30 
possibility of the future, then perhaps Prof. Perry’s gloomy 
picture of the decay of Great Britain may be falsified. The 
place of our coal-mines in our national assets would then be 
taken by the vast areas of sunlit land in our Colonial Empire, 
for fuel-production would then become a question of the number 
of acre-hours of sunlight available. 
I should like to add that what I have said in this letter does 
not at all lessen the urgency of Prof. Perry's plea for efficient 
engines ; in fact, I think that what I have pointed out tends to 
strengthen the demand for a great national effort at the sclution 
of these pressing problems. At present we are, in matters of 
energy, ‘‘robbing posterity’ while it is eminently desirable 
that we should discover a way—if there be one—of ‘ paying 
our way,” and I think that in fuel farming such a way may 
perhaps be found. WALTER ROSENHAIN. 
443 Gillott Road, Edgbaston, April 27. 
Mr. ROSENHAIN is mistaken as to the ignorance of inventors ; 
many engines have been invented and referred to in newspapers 
during the last thirty years for utilising solar heat. I may remark 
that such heat engines may be very efficient, because the avail- 
able temperature may be very high indeed. I have sometimes 
wondered why metallurgists neglected the possibility of obtaining 
very high temperature furnaces from the heat of the sun. As to 
the energy available, at p. 14 of my book on ‘* Steam” I say :-— 
“*On one square foot of Egypt the heat energy received in one 
year from the sun is about 10° foot pounds, or 500 horse-power 
hours.” This is nearly equivalent to the energy of a coating of 
coal all over Egypt one foot thick, and promises a future for 
the Sahara and other cloudless regions ot the earth. I there- 
fore admit that I did not give sufficient weight to this consider- 
ation of the direct heat from the sun, and I am very glad that 
Mr. Rosenhain has drawn attention to my neglect. 
J. PERRY. 
Experimental Mathematics. 
PROF. PERRY’S syllabus in practical mathematics has now 
been published two or three years, and the results of actual ex- 
perience of its working may have some interest. We have in 
this institute about three hundred students of mathematics, in- 
cluding boys in the day school as well as older evening students, 
who follow a course on the lines of Prof. Perry’s syllabus, and 
in both classes the adoption of the method has aroused an in- 
creased interest in the subject. This increase of interest seems 
to be due to the fact that the method is essentially experimental 
as well as deductive. Mathematics is treated as a science 
rather than, according to a common tradition, as an “arts” 
subject. The student is taught to investigate the facts for him- 
self by experiment in the form of actual plotting and measure- 
ment and numerical calculation, just as in the study of such a 
science as electricity he investigates a law for himself in the 
laboratory and, usually at a later stage, proves in his theoretical 
work that that law follows from his previous knowledge. This 
is not merely a question of illustrating elementary geometry, 
but the practice may be carried with advantage into what are 
usually considered quite advanced parts of his work. However 
well a student may know the analytical proofs involved, he 
greatly improves the firmness of his grasp by actually plotting, 
with various numerical values of the constants, curves to repre- 
sent such a case, for instance, as the small oscillations of a stiff 
spring, or the form of a bent beam. In pure mathematics, 
especially in differential geometry, many examples may be 
found, and, in fact, the method of conformal representation, 
which has been so fruitful in the theory of functions and its 
applications, is really an instance of this method. Besides in- 
creasing the average student’s interest in his work, these ‘‘ direct 
vision” methods, used systematically throughout a student’s 
course, give more solidity and a clearer definition to his ideas 
‘than it seems possible to attain by abstract reasoning alone. 
My special object, however, in writing is to insist on the 
value of the method as a logical training. We sometimes hear 
of the ‘invaluable logical training ” of Euclid with the implied 
assumption that other methods of treating mathematics are 
illogical. This view seems to ignore the fact that there is an 
inductive as well as a deductive logic. Ifa boy is taught from 
the beginning to verify all theorems by actual plotting and 
measurement, he trains, not only his logical powers of deductive 
reasoning in proving his theorem from its premises, but also his 
equally logical powers of inductive reasoning from observation 
NO. 1697, VOL. 66] 
NATURE 
[May 8, 1902 
and experiment. From the point of view of educational theory 
this seems a sounder method than to restrict his training to one 
form of logical reasoning to the neglect of the other. The de- 
ductive logic of the syllogism was the only form known in the 
time of Euclid, but it is scarcely necessary to say that inductive 
logic now holds a recognised place, and the whole development 
of modern experimental science has proceeded by its methods, 
John Stuart Mill, as is well known, devotes a very scanty 
consideration to syllogistic reasoning on the ground that 
“Formal Logic therefore, which . . . have represented as the 
whole of logic properly so called, is really a very subordinate 
part of it, not being directly concerned with the process of 
Reasoning or Inference in the sense in which that process is a 
part of the Investigation of Truth,” and that ‘“‘ The foundation 
of all sciences, even demonstrative or deductive sciences, is 
induction.” 
This may, perhaps, be the explanation of the difficulty which 
so many boys as well as older students feel in comprehending 
demonstrative geometry. Most teachers of evening students 
have met with men of considerable ability and some maturity of 
mind who have little or no difficulty with algebraical work, but 
can never comprehend the meaning of a proposition in Euclid. 
The syllogistic method of reasoning seems to find no avenue 
into their minds, although they can reason well enough from 
observed facts. Such people are usually set aside as having no 
mathematical gift, but all must have notions of space and time, 
and consequently of change and a rate of change, and if rigid 
deductive methods were so essential as is often supposed to the 
science which puts those notions into scientific form, they would 
scarcely be incomprehensible to so many. If anyone has the 
power of comprehending the facts of a science such as chemistry, 
he must have some power of putting that knowledge into 
scientific form, and so anyone whose experience is given in 
space and time can scarcely be quite without the power of un- 
derstanding the science which deals with those conditions of his 
experience. In fact, if students who seem to be without mathe- 
matical power are aliowed to approach the subject by an ex- 
perimental method, they find no difficulty in understanding it 
and may in time come to grasp the significance of deductive 
methods. In secondary schools of the classical and mathe- 
matical type, boys who are not on the science side are at present 
almost without the opportunity of developing their inductive 
logical powers, with the exception of the few who reach the 
stage where they can draw their own conclusions from the facts 
of philology or history. Experimental mathematics might in 
this case be made to supply the place of the missing experi- 
mental training. 
However one may admire the symmetry of an ideal rigid 
body of mathematical knowledge, built up in the mind of the 
learner so that each step is made to depend by flawless abstract 
reasoning on what has gone before, and so on down to necessary 
axioms at the foundations, such a process cannot be carried 
out in the practice of education. It is sometimes said that a 
student should not be allowed to use any process or to believe 
any theorem until he can render a complete and perfect reason 
for it. But ifa student is to follow such a method he should 
not be allowed to use 0°3, until he can justify his use of it from 
a knowledge of the meaning of a limiting value and of the 
criteria for the convergency of series, nor may he use V2 asa 
number until he has mastered the modern theory of irrational 
numbers and made up his mind whether to hold opinion with 
Dedekind and Weierstrass, that the conception of an irrational 
number is to be based on a purely arithmetical theory, or with 
Du Bois-Reymond, that it is essentially geometrical and insepar- 
ably connected with linear magnitude. It is obvious that no 
teacher can attempt such a course ; these difficulties are always 
passed over without proof. : 
In the method of practical mathematics, this practice is frankly 
recognised as legitimate and natural, and is systematically 
extended to other parts of mathematics. ‘ 
Whatever may be true of the superstructure, the fundamental 
notions of pure mathematics have not been built up by strict 
deductive process, but by a series of successive approximations 
to the truth, The conception of a limiting value is a case in 
point. Until the time of Cauchy, the existence of a limiting 
value was thought to be self-evident on geometrical grounds in 
such a case as that of the area of a polygon inscribed in a circle. 
Cauchy in his treatment of definite integrals recognised that 
it was necessary to prove that a definite limiting value existed 
in such a case, but it was only in 1883 that a completely neces- 
