In 1865, Rudolf Clausius, generally referred to only by his last name, Clausius, introduced what he called the "two fundamental principles of the mechanical theory of heat."

"Die Energie der Welt ist constant. Die Entropie der Welt strebt einen Maximum zu."
"The energy of the world is constant. The entropy of the world tends towards a maximum."

("The Fundamental Equations of the Mechanical Theory of Heat," in The Mechanical Theory of Heat, English translation, 1867, Ninth Memoir)

In his 1851 paper, "On the Moving Force of Heat," Clausius first discussed the conservation of energy (the first principle). While most thermodynamicists were still arguing that heat must be a fluid, called "caloric," Clausius clearly said heat is the motion of atoms, "the kind of motion we call heat." (The existence of atoms would not be accepted for another 50 years, when Albert Einstein explained the Brownian motion.)

Until recently it was the generally accepted view that Heat was a special substance, which was present in bodies in greater or less quantity, and which produced thereby their higher or lower temperature; which was also sent forth from bodies, and in that case passed with immense speed through empty space and through such cavities as ponder able bodies contain, in the form of what is called radiant heat. In later days has arisen the other view that Heat is in reality a mode of motion. According to this view, the heat found in bodies and determining their temperature is treated as being a motion of their ponderable atoms, in which motion the ether existing within the bodies may also participate; and radiant heat is looked upon as an undulatory motion propagated in that ether.

("On the Moving Force of Heat,"in The Mechanical Theory of Heat, chapter I)

In 1865, Clausius coined a word and a new symbol for the part of the heat that cannot be converted into mechanical work.

...the author proposed to call this quantity, after the Greek word τροπη, Transformation, the Entropy of the body...

If we denote the Entropy of the body by S we may put dQ/τ = S or otherwise dQ =τdS.

("Formation of the Two Fundamental Equations," in The Mechanical Theory of Heat, chapter V)

Clausius says he formed "entropy" from the Greek ην - "in," and τροπη, to turn, or transform. He modeled this new term based on "evolution," which was formed from Latin e or ex- "out" and volvere, to revolve or turn.

Note that the Greek ηντροπη begins with eta. Helmholtz and Gibbs later use the Greek letter η for entropy. We have no idea why Clausius used S (as did Boltzmann in his famous equation S = klogW)

What was original in Gibbs was the superiority of his entropy-temperature diagrams for visualizing the second law.

In his first memoir, Clausius repeats the work of his 1850 article, describing Carnot's work and the explanatory diagrams of Clapeyron.

When any body whatever changes its volume, the change is always accompanied by a mechanical work produced or expended. In most cases, however, it is impossible to determine this with accuracy, because an unknown interior work usually goes on at the same time with the exterior. To avoid this difficulty, Carnot adopted the ingenious contrivance before alluded to: he allowed the body to undergo various changes, and finally brought it into its primitive state; hence if by any of the changes interior work was produced, this was sure to be exactly nullified by some other change; and it was certain that the quantity of exterior work which remained over and above was the total quantity of work produced. Clapeyron has made this very evident by means of a diagram: we propose following his method with permanent gases in the first instance, introducing, however, some slight modifications rendered necessary by our maxim.

In the annexed figure let o e represent the volume, and e a the pressure of the unit-weight of gas when the temperature is t; let us suppose the gas to be contained in an expansible bag, with which, however, no exchange of heat is possible. If the gas be permitted to expand, no new heat being added, the temperature will fall. To avoid this, let the γασ during the expansion be brought into contact with a body A of the temperature t, from which it shall receive heat sufficient to preserve it constant at the same temperature. While this expansion by constant temperature proceeds, the pressure decreases according to the law of M., and may be represented by the ordinate of a curve a b, which is a portion of an equilateral When the gas has increased in volume from otto of, let the body A be taken away, and the expansion allowed to proceed still further without the addition of heat; the temperature will now sink, and the pressure consequently grow less as before. Let the law according to which this proceeds be represented by the curve b c. When the volume of the gas has increased from o f to o g, and its temperature is lowered from tto τ, let a pressure be commenced to bring it back to its original condition. Were the gas left to itself, its temperature would now rise; this, however, must be avoided by bringing it into contact with the body B at the temperature τr, to which any excess of heat will be immediately imparted, the gas being thus preserved constantly at τ. Let the compression continue till the volume has receded to h, it being so arranged that the decrease of volume indicated by the remaining portion h e shall be just sufficient to raise the gas from τ to t, if during this decrease it gives out no heat. By the first compression the pressure increases according to the law of M., and may be represented by a portion of another equilateral hyperbola. At the end the increase is quicker, and may be represented by the curve d a. This curve must terminate exactly in a; for as the volume and temperature at the end of the operation have again attained their original values, this must also be the case with the pressure, which is a function of both. The gas will therefore be found in precisely the same condition as at the commencement.

("On the Moving Force of Heat," in The Mechanical Theory of Heat, 1867, first memoir)

In his fourth memoir, Clausius introduces the equation for a quantity he will eventually name the entropy.

According to this, the second fundamental theorem in the mechanical theory of heat, which in this form might appropriately be called the theorem of the equivalence of transformations, may be thus enunciated:

If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generation of the quantity of heat Q of the temperature t from work, has the equivalence-value

Q/T,

and the passage of the quantity of heat Q from the temperature t, to the temperature t, has the equivalence-value

Q(1/T - 1//T),

wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected.

If to the last expression we give the form

Q/T - Q/T,

it is evident that the passage of the quantity of heat Q, from the temperature f to the temperature t has the same equivalence- value as a double transformation of the first kind, that is to say, the transformation of the quantity Q from heat at the temperature t into work, and from work into heat at the temperature t A discussion of the question how far this external agreement is based upon the nature of the process itself would be out of place here; but at all events, in the mathematical determination of the equivalence-value, every transmission of heat, no matter how effected, can be considered as such a combination of two opposite transformations of the first kind.

("Modified Form of the Second Fundamental Theorem," in The Mechanical Theory of Heat, 1867, fourth memoir)

In his last memoir, Clausius tells us about his new property entropy, and how he came to name it.

All the foregoing considerations had reference to changes which occurred in a reversible manner. We will now also take non-reversible changes into consideration in order briefly to indicate at least the most important features of their treatment.

In mathematical investigations on non-reversible changes two circumstances, especially, give rise to peculiar determinations of magnitudes. In the first place, the quantities of heat which must be imparted to, or withdrawn from a changeable body are not the same, when these changes occur in a non-reversible manner, as they are when the same changes occur reversibly. In the second place, with each non-reversible change is associated an uncompensated transformation, a knowledge of which is, for certain considerations, of importance.

In order to be able to exhibit the analytical expressions corresponding to these two circumstances, I must in the first place recall a few magnitudes contained in the equations which I have previously established.

One of these is connected with the first fundamental theorem, and is the magnitude U, contained in equation (I a) and discussed at the beginning of this Memoir; it represents the thermal and ergonal content, or the energy of the body. To determine this magnitude, we must apply the equation (Ia), which may be thus written,

dU = dQ - d w ; ........................................... (57),

or, if we conceive it to be integrated, thus:

U = U₀ + Q - w .......................... (58),

Herein U₀ represents the value of the energy for an arbitrary initial condition of the body, Q denotes the quantity of heat which must be imparted to the body, and w the exterior ergon which is produced whilst the body passes in any maimer from its initial to its present condition. As was before stated, the body can be conducted in an infinite number of ways from one condition to another, even when the changes are to be reversible, and of all these ways we may select that one which is most convenient for the calculation.

The other magnitude to be here noticed is connected with the second fundamental theorem, and is contained in equation (II a). In fact if, as equation (IIa) asserts, the integral ∫dQ/T vanishes whenever the body, starting from any initial condition, returns thereto after its passage through any other conditions, then the expression ^ under the sign of integration must be the complete differential of a magnitude which depends only on the present existing condition of the body, and not upon the. way by which it reached the latter. Denoting this magnitude by S, we can write

dS = dQ/T .......................... (59),

or, if we conceive this equation to be integrated for any reversible process whereby the body can pass from the selected initial condition to its present one, and denote at the same time by S₀ the value which the magnitude S has in that initial condition,

S = S₀ + ∫dQ/T................................................(60),

This equation is to be used in the same way for determining S as equation (58) was for defining U...

We might call S the transformational content of the body, just as we termed the magnitude U its thermal and ergonal content. But as I hold it to be better to borrow terms for important magnitudes from the ancient languages, so that they may be adopted unchanged in all modem languages, I propose to call the magnitude S the entropy of the body, from the Greek word τροπη, transformation. I have intentionally formed the word entropy so as to be as similar as possible to the word energy; for the two magnitudes to be denoted by these words are so nearly allied in their physical meanings, that a certain similarity in designation appears to be desirable.

("Convenient Forms of the Fundamental Equations," in The Mechanical Theory of Heat, 1867, ninth memoir)

Excerpts from The Mechanical Theory of Heat

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