• Engineer Nicolas Léonard Sadi Carnot, 1796
• Physicist and Jurist William Robert Grove, 1811
• Physicist and Engineer Ludwig August Colding, 1815

Sadi Carnot was born in Paris, and was the first son of the eminent military leader and geometer, Lazare Nicholas Marguerite Carnot. Lazare was the elder brother of Hippolyte Carnot, and uncle of Marie François Sadi Carnot (President of the French Republic 1887 - 1894). His father named him after the Persian poet Sadi of Shiraz, and he was always known by this third given name.

From the age of 16, he lived in Paris and attended the École polytechnique where he and his contemporaries, Claude - Louis Navier and Gaspard - Gustave Coriolis, were taught by such notable professors as Joseph Louis Gay - Lussac, Siméon Denis Poisson and André - Marie Ampère. After graduation he became an officer in the French army before committing himself to scientific research; Carnot's research went to make him one of the most celebrated of Fourier's contemporaries who were interested in the theory of heat. He served in the military until 1814; following the defeat of Napoleon in 1815, his father went into exile, and he later obtained permanent leave of absence from the French army. It was during his military leave that he spent time to write his book Reflections on the Motive Power of Fire.

When Carnot began working on his book the use of steam engines was relatively developed; notwithstanding this, there had been no real scientific study concerning steam engines. Steam engines had, however, risen to a widely recognized economic and industrial importance. Newcomen had invented the first piston - operated steam engine over a century before, in 1712; some 50 years after that, James Watt made his celebrated improvements which were responsible for greatly increasing the efficiency and practicality of engines. Compound engines (engines with more than one stage of expansion) had already been invented, and there was even a crude form of internal combustion engine, which Carnot was familiar with and which he described in some detail in his book. Significant progress had been made concerning engines, so there existed at the time some intuitive understanding of the workings of engines. Despite this, the scientific basis of their operation was almost nonexistent. In 1824 the principle of conservation of energy was still poorly developed and controversial, and an exact formulation of the first law of thermodynamics was still more than a decade away; what's more is that the mechanical equivalent of heat had not been identified and would remain unknown for another two decades. The prevalent theory of heat at the time was the caloric theory, which regarded heat as a sort of weightless and invisible fluid that flowed when out of equilibrium.

Engineers in Carnot's time had tried, by means such as highly pressurized steam and the use of fluids, to improve the efficiency of engines. In these early stages of engine development, the efficiency of a typical engine — the useful work it was able to do when a given quantity of fuel was burnt — was a mere 3%.

Carnot sought to answer two questions about the operation of heat engines: "Is the work available from a heat source potentially unbounded?" and "Can heat engines in principle be improved by replacing the steam with some other working fluid or gas?" He attempted to answer these in a memoir, published as a popular work in 1824 when he was only 28 years old. It was entitled Réflexions sur la Puissance Motrice du Feu ("Reflections on the Motive Power of Fire"). The book was plainly intended to cover a rather wide range of topics about heat engines in a rather popular fashion; equations were kept to a minimum and called for little more than simple algebra and arithmetic, except occasionally in the footnotes, where he indulged in a few arguments involving some calculus. He discussed the relative merits of air and steam as working fluids, the merits of various aspects of steam engine design, and even included some ideas of his own regarding possible improvements of the practical nature. The most important part of the book was devoted to an abstract presentation of an idealized engine that could be used to understand and clarify the fundamental principles that are generally applied to all heat engines, independent of their design.

Perhaps the most important contribution Carnot made to thermodynamics was his abstraction of the essential features of the steam engine, as they were known in his day, into a more general and idealized heat engine. This resulted in a model thermodynamic system upon which exact calculations could be made, and avoided the complications introduced by many of the crude features of the contemporary steam engine. By idealizing the engine, he could arrive at clear and indisputable answers to his original two questions.

He showed that the efficiency of this idealized engine is a function only of the two temperatures of the reservoirs between which it operates. He did not, however, give the exact form of the function, which was later shown to be ^{(T₁−T₂)}⁄_T1, where T₁ is the absolute temperature of the hotter reservoir. (Note: This equation probably came from Kelvin.) No thermal engine operating any other cycle can be more efficient, given the same operating temperatures.

The Carnot cycle is the most efficient possible engine, not only because of the (trivial) absence of friction and other incidental wasteful processes; the main reason is that it assumes no conduction of heat between parts of the engine at different temperatures. Carnot knew that the conduction of heat between bodies at different temperatures is a wasteful and irreversible process, which must be eliminated if the heat engine is to achieve maximum efficiency.

Regarding the second point, he also was quite certain that the maximum efficiency attainable did not depend upon the exact nature of the working fluid. He stated this for emphasis as a general proposition: "The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which the transfer of caloric takes place." For his "motive power of heat", we would today say "the efficiency of a reversible heat engine," and rather than "transfer of caloric" we would say "the reversible transfer of heat." He knew intuitively that his engine would have the maximum efficiency, but was unable to state what that efficiency would be.

He concluded:

The production of motive power is therefore due in steam engines not to actual consumption of caloric but to its transportation from a warm body to a cold body.

—Carnot

and

In the fall of caloric, motive power evidently increases with the difference of temperature between the warm and cold bodies, but we do not know whether it is proportional to this difference.

—Carnot

In Carnot's idealized model, the caloric heat converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Nevertheless, Carnot further postulated that some caloric is lost and is not converted into mechanical work. Hence, no real heat engine could realize the Carnot cycle's reversibility, and would consequently be less efficient.

Though formulated in terms of caloric, rather than entropy, this was an early rendition of the second law of thermodynamics.

Carnot’s book received very little attention from his contemporaries. The only reference to it within a few years after its publication was in a review in the periodical Revue Encyclopédique, which was a journal that covered a wide range of topics in literature. The impact of the work had only become apparent once it was modernized by Émile Clapeyron in 1834 and then further elaborated upon by Clausius and Kelvin, who together derived from it the concept of entropy and the second law of thermodynamics.

Carnot died during a cholera epidemic in 1832, at the age of only 36. Because of the contagious nature of cholera, many of Carnot's belongings and writings were buried together with him after his death. As consequence, only a handful of his scientific writings survived.

After the publication of Reflections on the Motive Power of Fire, the book quickly went out of print and for some time was very difficult to obtain. Kelvin, for one, had a difficult time in getting a copy of Carnot's book. In 1890 an English translation the book was published by R. H. Thurston; this version has been reprinted in recent decades by Dover and by Peter Smith, most recently by Dover in 2005. Some of Carnot's posthumous manuscripts have also been translated into English.

Carnot published his book in the heyday of steam engines. His theory explained why steam engines using superheated steam were better because of the higher temperature of the consequent hot reservoir. Carnot's theories and efforts did not immediately help improve the efficiency of steam engines; his theories only helped to explain why one existing practice was superior to others. It was only towards the end of the nineteenth century that Carnot's ideas, namely that a heat engine can be made more efficient if the temperature of its hot reservoir is increased, were put into practice. Carnot's book did, however, eventually have a real impact on the design of practical engines. Rudolf Diesel, for example, who was fascinated by Carnot's theories, went on to design the diesel engine, in which the temperature of the hot reservoir is much higher than that of a steam engine, resulting in an engine which is more efficient.

Born in Swansea in south Wales, Grove was the only child of John, a magistrate and deputy lieutenant of Glamorgan, and his wife, Anne née Bevan.

His early education was in the hands of private tutors, before he attended Brasenose College, Oxford, to study classics, though his scientific interests may have been cultivated by mathematician Baden Powell. Otherwise, his taste for science has no clear origin though his circle in Swansea was broadly educated. He graduated in 1832.

In 1835 he was called to the bar by Lincoln's Inn. In the same year, Grove joined the Royal Institution and was a founder of the Swansea Literary and Philosophical Society, an organization with which he maintained close links.

In 1829 at the Royal Institution Grove met Emma Maria Powles, and he married her in 1837. The couple embarked on a tour of the continent for their honeymoon. This sabbatical offered Groves an opportunity to pursue his scientific interests and resulted in his first scientific paper suggesting some novel constructions for electric cells.

In 1839, Grove developed a novel form of electric cell, the Grove cell, which used zinc and platinum electrodes exposed to two acids and separated by a porous ceramic pot. Grove announced the latter development to the Académie des Sciences in Paris in 1839. In 1840 Grove invented the first incandescent electric light, which was later perfected by Thomas Edison. Later that year he gave another account of his development at the British Association for the Advancement of Science meeting in Birmingham, where it aroused the interest of Michael Faraday. On Faraday's invitation Grove presented his discoveries at the prestigious Royal Institution Friday Discourse on 13 March 1840.

Grove's presentation made his reputation, and he was soon proposed for Fellowship of the Royal Society by such distinguished men as William Thomas Brande, William Snow Harris and Charles Wheatstone. Grove also attracted the attention of John Peter Gassiot, a relationship that resulted in Grove's becoming the first professor of experimental philosophy at the London Institution in 1841. Grove's inaugural lecture in 1842 was the first announcement of what Grove called the correlation of physical forces, in modern terms, the conservation of energy.

In 1842, Grove developed the first fuel cell (which he called the gas voltaic battery), which produced electrical energy by combining hydrogen and oxygen, and described it using his correlation theory. In developing the cell and showing that steam could be disassociated into oxygen and hydrogen, and the process reversed, he was the first person to demonstrate the thermal dissociation of molecules into their constituent atoms. The first demonstration of this effect, he gave privately to Faraday, Gassiot and Edward William Brayley, his scientific editor. His work also led him to early insights into the nature of ionization.

In the 1840s Grove also collaborated with Gassiot at the London Institution on photography and the Daguerreotype and calotype processes. Inspired by his legal practice, he presciently observed:

It would be vain to attempt specifically to predict what may be the effect of Photography on future generations. A Process by which the most transient actions are rendered permanent, by which facts write their own annals in a language that can never be obsolete, forming documents which prove themselves, — must interweave itself not only with science but with history and legislature.

— London Institution, 19 January 1842

In 1852 he discovered striae, dark bands that occur in electrical breakdown, and investigated their character, presenting his work in an 1858 Bakerian lecture.

In 1846, Grove published On The Correlation of Physical Forces in which he anticipated the general theory of the conservation of energy that was more famously put forward in Hermann von Helmholtz' Über die Erhaltung der Kraft (On the Conservation of Force) published the following year. His 1846 Bakerian lecture relied heavily on his theory.

James Prescott Joule had been inspired to his investigations into the mechanical equivalent of heat by comparing the mass of coal consumed in a steam engine with the mass of zinc consumed in a Grove battery in performing a common quantity of mechanical work. Grove was certainly familiar with William Thomson's theoretical analysis of Joule's experimental results and Thomson's immature suggestions of conservation of energy. Thomson's public champion, Peter Guthrie Tait was initially a supporter of Grove's ideas but later dismissed them with some coolness.

Though Groves's ideas were forerunners of the theory of the conservation of energy, they were qualitative, unlike the quantitative investigations of Joule or Julius Robert von Mayer. His ideas also shaded into broader speculation, such as the nature of Olbers's paradox, which he may have discovered for himself rather than through a direct knowledge.

... it is difficult to understand why we get so little light at night from the stellar universe, without assuming that some light is lost in its progress through space – not lost absolutely, for that would be an annihilation of force – but converted into some other mode of motion.

— On the Correlation of Physical Forces (1874)

Grove also speculated that other forms of energy were yet to be discovered "as far certain as certain can be of any future event."

The probability is that, if not all, the greater number of physical phenomena are, in one sense correlative, and that, without a duality of conception, the mind cannot form an idea of them: thus motion cannot be perceived or probably imagined without parallax or relative change of position ... in all physical phenomena, the effects produced by motion are all in proportion to the relative motion. ... The question of whether there can be absolute motion, or indeed any absolute isolated force, is purely the metaphysical question of idealism or realism.

— On the Correlation of Physical Forces (1874)

As soon as he became a Fellow of the Royal Society in 1840 Grove was a critic of the Society, deprecating its cronyism and the de facto rule of a few influential Council members. In 1843, he published an anonymous attack on the scientific establishment in Blackwood's Magazine and called for reform. In 1846 Grove was elected to the Council of the Royal Society, and was heavily involved in the campaign to modernize its charter, in addition to campaigning for the public funding of science.

A charter committee had already been established, and Grove joined it. Groves's fellow campaigners included Gassiot, Leonard Horner and Edward Sabine. Their principal objectives were for the number of new Fellows to be subject to an annual limit, and limitation of the power of nomination to the Council. The reformers' success in 1847 led to the resignation of several key conservatives and the establishment of Grove and his associates with domination of the Council. To celebrate, the reformers founded the Philosophical Club.

Though the Philosophical Club succeeded in ensuring that William Parsons, 3rd Earl of Rosse was appointed next President, they failed to get Grove appointed as Secretary. Grove continued to campaign for a single home for all the scientific institutions at Burlington House.

From 1846 Grove started to reduce his scientific work in favor of his professional practice at the bar, his young family providing the financial motivation; and in 1853 became a QC. The bar provided him with the opportunity to combine his legal and scientific knowledge, in particular in patent law and in the unsuccessful defense of poisoner William Palmer in 1856. He was especially involved in the photography patent cases of Beard v. Egerton (1845 – 1849), on behalf of Egerton, and of Talbot v. Laroche (1854). In the latter case Grove appeared for William Fox Talbot in his unsuccessful attempt to assert his calotype patent.

Grove served on a Royal Commission on patent law and on the Metropolitan Commission of Sewers.

In 1871 he was made judge of the Court of Common Pleas, and was appointed to the Queen's Bench in 1880. He was to have presided at the Cornwall and Devon winter assizes of 1884, which would have entailed him trying the notorious survival cannibalism case of R v. Dudley and Stephens. However, at the last minute he was substituted by Baron Huddleston, possibly because Huddleston was seen as more reliable in ensuring the guilty verdict that the judiciary required. Grove did sit as one of five judges on the final determination of the case in the Divisional Court of the Queen's Bench.

Grove was a careful, painstaking and accurate judge, courageous and not afraid to assert an independent judicial opinion. However, he was fallible in patent cases, where he was prone to become over - interested in the technology in question and to be distracted by questioning the litigants as to potential improvements in their devices, even going so far as to suggest his own innovations. He retired from the bench in 1887. His portrait was painted by Helen Donald - Smith in the 1890s.

Groves's daughter, Imogen Emily (died 1886), married William Edward Hall in 1866. His daughter Anna married Herbert Augustus Hills (1837 – 1907) and was mother to Edmond Herbert Grove - Hills ("Colonel Rivers"), and John Waller Hills.

Grove, his health perpetually troubled, died at home in London after a long illness. He is buried in Kensal Green Cemetery.

Grove became a Fellow of the Royal Society in 1840, and received their Royal Medal in 1847. He was Vice President of the Royal Institution in 1844. Receiving a knighthood in 1872, he was given an honorary degree by Cambridge University in 1879 and became Privy Councillor in 1887.

The lunar crater "Grove" is named in his honor. The Grove Fuel Cell Symposium and Exhibition is organized by Elsevier.

Born in Holbæk, Denmark, his father, Andreas Christian, had been an officer in the Danish privateer service. Ludwig's mother, Anna Sophie, was the daughter of a clergyman and imbued the household with a deeply religious sentiment. Around the time of Ludwig's birth, his father retired from seafaring and took up a position as a farm manager. He seems to have been particularly unsuited to such a profession and this, together with the upheavals of the Napoleonic Wars in Denmark, subjected the young Ludwig to a rather irregular childhood and schooling.

Hans Christian Ørsted was an old family friend and arranged for Colding to serve an apprenticeship under a craftsman in Copenhagen, Colding achieving the status of journeyman in 1836. Ørsted had, by this stage, become something of a mentor to the young Colding and encouraged him to enroll at the Copenhagen Polytechnic Institute. The Institute had been founded at Ørsted's initiative and he offered continual advice and support to the young Colding.

Colding graduated in 1841 and worked as a teacher before being appointed inspector of roads and bridges in Copenhagen in 1845. Colding's importance and influence grew until he was appointed state engineer for Copenhagen in 1857. He oversaw a vast range of public housing, transport, lighting and sanitation projects and gained a high reputation throughout Denmark and internationally. He retired from professional engineering in 1886.

Colding found time for private scientific work in fluid mechanics, hydrology, oceanography and meteorology as well as electromagnetism and thermodynamics. He was largely responsible for founding the Danish Meteorological Institute in 1872. However, he is best remembered for what he himself termed the "principle of imperishability of the forces of nature." Colding was influenced by D'Alembert's principle of "lost forces", Ørsted, the Naturphilosophie to which Ørsted subscribed and his own religious upbringing.

My first thought concerning the imperishability of the forces of nature I have ... borrowed from the view that the forces of nature must be related to the spiritual in nature, to the eternal reason as well as to the human soul. Thus it was the religious philosophy of life that led me to the concept of the imperishability of forces. By this line of reasoning I became convinced that just as it is true that the human soul is immortal, so it must also surely be a general law of nature that the forces of nature are imperishable.

Colding first fulfilled his ambition to work alongside Ørsted, who was conducting experiments on the compressibility of water, in 1839. He summarized this work with a review of other data on compression and friction of various materials in his first published scientific paper. In this work, he went on to state that "the quantities of heat evolved are, in every case, proportional to the lost moving forces" though he did not calculate a mechanical equivalent of heat as Joule was to do in the same year.

With Ørsted's support, a further series of quantitative experiments was sponsored by the Royal Danish Academy of Sciences and Letters, culminating in a report in 1847. By 1850, Colding had obtained a value for the mechanical equivalent of heat, some 14% lower than the modern value (4.1860 J·cal⁻¹) at a time when Joule had measured 4.159 J·cal⁻¹. A subsequent calculation by Colding in 1852 yielded a value only 3% below modern values.

Colding's thermodynamic work was neglected both in his native Denmark and internationally though, from an historical perspective, he seems to deserve no less credit in the development of the concept of energy than Joule or Mayer. However, his contributions to meteorology and the built environment of Copenhagen are notable in themselves.