|A supposed diagram of French physicist Guillaume Amontons 1699 fire wheel, wherein a fire sequentially heated air in the outer chambers, which then forced water to move between the inner segmented wheel's cavities. |
The “fire wheel”, or moulin à feu (mill of fire), was a theoretical device conceived by French physicist Guillaume Amontons and first documented in a memoir of 1699 entitled “A Commodious Way of Substituting the Action of Fire Instead of Men and Horses to Move Machines” delivered in June, the same month of Thomas Savery’s demonstration of his Miner’s friend to the Royal Society, to the Royal Academy of Sciences in Paris. 
The original memoir details the construction of Amontons’ machine; also Martin and Chambers's Abridgment of the Philosophical History and Memoirs of the Royal Academy of Sciences at Paris, volume i, gives a full account of it, with a figure (which may or may not be the one depicted, adjacent).
The Papin engine (1690) and the fire wheel (1699) are generally considered to be the two main theoretical forerunners to the working steam engine, i.e. the Savery engine (Miner's friend), Newcomen engine, and others to follow. 
Amonton’s fire wheel is said to have been the first attempt for producing “circular motion” by means of fire, otherwise than by the aeolipile (250BC) or the fly of a smoke-jack, prior, and the Watt sun and planet gear (1871) to follow.
Amontons' fire-wheel, as he called it, consists of a number of close buckets, or chambers, placed in the circumference of a hollow wheel, and communicating with each other by valves opening in one direction; and a sufficient quantity of water is put into these buckets to fill about one half of the number: another circle of similar buckets, but of larger dimensions, are placed on the outside of the circle of the former buckets; these large buckets contain air, and each one has a pipe conducted from it to one of the waterbuckets which are nearer to the center: a part of the circumference of the wheel, which is about the level of the center, is exposed to the fire of a furnace, so that each air-bucket that passes will be heated; and also the lower part of the wheel is immersed in a cistern of cold water, so as to cool the fame bucket again. 
The action of the machine may easily be understood. The air contained in the large bucket which is opposite the fire becomes heated and expanded, and by the pipe of communication it enters into that waterbucket which is at the lower side of the wheel, and pressing upon the surface of the water therein, causes it to mount up through the other chambers, in the direction in which the valves open from one chamber to the next; the water, being thus accumulated in the chambers at one side of the wheel, will give it a preponderating power to turn round upon its axis.
This motion brings another air-bucket opposite to the fire, and the air therein expands in its turn, and again elevates the water in the interior chambers as much as it had descended by the motion of the wheel; a continual succession is thus kept up, and the air-buckets which have pasted the fire descend into the cold water, and the air is thereby cooled and reduced to its former bulk. By the communication with the water-buckets, the pressure of the expanded air is removed from within them, and puts them in a situation to repeat their action.
This machine is ingenious, and if a better application of fire, by rarefying water into steam, had not been discovered, it is possible that the invention of Amontons might have been further prosecuted. From his computations it would appear, that the machine he proposed would act with a considerable power; but as he exhibited no working model, or actual trial, it was never proved that the machine, if put into practice, would be capable of producing anything near the effect promised by his calculations.
Other fire wheels
Leupold, in his "Theatrum Hydraulicarum," 1724, proposed an improved form of this fire-wheel; and steam-engines have been since made with mercury, or fluid metal, contained within a hollow wheel, which is to be always kept on one side with the mercury by the force of the steam: they have not been found to equal other modes of applying the force of steam. 
Some of the fortuitous repercussions of Amontons’ failed attempt to create a ‘fire wheel’, that used the heat of a fire to expand air and make it move a wheel, are said to have been some of his experimental work in thermometry and on the development of the ideal gas laws and research into the nature of cold, some of used to create better thermometers.
1. Amontons, Guillaume. (1699). “A Commodious Way of Substituting the Action of Fire Instead of Men and Horses to Move Machines”, communicated to the Royal Academy of Sciences in Paris, June.
2. Rees, Abraham. (1819). “Section: M. Amonton’s Fire-Wheel”, The Cyclopedia, pg. 40. Longman, Hurst, Rees, Orme & Brown.
3. Amonton Firewheel of 1698 – GenuineIdeas.com.