Sadi Carnot’s contribution to the second law of thermodynamicsDon S. Lemons and Margaret K. Penner Citation: Am. J. Phys. 76, 21 (2008); doi: 10.1119/1.2794346 View online: http://dx.doi.org/10.1119/1.2794346 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v76/i1 Published by the American Association of Physics Teachers Related ArticlesA Wealth of Numbers: An Anthology of 500 years of Popular Mathematics Writing. Am. J. Phys. 80, 745 (2012) Seven Tales of the Pendulum. Am. J. Phys. 80, 747 (2012) The House of Wisdom: How Arabic Science Saved Ancient Knowledge and Gave Us the Renaissance. Am. J. Phys. 80, 648 (2012) An uninvited guest: The positron in early 1930s physics Am. J. Phys. 80, 534 (2012) A Short History of Physics in the American Century Am. J. Phys. 80, 458 (2012) Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

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Sadi Carnot’s contribution to the second law of thermodynamicsDon S. Lemons and Margaret K. PennerBethel College, North Newton, Kansas 67117

�Received 6 June 2006; accepted 13 September 2007�

We identify an operative principle in Sadi Carnot’s only publication that is closely related to adistinct version of the second law of thermodynamics. Although Carnot did not propose the secondlaw of thermodynamics, he assumed its equivalent in proving Carnot’s theorem. We show that, inthe absence of the first law, Carnot’s assumption is equivalent to Clausius’ version of the second law.Both Carnot’s assumption and Clausius’ version, in the absence of the first law, are more restrictivethan Kelvin’s statement of the second law. © 2008 American Association of Physics Teachers.

�DOI: 10.1119/1.2794346�

I. INTRODUCTION

Many textbooks and popular histories of thermodynamicsacknowledge Sadi Carnot for inventing the concept of a re-versible cycle, in particular, the Carnot cycle, and for prov-ing Carnot’s theorem—no heat engine operating betweentwo heat reservoirs can be more efficient than a reversibleheat engine.1 But rarely do textbooks directly associate Car-not with the discovery of the second law of thermodynamicsand they never attribute to him a definite statement of thesecond law. Instead, textbooks refer to Thomson �LordKelvin� and to Clausius for the first statements of the secondlaw.2

Yet primary and scholarly secondary sources suggest thatCarnot did contribute directly to the development of the sec-ond law. Barnett characterizes Carnot as “founding withoutformally proposing” the second law of thermodynamics.3

Kestin credits Carnot with “the first verbal formulation of thesecond law.”4 Clausius and Thomson were powerfully moti-vated by Carnot’s theorem; they regarded Carnot’s theoremas true but ultimately were not able to accept all the elementsof Carnot’s proof. Their first task upon formulating their ver-sions of the second law was to prove Carnot’s theorem.5,6

Current thermodynamics texts also use the second law ofthermodynamics to prove Carnot’s theorem.7 Yet Carnot wasable to prove his theorem without formally advancing thesecond law. How did Carnot prove his theorem? In whatsense did Carnot provide a foundation for the second law?

Our approach is to discuss the first 12 pages of Carnot’sonly publication, “Reflections on the motive power of heatand the machines fitted to develop that power,”8 and to imag-ine the intellectual context in which Carnot and other scien-tists and engineers worked in 1824. We find that Carnot useda principle that is very closely related but not identical to astatement of the second law of thermodynamics. Carnot’soperative principle and other assumptions, which were quitenatural in 1824, allowed Carnot to complete his proof. Ex-pressing Carnot’s operative principle in a language that nei-ther assumes nor denies energy conservation produces whatwe call Carnot’s version of the second law or, for conve-nience, Carnot’s second law. Although Carnot never ad-vanced this version of the second law, we show that he as-sumed its logical equivalent.

This paper may tax the imaginations of its readers becausewe ask them to suspend their belief in the first law of ther-modynamics. We realize that such a suspension is extremelydifficult. But Carnot’s “Reflections” appeared in 1824 before

Joule’s experiments began to compel acceptance of the law

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of conservation of energy in the late 1840s. In 1824 Carnotsubscribed, as did many other scientists and engineers, to theconcept of caloric—an ingenerate, indestructible, and prob-ably massless fluid that in flowing from one body to anotherusually cooled the first and heated the second.9 We do notask readers to accept the concept of caloric uncritically, butwe do ask them to enter into the intellectual world of Car-not’s “Reflections”—a world in which energy was not awell-developed concept.10

Historical accuracy is one reason to carefully considerCarnot’s contribution to the second law, but this reason mightnot be high priority for physics students and teachers. Axi-omatic presentations of thermodynamics consciously adoptan ahistorical approach.11 Physics students and teachers haveanother reason for considering the historical evolution ofthermodynamics: Its history encourages us to confront alarger context of ideas. This larger context constitutes aspace of possible ideas that constrains the way we can thinkabout thermodynamics.12 Carnot, the calorist, realized cer-tain ideas out of this space; Clausius and Thomson, as ener-gists, realized others.

This possibility space allows us to formulate versions ofthe second law free from either the conservation of caloric orthe conservation of energy, that is, free of the first law ofthermodynamics. If the logical content of any statement con-sists of all that can be deduced from it, studying deductionsfrom the second law, apart from the first, may help us betterunderstand the second law’s meaning.

Some may wonder whether it is possible to conceptualizethe content of the second law without, at least implicitly,assuming the first law or an alternative to the first law suchas the law of conservation of caloric. That we and others13 doso provides an empirically based answer to this question.Nonetheless, the question is a good one. It is not possible toconsider consequences of the second law apart from the ze-roth law of thermodynamics, because the latter makes theconcept of empirical temperature meaningful and statementsof the second law refer to higher and lower empirical tem-peratures. Statements of the second law also exploit the con-cepts of work and heat. Work can always be reduced to apurely mechanical concept, and heat can be given a preciseoperational definition in terms of purely calorimetricconcepts.14 But there is no necessary linkage between thefirst and second laws of thermodynamics.

Our paper is organized as follows: Section II identifiesCarnot’s operative principle and the version of the secondlaw that is closely related to it. Section III points out that not

all formulations of the second law are logically equivalent.

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In particular, we show that what we call Carnot’s second lawis more restrictive than Thomson’s second law and is logi-cally equivalent to Clausius’s second law. Section IV brieflysummarizes our conclusions.

II. CARNOT’S PRINCIPLE AND SECOND LAW

Carnot’s first proof of Carnot’s theorem—no heat engineoperating between two heat reservoirs can be more efficientthan a reversible heat engine—is in the Thurston translationof Carnot’s “Reflections” �Ref. 8, pp. 11–12�. Because thetheorem usually requires some form of the second law for itsproof, we look into the first 12 pages of Ref. 8 for its opera-tional equivalent.

After an interesting cosmological and geopolitical intro-duction Carnot gives his main theme on p. 6 in the 19thparagraph.8 Carnot remarks that “The production of motionin steam-engines is always accompanied by a circumstanceon which we should fix our attention. This circumstance isthe re-establishing of equilibrium in the caloric; that is, itspassage from a body in which the temperature is more or lesselevated, to another in which it is lower.”15

So taken was Carnot with this idea, that in a heat enginethe caloric passes “from a body in which the temperature ismore or less elevated, to another in which it is lower,” herepeated it six times in the six subsequent paragraphs.16 Weunderstand his phrase “the re-establishing of equilibrium inthe caloric” to mean the passage of caloric “from a body inwhich the temperature is more or less elevated, to another inwhich it is lower.” Evidently Carnot regarded the mere pas-sage of caloric from a hotter body to a colder one sufficientto produce work in a heat engine. The sentences in whichthese repetitions occur are “We easily recognize in the op-erations that we have just described the re-establishment ofequilibrium in the caloric, its passage from a more or lessheated body to a colder one.” “The production of motivepower is then due in steam-engines not to an actual con-sumption of caloric, but to its transportation from a warmbody to a cold body,…” “According to this principle, theproduction of heat alone is not sufficient to give birth to theimpelling power: it is necessary that there should also becold; without it, the heat would be useless.” “Wherever thereexists a difference of temperature, wherever it has been pos-sible for the equilibrium of the caloric to be re-established, itis possible to have also the production of impelling power.”“We have shown that in steam-engines the motive-power isdue to a re-establishment of equilibrium in the caloric; thistakes place not only for steam engines, but also for everyheat-engine, that is, for every machine of which caloric is themotor.” “These changes are not caused by uniform tempera-ture, but rather by alternations of heat and cold.”

Figure 1�a� illustrates Carnot’s simplest heat engine—onethat produces work by transferring caloric from a hotter heatreservoir to a cooler one. A rectangle represents a heat res-ervoir, that is, an object with indefinitely large heat capacity;a circle stands for a heat engine that has experienced a cycle;and an arrow indicates the direction of non-vanishing heat orwork transfer. Because our purpose is to isolate a principleequivalent to the second law of thermodynamics, that is, Car-not’s principle, from conservation of caloric and conserva-tion of energy, we generalize Carnot’s simplest heat engineby transforming it as shown in the movement from Fig. 1�a�to Fig. 1�b�. In Fig. 1�b� no assumption is made about the

relative sizes of the quantities QH, QC, and W—only that

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they are all positive. If caloric is conserved, QH=QC=Q; ifenergy is conserved, QH=QC+W; without either we knowonly that QH�0, QC�0, and W�0.

If, according to Carnot’s principle, the simplest cyclic heatengine is one that must extract heat from one reservoir andreject heat to another cooler reservoir, cyclic heat enginesthat produce work while exchanging heat with fewer thantwo reservoirs are too simple to be possible. Such heatengines, forbidden by Carnot’s principle, are illustrated inFig. 2.

Carnot would have found the explicit prohibition of heatengine cycles that violate conservation of caloric, such asthose in Figs. 2�a� and 2�b�, unnecessary. According to thisview, Carnot’s real original contribution lies in forbiddingheat engine cycles that do not exchange heat yet producework �see Fig. 2�c��.17 Carnot exploited this prohibition ofthis kind of perpetual motion machine to prove Carnot’stheorem. His proof is indirect because it assumes the con-trary of Carnot’s theorem and absurdly concludes that“…this would be not only perpetual motion, but an unlimitedcreation of motive power without consumption either of ca-loric or of any other agent whatsoever.”18

Carnot’s indirect proof also involves another absurdity—aviolation of energy conservation. The first law of thermody-namics would make redundant the prohibitions in Figs. 2�b�and 2�c�, and so reduce Carnot’s principle to the prohibitionin Fig. 2�a�, which is equivalent to Thomson’s statement ofthe second law of thermodynamics.

Carnot’s proof of his theorem makes two assumptions:conservation of caloric, which in effect prohibits the cyclesin Figs. 2�a� and 2�b�, and the absurdity of a cycle that pro-duces work out of nothing, that is, the cycle in Fig. 2�c�. Butwhatever Carnot’s motivation, these assumptions are logi-cally equivalent to holding both conservation of caloric andconservation of energy in abeyance, and adopting all three

Fig. 1. �a� Carnot’s simplest heat engine with caloric conserved. �b� Carnot’ssimplest heat engine assuming neither that caloric is conserved, QH=QC

=Q, nor that energy is conserved, QH=QC+W. By supposition, Q�0, QH

�0, QC�0, and W�0.

Fig. 2. Cyclic heat engines that are too simple to be possible. Each one

violates Carnot’s principle.

22Don S. Lemons and Margaret K. Penner

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prohibitions of the cycles in Fig. 2, as a new law—the sec-ond law of thermodynamics. Recasting the prohibitions ofthe cycles in Fig. 2 into a verbal statement produces what wecall Carnot’s statement of the second law of thermodynam-ics: A heat engine whose only final result is to produce workand exchange heat with fewer than two heat reservoirs isimpossible.19 Although Carnot never advanced this versionof the second law of thermodynamics, he adopted its logicalequivalent in his proof of his theorem.

III. THE CLAUSIUS AND THOMSON STATEMENTSOF THE SECOND LAW

Soon after James Joule established the equivalence ofwork and heat, Clausius5 in 1850 and Thomson �LordKelvin�6 in 1851 proposed distinct versions of the secondlaw of thermodynamics.20 Figure 3�a� illustrates Clausius’ssecond law: A process whose only final result is to extractheat from a reservoir and reject heat to a hotter reservoir isimpossible. Clausius’s second law, stripped of first law con-tent, forbids cyclic heat transfers of the kind diagrammed inFig. 3�a� for QH�0 and QC�0, and not only for first lawcompliant, that is, QH=QC, heat transfers.

Figure 3�b� illustrates Thomson’s second law: A processwhose only final result is to extract heat from a single reser-voir and produce work is impossible. Thomson’s second law,stripped of first law content, forbids cyclic heat engines ofthe kind in Fig. 3�b� for any Q�0 and W�0 and not onlyfor first law compliant, that is, Q=W, cyclic heat engines.Planck corrected Thomson’s original 1851 statement of thesecond law by insisting that the word “only” be included.21

Fig. 4. �a� A heat transfer that violates Clausius’s second law and an allowedheat engine. �b� The combined engine resulting from the adjustment QC

=QC� . The combined engine violates Carnot’s second law whatever the sign

Fig. 3. �a� Cyclic heat transfer forbidden by Clausius’s statement of thesecond law. �b� Cyclic heat engine forbidden by Thomson’s statement of thesecond law.

or value of QH� −QH.

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Apart from the first law, Thomson’s second law prohibitsfewer types of cycles than Carnot’s second law and, for thisreason, is less restrictive than Carnot’s second law. In addi-tion, the Clausius and Carnot statements of the second laware logically equivalent, but some effort is required to showthis.22 To prove that Carnot’s second law and Clausius’s sec-ond law are logically equivalent we first show that Clausius’ssecond law implies Carnot’s and then show that Carnot’ssecond law implies Clausius’s. Together these demonstra-tions establish the logical equivalence of the Carnot andClausius statements.

We prove that Carnot’s second law implies Clausius’s sec-ond law by showing that a denial of Clausius’s second lawleads to a denial of Carnot’s second law. If Clausius’s secondlaw can be denied, heat QC�0 can be extracted from onereservoir at temperature TC and heat QH�0 rejected to ahotter reservoir at temperature TH�TC as shown in cycle 1of Fig. 4�a�. Combine this supposed heat transfer with anallowed heat engine23 of the type shown in Fig. 1�b� andillustrated by cycle 2 in Fig. 4�a�, which extracts heat QH�from the hotter reservoir at temperature TH, produces workW, and rejects heat QC� to the colder reservoir at temperatureTC. Adjust QC=QC� to produce the combined engine 1+2illustrated in Fig. 4�b�. �Given two engines we can alwaysmake one and only one such adjustment by allowing eachengine to complete an appropriate number of cycles.24� Thecombined heat engine 1+2 exchanges heat with at most onereservoir �whatever the sign or value of QH� −QH� and pro-duces work. For this reason, it contradicts Carnot’s secondlaw.

To show that Clausius’s second law implies Carnot’s sec-ond law we again prove the contrapositive, that is, violating

Fig. 5. �a� A heat engine that violates Carnot’s second law and an allowedrefrigerator. �b� The combined heat transfer resulting from the adjustmentW=W�. The combined heat transfer violates Clausius’s second law.

Fig. 6. �a� A heat engine that violates Carnot’s second law and an allowedrefrigerator. �b� The combined heat transfer resulting from the adjustment

W=W�. The combined heat transfer violates Clausius’s second law.

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Carnot’s second law leads to a violation of Clausius’s secondlaw. There are three ways to violate Carnot’s second law. Wewill chose one and show the reader how to complete theproof when choosing the other two.25 Suppose, contrary toCarnot’s second law, an engine can produce work W by ex-tracting heat Q from a single reservoir at temperature TC asshown in Figs. 2�a� and 5�a�. Combine this supposed heatengine with an allowed refrigerator24 that consumes workW�, extracts heat QC from the colder reservoir at temperatureTC, and rejects heat QH to a hotter reservoir at temperatureTH�TC. Adjust W=W� so that the combined engine merelyextracts heat Q+QC�0 from the reservoir at temperature TCand rejects heat QH�0 to the hotter reservoir at temperatureTH as illustrated in Fig. 5�b�. This combined cycle 1+2 di-rectly violates Clausius’s statement of the second law for anyvalues Q�0, QH�0, and QC�0.

To complete the proof we must show that violating Car-not’s second law in the two remaining ways, that is, assum-ing the forbidden cycles found in Figs. 2�b� and 2�c�, alsoleads to violations of Clausius’s statement of the second law.These inferences follow closely the pattern shown in Fig. 5.Figures 6 and 7 illustrate the logic sufficient to complete theproof.

IV. CONCLUSION

We have identified in Carnot’s essay8 a principle that, withconservation of caloric, played the role of the second law ofthermodynamics. This principle states that the simplest pos-sible cyclic heat engine is one that produces work by extract-ing heat from one heat reservoir and rejecting heat to acooler heat reservoir. In terms of more contemporary lan-guage and separated from the content of the first law of ther-modynamics or any alternative to it, this principle constitutesthe following statement of the second law of thermodynam-ics: A heat engine whose only final result is to produce workand exchange heat with fewer than two heat reservoirs isimpossible. Although Carnot never advanced this version ofthe second law of thermodynamics, he assumed its logicalequivalent. We have also shown that what we call Carnot’sstatement of the second law is, apart from the first law, logi-cally equivalent to Clausius’s version of the second law andboth of these are more restrictive than Thomson’s.

ACKNOWLEDGMENTS

The authors acknowledge the helpful comments of GalenGisler, Dwight Neuenschwander, Ralph Baierlein, and a

Fig. 7. �a� A heat engine that violates Carnot’s second law and an allowedrefrigerator. �b� The combined heat transfer resulting from the adjustmentW=W�. The combined heat transfer violates Clausius’s second law.

referee.

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1See, for example, C. J. Adkins, Thermodynamics �McGraw-Hill, London,1978�, pp. 59–61.

2E. Fermi, Thermodynamics �Dover, New York, 1936�, pp. 30–31, andRef. 1, p. 56.

3M. K. Barnett, “Sadi Carnot and the second law of thermodynamics,”Osiris 13, 333–357 �1958�.

4Joseph Kestin, The Second Law of Thermodynamics �Dowden, Hutchin-son and Ross, Stroudsburg, PA, 1967�, p. 12.

5Rudolph Clausius, “On the motive power of heat, and on the laws whichcan be deduced from it for the theory of heat,” found in Reflections on theMotive Power of Fire, edited by E. Mendoza and translated by W. F.Magie �Dover, New York, 1960�, pp. 107–152.

6W. Thomson, “On the dynamical theory of heat, with numerical resultsdeduced from Mr. Joule’s equivalent of a thermal unit, and M. Regnault’sobservations on steam,” excerpted in Ref. 4, pp. 110–111.

7See Ref. 4.8Sadi Carnot, Reflections on the Motive Power of Fire, edited by E. Men-doza and translated by R. H. Thurston �Dover, New York, 1960�, pp.1–59.

9The view that Carnot employed the caloric theory of heat in the “Reflec-tions” has been adopted by most expositors and contested by a few. Foran extended discussion and further references see T. S. Kuhn, “La Mer’sversion of ‘Carnot’s cycle’,” Am. J. Phys. 23, 387–389 �1955�; T. S.Kuhn, “Carnot’s version of ‘Carnot’s cycle’,” Am. J. Phys. 23, 91–95�1955�; V. K. La Mer, “Some current misinterpretations of N. L. SadiCarnot’s memoir and cycle II,” Am. J. Phys. 23, 95–102 �1955�; and V.K. La Mer, “Some current misinterpretations of N. L. Sadi Carnot’s mem-oir and cycle,” Am. J. Phys. 22, 20–27 �1954�.

10Carnot’s posthumously published notes reveal that Carnot came to be-lieve, within a few years following the publication of the “Reflections,”that energy rather than caloric was conserved. See Ref. 8, pp. 60–69.

11 Caratheodory’s axiomatization of thermodynamics is one example of anahistorical presentation of thermodynamics; Callen’s postulational ap-proach is another. For the former see C. J. Adkins, Thermodynamics�McGraw-Hill, London, 1978�, pp. 93–104; for the latter see H. B.Callen, Thermodynamics �Wiley, New York, 1962�.

12The concept of “a space of possible ideas” that shapes our thinking abouta subject is borrowed from Ian Hacking, The Emergence of Probability�Cambridge U. P., Cambridge, 1975�, p. 9.

13See H. Schamp, “Independence of the first and second laws of thermo-dynamics,” Am. J. Phys. 30, 825–829 �1962�.

14Qualitatively heat can be defined as that which changes the state of athermodynamic system without work interactions or mass transfer.

15Reference 8, p. 6.16Reference 8, pp. 7–9.17A statement prohibiting the cycle in Fig. 2�c� is what Kestin meant as

“the first verbal formulation of the second law.” See Ref. 4.18Reference 8, p. 12.19J. T. Vanderslice, H. W. Schamp, and E. A. Mason adopt this version of

the second law in their text Thermodynamics �Prentice Hall, EnglewoodCliffs, NJ, 1966�, p. 29, and loosely, but we believe misleadingly, char-acterize it as “similar to the statement given by William Thomson in1851–1852.”

20That Thomson proposed his statement of the second law independently ofClausius’s statement of the second law has been disputed by Levi Tansjoin “Comment on the discovery of the second law,” Am. J. Phys. 56,179–182 �1988�. Whether or not Thomson’s discovery was independentof Clausius’s discovery, the Thomson statement of the second law isverbally and logically distinct from the Clausius statement.

21M. Planck, Treatise on Thermodynamics �Dover, New York, 1945�, pp. 82ff.

22Seventy years ago the Nobel Laureate Percy Bridgman observed, “Therehave been nearly as many formulations of the second law as there havebeen discussions of it.…�and� not all these formulations can be exactlyequivalent, but it is possible to distinguish stronger and weaker forms.”See Percy Bridgman, The Nature of Thermodynamics �Harvard U. P.,Cambridge, MA, 1941�, p. 116. Bridgman adds, in the ellipsis of thisquotation, the seemingly adventitious remark, “I question whether suchan examination would be of great physical interest.” We disagree.

23Thermodynamics texts always assume that some cyclic processes arepossible in the course of proving that others are impossible. See, forinstance, Ref. 2, Fermi, pp. 31, 34–35, and Ref. 1, Adkins, pp. 56–57. We

do the same in assuming that a heat engine and a refrigerator are possible

24Don S. Lemons and Margaret K. Penner

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with some combination of values QH�0, QC�0, and W�0.24There is some unknown relation �for example, energy or caloric conser-

vation or some other relation� among QH, QC, and W that, given any twoquantities, determines the third. This unknown relation precludes us frommaking more than one adjustment. In general n−1 adjustments of heat or

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work can be made among quantities describing n engines. See Ref. 2,Fermi, p. 37.

25These deductions are assigned as Problem 3 of Chap. 5 in Ref. 19, and E.A. Mason, Thermodynamics �Prentice Hall, Englewood Cliffs, NJ, 1966�,p. 47.

Overbeck’s Cross Pendulum. This piece of apparatus, at Case Western Reserve University, is listed in the 1900catalogue of Max Kohl of Chemnitz, Germany as Overbeck’s cross pendulum. I can see two uses for this apparatus,which has an axle with masses that can be set out various distances from it. If the masses are adjusted so that thecenter of mass is not at the axis, the system will oscillate as a physical pendulum. The location of the center of massis calculated and the moment of inertia found either by calculation or by experiment, and the resulting calculatedperiod compared with experiment. Or, the masses can all be set out the same distance from the axle, and the calculatedmoment of inertia compared with that found by winding a string around the axle and observing the motion of a fallingmass fastened to the other end of the string. �Photograph and Notes by Thomas B. Greenslade, Jr., Kenyon College�

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