equal. The definition to which Mr Stewart alludes, is, in fact, Euclid's eighth axiom, that magnitudes which coincide with each other are equal. But we apprehend, that, with the help of these hypotheses, and this definition, or ax

iom, the sole inference that can be legitimately drawn is, that the two given sides, and the given angle, which, by the hypothesis, are stated to be equal, are found to be so, by their coinciding on superposition.

A little examination and reflection will, we think, convince us, that in the case of this proposition, the thing proved simply amounts to this:-that where two lines have the same limits, they are equal: for two sides of the triangles, and the included angles, being supposed equal, the limits of the third side, in each triangle, are, by this very supposition, positively fixed; and if we suppose that the remaining sides are not respectively equal, we must, at the same time, suppose that the hypothesis is altered in some one respect. Similar remarks might be made on that part of the theorem which relates to the equality of the remaining angles. If these observations be well founded, it would follow, that all mathematical evidence resolves itself ultimately into the perception of identity. This opinion, we are aware, has been held by some writers, and is strongly opposed by Mr Stewart. He thinks that it is founded on the error of using the terms, identity and equality, as synonymous and convertible terms, and he endeavours to prove that they are not. But, in the only strict and proper meaning which can be attached to them in mathematical reasoning, they undoubtedly are synonymous and convertible. Let us take, for example, the fourth proposition, and confine ourselves to the equality of the third side. Mathematics is conversant alone with magnitude and figure: if, therefore, two lines are equal in length, they are, in a strict mathematical sense, identical. Mathematics know no other identity. In every sense, identity is a metaphysical idea; and Mr Stewart's mistake arises from inferring, that because equality is not the same as metaphysical identity, therefore it is not the same as mathematical identity; but identity is a term which ought not to be admitted into mathematical demonstration.

We have remarked, that the truths of Mathematics may be proved by induction, as well as by demonstration, in whatever that may consist. We are indeed expressly told by Proclus, in his Commentary on Euclid, that the general theorem of the equality of the VOL. XVII.

three angles of a triangle to two right angles, was the result of a previous discovery of this equality in all the kinds of triangles. And there is good reason to believe, that the celebrated and most important binomial theorem of Newton was entirely the result of induction. "There is no reason to suppose," observes Mr Stewart, "that he ever attempted to prove the theorem in any other way; and yet there cannot be a doubt, that he was as firmly satisfied of its being universally true, as if he had examined all the different demonstrations of it which have since been given." Mr Stewart adds, that considerable use is made of the method of induction, by Dr Wallis, in his Arithmetica Infinitorum; and this innovation, in the established forms of mathematical reasoning, gave great offence to some of his contemporaries; in particular to M. de Fermat, one of the most distinguished geometers of the 17th century. The ground of the objection was not any doubt of the conclusions obtained by Dr Wallis, but because Fermat was of opinion, that this truth might have been established by a more legitimate and elegant process.

It is rather singular, that I.a Place should have given his sanction to inductive reasoning, and that he should have particularly noticed a striking instance of its failure by that very Fermat, who did not object to its employment from any doubt of the truth of the conclusions to which it leads.

We allude to that passage of La Place's Essai Philosophique sur les Probabilités, where he cites, as an example that induction sometimes leads to inaccurate results, the theorem of Fermat on prime numbers. The induction on which he rested his theorem he had carried to a considerable extent; and hence he inferred that the truth of the theorem might be depended upon in all cases, and to whatever extent the induction was pushed. In short, he maintained that his theorem would always lead to a prime number, because, in all cases that he had tried, it had done so. Euler, however, proved that the theorem failed in producing a prime number, when the process was carried to a certain point, and thus exhibited an instance of the failure of induction in mathematicsa failure which it would not be easy to parallel in those sciences to which the

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