NAT ORE 52 
THURSDAY, APRIL 8, 1897. 
HEAT (AND OTHER MATTERS). 
Die Principien der Warmelehre, historisch-kritisch ent- 
wickelt, Von Prof. E. Mach. Pp. viii + 472. 
(Leipzig : J. A. Barth, 1896.) 
ROF. MACH has conceived this treatise in much 
the same spirit as his “ Mechanik.” Both books 
are partly based upon his lectures: neither of them pre- 
tends to give a complete account of the subject in its 
‘details and applications. The aim is rather to describe 
the principal facts and the development of ideas, to 
explain their connection and to examine critically the 
principles which underlie the study of thermal phenomena. 
The contents of the book may be roughly divided into 
three parts, of which the first two deal with heat and 
thermodynamics. In these the general arrangement is a 
compromise between the strictly historical method and 
that of the ordinary text-book ; nowhere is any attempt 
made to secure the completeness and detail of a technical 
treatise. Subjects which have been fully discussed by 
other authors (suchas thermo-chemistry and the dynam- 
ical theory of gases) are omitted, or only briefly referred 
to. In the first part the principal subjects treated of 
are thermometry, the conduction of heat, radiation, calori- 
metry and the calorimetric properties of gases. In each 
case there is a historical survey, followed by a critical 
discussion of the ideas involved in the development of 
the subject. The chapters on thermometry open in 
quite the orthodox way. We meet our old friends the 
three basins of water—cold, lukewarm and hot—and are 
warned of the danger of relying implicitly upon the un- 
controlled verdict of our senses. In the next chapter 
(a critique of the idea of temperature) we are further 
warned that we have nothing to do with metaphysical 
ideas as toa “true” or “natural” scale of temperatures ; 
ithe object of thermometry being to indicate the thermal 
state of a body by a number with reference to a scale 
which can be produced and reproduced with certainty 
and accuracy. Every distinct pyrometric method in- 
volves a different definition of temperature ; and the 
temperatures measured by any one of these methods 
have just the same significance as numbers in an inven- 
tory. Wenow come to two chapters which give us a 
foretaste of the freedom of treatment in which the author 
indulges so fully in the last third of the book. In one 
of these he answers the questions, ‘*‘ What are numbers ?” 
and ‘What are names?”; the other is devoted to a 
discussion of Das Continuum. We next follow the slow 
growth of quantitative conceptions as to the conduction 
of heat, beginning with the case of a bar heated at one 
end. First we have Amontons’ incorrect assumption 
that the temperature increases proportionately to the 
‘distance from the cold end. Lambert has a clearer con- 
ception of the permanent state, and finds that the excess 
of temperature above that of the surrounding medium 
follows an exponential law: but the way in which he 
deduces the result is incorrect. Biot, making use of 
Newton’s law of cooling, first gives a correct theoretical 
and experimental investigation of this case. The more 
general treatment of conduction is due to Fourier, whose 
«Théorie analytique de la Chaleur” has been of the 
NO. 1432, VOL. 55] 
greatest importance in the development and transform- 
ation of the methods of mathematical physics. Con- 
sidered from the physical standpoint, Fourier’s theory 
has the merit of not being based (as is the kinetic theory 
of gases) upon an hypothesis, but upon an assumption 
proved by experiment to be correct, viz. that the flow 
of heat is proportional to the rate of change of tem- 
perature along the line of flow. His mathematical treat- 
ment is highly original and exquisite in form. At the 
same time his work well illustrates the beneficial re- 
action of investigations in various branches of physics 
and mathematics. The way for Fourier had been pre- 
pared by the researches of Taylor and others on the 
vibrations of stretched strings and the nature of partial 
differential equations (the results of which are here re- 
produced in somewhat modernised form). In turn, the 
powerful methods which he devised for solving problems 
on the conduction and radiation of heat have proved 
invaluable in the study of electricity, diffusion and 
hydrodynamics. 
Among the founders of thermodynamics, Sadi Carnot 
occupies an unique position—one which must be rather 
unintelligible to those who hold that this science owes 
its birth to the conception of heat asa “‘ mode of motion.” 
In 1798 Rumford published his “Inquiry,” and came to 
the conclusion that heat “cannot possibly be a material 
substance.” Soon afterwards Davy, from further ex- 
periments, had deduced that “heat . . . may be defined 
a peculiar motion. . . .” And long before these two ex- 
perimenters there had been speculations’ to the same 
effect. Yet Carnot’s “ Réflexions sur la puissance motrice 
du feu” (1824) throughout assumes the materiality (or 
indestructibility) of heat. In his Gedankenexperiment, 
the reversible cycle, it is assumed that the heat lost by 
the working substance in one set of operations is pre- 
cisely compensated by the amount absorbed in the other 
set. “Ce fait n’a jamais été révoqué en doute. ... Le 
nier, ce serait renverser toute la théorie de la chaleur.” 
In spite of this, of the incorrect definition of the third 
operation in the cycle, of the difficulty of accepting step 
by step his reasoning, his conclusions are correct, and 
his method has proved profoundly suggestive. How 
happy his choice of the reversible cycle was, we can 
only appreciate by considering the very imperfect state 
of knowledge at that time as to the thermal properties 
of bodies: by requiring that the working substance 
should finally be brought back to its primitive physical 
condition, he ensures that it shall contain the same 
amount of heat after the cycle as before, and thus skil- 
fully avoids a gap which could not then be bridged. In 
1834 Clapeyron recalled attention to Carnot’s essay, and 
gave appropriate graphical and analytical expositions 
of his method ; but he still adheres to the hypothesis of 
the invariability of the quantity of heat (gained and lost 
in the cycle), and does not advance the essential ideas 
beyond the stage in which Carnot left them. These 
ideas first fell upon good ground in our own country, and 
bore fruit in Thomson’s remarkable paper, “On an 
Absolute Thermometric Scale,” published in 1848. 
In one respect fate did not deal kindly with Carnot. 
He admits (in words following those quoted above) that 
the foundations on which the theory of heat rests 
require careful examination, and that several experi- 
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