In thermodynamics, Carnot’s principle, aka “principle of Carnot”, is an 1824 postulate, made by French physicist Sadi Carnot, that in the construct of any generalized heat engine the motive power of heat is independent of the material of the working substance; depending solely on the temperatures between which the caloric (indestructible heat particles) is transported. [1]

Overview
In 1849, Irish physicist William Thomson restated a condensed version of Carnot’s principle as follows:

“At the end of a cycle of operations, when a body is left in precisely its primitive physical condition, if it has absorbed any heat during on part of the operations, it must have given out again exactly the same amount during the remainder of the cycle … the truth of this principle is considered as axiomatic by Carnot, who admits it as the foundation of his theory.”

After further digression, Thomson then concludes by stating that: [2]

“In the present state of science, no operation is known by which heat can be absorbed into a body without either elevating its temperature or becoming latent, and producing some alteration in its physical condition; and the fundamental axiom adopted by Carnot may be considered as still the most probable basis for an investigation of the motive power of heat; although this, and with it every other branch of the theory of heat, may ultimately require to be reconstructed upon another foundation, when our experimental data are more completed. On this understanding, and to avoid a repetition of doubts, I shall refer to Carnot’s fundamental principle, in all that follows, as if its truth were thoroughly established.”

The central issue of contention in Carnot’s principle, addressed by Thomson, and later by German physicist Rudolf Clausius (1850), is the incompatibility between Carnot’s assumption of the equivalence of caloric absorbed and emitted and during one engine cycle and the newly forming logic embodied in the mechanical equivalent of heat, as introduced by English physicist James Joule (1843) and others, which notes, based on experiment, that heat can be converted into work and vice versa; in the sense that, in a reformulation of Carnot’s principle, during one engine cycle, some of the caloric must be converted or transformed into non-recoverable internal system work.

Clausius
In 1850, German physicist Rudolf Clausius begins his famous paper "On the Motive Power of Heat: and on the Laws which can be Deduced from It", by quoting Thomson in his 1849 paper as saying: “if we abandon his principle [of Carnot], we meet with innumerable other difficulties—insuperable without further experimental investigation—and an entire reconstruction of the theory of heat from its foundation.” Then Clausius goes on to say that: "I do not think the difficulties are so serious as Thomson suggests [rather] we should not be daunted by these difficulties but should familiarize ourselves as much as possible with the consequences of the idea that heat is a mode of motion.” Clausius concludes:

“A careful examination shows that the new method does not stand in contradiction to the essential principle of Carnot, but only to the subsidiary statement that no heat is lost.”

Clausius then, over the next fifteen-years, reformulated Carnot's principle (essentially be replacing the hypothesis of caloric by that of entropy) into the second principle of the mechanical theory of heat or the second law of thermodynamics, in the modern sense. In short, according to Clausius' refomulation, heat is lost in the body and that loss is quantified by an increase in entropy in the working body.

Carnot's theorem
A corollary of Carnot’s principle, often called "Carnot's theorem" (although the two are somewhat intertwined), is that any two idealized (reversible) Carnot engines working between the same temperature differences (ΔT = TH - TC) will always have the same efficiency η :

$\eta = \frac{\Delta T} {T_H}\$

and that this is the maximum efficiency that any generalized heat engine can obtain. [4] This is sometimes called Carnot's rule. [5]

References
1. (a) 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.
(b) Daniell, Alfred. (1904). A Text Book of the Principles of Physics (term: Carnot’s principle, pg. 397). MacMillan and Co.
2. Thomson, William. (1849). “An Account of Carnot’s Theory of the Motive Power of Heat – with Numerical Results Deduced from Regnault’s Experiments on Steam”, (127-203) Transactions of the Edinburgh Royal Society, xiv.; Annales de Chime, xxxv. 1852.
3. (a) Thomson, William. (1849). " An Account of Carnot’s Theory of the Motive Power of Heat; with Numerical Results Deduced from Regnault’s Experiments on Steam", Math. and Phys. Papers. Vol. i, pg. 119, note.
(b) Clausius, Rudolf. (1850). "On the Motive Power of Heat, and on the Laws which may be deduced from it for the Theory of Heat", Communicated in the Academy of Berlin, Feb.; Published in Poggendorff's Annalen der Physick, March-April. LXXIX, 368, 500.
4. (a) Eu, Byung C. (2002). Generalized Thermodynamics: the Thermodynamics of Irreversible Processes and Generalized Hydrodynamics (4.1: Carnot’s Theorem, pgs. 32-33). Springer.
(b) Finn, Colin B.P. (1993). Thermal Physics (4.5: Carnot’s Theorem, pgs. 59-60). CRC Press.
5. Carnot's theorem (thermodynamics) - Wikipedia.