_ correctly describable under the title 
_ Teach the Calculus.” We are sorry to find the present 
_ methods of treatment of infinitesimals, 
NATURE 837 
represents the error introduced by measuring the change 
in a finite interval situated all on one side of the value x. 
In fact, instead of being small quantities which may 
be included or omitted in this way, these “‘ second order 
quantities ” are really rather of the nature of errors 
which must be taken off if we wish to study a con- 
tinuous process closely at every instant. For example, 
they represent the correction that would have to be 
applied to obtain the velocity of a train at any instant 
from the average velocity in an interval of one, two, or 
more seconds after that instant. 
Now, it is customary among mathematicians to use 
éx to denote a finite variation in which second-order 
quantities may be involved, and dx to denote the 
limiting form. But Mr. Brewster in his preface says, 
“The difference between any two values of x is an 
easy idea to grasp, and the use of dx or dx (it does 
not matter much which) emphasises the fundamental 
meaning of a differential.” .. . “ My advice would be 
to regard dx and dx as the same thing provided dx is 
taken very small, and to be satisfied with a common- 
sense explanation of the omission of terms of the 
second order.” And on p. 23, speaking of dy/dx and 
dy/5x, he says, “ You can get on quite well without 
bothering to distinguish between the two.” 
Again, surely it would be more in accordance with 
common sense if some account were also taken of what 
happens to a function before the variable reaches the 
given value x, and if this were done with the function 
'x® we should get a second value of the variation, say 
June 23, 1923] 
metal retorts used seem to undergo rather drastic 
treatment, and it would be interesting to know their 
length of life. 
The chapter on “ Surface Combustion,” by Mr. A. E. 
Blake, reports mainly progress obtained with the impact 
type of burner, and is followed by one on the “ Future 
of the Artificial Gas Industry,” and by others of a 
general character, such as “ Fuel Conservation,” and 
“Some Problems in the Utilisation of Fuel,” both these 
making interesting reading. An appendix deals with 
methods for the analysis of coal and fuel oils. 
The book is well printed and generously illustrated 
throughout. It is certain to be very useful, not only 
in America but also in other countries, particularly if 
read with the discrimination suggested above. 
Joun W. Coss. 













The Teaching of the Calculus. 
Common Sense of the Calculus. By G. W. Brewster. 
Pp. 62. (Oxford: Clarendon Press; London: 
Oxford University Press, 1923.) 2s. net. 
ieee time to time in recent years, small books 
have appeared which would be more or less 
“How Not to 
volume to be no exception to this rule, at least in its 
Its main peculiarity is the way “quantities of the 
Second order” crop up continually, and the way in 
which readers are led to believe that it does not matter 
much whether these are put in or left out. The main 
advantage of this kind of treatment is that students 
who have neglected their class-work and absorbed 
such a book for their examination are easily detected 
by their examiner, as they are certain to do something 
against which they were warned in class. 
If we take y=x? and define dy as the difference 
between x? and (« + x)? we undoubtedly get 
4 dy = 2xdx + dx?. 
But surely it would be more in accordance with most 
People’s ideas of common sense if instead of bringing 
in the notion of “ second order quantities ” the author 
had pointed out that this dy represents the change 
taking place in the value of the Square in an interval 
which begins with the value of x and continues for a 
distance 6x beyond that value, It would also surely 
be easier for a student to see that the greater the 
interval dx the more does this variation fail to give a 
_ Correct idea of the manner in which the function was 
_ Varying round about the instant that it passed the 
_ value x. Also, as the term dx2 increases in relative 
importance when éx is made larger, it would not be 
difficult for a reader to infer that this Squared difference 
NO. 2799, VOL. 111] 



dly = awdx — dx2, 
The differential equation dy=2xdx has now a precise 
meaning, as it describes a variation which always lies 
intermediate between the first and second estimates, 
however small the interval dx. As a matter of fact, 
fluctuations in statistics, such as increases of population, 
Tise or fall of stock exchange quotations, ave always and 
must always be estimated by comparing the value on 
any day or year with a previous value. 
The same mistake is made in dealing with integration 
as applied, for example, to areas. Mr. Brewster’s 
figures replace the actual area of a curve by a series of 
rectangles the left-hand sides of which are ordinates of 
the curve. Why does he not try taking the right-hand 
sides as well? If he would only shove his rectangles 
one space to the left, taking off the first and adding 
one at the end, he would have spared all his arguments 
by showing that the true area lies between two limits 
the difference of which can be made as small as we like 
by making the slices thin enough. 
There is undoubtedly a great demand for a book 
that will introduce the notions of a differential co- 
efficient and an integral and illustrate them with 
applications not involving any other functions than 
2BI 
