In thermodynamics, a transformation is the process by which a working body is “transformed” through a specific path (path dependent or path independent), either in one direction or cyclically, being defined by an initial state and a final state.  A transformation is said to be reversible if the forward process “compensates” the reverse process; if not, the process is said to be irreversible. The mathematical quantity entropy dQ/T, or the quantity of heat dQ divided by the absolute temperature T of the body in the region of heat flow, represents the energetic value of one direction of the "transformation content" of the working body, which accounts for internal system, atomic-molecular, heat-conversion-to-work actions, resultant due to doctrine of the mechanical equivalent of heat, as heat flows through it. 
The term ‘transformation’ was first used by French physicist Sadi Carnot in 1824 in his Reflections on the Motive Power of Fire. In particular, after detailing the now-famous seven step Carnot cycle, in which a through a cycle of operations, a quantity of “caloric” (an indestructible particle of heat) is passed from a hot body A, first into the intermediary “working substance”, thus causing the substance to expand and exert effort (work) on the piston, and from thence into a cooler body B, he states that “the air we have used to develop the motive power is restored at the end of each cycle of operations exactly to the state in which it was found at the first round.” In an asterisked footnote to this remark, however, Carnot details his opinion that:
“We tacitly assume in our demonstration, that when a body (working body) has experienced any changes, and when after a certain number of transformations it returns to precisely its original state, that is, to the state considered in respect to density, to temperature, to mode of aggregation (atomic ordering)—let us suppose, I say, that this body is found to contain the same quantity of heat that it contained at first.”
The difficultly here, as German physicist Rudolf Clausius spent over 25-years on (in his Mechanical Theory of Heat), in the modern view, is that a body cannot contain heat, but rather heat, as a form of movement of the particles of the body, can only exist as energy in transit. Carnot being schooled in the caloric theory of French chemist Antoine Lavoisier, however, elaborates further on his views, in a defining sentence that greatly catches the attention of Clausius, by stating that:
“We shall assume that the quantities of heat absorbed and emitted in these different transformations compensate each other exactly.”
On this supposition, Carnot states very distinctly that: “this fact has never been held in doubt; admitted at first without reflection, it has since been verified in many instances by experiments with the calorimeter.” He concludes “to deny this would overthrow the whole theory of heat, which rests on it as a basis.”
Lastly, as an end note to these declarations, Carnot states, in passing, that “the main principles on which the theory of heat rests require the most careful examination … many experimental facts appear almost inexplicable in the present state of this theory.” These incongruent experimental facts to which Carnot refers, in relation to his model of the production of motive power by heat, are in likely reference to the now famous 1798 cannon-boring experiments of American-born English physicist Benjamin Thomson, in which work of the boring process produced frictional heat, and the 1799 ice-rubbing experiment of British chemist Humphry Davy, in which it was shown that by rubbing ice cubes together in a room colder than the freezing point of water can cause heat to be generated, due to friction, thus melting the ice.
In any event, these “main principles” spoken of here, were later developed by Clausius, between 1850 and 1865, into the form of what he called the first main principle of the mechanical theory of heat and the second main principle of the mechanical theory of heat, later to be called the first and second laws of thermodynamics, respectively.
In the second main principle, Clausius greatly elaborated on the view that, in reality, “in the production of work a corresponding quantity of heat is consumed (such as when heat transforming in the system is converted into irreversibly lost intermolecular work and friction), and that in consequence the quantity of heat given out to the surrounding space during the cyclical process is less than that received from it.” To represent this difference mathematically, in that according the modern theory of heat in contrast to that of the older caloric theory there exist uncompensated forward and reverse transformations, Clausius used the quantity dQ/T to represent the measure of one direction of the transformation and called it the “entropy of the body” S, after the word the Greek word τροπη (transformation).
● Uncompensated transformation
● Transformation content
● Positive transformation
● Negative transformation
● Equivalence-value of all uncompensated transformations
1. Perrot, Pierre. (1998). A to Z of Thermodynamics. Oxford: Oxford University Press.
2. Carnot, Sadi. (1824). “Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power.” Paris: Chez Bachelier, Libraire, Quai Des Augustins, No. 55.
3. Clausius, Rudolf. (1879). The Mechanical Theory of Heat, London: Macmillan & Co. (second edition), original.