# S

A snapshot of how Thims (2020) arrived at the conclusion, following his reading of Cardwell (1971), who refers to Thomson's May 1854 publications about Q/T, that the "S" symbol of entropy, introduced by Clausius (1865), is symbolic the the Greek letter Σς, in reference to Clausius' theory of the "sum" (Σ or ς) of the equivalence values of all uncompensated transformations as being positive, or greater than zero, in a non-reversible (real) Clausius cycle.

In letters, S (TR:22) (LH:2) (TL:24), from the Greek letter Sigma (Σίγμα) (NE:254), symbols: Σ, σ, or ς, is the 20th letter of the Greek alphabet, and the 19th letter of the English alphabet.

## Overview

### Secret name

The numerical equivalent value of Sigma (Σίγμα) (NE:254), according to Kieren Barry (1999), renders as isopsephy equivalent to the word "change" (παρ-αλλαγη) (NE:254), who cites, via example, this Greek term used in James 1:17.[1] This lends loosely tentative credence to the Sigma, summa, summation conjecture, behind the entropy symbol "S", discussed below, in respect to Clausius's entropy function being replacement upgrade for the Lavoisier-Carnot model of "change", as quoted below:

“This ‘transmission’ Carnot regards as the change of heat corresponding to the work produced. He says expressly, that no heat is lost in the process, that the ‘quantity’ remains un-changed; and he adds, ‘This is a fact which has never been disputed; it is first assumed without investigation, and then confirmed by various calorimetric experiments. To deny it, would be to reject the entire theory of heat, of which it forms the principal foundation’.”
Rudolf Clausius (1850), “On the Moving Force of Heat” (pg. 2); in: The Mechanical Theory of Heat (pg. 15)

Clausius, we note, would go on, over the next 25-year to not necessarily "reject" the entire Lavoisier theory of heat, but to overhaul it, with his new entropy model of heat.

## Entropy

In 1824, Sadi Carnot was using small “s” as a symbol for heat.

In 1865, Rudolf Clausius renamed the heat quantity over temperature function: (later: ), formerly had associated with the symbol “N”, in reference to the sum of the uncompensated transformation equivalents at the end of a Clausius cycle, with the new symbol S; which he explained as follows:

“I propose to call S the entropy of the body, from the Greek word τροπή, transformation.”
— Rudolf Clausius (1865), The Mechanical Theory of Heat (pg. #)

On 8 Nov 1877, Clausius wrote a letter Maxwell about the origin of the term entropy; a rough translation[2] of the letter, from an annotated edition of Maxwell's Collected Works[3], is as follows:

German English
Hochgeehrter Herr, / Ich beeile mich, Ihre Anfrage wegen des Namens Entropie zu beantworten, indem ich das grosste Gewicht darauf lege, mit Ihnen nicht nur in BeZug auf die Benennungen in Debereinstimmung zu sein. / Den Namen Entropie habe ich eingefuhrt in einer Abhandlung Von 1865, welche ich Ihnen beifolgend unter Kreuzband ubersende. Die Stelle befindet sich auf Seite 390; es ist aber zweckmassig, zum besseren Verstandnisse auch dir rarkehrgehenden Seiten. etwa von 5.387 an. zu lesen. Die frühere Abhandlung von 1862. welche dort erwahnt ist, fuge ich ebenfalls / Beide Abhandlungen sind auch in meiner Abhandlungen -Sammlung als Abhandlung VI und IX abgedruckt und befinden sich auch in der von Hirst herausgegebenen englischen Leber-setzung: wo die Stelle uher den Namen Entropie sich auf p357 befindet. / Mit der Versicherung der aus gezeichnetsten Hochachtung verbleibe ich / ganz der Ihrige / R. Clausins.'

In explaining the origin of the term 'Entropie' Clausius made reference to his paper 'lieber verschiedene fur die Anwendung bequeme Form der Hauptgleichungen der mechanischen Warrnecheorie, Ann. Pkrt., 125 (1865): 353-4011 esp. 390: 'nach dem griechischen SVorte rporre). die Verwandlung. die Enfree des Korpers zu nennen': in Clausitts, 771e Mechaniral Thron ofReal,:trans.) T. A. Hirst (London, 18671: 327-65, esp. 357: and see note (5). He also referred to his paper lieber die Anwendung des Satzes von der Acquivatenz der Verwandlung auf die innere Arbeit'. Ann. Ein.. 116118621: 73-112 (in The Aledurniral Preory ofneat 215-50)- sec Number 675. Glausius citcd publitation of diese papers in bis Abhandlungen mbar ehe merharusche Iriumetheone, 2 ‘vls. (Braunschweig. 186-1-7).

Dear Sir, / I hasten to answer your question about the name entropy, placing the greatest emphasis on not only being in agreement with you with regard to the designations. / I introduced the name entropy in a treatise from 1865, which I am sending you under the cruciate ligament. The location is on page 390; It is, however, expedient, for a better understanding, to also find pages that are coming to you. from around 5,387. to read. I also add the earlier treatise from 1862, which is mentioned there / Both treatises are also printed in my collection of treatises as treatises VI and IX and are also in the English liver composition published by Hirst: where the passage comes from the name Entropy is on p357. / With the assurance of the utmost respect I remain / entirely yours / R. Clausins. '

In explaining the origin of the term 'entropy' Clausius made reference to his paper 'prefers various convenient forms of the main equations of mechanical warning theory, Ann. Pkrt., 125 (1865): 353-4011 esp. 390: 'after the Greek word rporre). The transformation. to call the enfree of the body ': in Clausitts, 771e Mechaniral Thron ofReal,: trans.) TA Hirst (London, 18671: 327-65, esp. 357: and see note (5). He also referred to his paper prefer die Application of the principle of the acquisition of metamorphosis to inner work '. Ann. A .. 116118621: 73-112 (in The Aledurniral Preory ofneat 215-50) - sec Number 675. Glausius citcd publication of these papers in bis treatises mbar ehe merharusche Iriumetheone, 2 'from left (Braunschweig. 186-1-7).

While we don't necessarily any symbol origin talk here, we do not that Clausius refers Maxwell to his paper on the "Application of the Principle of the Acquisition of Metamorphosis to Inner Work".

### Battino

In 1997, Rubin Battino, Laurence Strong, and Scott Wood, three American chemistry professors, were asked by a student: "I understand the use of H for enthalpy, since it is heat related, but where does S for entropy come from?", to which the group was stumped? Subsequently, the did some historical investigation, but in the end concluded that it was "arbitrarily chosen symbol" by Clausius. Their resulting Journal of Chemical Education article, about why Clausius choose the symbol "S" for the function , comments as follows:

“In German, Clausius called it the entropie, based on the Greek word meaning transformation or turning. So, the entropy represented the ‘transformation-contents’ or verwandlungsinhalt (German) [verwandlungs = transformation; inhalt = content] of a system. For this new function related to , he used the symbol S, but it is not clear why he chose this letter? A reading of his paper indicates that it was an arbitrarily chosen symbol. This was apparently done like stating: ‘Let X represent ....’.”
— Rubin Battino (1997), “A Brief History of Thermodynamics Notation” (pg. 304)[4]

That Clausius "arbitrarily" choose the most-important symbol of his magnum opus, after devoting so much discussion to his other symbols, such as "U" or energy, hardly seems feasible?

In 2001, Irmgard Howard, in his “S is for Entropy. U is for Energy. What Was Clausius Thinking?”[5], inspired by Battino's article, went through the Clausius memoirs and one of the Thomson 1852 articles, attempt ferret out any place an "s" symbol was used, commenting along the way:

“One can argue that it took Clausius 11-years (1854-1865) to symbolize and name an integral, 15-years (1850-1865) to encapsulate into a single word the physical meaning of the second law, and possibly 25-years (1850-1865) to do the same with the first law.”
— Howard Irmgard (2001), “S is for Entropy. U is for Energy. What Was Clausius Thinking?” (pg. 507)

but in the end arriving at no opinion or commentary as to where the "S" symbol arose, other than saying that it was simply dropped in by Clausius in 1865.

### Thims

A scan of Libb Thims 2020 note-filled copy of Clausius' 1865 Mechanical Theory of Heat (pg. 127), wherein Thims, while drafting the Clausius chapter of his HCT, after reading Donald Caldwell's 1971 From Watt to Clausius (pgs. 258-59), wherein it is pointed out that Clausius' Dec 1854 formulation of sums of δQ/T, came AFTER Thomson's May 1854 formulation of sums of Q/T, illustrating that Thims had arrived at the S = Summa etymology of the symbol "S" introduced by Clausius.

In 2009, Libb Thims conjectured that the symbol "S" was chosen in honor of "S. Carnot", per reason that Sadi Carnot is frequently cited by Clausius in this format.[6] Thims, at some point herein, added this to the entropy etymology section of Wikipedia, where this conjecture has proliferated, therefrom. Thims later recanted on this conjecture..

#### S | Heat?

In 2015, Thims, noting that Carnot was using small "s" for heat, back-tracked on his early "done in honor of S. Carnot" conjecture.[7] Carnot may have employed this unit, in the sense of "specific heat" or specific heat capacity", which stems from Lavoisier and earlier terminology. In 2018, Thims posting at Quora about the small "s" for heat symbol use of Carnot.[8] This conjecture, however, did not seem to hold weight in the long run? Thims later, e.g. tracked down that the Q symbol for heat comes from Lavoisier's Memoir on Heat (1783) meaning "quantity of heat".

#### S | State?

One possibility is that Clausius introduced the letter "S" in reference to the change in "state" that occurs in the working body at the end of the Clausius cycle; which is the crux of the entire problem, as stating in his first article:

“Thomson mentions distinctly the obstacles which lie in the way of an unconditional acceptance of Carnot's theory, referring particularly to the investigations of Joule, and dwelling on one principal objection to which the theory is liable. If it be even granted that the production of work, where the body in action remains in the same state after the production as before, is in all cases accompanied by a transmission of heat from a warm body to a cold one, it does not follow that by every such transmission work is produced, for the heat may be carried over by simple conduction; and in all such cases, if the transmission alone were the true equivalent of the work performed, an absolute loss of mechanical force must take place in nature, which is hardly conceivable.”
— Rudolf Clausius (1850), “On the Moving Force of Heat and the Laws of Heat which may be Deduced Therefrom” (pgs. 16-17)[9]

In other words, Clausius entire effort, in founding thermodynamics, was to show that a change does occur in the "state" of the working body, therein overthrowing Lavoisier and his caloric model of heat. In this sense, S would have been introduced, in Clausius's mind, without saying so directly in print, as being symbolic of the "transformation content" of the "state" or change of state of the working body, in respect to heat transforming into work and work transforming into heat, with relation to so-called "interior work" going on inside the working body, done by the molecules of the system on each other, irreversibly.

#### S | Summa = Σ = ∫

In 2020, Thims, in his draft §25: Clausius chapter, of his HCT manuscript, conjectured that Clausius employed the "S" symbol in the sense of the mathematical symbol for "sum", thematic to the summation (Sigma) Σ symbol or the summa integral symbol, both of which employed in his 1854 to 1862 characteristic function formulations for what we now think of as entropy.[10]

In 1854, Clausius, in the development of his "characteristic functions", defined “N”, the precursor symbol to "S", as the “sum” of the or total value of all of the equivalence values of the transformations, involve in the Clausius cycle, which he defined formulaically as follows:

${\displaystyle N=\sum {\frac {\delta Q}{T}}}$

Then, in 1856, Clausius, in the development of his "characteristic functions", defined N as "equivalence value of all the uncompensated transformations"[11], which he defined formulaically as follows:

${\displaystyle N=-\int {\frac {\delta Q}{T}}}$

At this point, we note, in respect to mathematical notation usage, that Leibniz (1675) was the one who introduced the Greek alphabet letter "sigma" (Σ, σ, ς) equals "sum" equals, which he rescripted into the "integral" notation of calculus.[12] Hence, in this year, we can conjecture, Clausius had the letter "S" in his mind, in respect to his so-called "theorem of equivalence of transformations".[13]

In 1862, Clausius defined the following expression:[14]

${\displaystyle \int {\frac {\delta Q}{T}}\geqq 0}$

by stating that this was algebraic "sum" of the of all "positive transformations" (TR:5)[15] and "negative transformations" (TR:5)[16], occurring in a cyclical process, which can only be positive.

Then, in 1865, he introduces the "S" symbol as follows:

${\displaystyle dS={\frac {\delta Q}{T}}}$

calling this the "entropy" and alluding to the former formulas as "entropy increase".

##### Entropy = S

The following is the nutshell synopsis of how Clausius, in the years 1854 to 1865, arrived at "S" for the symbol of "entropy":

${\displaystyle N={\frac {Q_{1}}{T_{2}}}+{\frac {Q_{2}}{T_{2}}}+{\frac {Q_{3}}{T_{3}}}\dots }$
${\displaystyle N=\sum {\frac {Q}{T}}}$
${\displaystyle N=\int {\frac {dQ}{T}}}$
${\displaystyle {\text{Entropy}}=S}$

Clausius, in short, never gives the reason why he choose the letter "S" for entropy, but given a digestion of the first and second volumes of his collected memoirs on thermodynamics, it seems to the the case that "S" is short for the phrases:

• S = SUM [S] of the equivalence values (aka entropy)
• S > 0 = Sum [S] of the equivalence values of all uncompensated transformations (aka entropy increase)

The jump from the symbol "N" to the symbol "S", however, is not easily summarized in short.

## End matter

### References

1. James 1:17 (NIV) – BibleGateway.
2. Entropy (etymology) – Hmolpedia 2020.
3. Maxwell, James. (1995). The Scientific Letters and Papers of James Clerk Maxwell: Volume II, 1862-73 (editor: Peter Harman) (pg. #). Cambridge.
4. Battino, Rubin; Wood, Scott E.; Strong, Laurence E. (1997). “A Brief History of Thermodynamics Notation” (abs) (pdf), Journal of Chemical Education, 74(3):304-05.
5. Howard, Irmgard K. (2001). “S is for Entropy. U is for Energy. What Was Clausius Thinking?” (pdf), Journal of Chemical Education, 78(4):505-08.
6. Revision #6 of the "S" Hmolpedia article (5 May 2009).
7. Edit version #8 of article "S" of Hmolpedia (15 Jun 2015).
8. Thims, Libb. (2018). “Answer to: Why is Entropy Called S?”, Quora, Mar 15.
9. Clausius, Rudolf. (1865). The Mechanical Theory of Heat (translator: Thomas Hirst) (§1:14-89). Macmillan, 1867.
10. Thims, Libb. (65AE). Human Chemical Thermodynamics (pdf). Publisher.
11. Equivalence values of all uncompensated transformations – Hmolpedia 2020.
12. Symbols – Hmolpedia 2020.
13. Theorem of the equivalence of transformations – Hmolpedia 2020.
14. Clausius, Rudolf. (1865). The Mechanical Theory of Heat (translator: Thomas Hirst). Macmillan, 1867.
15. Positive transformation  - Hmolpedia 2020.
16. Negative transformation – Hmolpedia 2020.