Commentary

Carnot states this is self evident, and offered no proof.

Proof is quite simple. Take any vapor, divide into constrained volume. Apply the temperature difference, a pressure difference must be created. The basis for all vapor engines is achieved.

Proof: The proof of this is simple. Work is only produced via Volume Expansion. Any other operation of a heat engine cannot aid in Energy Conversion.

Also, same page, section 1 would be struck through, if this was discussing only maximum energy conversion ratio (deliberately avoiding "efficiency"), but the topic of this paragraph is maximum power, which is energy times the ratio of energy conversion.

Proof: The proof of this is again simple and the argument the same. Work is only produced via Volume Expansion. Any other operation of a heat engine cannot aid in Energy Conversion. Disbursing or diffusing heat is waste of energy that could have been converted to mechanical Work.

Also, same page, section 1 would be struck through, if this was discussing only maximum energy conversion ratio, but the topic of this paragraph is maximum power, which is energy times the ratio of energy conversion.

In page 56, he states the same idea more clearly. (If the reader understands "re-establishment of equilibrium" means the heat leaves the machine).

"Reflections", page 112, Sadi Carnot
Chapter Motive Power of Heat
   (1) The temperature of the fluid should be made
   as high as possible, in order to obtain a great fall
   of caloric,, and consequently a large production of
   motive power.
   (2) For the same reason the cooling should be
   carried as far as possible.
   (3) It should be so arranged that the passage
   of the elastic fluid from the highest to the lowest
   temperature should be due to increase of volume;
   that is, it should be so arranged that the cooling of
   the gas should occur spontaneously as the effect of
   rarefaction. The limits of the temperature to
   which it is possible to bring the fluid primarily,,
   are simply the limits of the temperature obtainable
   by combustion ; they are very high.

Proof again relies on the Law of Conservation of Momentum and Energy.

Carnot and Kelvin both clearly considered all Vapor engines reversible and hence capable of maximum heat conversion to motion. (And it is true all Engine cycles are as reversible as the Carnot Cycle.) It is interesting that Carnot also maintained the motive energy was independent of the machine.

Its hard to say what they considered, as they clearly did not consider that any heat engine in existence was not reversible. Kelvin mentions Thermo-electric devices as heat devices, but clearly they are not directly creating motion. Kelvin made no analysis of thermo-electricity in "Reflections".

While it is speculation, both men also adhered to the principle of conservation of energy, so it seems they considered reversibility of engine cycles proof of concepts, and did not seriously consider the possibility of another type of heat engine, so did not address it as a relevant prediction to make. In almost 200 years, no other heat engines have shown up.

Proof lies in the text. For a machine to work with heat as fuel, it must produce a usable mechanical effect. Only vapors produce a significant effect from heat.

This was a bold prediction, but has withstood the test of time. It was made prior to 1824 when Sadi Carnot initially published his work in French. "Reflections" referred to was published decades later in English translation and the version of Reflections referred to was copyright 1897. In 2012, the claim stands.

The material doing all the work is a Vapor. Carnot points out the machine depends on the indestructible property of a vapor. Solids and to a lesser extent liquids can be broken down by mechanical effect. Vapors are "indestructible" up to temperatures where molecules start coming apart.

page 111 continued.
The elastic fluids, gases or vapors, are the means
really adapted to the development of the motive
power of heat. They combine all the conditions
necessary to fulfil this office. They are easy to
compress ; they can be almost infinitely expanded ;
variations of volume occasion in them great
changes of temperature; and, lastly, they are very
mobile, easy to heat and to cool, easy to transport
from one place to another, which enables them to
produce rapidly the desired effects. We can easily
conceive a multitude of machines fitted to develop
the motive power of heat through the use of
elastic fluids ; but in whatever way we look at it,
we should not lose sight of the following principles:

Proof is the shape of the pressure curve and math. As stated, each of these volume changes begins at the same temperature, so will always produce the same temperature change.

While it falls short of producing the formula by which heat in vapor accumulates over progressive change, it identifies the dominant cause of Pressure and hence Work as the geometric change in volume must be met by an inverse change in density and hence Pressure.

The text continues to attribute the heat to "specific heat", but is otherwise correct, and the conclusion is correct. Pressure curves are logarithmic with volume.