An extra-labeled "entropy pied piper", original from cover of Harold Morowitz’ 1992 book Entropy and the Magic Flute, showing various entropy formulations.
In thermodynamics, entropy formulations refers to various mathematical formulas employed to quantify heat, heat multiplied by a thermodynamic function (e.g. the inverse of the absolute temperature), or entropy.

Overview
Historically, a number of equations have been used to quantify the 1850 statement, by German physicist Rudolf Clausius, that “an expression was needed” to account for the experimental fact that "loss of heat occurs when work is done", in the working body, during one heat engine cycle, in that the forward expansion and reverse contraction of the body is not reversible; an action that French physicist Sadi Carnot, in 1824, had assumed did not occur, being that he assumed heat consisted of indestructible caloric particles. [1] A chronological tabulation of some of these various entropy formulations are listed below.

The fact that there are so many formulations of entropy, which are often said to be equivalent, is exemplified by the cover of American biophysicist Harold Morowitz’ 1992 book Entropy and the Magic Flute, cropped cover section shown adjacent, which shows a pied piper playing a magic flute to the "tune of entropy", with the notes or chords of:

$S= k \ln W = \frac{dQ}{T} = -k \sum f_i \ln f_i \,$

signifying Boltzmann entropy (1906), Clausius entropy (1854), and Gibbs entropy (1901), a tune that supposedly is leading all the scientists down a swirling black hole.

 # Name Date Formula Description Fire 350BC $\Delta \,$ Aristotle: there are four elements out of which all is made: earth, air, water, fire. Each of the four earthly elements has its natural place; the earth at the center of the universe, then water, then air, then fire. When they are out of their natural place they have natural motion, requiring no external cause, which is towards that place; so bodies sink in water, air bubbles rise up, rain falls, flame rises in air. [15] Sulphur c.790 Geber: to further explain the phenomena of combustion and metallic properties, metals (copper, iron, gold, etc.) were said to be formed out of two elements: sulphur, ‘the stone which burns’, which characterized the principle of combustibility, and mercury, which contained the idealized principle of metallic properties. This evolved into the Arabic three principles: sulphur giving flammability or combustion, mercury giving volatility and its opposite, and salt giving solidity. Sulphur 1524 Paracelsus: in an attempt at a unification of Aristotle’s four element theory and Geber’s three principles, reasoning that the latter appeared in the former’s bodies, justifying this by recourse to a description of how wood burns in fire: Mercury included the cohesive principle, so that when it left in smoke the wood fell apart. Smoke represented the volatility (the mercury principle), the heat-giving flames represented flammability (sulphur), and the remnant ash represented solidity (salt). Terra pinguis 1669 Johann Becher: in modification on Paracelsus’ theory, it was argued that the constituents of bodies are air, water, and three types of earth: terra fluida, the mercurial element, which contributes fluidity and volatility; terra lapida, the solidifying element, which produces fusibility or the binding quality; and terra pinguis, the fatty element, which gives material substance its oily and combustible qualities. These three earths correspond with Geber’s three principles. A piece of wood, for instance, is composed of ash and terra pinguis; when the wood is burnt, the terra pinguis is released, leaving the ash. In other words, in combustion the fatty earth burns away. Phlogiston 1703 ϕ Georg Stahl: on a modification of Becher’s three earths theory, terra pinguis was renamedas phlogiston, from the Ancient Greek phlogios for ‘fiery’, which was said to be the “matter and principle of fire, and not fire itself” that escapes from burning bodies with a rapid whirling motion, and is contained in all combustible bodies and also in metals, which can be burnt to “calces”.In phlogiston theory, calx is a residual substance, sometimes in the form of a fine powder, that is left when a metal or mineral combusts or is calcinated due to heat. Calx, especially of a metal, is now properly defined as an ‘oxide’, i.e. a chemical compound containing an oxygen atom and other elements. In the phlogiston theory, the calx was the true elemental substance, having lost its phlogiston in the process of combustion. The phlogiston was said to have the property that it could be restored to the original substance by supplying a replacement phlogiston from any material containing it, such as oil, wax, charcoal, or soot, which was thought to be nearly pure phlogiston. Phlogiston was thought of as a material entity, sometimes considered as the matter of fire, sometimes as a dry earthy substance (soot), sometimes as a fatty principle, such as in sulphur, oils, fats, and resins, and sometimes as invisible particles emitted by a burning candle; contained in animal, vegetable, and mineral bodies. It could be transferred from one body to another. Caloric 1787 Antoine Lavoisier: a difficulty in Stahl’s theory was that calx becomes lighter when reduced to a metal by taking up phlogiston. If the phlogiston had mass, then when added to something, it should make it heavier.To remedy this issue, an experiment (Lavoisier 1786) proved that the matter of heat is weightless. Specifically, phosphorus burnt in air in a closed flask, with no appreciable change in weight. A new particle was proposed, called “caloric”, replacing phlogiston, defined as ‘the repulsive cause, whatever that may be, which separated the particles of matter from each other.’ In this sense, heat was said to be an elastic, fluid-like, substance (called caloric), whereby the sensation of warmth was the accumulation of this substance in the interstices of the particles of matter and that, to explain Boerhaave’s law (1720), the more caloric particles a body had in its pores the more it would expand in volume; and conversely when caloric particles were taken out of a body, the body would contract. Motion 1798 Benjamin Thomson: heat produced, via friction, in the boring of a cannon, which was used to make water boil in 2.5 hours time, gave “farther insight into the hidden nature of heat; and to enable us to form some reasonable conjectures respecting the existence, or non- existence, of an igneous fluid.” “What is heat? Is there anything as igneous fluid? Is there anything that can with propriety be called caloric? That heat generated by friction [in the boring experiments] appeared, evidently, to be inexhaustible, [it] cannot possibly be a material substance; … it appears to me to me to be extremely difficult, if not quite impossible, to form any distinct idea of anything capable of being excited and communicated in the manner heat was excited and communicated in these experiments, except it be MOTION.” [17] Caloric 1824 Carnot: “we shall assume that the quantities of caloric absorbed and emitted in these different transformations compensate each other exactly.” [2] Mechanical equivalent of heat 1843 Joule: showed that heat Q and work W were inter-convertible according to a formulaic expression: “the grand agents of nature are … indestructible; and that wherever mechanical force is expended, an exact equivalent of heat is always obtained.” [18] ? 1850 Clausius: stated an "expression needed" to amend Carnot's view (above) that "no permanent change occurs in the condition of the working body." [1] 1. Equivalence-value 1854 $\frac{Q}{T}$ Clausius: 2. Thermodynamic function 1854 $\Phi \,$ Rankine: 3. Equivalence-value of all uncompensated transformations 1856 $-N\,$ Clausius: 4. Disgregation 1862 Clausius: 5. Entropy 1865 $dS=\frac{dQ}{T}$ Clausius: the symbol S is the "transformation-content" of the working body, but termed "entropy", so to have similarity to the word energy. 6. $S = S_{trans} + S_{rot} + S_{vib} + S_{elec} \,$ Clausius (?): entropy of an ideal gas. [8] 7. 1872 $E \,$ Boltzmann: [7] 8. Entropy 1873 $d\eta=\frac{dH}{t}\,$ Gibbs: the quantity η is the entropy, dη is the differential of entropy, H is the heat received by the body in passing from one state to another, and t the absolute temperature of a body in a given state. [5] 9. 1875 $~ dS = \frac{\delta Q}{T} ~$ Carl Gottfried Neumann: [10] 10. 11. Boltzmann entropy $S_B = - N k_B \sum_i p_i \log p_i \,$ [9] 12. Gibbs entropy 1901 $S = -k_B\,\sum_i f_i \ln \,f_i$ Gibbs: [14] 13. c.1906 $S = k \ln W\,$ The expression S = k ln W is said to have been introduced by Planck. 14. 15. 16. c.1935 $S = k \log W \!$ Chiseled onto Boltzmann's tombstone in circa 1935 (as pictured adjacent). 17. Prigogine entropy 1945 $dS= d_e S + d_i S\,$ Prigogine: (a) The entropy of a system is an extensive property: if the system consists of several parts, the total entropy is equal to the sum of the entropies of each part.(b) The change in entropy can be split into two parts: denoting deS as the flow of entropy, due to interactions with the exterior, and diS the contributions due to changes inside the system. [3] 18 Shannon entropy 1948 $H = -K\sum_{i=1}^np_i\log p_i\,\!$ Shannon: “we shall call H = – Σ pi log pi the entropy of a set of probabilities p1, ..., pn.” [4] 19. Von Neumann entropy 1955 $S(\rho) \,=\,-{\rm Tr} (\rho \, {\rm ln} \rho),$ John Neumann: [11] 20. 21. Black hole entropy 1972 $S_{BH} = \frac{kA}{4\ell_{\mathrm{P}}^2}$ Jacob Bekenstein and Stephen Hawking: [13] 22. 23. Tsallis entropy 1988 $S_q(p) = {1 \over q - 1} \left( 1 - \int p^q(x)\, dx \right)$ Constantino Tsallis: [12] 24. 25. 26. 27. 28. Human entropy 2007 $S = S_P + S_O + S_I + S_S + S_N\,$ Thims: entropy of a system of ideal human molecules. [6] 29.

Entropy (etymology)

References
1. 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.
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. (a) Prigogine, Ilya. (1945). Etude Thermodynamics des Phenomenes Irreversibles (Study of the Thermodynamics of Irreversible Phenomenon). Presented to the science faculty at the Free University of Brussels (1945); Paris: Dunod, 1947.
(b) Prigogine, Ilya. (1955). Introduction to Thermodynamics of Irreversible Processes, (pg. 16). New York: Interscience Publishers.
4. Shannon, Claude E. (1948). "A Mathematical Theory of Communication" (pg. 11), Bell System Technical Journal, vol. 27, pp. 379-423, 623-656, July, October.
5. Gibbs, J. Willard. (1873). "Graphical Methods in the Thermodynamics of Fluids" (pgs. 1-2), Transactions of the Connecticut Academy, I. pp. 309-342, April-May.
6. Thims, Libb. (2007). Human Chemistry (Volume One) (pg. 271) (preview) (Google books). Morrisville, NC: LuLu.
7. Boltzmann, Ludwig. (1872). "Further Studies on the Thermal Equilibrium of Gas Molecules", in Sitzungsberichte der Akademie der Wissenschaften, Mathematische-Naturwissenschaftliche Klasse (pgs. 275-370), Bd. 66, Dritte Heft, Zweite Abteilung, Vienna: Gerold.
8. Barrow, Gordon M. (1988). Physical Chemistry (pg. 202). McGraw-Hill.
9. Boltzmann entropy - Wikipedia.
10. (a) The symbol đ (d-hat), or δ (delta), in the modern sense, supposedly, originated from the work of German mathematician Carl Gottfried Neumann in his 1875 Lectures on the Mechanical Theory of Heat (Vorlesungen über die mechanische Theorie der Wärme), indicating that Q and W are path dependent, and are thus inexact differentials.
(b) Laider, Keith, J. (1993). The World of Physical Chemistry. Oxford University Press.
11. (a) Von Neumann, John. (1955). Mathematical Foundations of Quantum Mechanics (Mathematische Grundlagen der Quantenmechanik). Berlin: Springer.
(b) Von Neumann entropy – Wikipedia.
12. Tsallis entropy – Wikipedia.
13. Black hole thermodynamics – Wikipedia.
14. Gibbs, J. Willard (1901). Elementary Principles in Statistical Mechanics - Developed with Special Reference to the Rational Foundation of Thermodynamics. New York: Dover (reprint).
15. Aristotle. (350BC). Physics. Greece.
16. Becher, Johann. (1669). Physica Subterranea. Germany.
17. (a) Thomson, Benjamin. (1798). “An Inquiry Concerning the Source of Heat which is Excited by Friction”. Philosophical Transactions. Vol. XVIII, pg. 286.
(b) Thomson, Benjamin. (1798). “An Inquiry Concerning the Source of Heat which is Excited by Friction” in The Complete Works of Count Rumford, (pgs. 469-93). Oxford University Press, 1870.
18. Joule, James P. (1845). "On the Mechanical Equivalent of Heat", Brit. Assoc. Rep., trans. Chemical Sect, p.31, read before the British Association at Cambridge, June.