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Front Page Titles (by Subject) Analogy to the Theory of the Lever. - The Theory of Political Economy

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Subject Area: Economics
Topic: General Treatises on Economics

## Analogy to the Theory of the Lever. - William Stanley Jevons, The Theory of Political Economy [1871]

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The Theory of Political Economy (London: Macmillan, 1888) 3rd ed.

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### 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.