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• Preface to the First Edition (1871)
• Preface to the Second Edition (1879)
• Preface to the Third Edition By Harriet Jevons
• Errata
• Chapter I: Introduction
• Mathematical Character of the Science.
• Confusion Between Mathematical and Exact Sciences.
• Capability of Exact Measurement.
• Measurement of Feeling and Motives.
• Logical Method of Economics.
• Relation of Economics to Ethics.
• Chapter II: Theory of Pleasure and Pain
• Pleasure and Pain As Quantities.
• Pain the Negative of Pleasure.
• Anticipated Feeling.
• Uncertainty of Future Events.
• Chapter III: Theory of Utility
• Definition of Terms.
• The Laws of Human Want.
• Utility Is Not an Intrinsic Quality.
• Law of the Variation of Utility.
• Total Utility and Degree of Utility.
• Variation of the Final Degree of Utility.
• Disutility and Discommodity.
• Distribution of Commodity In Different Uses.
• Theory of Dimensions of Economic Quantities.
• Actual, Prospective, and Potential Utility.
• Distribution of a Commodity In Time.
• Chapter IV: Theory of Exchange
• Importance of Exchange In Economics.
• Ambiguity of the Term Value.
• Value Expresses Ratio of Exchange.
• Popular Use of the Term Value.
• Dimension of Value.
• Definition of Market.
• Definition of Trading Body.
• The Law of Indifference.
• The Theory of Exchange.
• Symbolic Statement of the Theory.
• Analogy to the Theory of the Lever.
• Impediments to Exchange.
• Illustrations of the Theory of Exchange.
• Problems In the Theory of Exchange.
• Complex Cases of the Theory.
• Competition In Exchange.
• Failure of the Equations of Exchange.
• Negative and Zero Value.
• Equivalence of Commodities.
• Acquired Utility of Commodities.
• The Gain By Exchange.
• Numerical Determination of the Laws of Utility.
• Opinions As to the Variation of Price.
• Variation of the Price of Corn.
• The Origin of Value.
• Chapter V: Theory of Labour
• Definition of Labour.
• Quantitative Notions of Labour.
• Symbolic Statement of the Theory.
• Dimensions of Labour.
• Balance Between Need and Labour.
• Distribution of Labour.
• Relation of the Theories of Labour and Exchange.
• Relations of Economic Quantities.
• Various Cases of the Theory.
• Joint Production.
• Over-production.
• Limits to the Intensity of Labour.
• Chapter VI: Theory of Rent
• Accepted Opinions Concerning Rent.
• Symbolic Statement of the Theory.
• Illustrations of the Theory.
• Chapter VII: Theory of Capital
• The Function of Capital.
• Capital Is Concerned With Time.
• Quantitative Notions Concerning Capital.
• Expression For Amount of Investment.
• Dimensions of Capital, Credit and Debit.
• Effect of the Duration of Work.
• Illustrations of the Investment of Capital.
• Fixed and Circulating Capital.
• Free and Invested Capital.
• Uniformity of the Rate of Interest.
• General Expression For the Rate of Interest.
• Dimension of Interest.
• Peacock On the Dimensions of Interest.
• Tendency of Profits to a Minimum.
• Advantage of Capital to Industry.
• Are Articles In the Consumers' Hands Capital?
• Chapter VIII: Concluding Remarks
• The Doctrine of Population.
• Relation of Wages and Profit.
• Professor Hearn's Views.
• The Noxious Influence of Authority.
• Appendix I
• Appendix Ii

Analogy to the Theory of the Lever.

I have heard objections made to the general character of the equations employed in this book. It is remarked that the equations in question continually involve infinitesimal quantities, and yet they are not treated as differential equations usually are, that is integrated. There is, indeed, no reason why the process of integration should not be applied when it is required, and I will here show that the equations employed do not differ in general character from those which are really treated in many branches of physical science. Whenever, in fact, we deal with continuously varying quantities, the ultimate equations must lie between infinitesimals. The process of integration, if I understand the matter aright, only ascertains other equations, the truth of which follows from the fundamental differential equation.

The mode in which mechanies is usually treated in elementary work tends to disguise the real foundation of the science which is to be found in the so-called theory of virtual velocities. Let us take the description of the lever of the first order as it is given in some of the best modern elementary works, as, for instance, in Mr. Magnus's Lessons in Elementary Mechanics, p. 128. We here read as follows:—

"Let AB be a lever turning freely about C, the fulcrum, and let P be the force applied at A, and W the force exerted, or resistance overcome, or weight raised at B. Suppose the lever turned through the angle ACA', then the work done by P equals P × are AA', and work done by W equals W × arc BB', if P and W act perpendicularly to the arm. Therefore, by the law of energy,

P × AA' = W × BB', and since

we have P × AC = W × BC,

or, P × its arm = W × its arm."

Now, in such a statement as this, we seem to be dealing with plain finite quantities, and there is no apparent difficulty in the matter. In reality the difficulty is only disguised by assuming that P and W act perpendicularly to the arm through finite arcs. This condition is, indeed, carried out with approximate exactness in the problem of the wheel and axle,1 which may be regarded as combining together an infinite series of straight levers, coming successively into operation. In this machine, therefore, the weights, roughly speaking, always act perpendicularly to arms of invariable length. But, in the generality of cases of the lever, the theory is only true for infinitely small displacements, and no sooner has the lever begun to move through any finite arc AA', than it ceases to be exactly true that the work done by P equals P × arc AA'. Nevertheless, the theory is quite correct as applied to the lever considered statically, that is, as in a state of rest and equilibrium, because the finite arcs of displacement, when it really is displaced, are exactly proportional to the infinitely small arcs, known as virtual velocities, through which it would be displaced, if instead of being at rest, it suffered an infinitely small displacement.

It is curious, moreover, that, when we take the theory of the lever treated according to the principle of virtual velocities, we get equations exactly similar in form to those of the theory of value as established above. The general principle of virtual velocities is to the effect that, if any number of forces be in equilibrium at one or more points of a rigid body, and if this body receive an infinitely small displacement, the algebraic sum of the products of each force into its displacement is equal to zero. In the case of a lever of the first order, this amounts to saying that one force multiplied into its displacement will be neutralised by the other force multiplied into its negative displacement. But inasmuch as the displacements are exactly proportional to the lengths of the arms of the lever, we obtain as a derivative equation, that the forces multiplied each by its own arm are equal to each other. No doubt in the quotation given above, P × AC = W × BC is an equation between finite quantities; but the real equation derived immediately from the principle of virtual velocities, is P × AA' = W × BB', in which P and W are finite, but AA' and BB' are in strictness infinitely small displacements. Let us write this equation in the form

then as we also have

we can substitute; hence

I dwell upon this matter at some length because we here have exactly the forms of the equations of exchange. As we have seen, the original equation is of the general form

where fx and yy represent finite expressions for the degrees of utility of the commodities Y and X, as regards some individual, and dy and dx are infinitesimal quantities of those commodities exchanged. But these infinitesimals may in this case at least be eliminated, because, in virtue of the Law of Indifference, they are exactly proportional to the whole finite quantities exchanged. Hence for we substitute . We may write the equations one below the other, so as to make the analogy visible—thus

To put this analogy of the theories of exchange and of the lever in the clearest possible light, I give below a diagram, in which the several economic qualities are represented by the parts of the diagram to which they correspond or are proportional.

Now in statical problems no such process as integration is applicable. The equation lies actually between imaginary infinitesimal quantities, and there is no effect to be summed up. Yet there is no statical problem which is not subject to the principle of virtual velocities, and Poisson, in his Traité de Mécanique,

which commences with statical theorems, asserts explicitly,1 "Dans cet ouvrage, j'emploierai exclusivement la méthode des infiniment petits."

[[1]]See Magnus's Lessons, sec. 91.

[[1]]Seconde Édition, Paris, 1833, sec. 12, vol. i. p. 14.