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A note on entropy

Rudolf Clausius coins the term entropy (from his Ninth Memoir, 1865).

Sadi Carnot's most important single idea may have been the completely reversible thermodynamic process. It led to the concept of entropy, whose meaning is indicated below.

The word entropy ("transformed energy") was coined by Clausius in 1865 to refer to a variable in his mathematical reasoning. It stands for something — quantity of heat divided by temperature — that is not directly accessible to the human senses and which (normally) cannot even be measured. Accordingly, even amongst those who have to use the concept professionally, there can exist a rather hazy understanding of entropy.

In the same paper Clausius summarised the laws of thermodynamics as follows:

1. The energy of the universe is constant.
2. The entropy of the universe tends to a maximum.

For present purposes 2. can be understood as follows:

Energy of all types [including heat] changes from being localized to becoming dispersed or spread out, if it is not hindered from doing so. The overall process is measured by the increase in thermodynamic entropy.

Another way to think of it is as follows. Energy cannot be created or destroyed. But it can become less available for use. Entropy is a measure of

Thermodynamics educator Frank L. Lambert summarised it thus:

Energy of all types [including heat] changes from being localized to becoming dispersed or spread out, if it is not hindered from doing so. The overall process is measured by the increase in thermodynamic entropy.

• Electromagnetism
• Introduction to electromagnetism

• Entropy
• Introduction to entropy

• Evolution
• Introduction to evolution

• Gauge theory
• Introduction to gauge theory

• Genetics
• Introduction to genetics

• General relativity
• Introduction to general relativity

• Mathematics of general relativity
• Introduction to the mathematics of general relativity

• M-theory
• Introduction to M-theory

• Quantum mechanics
• Introduction to quantum mechanics

• Viruses
• Introduction to viruses

Wikipedia talk:Make technical articles understandable#Question 2: retiring the introductory articles

Sadi Carnot's most important single idea may be the completely reversible thermodynamic process. It led to the concept of entropy. The word entropy ("transformed energy") was coined by Clausius (1865) to refer to a variable in his mathematical reasoning. In the same paper he summarised the laws of thermodynamics as follows:

1. The energy of the universe is constant.
2. The entropy of the universe tends to a maximum.

The concept entropy, though important in thermodynamics, is not easy to understand and is not necessary for an intuitive understanding of Carnot's theory. There are many formulations of the Second Law that do not mention entropy at all, including the original Clausius and Thomson versions.

It has been found that introductory and advanced physics students have trouble understanding entropy. It has been said that the vast majority of engineers either avoid it altogether or use it as a tool "without an in-depth physical understanding". In general, "It is fair to say that no one really knows what entropy is".

For thermodynamics educator Frank L. Lambert entropy has been taught badly to chemistry students for a century (it may be their "most feared" topic), mainly because specialists have not explained it well to those who have to do the teaching. Misleading metaphors have proliferated.

The definition, “entropy is disorder”, used in all US first-year college and university textbooks prior to 2002, has been deleted from 15 of 16 new editions or new texts published since 2002. Entropy is not ‘disorder’ nor is entropy change a change from order to disorder. (Messy papers on a desk or shuffled cards are totally irrelevant to thermodynamic entropy.)

The second law that can be simply stated and is based on fundamental thermodynamics is Energy of all types changes from being localized to becoming dispersed or spread out, if it is not hindered from doing so. The overall process is measured by the increase in thermodynamic entropy.

The Carnot cycle was deliberately contrived to be fully reversible i.e. none of the heat is allowed to escape without the opportunity to do work, and the process is fully recoverable. The dispersion of heat is indeed 'hindered'. Hence the change in entropy is zero. All real engines (and other thermal processes) disperse heat and so increase entropy.

Another way to think about entropy is as a measurement of the availability of useful energy in a system. While energy cannot be created or destroyed, as the system approaches equilibrium the energy of that system becomes less available for use.

Sources

• Ben-Naim, Arieh (2019). "Entropy and Information Theory: Uses and Misuses". Entropy. 21 (12): 1170. doi:10.3390/e21121170.{{cite journal}}: CS1 maint: unflagged free DOI (link)

• Brundage, Mary Jane; Meltzer, David E.; Singh, Chandraleka (2024). "Investigating introductory and advanced students' difficulties with entropy and the second law of thermodynamics using a validated instrument". Physical Review Physics Education Research. 20 (2): 020110. doi:10.1103/PhysRevPhysEducRes.20.020110. Retrieved 24 April 2026.{{cite journal}}: CS1 maint: article number as page number (link)

• Čápek, Vladislav; Sheehan, Daniel P. (2005). Challenges to the Second Law of Thermodynamics: Theory and Experiment (PDF). Dordrecht: Springer Netherlands. ISBN 1-4020-3016-9. Retrieved 23 April 2026.

• Clausius, Rudolf (1867). T. Archer Hirst (ed.). The Mechanical Theory of Heat with its application to the steam-engine and to the physical properties of bodies. London: John van Voorst. Retrieved 24 April 2026.

• Foley, Andrew C. (2007). "Escape From Carnot: A New Way To Introduce The Mysterious Property, Entropy" (PDF). ASEE Annual Conference & Exposition. Honolulu, Hawai. pp. 12.687.1-12.687.11. doi:10.18260/1-2--1584. ISSN 2153-5965. Retrieved 22 April 2026.

• Jeppsson, Frederick; Haglund, Jesper; Strömdahl, Helge (2011). "Exploiting language in teaching of entropy" (PDF). Journal of Baltic Science Education. 10 (1): 27–35. ISSN 1648-3898. Retrieved 22 April 2026.

• Kragh, Helge; Weininger, Stephen J. (1996). "Sooner Silence than Confusion: The Tortuous Entry of Entropy into Chemistry". Historical Studies in the Physical and Biological Sciences. 27 (1): 91–130. JSTOR 27757770.

• Lambert, Frank L. (1999). "Shuffled cards, messy desks, and disorderly dorm rooms—Examples of entropy increase? Nonsense!" (PDF). Journal of Chemical Education. 76 (10): 1385. doi:10.1021/ed076p1385. Retrieved 22 April 2026.

• Lambert, Frank L. (2006). "A modern view of entropy" (PDF). Chemistry. 15 (1): 13–21. Retrieved 23 April 2026.

• Norton, John D. (2022). "How analogy helped create the new science of thermodynamics". Synthese. 200 (4): 1–42. JSTOR 10.2307/27324886.

• Popovic, Marko (2017). "Researchers in an entropy wonderland: A review of the entropy concept". arXiv. doi:10.48550/arXiv.1711.07326. Retrieved 22 April 2026.

• Raviv, Daniel; Barb, Daniel Ryan (2020). "A Visual and Intuitive Approach to Teaching and Learning the Concept of Thermodynamic Entropy". ASEE Virtual Annual Conference Content Access. American Society for Engineering Education.

History

Thames embankments at Westminster, 1641 (Wenceslaus Hollar, Yale Center for British Art). Notice the watermen's stairs.

At Limehouse, 1750 (John Boydell, YCBA.) Notice the ladders.

Nomenclature

The "Thames Embankment" (capitalised) is usually taken to refer to certain riverside roads and associated infrastructure in Central London — specifically the Victoria, Albert and Chelsea Embankments — built in the Victorian era under the supervision of Sir Joseph Bazalgette. In a general sense, however, a river embankment means any artificial wall or structure that prevents it overflowing. Thus the river Thames had been artificially embanked for many centuries before the Bazalgette works, and on a much larger scale, though the fact is less well known and the contrary is sometimes asserted or implied. "Embanking the Thames is a very old practice". Some of those embankments were designed by professionals; but most were instances of vernacular architecture. The Thames Embankment (capitalised) replaced vernacular embankments that had long existed in those parts of the river.

Vernacular embankments

At least one major embankment - in the City - can be traced back to the Roman era. The medieval embankment at the Tower of London is a tidal defence probably over 700 years old. Between Teddington in the west and Sheerness and Shoeburyness to the east there are more than 300 km of sea walls and embankments: some are private property, some are owned by the Environment Agency. They were upgraded in the 1980s, but most have medieval origins.

Before human intervention the pristine river Thames was shallow and wide. The original embankments had been made by local people who wanted to reclaim land along its marshy foreshore, including in London; hence they had built artificial banks to prevent the river overflowing their newly won lands at high tide. At first, quite low ones sufficed, since in that era tides were modest.

The reclamation process was repeated and cumulative. James Walker (engineer) thought that over the centuries the anonymous builders had gradually constrained and confined the natural river, to such an extent that it was now a fifth of its original width; the water flowing more rapidly and had scoured out a deeper channel, making it navigable by large ships. Gustav Milne thought the river has been transformed into a tidal canal: at the time of the Roman invasion it was very much shallower, particularly towards the south, being 1 km wide near London.

Increasing tides: raising the banks

James Walker, Scottish engineer who laid down the general lines of the Embankment

Progressively higher tides were recorded in the Thames, mainly because the land is sinking relative to the sea. They started to cause serious flooding from c. 1300 onwards and therefore the artificial banks had to be raised further to match. The process was gradual, but cumulative over the centuries, so that later observers described the embankments as "mighty" and "stupendous". It was important to keep embankments in good working order since a breach of the river wall could flood a neighbourhood. Therefore the English kings took to appointing land drainage authorities, called commissions of sewers, with power to compel owners to keep up their banks. The first of these was convened in 1280: at West Ham. Serious breaches did occur - at Bermondsey, Southwark, Greenwich, Woolwich, Plumstead, Barking and Stepney, for example. By the early Victorian era, however, a high standard of integrity had been achieved and significant breaches were rare.

Surge tides

Despite this, there were occasional freak (or surge) tides, so large that, though not breaching the embankments, they went over their tops. Therefore James Walker recommended a practice of making walls to a 'safe' height 4 feet above the highest normal tides. Walker's 4 foot standard was widely accepted; Bazalgette said he adopted it himself without question as the safe line for the Victoria and Albert Embankments. Later, Bazalgette realised it was too low. For the Chelsea Embankment he increased it to 5. In the long run even this was not enough, for the trend continued. In the 1928 Thames flood so much water overflowed the Embankment that a number of Londoners were drowned in their basements

Since then the Thames walls have been raised still further, in central London most recently in 1973; and since its inauguration in 1982 the Thames Barrier, at first rarely closed, has had to be shut with increasing frequency.

Professionally designed embankments

Embankment at Greenwich Hospital (Canaletto, 1752, Royal Museums Greenwich)

A few embankments, professionally designed by architects or engineers, had been built for public purposes. One of the first was a 40-foot wide embankment road from the Tower to the Temple; it lasted for 100 years before it was choked off by illegal encroachments. It was proposed by Sir Christopher Wren as part of his plans for rebuilding London after the Great Fire. Robert Mylne, when erecting Blackfriars Bridge, had built a half-mile embankment (1767); it did not include a road, but by straightening out the river it prevented the accumulation of "extremely offensive" mud that had blocked access to the wharves.

Robert Smirke had built a 1,500 foot embankment at Millbank, then one of London's few Thames-side roads; he pioneered the use of concrete foundations. Greenwich Hospital had a 1,000 foot embankment. The Houses of Parliament had their own 1,000 embankment: by James Walker. The London dock companies had built several miles of embankments. A scheme to embank the north shore between Vauxhall and Battersea Bridges was only partly completed for lack of funds, but the embankment and roadway - now the Grosvenor Road - were formed as far as Chelsea Hospital Gardens: .

Further information: Embanking of the tidal Thames

Defects of existing embankation

Although the vernacular embankments of the Thames impressed Victorian engineers by their sheer scale, they had been built by local people who were not working to any general scheme or plan. They had several defects.

Irregularity

The river Thames in central London c. 1840: irregular embankments, few riverside roads. The river itself is the main highway - hence numerous watermen's stairs. Notice enormous mudbank (top left) accumulating near Waterloo Bridge. (G.F. Cruchley, 1839: David Rumsey Historical Map Collection.)

Dilapidated river wall at Lambeth c. 1860, subsumed into Albert Embankment. Notice backyard lavatories discharging onto foreshore. (Historic England.)

The banks of the Thames in London were irregular and uneven (see map). Some parts of the river were twice as wide as nearby parts. In the wide parts, mud tended to accumulate, interfering with the navigation. Partly composed of matter discharged from London's sewers, it had a red colour, was "in a state of constant fermentation", and considered a health hazard. The map reveals a 27-acre mudbank accumulating in the vicinity of Waterloo and Blackfriars Bridges: in parts it exceeds half the river's width.

Therefore in 1840 James Walker prescribed general lines to which all future embankation works ought to conform. They were followed by Bazalgette in designing the Thames Embankment.

Lack of riverside roads

Although in theory the banks of the Thames were public highways, in practice the law was ignored and the banks were treated as valuable private property. Thus with few exceptions there were no public roads alongside the river in central London.

It meant that the main public highway to the north of the river was the Strand, which became highly congested. There was nowhere to build an intercepting sewer without causing major traffic disruption.

Local exceptionalism

The main incentive for making good embankments and keeping them in repair was flood prevention. But in some locations it was not critical.

Some premises (e.g. boatbuilders' yards) did not want an embankment at all: they preferred to launch their boats on a sloping shore.

In other locations the embankment might be little more than a brick wall, perhaps shoddily maintained (illustration). Generally, the embankments were made of clay, and the wash of steamboats tended to start breaks; some owners did not do enough to defend them.

Innovations and achievements of Bazalgette embankments

The new embankments provided more than 3 miles of spacious public roadways alongside the riverfront in central London, sometimes with ornamental gardens. Their river walls, faced with granite backed with Portland cement concrete, were resistant to steamboat wash. They subsumed some of the existing embankments, straightening out numerous irregularities. A matter of civic pride, they transformed an unsightly commercial riverfront into "an architectural monument that still impresses visitors from around the world". Of more pressing importance, the Victoria Embankment provided an opportunity to construct a much-needed intercepting sewer to deal with a growing sanitary crisis. As a bonus it enclosed the Metropolitan District Railway, London's second underground line.

Chelsea embankment before and after Bazalgette

1.

2.

1. View of "The Adam & Eve" inn amongst terraced houses on the embankment (Walter Greaves, etching with drypoint, British Museum)
2. Walter Greaves and Alice Greaves on the Embankment (Walter Greaves, oil on canvas, Tate Britain)

Previously when Thames foreshore had been reclaimed by embankation, the newly recovered land had become the private property of the builders. The Victoria Embankment had recovered 37 acres and there had been numerous proposals to pay for the scheme by privatising the land. However "it had been left to a public body, having funds, to expend those funds for the public good", said Sir James Bazalgette, which he thought a matter for congratulation.

The Victoria, Albert and Chelsea Embankments require to be considered separately since these three works were built neither for a uniform purpose nor to a uniform design, as Bazalgette himself pointed out. For example the Albert Embankment contained no sewer and was not very effective for preventing floods, not having been intended for that purpose.

Sewage disposal

The Victoria Embankment in section, showing 1. Subway for service pipes 2. Interceptor sewer 3. Metropolitan District Railway 4. Pneumatic Railway (abandoned)

The most pressing problem was sewage disposal, which started to become acute around the middle of the nineteenth century, partly owing to a mistaken government policy.

London's sewers had always discharged into the Thames but they were meant for surface runoff water only. Human excreta had to be collected in cesspits, which were periodically cleaned out by gangs of men who sold the product for agricultural fertiliser; it was illegal to dump it in the sewers. (The word sewage with an unpleasant sanitary connotation is not attested before 1834.) Thus the river Thames, though flowing past the world's largest city, was reasonably clean, and supported a substantial fishing industry.

The cesspits stank, however; there were outbreaks of cholera; and a mistaken medical theory held the disease was spread by bad air. Householders, impelled by the smell and believing the medical theory, increasingly fitted flush toilets. The sewer companies were willing to tolerate it since they thought the flushing action helped to keep their lines clear. Hence the practice changed and after 1815 it became first permissible, then legally mandatory for public health reasons, to discharge organic waste into the London sewers. Since these had always outfallen into the Thames it came to be that 600,000 tons of raw sewage were dumped in the river daily.

Bazalgette's Metropolitan Board of Works proposed a scheme of intercepting sewers which would discharge effluent further down the river. There was no major engineering problem south of the river

Sources

• Bazalgette, Edward (1878). "The Victoria, Albert and Chelsea Embankments of the River Thames". Minutes of Proceedings of the Institution of Civil Engineers. LIV (IV): 1–60. Retrieved 27 Oct 2025.

• Bazalgette, Joseph (1878a). "The Victoria, Albert and Chelsea Embankments of the River Thames (discussion)". Minutes of Proceedings of the Institution of Civil Engineers. LIV (IV): 30. Retrieved 27 Oct 2025.

• Bowen, Anthony J.; Pinless, Sally J. (1974). "Effects of bank raising along the Thames". Coastal Engineering. ASCE Press. pp. 2471–2482. doi:10.1061/9780872621138.148. Retrieved 20 November 2025.

• Dale, Hylton B. (1922). "The Worshipful Company of the Woodmongers and the Coal Trade of London". Journal of the Royal Society of Arts. 70 (3648): 816–823. JSTOR 41355975.

• Galloway, James A.; Potts, Jonathan S. (2007). "Marine Flooding in the Thames Estuary and Tidal River C. 1250-1450: Impact and Response". Area. 39 (3): 370–379. JSTOR 40346052.

• Horner, R.W. (1979). "The Thames Barrier Project". The Geographical Journal. 145 (2): 242–253. JSTOR 634390.

• Lavery, Sarah; Donovan, Bill (2005). "Flood Risk Management in the Thames Estuary Looking Ahead 100 Years". Philosophical Transactions: Mathematical, Physical and Engineering Sciences. 363 (1831, The Big Flood: North Sea storm surge). Royal Society: 1455–1474. JSTOR 30039664.

• Porter, Dale H. (1998). The Thames Embankment: Environment, Technology and Society in Victorian London. Akron, Ohio: University of Akron Press. ISBN 1-884836-29-1.

• Redman, John Baldry (1878). "The Victoria, Albert and Chelsea Embankments of the River Thames (discussion)". Minutes of Proceedings of the Institution of Civil Engineers. LIV (IV): 33–38. Retrieved 27 October 2025.

• Spurrell, F.C.J. (1885). "Early Sites and Embankments on the Margins of the Thames Estuary" (PDF). The Archaeological Journal: 269–303 + 2 maps. Retrieved 10 November 2025.

• Walker, James (1845). "No. 276: Report of James Walker". Royal Commission on Tidal Harbours: Appendix to Second Report. pp. 542–549. Retrieved 4 November 2025.

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Southwark

n. The London Bridge area around AD 50. "Southwark was a low-lying expanse of sandy riverine islands, or eyots, interspersed with braided channels; a shifting landscape with extensive mudflats inundated at high tide". Modern features located for comparison:

Turn performance

Although the Hawker Hurricane did not have the speed or climb rate of the Supermarine Spitfires or the Messerschmidt Bf 109, its lower wing loading meant that it was able to turn in a tighter circle, an important advantage in some phases of fighter combat. It has sometimes been claimed the Bf 109 could turn inside the Hurricane, a matter considered below.

Significance

Fighter aircraft have been classified into turn performance and energy performance types; the latter have higher speeds and can climb faster; but the former have better instantaneous turn, and a tighter sustained turn radius.

From the early 1930s onwards, in what aeronautical engineer Georg Hans Madelung called the "speed craze", fighter designers prioritised maximum speeds and climb rates. Not until much later was there a general requirement for a better balance of performance, reducing wing loads to get extra agility in the turn.

In the Battle of Britain the Hurricanes were assigned to a defensive role i.e. they were encouraged to seek out enemy bombers not fighters. The bombers were escorted by fighters, often Messerschmidt Bf 109s or Bf 110s, whose task, therefore, was to attack any incoming Hurricanes. A standard fighter manouevre, taught to most fighter pilots, was to attack from astern, preferably by ambush from above. If the attacker was not seen coming in and could get close enough the result was probably a victory. If, however, he was seen in time, the basic evasive manouevre was the break, in which the defending aircraft turned hard in the direction of the attacker, if possible getting onto his tail.

Wrote Pilot Officer Geoffrey Page:

In the Hurricane we knew that the Me 109 could out-dive us, but not out-turn us. With that knowledge, one obviously used the turning manouevre rather than trying to beat the man at the game in which he was clearly superior. With a 109 sitting behind you, you'd stay in a really tight turn, and after a few turns the position would be reversed and you'd be on his tail.

Turning radius of Hurricane and other Battle of Britain fighters

An aircraft's minimum turning radius cannot be quoted as a single number since it varies with altitude and loading. Accurate historical data are sparse because they were not published at the time, presumably for security reasons, or later for lack of archival research.

In May 1940 a Messerschmidt Bf 109E3 captured by French forces was tested against a Hurricane, both flown by British pilots. The pilots reported that the Messerschmidt was faster than the Hurricane, and could out-climb and out-dive it, but could not turn as well. In climbing and diving turns at high speed, although the Bf 109 was placed line astern to start with, it could not keep the Hurricane in its gunsights and the latter was able to get onto its tail in only four turns. However, these trials were restricted to below 15,000 feet owing to lack of oxygen equipment.

Likewise, a RAF Fighter Command report stated that "the Hurricane will easily out-turn the Spitfire in a simple tailchase, and bring guns to bear in two or three turns".

However in 1967, in The Hawker Hurricane I (Profile Publications, No. 111, an 18-page booklet), aviation author Francis K. Mason wrote:

Much has been written of the relative manouevrability of the four principal Battle of Britain fighters. Published here for the first time are comparative turning radii attainable, resolved to a combat altitude of 10,000 feet—the most common area of combat chosen by the Hurricane pilots in 1940 ... if and when the choice was theirs. True airspeed, 300 m.p.h.

	CL	1⁄2 ρ v2	Wing loading at half-fuel weight	"Available g"	Turning Radius
Hurricane I	1.0	170	22 lb./sq. ft.	7.5	800 feet
Spitfire I	1.0	170	24 lb./sq. ft.	7.0	880 feet
Bf 109E	1.0	170	25 lb./sq. ft.	8.1	750 feet
Bf 110C	1.0	170	32 lb./sq. ft.	5.2	1,210 feet

Mason did not state his sources. His numbers were repeated in Len Deighton's 1977 book Fighter, achieving wider publicity. They were criticised by Ackroyd and Lamont of the Aerospace department of Manchester University, who said Mason had arrived at them by calculation, which calculations were flawed.

According to them, to arrive at those values Mason had assumed the aircraft were flown at 300 m.p.h. while in a sustained angle of bank so steep - about 80° - that the g forces would have been "beyond the limits of any pilot at that time and most probably beyond the structural limits of the aircraft". Further, none of the aircraft engines could have delivered the required thrust, since far more power is required to overcome aircraft drag in a very steep turn.

Recalculating for speeds actually achievable with those engines, and assuming the aircraft were flown at maximum power as near to the stall as possible, again at 10,000 feet, Ackroyd and Lamont arrived at the following among other parameter values:

	n (≅"available g")	Angle of bank	Turning radius
Hurricane I (Late)	2.80	69.1°	202 m (663 ft)
Spitfire IA	2.74	68.6°	209 m (686 ft)
Messerschmidt Bf109E-3	2.67	68.0°	260 m (853 ft)
Messerschmidt Bf 110C-4	2.64	67.8°	256 m (840) ft)

They accepted those were still only estimates, but believed them to provide a reasonable assessment. They tended to be confirmed by a contemporaneous flight report on a captured Bf109 E-3.

Sources

• Ackroyd, J.A.D.; Lamont, P.J. (2000). "A comparison of turning radii for four Battle of Britain fighter aircraft". The Aeronautical Journal. 104 (1032): 53–8. doi:10.1017/S0001924000017784.

• Ackroyd, J.A.D (2016). "The aerodynamics of the Spitfire". The Journal of Aeronautical History. 6: 59–86. Retrieved 20 August 2024.

• Caygill, Peter (2005). Flying to the Limit: Testing World War II Single-Engined Fighter Aircraft. Pen and Sword Books. ISBN 9781783409358.

• Deighton, Len (1977). Fighter: The True Story of the Battle of Britain. London: Jonathan Cape. ISBN 0-224-01422-6.

• Delve, Ken (2024). The Story of the Spitfire: An Operational and Combat History. Yorkshire and Philadelphia: Air World. ISBN 978-1-03615--004-4.

• Dibbs, John; Holmes, Tony; Riley, Gordon (2017). Hurricane: Hawker's Fighting Legend. Bloomsbury. ISBN 9781472822963.

• Madelung, G.O. (1978). "Characteristics of Fighter Aircraft". Journal of Aircraft. 15 (3): 129–133. doi:10.2514/3.58329.

• Mason, Francis K. (n.d.). The Hawker Hurricane I. Vol. 111. Leatherhead, Surrey: Profile Publications.{{cite book}}: CS1 maint: year (link)

• Rigby, David J. (1990). "Hawker's Wonderful Hurricane". Air Power History. 37 (2): 26–28. JSTOR 26271114.

• Shaw, Robert L. (1987). Fighter Combat: Tactics and Maneuvering (PDF). Annapolis, Maryland: Naval Institute Press. ISBN 0-87021-059-9. Retrieved 25 August 2025.

• Spick, Mike (2021). Luftwaffe Fighter Aces: The Exploits and Tactics of Germany's Greatest Pilots. New York: Frontline Books. ISBN 978-1-5107-5436-2.

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A Royal Commission was appointed to inquire into the system under which sewage was discharged into the Thames by the MBW, and whether any evil effects resulted therefrom. The Commission reported (February 1884) that although the sewerage works were highly creditable and had been of great benefit to the metropolis, there were a number of problems. Sewage from the northern outfall was being partly discharged over the foreshore and not through submerged pipes as originally intended. Crude sewage was being discharged throughout the year without attempting to separate the solids. The tide was carrying effluent a long way up from the outfalls, almost as high as Teddington, oscillating back and forth for a long period before getting out finally to sea. In hot and dry weather there was a serious nuisance, the smell being very offensive. For 15 miles below the outfalls, fish had disappeared.

Royal Commission (1884). "Metropolitan Sewage Discharge". The British Medical Journal. 1 (1207): 333–4. JSTOR 25265644.

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In the followinf explanation no scientific knowledge is assumed, beyond that heating a gas will make it expand (as in a hot air balloon); cooling it will make it contract; compressing it will generate heat (as in a bicycle pump); expanding it will absorb heat.

When the piston rises it can be imagine to lift a weight i.e. to perform work. This work is available for use by falling in the opposite direction.

Dod Street and socialist politics

Dod Street, on the corner by Burdett Road Bridge and whose factories overlooked the Cut, was built and named by 1861. It first attracted public attention in the smallpox hospital controversy (above) and then as a site for Sunday political meetings. Socialists like John Burns, Amie Hicks, Henry Hyndman Eleanor Marx, William Morris and George Bernard Shaw, among others, were speakers there.

Socialist meetings were held at Dod Street. Speakers included (left to right) John Burns, Eleanor Marx, George Bernard Shaw and William Morris

Dod Street gave rise to the expression "the Dod Street trick" used in socialist politics. The police felt these meetings were subversive and sought to prevent them by arresting demonstrators for highway obstruction. Since there was no traffic to obstruct — it was a street of canal-side factories on a Sunday — the police were perceived as denying freedom of speech.

The Dod Street trick, devised to counter this, was thus described by Bernard Shaw:

Find a dozen... who are willing to get arrested at the rate of one a week by speaking in defiance of the police. In a month or two, the repeated arrests, the crowds which they attract, the scenes which they provoke, the sentences passed by the magistrates... and the consequent newspaper descriptions, rouse sufficient public feeling to force the Home Secretary to give way whenever the police are clearly in the wrong,

which is what happened. Public indignation gathered enormous crowds of people — only a very few of which were socialists — and they were let alone.

Paraguayan government publications

• "Una ocurrencia desagradable". Semanario de Avisos y Conocimientos Útiles (in Spanish). No. 56. Asunción. 5 August 1854. Retrieved 12 June 2026.

• "Documentos oficiales". Semanario de Avisos y Conocimientos Útiles (in Spanish). No. 62. Asunción. 16 September 1854. Retrieved 13 June 2026.

• "El Water Witch Americano". Semanario de Avisos y Conocimientos Útiles (in Spanish). No. 82. Asunción. 3 February 1855. Retrieved 13 June 2026.

Driving the piston

• Step 1. Connect the hot reservoir to the fluid; let the heat expand it and drive the piston. Since the expansion will counteract the fluid's tendency to get hot, it can be adjusted for no change of temperature at all. Called "isothermal expansion", it will do useful work yet not waste heat pointlessly raising the fluid's temperature. When this will go no further:
• Step 2. Let the fluid keep on expanding, but cut off the heat supply. (This is called "adiabatic expansion".) Expansion will make the fluid cool down, and we will gain yet more work.

Retracting the piston

The engine has now delivered all its work; the fluid has reached its coolest temperature. Next the piston must pushed back, ready for the next cycle. This is going to consume some work, but not as much as we have already gained.

• Step 3. Squeeze the piston to compress the fluid. This will generate some heat; so apply the cool reservoir to absorb it without raising the fluid's temperature. (This is called '"isothermal compression".) When this will go no further:
• Step 4. Continue compressing, but disconnect the cool reservoir. Since the heat has nowhere else to go it will restore the fluid to its original high temperature. (This step is called "adiabatic compression".)

The reason more work is gained during expansion (Steps 1 and 2) than is consumed during compression (Steps 3 and 4) is because the fluid is hotter. It requires less effort to re-compress the same quantity of fluid when cooler.}}

5. Nothing can be more efficient than a Carnot cycle

Lastly, he proved that no more efficient cycle can possibly be devised. For, if it were possible to to have an engine even better than a Carnot engine, the better engine could drive the Carnot backwards, getting perpetual motion, which is absurd.

What Carnot demonstrated

Almost by pure reasoning, summarised later below, Carnot established the following propositions.

1. There is an ideal engine whose efficiency cannot be improved. As an exercise, Carnot devised the ultimately efficient engine and proved that no engine can be more efficient, even in theory. Though a mathematical ideal, it shows there is indeed a fundamental limit to engine efficiency. Today it is called the Carnot engine in his honor.

2. To be ideal it must be fully reversible. He also proved that, for any engine to be ideal, it must be what is termed reversible i.e. if driven backwards it will behave as the ideal refrigerator and exactly restore the initial conditions. Carnot showed that a reversible engine, working between two given temperatures, and receiving at the higher temperature a given quantity of heat, performs at least as much work as any other engine whatever working under the same conditions. It follows from this that all reversible engines, whatever be the working fluid employed, have the same efficiency under the same conditions.

3. There is no magic working substance. As a consequence, there was little to be gained by experimenting with exotic substances for none was intrinsically better. In practice air was the only promising substitute for steam. "Air could be heated directly by combustion carried on within its own mass", wrote Carnot; in other words, the internal combustion engine.

4. Efficiency depends on the temperature gap.

Crucially, his work showed that no engine, not even his ideal engine, can convert all of the heat into mechanical work. Some heat "escapes" without doing any work at all. This cannot be because of defective insulation for there are no imperfections in this engine. It suggests that some fundamental law of nature forbids us to use all of the heat. There can be such a thing as unavailable heat.

He showed that the amount of heat that is inevitably lost - even in the ideal engine - is defined by the temperatures of its coolest and hottest parts, and nothing else. Since it is not practical to get these very far apart, it turns out to be a major limitation.

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Carnot engine: intuitive explanation

Significance and importance

Carnot's engine is of fundamental importance in the history and reasoning of science. According to Nobel laureate Richard Feynmann, "The science of thermodynamics began with an analysis, by the great engineer Sadi Carnot, of the problem of how to build the best and most efficient engine". It has been said that "Not knowing the Second Law of Thermodynamics is equivalent to never having read a work by Shakespeare”.

Carnot's Reflections on the Motive Power of Heat (1824) is a short book addressed to practical engineers; it is written in popular language. Yet, as taught in physics courses today, many students have trouble intuiting his ideas. The following is a non-technical explanation.

In Carnot's day, engines were fairly typically employed for pumping water out of mines. Customers had a practical question: how much the fuel was going to cost them. The "most efficient" engine was the one that, burning a standard quantity of fuel, could hoist a given load to the greatest height.

Carnot's concept

Heat engines

"Every one knows that heat can produce motion", began Carnot. There are many kinds of heat engines but in his day most were steam engines. Indispensable to the Industrial Revolution, they had been greatly improved by British engineers, said Carnot, without really understanding the theory of what they were doing. Other working fluids had been tried instead of steam, for instance, air. Carnot pointed out that anything that exerts a force when heated and cooled might be used, even a solid metallic bar.

Regardless of such secondary details, his question was: Can heat engines be improved indefinitely? Or shall we run up against a fundamental limit beyond which it is impossible to go?

His achievements summarised

As an exercise, Carnot devised the ideal engine i.e. it can be proved that no engine can be more efficient, even in theory. Today it is called the Carnot engine in his honor.

Crucially, he proved that even his ideal engine cannot convert all of the heat into mechanical work. Some heat escapes, without doing any. This cannot be because of imperfections in the engine's construction, for there are none. It suggests that some fundamental law of nature is at work which forbids us to use all of the heat. There can be such a thing as unavailable heat.

Carnot also showed that, for an engine to be ideal, it must, if run backwards, behave as the ideal refrigerator.

He also proved that the amount of heat that is inevitably lost, even in the ideal engine, is defined by the temperatures of its coolest and hottest working parts, and nothing else. It turns out to be a major limitation indeed. (The best automobile engines are only about 40% efficient.)

Carnot reasoned as follows.

1. Heat, without cold, cannot generate motive power

Carnot showed, first, that heat by itself can never produce motion: it must also have a cooler place to go to. The common steam engine had a hot place (the furnace) and a cool place (the condenser); but he showed it must be true for all heat engines that could possibly be devised. He did it by imagining an engine with no cool place at all i.e. engine and surroundings are uniformly hot. Then the working fluid (steam, or whatever) can never cool down; the piston cannot retract; such an engine can deliver no power. "It is necessary that there should also be cold; without it, the heat would be useless".

Carnot supplied an analogy: to get power from a water-wheel, water, by itself, is useless: it must be able to `'fall`' to a lower place.

That heat engines derive their power by exploiting the difference in temperature between a hot place and a cool place was not so obvious. The insight was afterwards used to formulate the Second Law of Thermodynamics.

2. A heat engine can be run in reverse and will behave as a refrigerator

Next, Carnot reasoned that - like the water-wheel - the heat engine can be run backwards. Instead of exploiting the "fall" to get useful mechanical effort, we could expend the identical effort to drive the fluid "upwards".

Specifically, by forcing the engine backwards, we can make heat go from the cool place to the hot place, contrary to what naturally happens. The cool place will be made even cooler (as in a refrigerator) and the hot place will be made hotter. In effect, Carnot had invented the heat pump.

(This insight - that to convey heat from a cool to a warm place requires the expenditure of mechanical effort, lies at the heart of another way of stating the Second Law of Thermodynamics.)

3. The ideal engine should be fully reversible and restore the initial conditions

Carnot's next insight was to appreciate that, for the engine to be truly ideal - i.e. impossible to improve even in theory - it must be capable of running in both directions with equal facility. As a motor, it will transfer a certain amount of heat while lifting a weight a certain distance. Run backwards as a refrigerator, it will exactly restore the status quo.

4. The ideal engine's efficiency depends on just two parameters, and nothing else

Lastly, Carnot devised an engine cycle which would minimise heat waste. Furthermore, he proved that no better engine cycle for the purpose can be devised.

The details may not matter to the non-scientific reader. For completeness they are as follows.

According to well known laws of physics, tranferring heat into a body of gas will partly raise its temperature, and partly cause it to expand. (The heat energy is, so to speak, partitioned between those two destinies.) However, we can arrange for extreme cases. In one of these, the body of gas will be constrained, as in a bottle;; it cannot expand; so all of the heat energy will be devoted to increasing its temperature. That is exactly what we do not want. We want the opposite extreme. We want the gas to be allowed to expand, driving the piston - giving us useful work - while holding its temperature constant. The technical name for this is isothermal expansion.

the gas will expand, pushing he piston, but the temperature will be absolutely steady. This is the thing to aim for,

unless it is forcibly prevented from doing so, in which case its temperature will be raised instead. The first alternative can give us useful mechanical work; the second does nothing for us and just wastes some of the heat. Therefore, the right design principle should be: never allow the temperature of the working fluid in the cylinder to change, unless it is caused by a change of volume (expansion or compression).

Carnot devised a four-step cycle according to that principle.

The details of the ideal cycle are shown in the diagram elsewhere in this article. For the non-technical reader they matter least. There are four steps.

• Step 1. Convert heat entirely into work, keep the temperature steady. (Called "isothermal expansion").
• Step 2.

Feed heat from the hot reservoir to the fluid in the cylinder; it will make the fluid expand against the piston. But keep the temperature steady. (This is called isothermal expansion. All of the heat is turned into useful work; none is wasted pointlessly heating up the fluid.)

• Step 2. Stop transferring heat and allow the fluid to go on expanding against the piston from the heat it already contains (this is called adiabatic expansion). It will make the fluid cool down.
• Step 3. Mechanically move the piston to compress the fluid. This will generate some heat. Allow this to escape to the cool reservoir without raising the fluid's temperature. (This step, called isothermal compression, will cost us some mechanical work, but not as much as we gained by steps 1 and 2.)
• Step 4. Disconnect the cool reservoir and continue to compress the fluid with the piston. This will restore its original high temperature.

An engine that works on this cycle is today called a Carnot engine. A simple thought experiment showed that no engine can be more efficient than this. For, if one existed, it could drive the Carnot engine backwards, and

Sources

Kerker 1960

Interpretations

Recruitment tool

Jovita Feitosa turned up an opportune moment for the war effort, since recruitment was faltering. It has been argued that she was made a tool of recruitment propaganda and used to manipulate public opinion (though she herself had not intended this). Here was a country girl from a remote part of the Empire who had disguised herself as a man in order to enlist: an example to encourage timorous male volunteers and shame draft-dodgers. That was why she was fêted and accompanied by newspaper reporters everywhere she went - her tour through the northern provinces has been described as a "veritable circus" - and granted an audience by the Emperor Pedro II himself.

She began to divide public opinion, however; there were many discordant voices, and some even questioned her motives for joining, saying she did it to follow a lover. It was asked how she had been made a sergeant, quite a difficult promotion for male soldiers and one never granted to raw recruits. There was anyway a proper role for women in war but it did not include combat. The author and soldier Alfredo D'Escragnolle Taunay wrote that "she should have remembered that for a woman it is more noble to heal wounds than to open them". On 16 September 1865 the war department ruled that to allow her to be a combatant was contrary to military regulations, though it did not forbid her to go to the theatre of war in some other role, e.g. nursing, which was a longstanding Brazilian tradition. (According to historian Francisco Doratioto, she did become a nurse; some sources say she did go to Paraguay in that role, between August and December 1865.) Upon ceasing to be useful to the authorities she was allowed to drop out of the news; hence her subsequent history is obscure. Returning to her home province, she was not well received by her father. Her suicide was reported in the local newspaper on 16 November 1867. She received a pauper's funeral. Some said she returned to Rio de Janeiro, where she was abandoned by her lover, an English engineer; others, that she died in a house fire.

Archetype

The figure of Jovita can also be seen as an archetype that recurs in Western culture, namely the warrior-maiden; she has been called the Brazilian Joan of Arc.

Main article: Women warriors in literature and culture

Sources

Carvalho, José Murilo de (2022). Jovita Alves Feitosa: Voluntária da pátria, Voluntaria da morte (in Portuguese). São Paulo: Chão. ISBN 978-6580341009.

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Oppose. The current title has been in place since 2005. Now it is claimed that War of the Triple Alliance is "the widely used term" in English — that Paraguayan War is hardly used. This is simply wrong. Paraguayan War is overwhelmingly the preferred usage in the English language, certainly in serious scholarly writing.

The JSTOR library is a database of nearly all recent high-quality scholarly articles in the English language. The facts speak for themselves:

Articles in JSTOR database that mention the quoted phrase:

	"paraguayan war"	"war of the triple alliance"	"triple alliance war"
in the article's title	59 (32)	7 (6)	2 (2)
at least once in body	1,197 (831)	702 (429)	180 (126)

(Source: JSTOR, interrogation of search engine provided, 17 May 2026. Figures in brackets are articles that can be accessed now with a JSTOR subscription; bold figures may need another subscription to relevant journal.)

The numbers in the second column greatly understate the case, however; for nearly always the title supplies the missing context: it explicitly goes on to tell the reader the article is about Paraguay. For instance, "Draft Dodgers, War Resisters and Turbulent Gauchos: The War of the Triple Alliance against Paraguay"; "The Paraguayan Image of the War of the Triple Alliance". I have found only one exception ("The War of the Triple Alliance: Three Explanatory Models" by Diego Abente) but that article was published in the Latin American Research Review where the context was obvious. As an international encyclopaedia we lack that context; Wikipedia is widely read on five other continents where the war may be unknown.

The Google Scholar database (though it contains much junk as well as sound scholarship) paints a broadly similar picture:

Google Scholar hits

"paraguayan war"	"war of the triple alliance"	"triple alliance war"
3,880	2,610	814

Likewise, there are clearly more books with "Paraguayan War" in the title, than "War of the Triple Alliance. (Source: Google Books, interrogate intitle:field.)

It stands to reason: there have been several triple alliances in human history; indeed most were European, and one was North American. The usage "war of triple alliance" without specified context is narrow and parochial.

"War of the Triple Alliance" is not accurate anyway. The War actually began in 1864 when Paraguay declared war on Brazil; there was no triple alliance until Paraguay declared war on Argentina the next year, driving these traditional enemies into one another's arms.

Editors should perhaps be aware that seemingly innocuous disputes about the war's proper title can carry a lot of local ideological baggage. Even after 150 years this war is hotly debated. This is no present concern of Wikipedia.

vv

According to Jeffers, the fort then fired two blank shots, followed by a live round. The live shot killed the helmsman, Samuel Chaney. Water Witch retaliated with her three howitzers, but apparently did little damage. His way ahead uncertain because of a risk of grounding, Jeffers reversed course and ran the gauntlet again. The fort hulled Water Witch ten times, destroyed two boats and damaged a paddle wheel. Water Witch limped into an Argentine port.