of temperature of two bodies or contiguous parts of one body. It is only very recently indeed that Forbes has shown that the conductivity of a body for heat diminishes as its temperature increases; and thus that the details of Fourier's solutions are not strictly accurate when great differences of temperature are involved. But, besides the fact that Fourier has shown how to adapt his methods to any experimental data, the solutions he has given are approximate enough for application to many of the most interesting cases, such as the secular cooling of the earth, underground temperature as depending on solar radiations, etc. By this publication, Fourier has reduced the treatment of any question involving transference of heat by conduction or radiation to a perfectly definite form; and must therefore stand, in the history of the subject, as one of its greatest promoters.

Very different in form and object from the systematic treatise of Fourier, is the profound and valuable work of Carnot, published in 1824.1 The author endeavours to determine how it is that heat produces mechanical effect, and though some of his assumptions are not correct, he investigates the question in an exceedingly able and instructive manner. Starting with a correct principle, which, obvious as it is, has been sadly neglected by many later writers, he is led into error by assuming the materiality of heat. But with true philosophical caution he avoids committing himself to this hypothesis, though he makes it the foundation of his attempt to discover how work is produced from heat. He says:

"If a body, after having experienced a certain number of transformations, be brought identically to its primitive physical state as to density, temperature, and molecular constitution, it must contain the same quantity of heat as that which it initially possessed; or, in other words, the quantities of heat lost by the body under one set of operations are precisely compensated by those which are absorbed in the others. This fact has never been doubted; it has at first been admitted without reflection, and afterwards verified, in many cases, by calorimetrical experiments. To deny it would be to overturn the whole theory of heat, in which it is the fundamental principle. It must be admitted, however, that the chief foundations on which the theory of heat rests would require a most attentive examination. Several experimental facts appear nearly inexplicable in the actual state of this theory."

This fundamental principle of Carnot is still evidently axiomatic, as we know of no case in which heat can be communicated to a body, or abstracted from it, without altering its

1 We are indebted for our knowledge of Carnot to an excellent paper-" An Account of Carnot's Theory of the Motive Power of Heat," etc., by W. Thomson. Trans. R.S.E. 1849.

temperature, its volume, its form, or its molecular constitution. In fact, it is entirely upon our confidence in the accuracy of this idea that our means of measuring temperature by thermometers depend. If we had not, for instance, experimental proof that a mass of mercury has always the same volume at the same temperature, our mercurial thermometers, supposing glass to be perfect in this respect, would be worse than useless,-they might be deceptive.

Thus, from Carnot's point of view, it is evident that the motive power of heat depends upon its being transferred from one body to another through the medium by whose change of volume or form the external mechanical effect is produced, as this medium is supposed to remain at the end of the operation in precisely the same state as at the commencement. Thus for the production of mechanical effect, we are to look to the successive communication of heat to, and abstraction of heat from, the particular medium employed; and to illustrate this it is natural to consider the steam-engine as the most stupendous practical application of the principle.

Carnot's reasoning may easily be made intelligible without mathematical details. In the simple case we shall take, all that is attempted is to show that in the ascent of the piston in the cylinder, more work is done against external forces than is required to be done by them to produce the descent and restore the piston to its first position. And in order that Carnot's axiom may be applied with strictness, and yet with simplicity, it is better to consider a hypothetical, than the actual, case.

Suppose we have two bodies, A and B, whose temperatures, S and T, are maintained uniform, A being the warmer body, and suppose we have a stand, C, which is a non-conductor of heat. Let the sides of the cylinder and the piston be also nonconductors, but let the bottom of the cylinder be a perfect conductor; and let the cylinder contain a little water, nearly touching the piston when pushed down. Set the cylinder on A; then the water will at once acquire the temperature S, and steam at the same temperature will be formed, so that a certain pressure must be exerted to prevent the piston from rising. We shall take this condition as our starting-point for the cycle of operations.

First, Allow the piston to rise gradually; work is done by the pressure of the steam which goes on increasing in quantity as the piston rises, so as always to be at the same temperature and pressure. And heat is abstracted from A, namely, the latent heat of the steam formed during the operation.

Second, Place the cylinder on C, and allow the steam to raise the piston farther. More work is done, more steam is formed,

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but the temperature sinks on account of the latent heat required for the formation of the new steam. Allow this process to go on till the temperature falls to T, the temperature of the body B.

Third, Now place the cylinder on B; there is of course no transfer of heat. But if we now press down the piston, we do work upon the contents of the cylinder, steam is liquefied, and the latent heat developed is at once absorbed by B. Carry on this process till the amount of heat given to B is exactly equal to that taken from A in the first operation, and place the cylinder on the non-conductor C. The temperature of the contents is now T, and the amount of caloric in them is precisely the same as before the first operation.

Fourth, Press down the piston farther, till it occupies the same position as before the first operation; additional work is done on the contents of the cylinder, a farther amount of steam is liquefied, and the temperature rises.

Moreover, it rises to S exactly, by the fundamental axiom, because the volume occupied by the water and steam is the same as before the first operation, and the quantity of caloric they contain is also the same as much having been abstracted in the third operation as was communicated in the first-while in the second and fourth operations the contents of the cylinder neither gain nor lose caloric, as they are surrounded by nonconductors.

Now, during the first two operations, work was done by the steam on the piston, during the last two work was done against the steam; on the whole, the work done by the steam exceeds that done upon it, since evidently the temperature of the contents, for any position of the piston in its ascent, was greater than for the same position in the descent, except at the initial and final positions, where it is the same. Hence the pressure also was greater at each stage in the ascent than at the corresponding stage in the descent, from which the theorem is evident.

Hence, on the whole, a certain amount of work has been communicated by the motion of the piston to external bodies; and, the contents of the cylinder having been exactly restored to their primitive condition, we are entitled to regard this work as due to the caloric employed in the process. This we see was taken from A and wholly transferred to B. It thus appears that caloric does work by being let down from a higher to a lower temperature. And the reader may easily see that if we knew the laws which connect the pressure of saturated steam, and the amount of caloric it contains, with its volume and temperature, it would be possible to apply a rigorous calculation to the various processes of the cycle above explained, and to

express by formulæ the amount of work gained on the whole in the series of operations, in terms of the temperatures (S and T) of the boiler and condenser of a steam-engine, and the whole amount of caloric which passes from one to the other.

We wish to avoid formulæ as far as possible, and shall not give any here; since although the above process is exceedingly ingenious and important, it is to a considerable extent vitiated by the assumption of the materiality of heat which is made throughout. To show this, it is only necessary to consider the second operation, where work is supposed to be done by the contents of the cylinder expanding without loss or gain of caloric, a supposition which our present knowledge of the nature of heat shows to be incorrect. But it is quite easy, as we shall soon see, to make the necessary corrections in accordance with the true theory of heat; and it is but bare justice to acknowledge that Carnot himself was by no means satisfied with the caloric hypothesis, and insinuates, as we have already seen, more than a mere suspicion of its correctness.

But we owe Carnot much more than this, as we proceed to show; and we shall defer to a later portion of our article an examination of the curious particulars in which his results for the steam-, or air-, engine differ from those now received.

If we carefully examine the above cycle of operations we easily see that they are reversible, i.e., that the transference of the given amount of caloric back again from B to A, by performing the same operations in the opposite order, requires that we expend on the piston, on the whole, as much work as was gained during the direct operations. This most important idea is due also to Carnot, and from it he deduces his test of a perfect engine, or one which yields from the transference of a given quantity of caloric from one body to another (each being at a given temperature) the greatest possible amount of work. And the test is simply that the cycle of operations must be rever

sible.

To prove it we need only consider that, if a heat-engine M could be made to give more work by transferring a given amount of caloric from A to B, than a reversible engine N does, we may set M and N to work in combination, M driven by the transfer of heat, and in turn driving N, which is employed to restore the heat to the source. The compound system would thus in each cycle produce an amount of work equal to the excess of that done by M over that expended on N, without on the whole any transference of heat, which is of course absurd.

The remarkable consequences deduced by Thomson, by a combination of the methods and results of Fourier and Carnot, with reference to the dissipation of heat, and the final trans

formations of all forms of energy, though properly belonging to this part of the development of our subject, are left to a future page, so that we may keep as closely as possible to the chronological order, in presenting the most important additions to the science.

A little before the publication of Carnot's work, a second method of procuring work from heat was discovered by Seebeck. It consists in the production of electricity by the action of heat on heterogeneous conducting matter, and the employment of the current to drive an electro-magnetic engine. It is not alluded to by Carnot; and it will tend greatly to the simplicity of this explanatory narrative if we defer to a second article the consideration of the other physical agents which the grand principle of conservation of energy has shown to be so intimately related to heat. We shall, therefore, confine ourselves as strictly as possible to the relation between heat and mechanical effect, which is, however, only one branch of the dynamical theory.

For nearly twenty years after the appearance of Carnot's treatise little appears to have been done with reference to the theory of heat. Clapeyron, in 1834, recalled attention to Carnot's reasoning, and usefully applied the principle of Watt's diagram of energy to the geometrical exhibition of the different quantities involved in the cycle of operations by which work is derived from heat by the temporary changes it produces in the volume or molecular state of bodies.

Then there appeared, almost simultaneously, a group of four or five speculators or experimenters whose relative claims have been since pressed, in some cases, with considerable violence. The work of one of these, Rebenstein, we have not seen; that of another, Colding, is in Danish. Of the others, Séguin and Mayer, it seems not very difficult to estimate the claims so far as the discovery either of the true theory, or the mechanical equivalent, of heat is concerned. Séguin in 1839, and Mayer in 1842, gave as values of the mechanical equivalent, the first 363 kilogrammètres, or in terms of the ordinary British units, 660 foot-pounds; the second the almost identical numbers 365 or 663. It is curious also to observe that the methods employed were almost identical: that of Séguin being founded on the principle that the work given out by any body dilating, and thereby losing heat, is the equivalent of the heat lost; while that of Mayer is, that the heat developed by compression is the equivalent of the work expended in compressing the body. Neither makes the slightest limitation as to the nature of the substance to be experimented on, both their statements are perfectly general; and, we may add, not only inaccurate,