HEAT, a general term applied to that branch of physical science which deals with the effects produced by heat on material bodies, with the laws of transference of heat, and with the trans formations of heat into other kinds of energy. The object of the present article is to give a brief sketch of the historical de velopment of the science of heat, and to indicate the relation of the different branches of the subject, which are discussed in greater detail with reference to the latest progress in separate articles.
1. The term heat is employed in ordinary language in a number of different senses. This makes it a convenient term to employ for the general title of the science, but the different meanings must be carefully distinguished in scientific reasoning. For the present purpose, omitting metaphorical significations, we may dis tinguish four principal uses of the term :—(a) sensation of heat; (b) temperature, or degree of hotness; (c) quantity of thermal energy; (d) radiant heat, or energy of radiation.
(a) From the sense of heat, aided in the case of very hot bodies by the sense of sight, we obtain our first rough notions of heat as a physical entity, which alters the state of a body and its condition in respect of warmth, and is capable of passing from one body to another. By touching a body we can tell whether it is warmer or colder than the hand, and, by touching two similar bodies in suc cession, we can form a rough estimate, by the acuteness of the sensation experienced, of their difference in hotness or coldness over a limited range. If a hot iron is placed on a cold iron plate, we may observe that the plate is heated and the iron cooled until both attain appreciably the same degree of warmth; and we infer from similar cases that something which we call "heat" tends to pass from hot to cold bodies, and to attain finally a state of equa ble diffusion when all the bodies concerned are equally warm or cold. Ideas such as these derived entirely from the sense of heat, are, so to speak, embedded in the language of every nation from the earliest times.
(b) From the sense of heat, again, we naturally derive the idea of a continuous scale or order, expressed by such terms as summer heat, blood heat, fever heat, red heat, white heat, in which all bodies may be placed with regard to their degrees of hotness, and we speak of the temperature of a body as denoting its place in the scale, in contradistinction to the quantity of heat it may contain.
(c) The quantity of heat contained in a body obviously de pends on the size of the body considered. Thus a large kettleful of boiling water will evidently contain more heat than a teacupful, though both may be at the same temperature. The temperature does not depend on the size of the body, but on the degree of concentration of the heat in it, i.e., on the quantity of heat per unit mass, other things being equal.
It may be taken as axiomatic that a given body in a given state under given conditions (e.g., a pound of water in the liquid state at freezing point under atmospheric pressure) must always contain the same quantity of heat, and that different quanti ties of the same substance in the same state under the same conditions must contain quantities of heat proportional to their several masses. But it is for experiment to determine how the heat-content varies for any given substance in different states, or for different substances in similar states, and how it is affected by variations of temperature and pressure in each case.
(d) It is a matter of common observation that rays of the sun or of a fire falling on a body warm it, and it was in the first in stance natural to suppose that heat itself somehow travelled across the intervening space from the sun or fire to the body warmed, in much the same way as heat may be carried by a current of hot air or water. But we now know that energy of radiation is not the same thing as heat, though it is converted into heat when the rays strike an absorbing substance. The term "radiant heat," however, is generally retained, because radiation is commonly measured in terms of the heat it produces, and because the trans ference of energy by radiation and absorption is the most im portant agency in the diffusion of heat.
2. Evolution of the Thermometer.—The first step in the development of the science of heat was necessarily the invention of a thermometer, an instrument for indi cating temperature and measuring its changes. The first requisite in the case of such an instrument is that it should always give, at least aproximately the same indica tion at the same temperature. The air thermoscope of Galileo, illustrated in fig. 1, which consisted of a glass bulb contain ing air, connected to a glass tube of small bore dipping into a coloured liquid, though very sensitive to variations of temperature, was not satisfactory as a measuring instru ment, because it was also affected by varia tions of atmospheric pressure. The French doctor Rey describes in a letter dated 163 i a thermometer in which the expansion of water itself was used to indicate temperature, but it is not clear from his description whether the thermometer tube was left open or closed.
The type of thermometer familiar at the present time, con taining a liquid hermetically sealed in a glass bulb with a fine tube attached, was first brought into general use by the Grand Duke Ferdinand II. of Tuscany, and he is said to have possessed such instruments as early as 1654. They were much employed by the members of the Accademia del Cimento founded under his pro tection at Florence, and were long known as Florentine ther mometers. Alcohol was the liquid first employed, and the degrees, intended to represent thousandths of the volume of the bulb, were marked with small beads of enamel fused on the stem, as shown in fig. 2, which represents two thermometers as depicted in the Saggi di Naturali Experienze published by the Accademia del Cimento in 1666.
In order to render the readings of such instruments comparable with each other, it was necessary to select a fixed point or standard temperature as the zero or start ing point of the graduations. Instead of making each degree a given fraction of the volume of the bulb, which would be diffi cult in practice, and would give different values for the degree with different liquids, it was soon found to be preferable to take two fixed points, and to divide the interval between them into the same number of degrees. It was natural in the first instance to take the temperature of the human body as one of the fixed points. In 1701 Sir Isaac Newton pro posed a scale in which the freezing-point of water was taken as zero, and the temperature of the human body as 12°. About the same date (1714) Gabriel Daniel Fahrenheit proposed to take as zero the lowest temperature obtainable with a freezing mixture of ice and salt, and to divide the interval between this temperature and that of the human body into 12 °. To obtain finer graduations the number was subsequently increased to 96°.
The freezing-point of water was at that time supposed to be somewhat variable, because as a matter of fact it is possible to cool water several degrees below its freezing-point in the absence of ice. Fahrenheit showed, however, that as soon as ice began to form the temperature always rose to the same point, and that a mixture of ice or snow with pure water always gave the same temperature. At a later period he also showed that the tempera ture of boiling water varied with the barometric pressure, but that it was always the same at the same pressure, and might there fore be used as the second fixed point (as Edmund Halley and others had suggested) provided that a definite pressure, such as the average atmospheric pressure, were specified. The freezing and boiling-points on one of his thermometers, graduated as al ready explained, with the temperature of the body as 96°, came out in the neighbourhood of 3 2 ° and 212° respectively, giving an interval of 18o° between these points. Shortly after Fahren heit's death (1736) the freezing and boiling-points of water were generally recognized as the most convenient fixed points to adopt, hut different systems of subdivision were employed. Fahrenheit's scale, with its small degrees and its zero below the freezing-point, possesses undoubted advantages for meteorological work, and is still retained in most English-speaking countries. For general scientific purposes, however, the centigrade system, in which the freezing-point is marked o° and the boiling-point 10o°, is now almost universally employed, on account of its greater simplicity from an arithmetical point of view. For work of precision the fixed points have been more exactly defined (see THERMOMETRY), but no change has been made in the fundamental principle of graduation.
3. Comparison of Scales Based on Expansion.—Ther mometers constructed in the manner already described will give strictly comparable readings, provided that the tubes be of uni form bore, and that the same liquid and glass be employed in their construction ; but they possess one obvious defect from a theo retical point of view, namely, that the subdivision of the tem perature scale depends on the expansion of the particular liquid selected as the standard. A liquid such as water, which, when con tinuously heated at a uniform rate from its freezing-point, first contracts and then expands, at a rapidly increasing rate, would obviously be unsuitable ; but there is no a priori reason why other liquids should not behave to some extent in a similar way. As a matter of fact, it was soon observed that thermometers care fully constructed with different liquids, such as alcohol, oil and mercury, did not agree precisely in their indications at points of the scale intermediate between the fixed points, and diverged even more widely outside these limits. Another possible method, proposed in 1694 by Carlo Renaldeni (1615-1698), professor of mathematics and philosophy at Pisa, would be to determine the intermediate points of the scale by observing the temperatures of mixtures of ice-cold and boiling water in varying proportions. By this method, the temperature of 5o° C would be defined as that obtained by mixing equal weights of water at o° C and zoo° C; 2o° C, that obtained by mixing 8o parts of water at o° C with 20 parts of water at ioo° C and so on. Each degree rise of temperature in a mass of water would then represent the addition of the same quantity of heat. The scale thus obtained would, as a matter of fact, agree very closely with that of a mercury thermometer, but the method would be very difficult to put in practice, and would still have the disadvantage of depending on the properties of a particular liquid, namely, water, which is known to behave in an anomalous manner in other respects.
At a later date, the researches of Gay-Lussac (18o2) and Regnault (1847) showed that the laws of the expansion of gases are much simpler than those of liquids. Whereas the expansion of alcohol between o° C and ioo° C is nearly seven times as great as that of mercury, all gases (excluding easily condensible va pours) expand equally, or so nearly equally that the differences between them cannot be detected without the most refined ob servations. This equality of expansion affords a strong a priori argument for selecting the scale given by the expansion of a gas as the standard scale of temperature, but there are still stronger theoretical grounds for this choice, which will be indicated in dis cussing the absolute scale (§ 23). Among liquids mercury is found to agree most nearly with the gas scale, and is generally employed in thermometers for scientific purposes on account of its high boiling-point and for other reasons. The differences of the mercurial scale from the gas scale having been carefully determined, the mercury thermometer can be used as a secondary standard to replace the gas thermometer within certain limits, as the gas thermometer would be very troublesome to employ directly in ordinary investigations. For certain purposes, and especially at temperatures beyond the range of mercury ther mometers, electrical thermometers, also standardized by reference to the gas thermometer, have been very generally employed in re cent years, while for still higher temperatures beyond the range of the gas thermometer, thermometers based on the recently estab lished laws of radiation are the only instruments available. For a further discussion of the theory and practice of the measurement of temperature, the reader is referred to the article THER MOMETRY.
4. Among the most important effects of heat is that of changing the state of a substance from solid to liquid, or from liquid to vapour. All substances, with the exception of some unstable com pounds, are known to be capable of existing in each of the three states under suitable conditions of temperature and pressure. The transition of any substance, from the state of liquid to that of solid or vapour under the ordinary atmospheric pressure, takes place at fixed temperatures, the freezing and boiling-points, which are very sharply defined for pure crystalline substances, and serve in fact as fixed points of the thermometric scale. A change of state cannot, however, be effected in any case without the addi tion or subtraction of a certain definite quantity of heat. If a piece of ice below the freezing-point is gradually heated at a uni form rate, its temperature may be observed to rise regularly till the freezing-point is reached. At this point it begins to melt, and its temperature ceases to rise. The melting takes a considerable time, during the whole of which heat is being continuously sup plied without producing any rise of temperature, although if the same quantity of heat were supplied to an equal mass of water, the temperature of the water would be raised to the extent of nearly 8o° C.
Heat thus absorbed in producing a change of state without rise of temperature is called latent heat, a term introduced by Joseph Black, who was one of the first to study the subject of change of state from the point of view of heat absorbed, and who in many cases actually adopted the comparatively rough method described above of estimating quantities of heat by observing the time required to produce a given change when the substance was receiving heat at a steady rate from its surroundings. For every change of state a definite quantity of heat is required, without which the change cannot take place. Heat must be added to melt a solid, or to vaporize a solid or a liquid, and conversely, heat must be subtracted to reverse the change, i.e., to condense a vapour or freeze a liquid. The quantity required for any given change depends on the nature of the substance and the change considered, and varies to some extent with the conditions under which the change is made, but is always the same for the same change under the same conditions. A rough measurement of the latent heat of steam was made as early as 1764 by James Watt, who found that steam at 212° F, when passed from a kettle into a jar of cold water, was capable of raising nearly six times its weight of water to the boiling point. He gives the volume of the steam as being approximately 1,800 times that of an equal weight of water.
5. General Phenomena of Fusion.—There are two chief va rieties of the process of fusion, namely, crystalline and amorphous, which are in many ways distinct, although it is possible to find in termediate cases which partake of the characteristics of both. The melting of ice may be taken as a typical case of crystalline fusion. The passage from rigid solid to mobile liquid occurs at a definite surface without any intermediate stage or plastic condition. The change takes place at a definite temperature, the fusing or freezing point (abbreviated F.P.), and requires the addition of a definite quantity of heat to the solid, which is called the latent heat of fusion. There is also in general a considerable change of volume during fusion, which amounts in the case of ice to a contraction of 9%.
Some typical cases of amorphous solidification are those of silica, glass, plastic sulphur, pitch, alcohol and many organic liquids. In this type the liquid gradually becomes more and more viscous as the temperature falls, and ultimately attains the rigidity characteristic of a solid, without any definite freezing point or latent heat. The condition of the substance remains uniform throughout, if its temperature is uniform ; there is no separation into the two distinct phases of solid and liquid, and there is no sudden change of volume at any temperature.
The melting or freezing of a pure crystalline solid is character ized most clearly by the perfect constancy of temperature during the process. In fact, the law of constant temperature, which is generally stated as the first of the so-called "laws of fusion," does not strictly apply except to this case. The constancy of the F.P. of a pure substance is so characteristic that change of the F.P. is often one of the most convenient tests of the presence of foreign material. In the case of substances like ice, which melt at a low temperature and are easily obtained in large quantities in a state of purity, the point of f Lision may be very accurately determined by observing the temperature of an intimate mixture of the solid and liquid while slowly melting as it absorbs heat from surround ing bodies. In the majority of cases, however, it is more conven ient to observe the freezing point as the liquid is cooled. By this method it is possible to ensure perfect uniformity of temperature throughout the mass by stirring the liquid continuously during the process of freezing, whereas it is difficult to ensure uniformity of temperature in melting a solid, however gradually the heat is supplied, unless the solid can be mixed with the liquid. It is also possible to observe the F.P. in other ways, as by noting the tem perature at the moment of the breaking of a wire, of the stoppage of a stirrer, or of the maximum rate of change of volume, but these methods are generally less certain in their indications than the point of greatest constancy of temperature in the case of homo geneous crystalline solids.
Fusing Points of Common Metals in ° C Mercury . . —38.8 Antimony . . . 63o Potassium . . 62.5 Aluminium . . 655 Sodium . 97.6 Silver 962 Tin . . . . 231•9 Gold 1064 Bismuth . 269.2 Copper 1082 Cadmium . . 320•7 Nickel . 1452 Lead . . . . 327.7 Palladium . 1556 Zinc . . . . . 419•0 Platinum . . . • 1756 The table contains some of the most recent values of fusing points of metals determined (except the last three) with platinum thermometers. These points are often utilized as fixed points on the thermometric scale, especially for the calibration of ther mocouples.
It has been shown by H. A. Miers (Jour. Chem. Soc., 1906) that for a supersaturated solution in metastable equilibrium there is an inferior limit of temperature, at which it passes into the "labile" state, i.e. spontaneous crystallization occurs throughout the mass in a fine shower. This seems to be analogous to the fine misty condensation which occurs in a supersaturated vapour in the absence of nuclei (see VAPORIZATION) when the supersatura tion exceeds a certain limit.
The equilibrium temperature, at the surface of contact between the solid and liquid, depends only on the composition of the liquid phase and not at all on the quantity of solid present. The abscissa of the F.P. curve represents the composition of that portion of the original solution which remains liquid at any temperature. If instead of starting with a dilute solution we start with a strong solution represented by a point N, and cool it as shown by the vertical line ND, a point D is generally reached at which the solu tion becomes "saturated." The dissolved substance or "solute" then separates out as the solution is further cooled, and the concen tration diminishes with fall of temperature in a definite relation, as indicated by the curve CB, which is called the solubility curve. Though often called by different names, the two curves AC and CB are essentially of a similar nature. To take the case of an aqueous solution of salt as an example, along CB the solution is saturated with respect to salt, along AC the solution is saturated with respect to ice. When the point C is reached along either curve, the solution is saturated with respect to both salt and ice. The concentration cannot vary further, and the temperature re mains constant, while the salt and ice crystallize out together, maintaining the exact proportions in which they exist in the solu tion. The resulting solid was termed a cryohydrate by F. Guthrie, but it is really an intimate mixture of two kinds of crystals, and not a chemical compound or hydrate containing the constituents in chemically equivalent propor tions. The lowest temperature attainable by means of a freez ing mixture is the temperature of the F.P. of the corresponding cry ohydrate. In a mixture of salt and ice with the least trace of water a saturated brine is quickly formed, which dissolves the ice and falls rapidly in temperature, owing to the absorption of the latent heat of fusion. So long as both ice and salt are present, if the mixture is well stirred, the solution must necessarily become saturated with respect to both ' ice and salt, and this can only occur at the cryohydric tempera ture, at which the two curves of solubility intersect.
The curves in fig. 3 also illustrate the simplest type of freezing point curve in the case of alloys of two metals A and B which do not form mixed crystals or chemical compounds. The alloy cor responding to the cryohydrate, possessing the lowest melting point, is called the eutectic alloy, as it is most easily cast and worked. It generally possesses a very fine-grained structure, and is not a chemical compound. (See ALLOYS.) 8. Calorimetry by Latent Heat.—In principle, the simplest and most direct method of measuring quantities of heat consists in observing the effects produced in melting a solid or vaporizing a liquid. It was, in fact, by the fusion of ice that quantities of heat were first measured. If a hot body is placed in a cavity in a block of ice at o° C, and is covered by a closely fitting slab of ice, the quantity of ice melted will be directly proportional to the quantity of heat lost by the body in cooling to o° C. None of the heat can possibly escape through the ice, and conversely no heat can pos sibly get in from outside. The body must cool exactly to o° C, and every fraction of the heat it loses must melt an equivalent quantity of ice. Apart from heat lost in transferring the heated body to the ice block, the method is theoretically perfect. The only diffi culty consists in the practical measurement of the quantity of ice melted. Black estimated this quantity by mopping out the cavity with a sponge which was weighed before and after the operation. But there is a variable film of water adhering to the walls of the cavity, which gives trouble in accurate work.
In 178o Laplace and Lavoisier used a double-walled metallic vessel containing broken ice, which was in many respects more convenient than the block, but aggravated the difficulty of the film of water adhering to the ice. In spite of this practical difficulty, the quantity of heat required to melt unit weight of ice was for a long time taken as the unit of heat. This unit possesses the great advantage that it is independent of the scale of temperature adopted. At a much later date R. Bunsen (Phil. Mag., 1871), adopting a suggestion of Sir John Herschel's, devised an ice calorimeter suitable for measuring small quantities of heat, in which the difficulty of the water film was overcome by measuring the change in volume due to the melting of the ice. The volume of unit mass of ice is approximately 1.0920 times that of unit mass of water, so that the diminution of volume is 0.092 of a cubic centimetre for each gramme of ice melted. The method requires careful attention to details of manipulation, which are more fully discussed in the article on CALORIMETRY.
For measuring large quantities of heat, such as those produced by the combustion of fuel in a boiler, the most convenient method is the evaporation of water, which is commonly employed by engi neers for the purpose. The natural unit in this case is the quantity of heat required to evaporate unit mass of water at the boiling point under atmospheric pressure. In boilers working at a higher pressure, or supplied with water at a lower temperature, appro priate corrections are applied to deduce the quantity evaporated in terms of this unit.
For laboratory work on a small scale the converse method of condensation has been successfully applied by John Joly, in whose steam-calorimeter the quantity of heat required to raise the tem perature of a body from the atmospheric temperature to that of steam condensing at atmospheric pressure is observed by weighing the mass of steam condensed on it. (See CALORIMETRY.) 9. Thermometric Calorimetry.—For the majority of pur poses the most convenient and the most readily applicable method of measuring quantities of heat, is to observe the rise of tempera ture produced in a known mass of water contained in a suitable vessel or calorimeter. This method was employed from a very early date by Count Rumford and other investigators, and was brought to a high pitch of perfection by Regnault in his extensive calorimetric researches (Memoires de l'Institut de Paris, 1847) ; but it is only within comparatively recent years that it has really been placed on a satisfactory basis by the accurate definition of the units involved. The theoretical objections to the method, as compared with latent heat calorimetry, are that some heat is nec essarily lost by the calorimeter when its temperature is raised above that of the surroundings, and that some heat is used in heat ing the vessel containing the water. These are small corrections, which can be estimated with considerable accuracy in practice. A more serious difficulty, which has impaired the value of much care ful work by this method, is that the quantity of heat required to raise the temperature of a given mass of water I° C depends on the temperature at which the water is taken, and also on the scale of the thermometer employed. It is for this reason, in many cases, impossible to say, at the present time, what was the precise value of the heat unit, in terms of which many of the older results, such as those of Regnault, were expressed. These difficulties are dis cussed in the articles CALORIMETRY and THERMOMETRY. The unit generally adopted for scientific purposes is the quantity of heat required to raise 1 gramme of water I° C, and is called the gramme-calorie. English engineers usually state results in terms of the British Thermal Unit (B.Th.U.), which is the quantity of heat required to raise 'lb. of water I° F. (See CALORIE.) 10. Watt's Indicator Diagram; Work of Expansion.—The rapid development of the steam-engine (q.v.) in England during the latter part of the i8th century had a marked effect on the pi ogress of the science of heat. In the first steam-engines the working cylinder served both as boiler and condenser, a very wasteful method, as most of the heat was transferred directly from the fire to the condensing water without useful effect. The first improvement (about 1700) was to use a separate boiler, but the greater part of the steam supplied was still wasted in reheating the cylinder, which had been cooled by the injection of cold water to condense the steam after the previous stroke. In i 769 James Watt showed how to avoid this waste by using a separate condenser and keeping the cylinder as hot as possible. In his earlier engines the steam at full boiler pressure was allowed to raise the piston through nearly the whole of its stroke. Connection with the boiler was then cut off, and the steam at full pressure was discharged into the condenser. Here again there was unnecessary waste, as the steam was still capable of doing useful work. He subsequently introduced "expansive working," which effected still further econ omy. The connection with the boiler was cut off when a fraction only, say of the stroke had been completed, the remainder of the stroke being effected by the expansion of the steam already in the cylinder with continually diminishing pressure. By the end of the stroke, when connection was made to the condenser, the pressure was so reduced that there was comparatively little waste from this cause. Watt also devised an instrument called an indica tor (see STEAM ENGINE), in which a pencil, moved up and down vertically by the steam pressure, recorded the pressure in the cylinder at every point of the stroke on a sheet of paper moving horizontally in time with the stroke of the piston. The diagram thus obtained made it possible to study what was happening inside the cylinder, and to deduce the work done by the steam in each stroke. The method of the indicator diagram has since proved of great utility in physics in studying the properties of gases and vapours.
Fig. 4 represents an imaginary indicator diagram for a steam engine, taken from one of Watt's patents. Steam is admitted to the cylinder when the piston is at the beginning of its stroke at S, ST represents the length of the stroke or the limit of horizontal movement of the paper on which the diagram is drawn. The indi cating pencil rises to the point A, representing the absolute pressure of 6olb. per sq.in. As the piston moves outwards the pencil traces the horizontal line AB, the pressure remaining constant till the point B is reached, at which connection to the boiler is cut off. After cut off at B the steam expands under diminishing pressure, and the pencil falls gradually from B to C, following the steam pressure until the exhaust valve opens at the end of the stroke. The pressure then falls rapidly to that of the condenser, which for an ideal case may be taken as zero, following Watt. The work done during expansion is found by dividing up, as shown, into a number of small rectangles. The whole work done in the forward stroke is represented by the area ABCTSA, or by the average value of the pressure p over the whole stroke multiplied by the stroke 1. This area must be multiplied by the area of the piston a in square inches to get the work done per stroke in foot pounds.
I I. Thermal Efficiency.—The thermal efficiency of an engine is the ratio of the work done by the engine to the heat supplied to it. According to Watt's observations, confirmed later by Clement and Desormes, the total heat required to produce 'lb. of saturated steam at any temperature from water at o° C was approximately 65o times the quantity of heat required to raise 'lb. of water I ° C. Since 'lb. of steam represented on this assump tion a certain quantity of heat, the efficiency could be measured naturally in foot-pounds of work obtainable per lb. of steam, or conversely in the pounds of steam which are consumed per horse power-hour.
In his patent of 1782 Watt gives the following example of the improvement in thermal efficiency obtained by expansive work ing. Taking the diagram already given, if the quantity of steam represented by AB, or 30o cu.in. at 6olb. pressure, were em ployed without expansion, the work realized, represented by the area ABSF, would be 6,o0o/4=1,50o foot pounds. With expan sion to 4 times its original volume, as shown in the diagram by the whole area ABCTSA, the mean pressure (as calculated by Watt, assuming Boyle's law) would be 0.58 of the original pres sure, and the work done would be 6,000Xo•58=3,48o foot-pounds for the same quantity of steam, or the thermal efficiency would be 2.32 times greater. The advantage actually obtained would not be so great as this, on account of losses by condensation, back-pres sure, etc., which are neglected in Watt's calculation, but the mar gin would still be very considerable. Three hundred cu.in. of steam at 6olb. pressure would represent about •o245 of Ilb. of steam, or 28.7 B.Th.U., so that, neglecting all losses, the possible thermal efficiency attainable with steam at this pressure and four expansions (4 cut-off) would be 3480/28.7, or 121 foot-pounds per B.Th.U.
About 1820, it was usual to include the efficiency of the boiler with that of the engine, and to reckon the efficiency or "duty" in foot-pounds per bushel or cwt. of coal. The best Cornish pump ing-engines of that date achieved about 7o million foot-pounds per cwt., or consumed about 3.21b. per horse-power-hour, which is roughly equivalent to 43 foot-pounds per B.Th.U. The efficiency gradually increased as higher pressures were used, with more com plete expansion, but the conditions upon which the efficiency de pended were not fully worked out till a much later date. Much additional knowledge with regard to the nature of heat, and the properties of gases and vapours, was required before the problem could be attacked theoretically.