Marcu 24, 1904] 
that the thought shall precede the form, 
shall not conceal the thing symbolised. They insist that 
systematic and progressive problems based upon every-day 
experience and observation shall be, to a much greater 
extent, the materials of education. They demand that the 
several elementary mathematical subjects, from arithmetic 
to the calculus, shall develop side by side in the boy’s mind. 
They demand ‘that the mastery of these subjects shall be 
more the work of the judgment than of the memory. They 
demand that from first to last, at least during the secondary 
period, mathematical ability and the ability to think clearly, 
investigate closely and conclude correctly shall develop 
together, and to the extent that four well-spent years will 
on the average permit. Those who formulate these ideas 
contend that they lead to the correct mathematical training 
for all professions and all careers. 
The proposition that mathematics is the very bone 
sinew of an engineering course needs no discussion. It is 
everywhere conceded, The extent and nature of the mathe- 
matical element in the curriculum, however, are two decided 
fluents with curves of opposite slope. More mathematics 
but fewer kinds seems to be the tendency. The opinion 
appears to be gaining ground that the purely descriptive 
and highly specialised and professionalised elements in our 
technical courses should be reduced, while more subjects 
with a mathematical basis, with long unbroken continuity 
and bound together with a strong logical element should 
command the attention of the student to the end of his 
undergraduate period. 
Upon the question as to what mathematical subjects shall 
the undergraduate courses include in our technical colleges, 
opinions are decidedly at variance. Upon the four ordinary 
elementary subjects the sentiment is practically unanimous 
but these should be principally taught in the secondary 
and 
schools. The practical people, however, are inclined to 
relegate analytic geometry and the calculus to the scrap 
pile. To such subjects as vectors, theory of functions, 
theory of groups, they allow no place whatever. 
One cannot but feel that this verdict against analytic 
geometry and the elementary calculus—not to mention 
higher subjects—is a great pity. Especially does it seem 
true when we recall that instruction in these two lines 
forms the principal mathematical element of the second and 
third years of the ordinary technical course, and that the 
calculus itself is probably the most powerful and wonderful 
tool for investigation that the genius of man has ever 
contrived. 
Why do practical men almost unanimously place calculus 
among the dispensable elements of a technical curriculum ? 
The answer, of course, is very simple; they have never 
found any use for it, probably because they have never 
learned how to use it. Yet they dare not pronounce 
against it altogether. They know that Rankine and 
Maxwell were master mathematicians, and that through 
this mastery of the most powerful of tools they were able 
to do for terrestrial what Newton and Laplace did for 
celestial mechanics. In college the engineer has not learned 
to use the modern tool called the higher analysis ; it remains 
to him as foreign currency. Out of college he has not 
time to learn its use. 
The most effective teaching of the higher analysis will 
be possible only when reforms in mathematical instruction 
have permeated the principal secondary schools. 
The teacher should be saturated with his subject. Not 
only should he be strong and apt on the formal side, but 
more important still, its inner meaning should be clear to 
him and its close relation to the phenomena of the objective 
and subjective life. Some contend that the only man to 
whom the mathematics of a technical college can be en- 
trusted is an engineer. Does that make any difference ? 
Rather are not these the essential questions? Does the 
man know his subject? In his teaching can he assemble 
from engineering and other records the material that will 
vitalise his work? Is he in sympathy with engineering 
essentials and ideals? 
Throughout the college course the teaching should be 
mainly concrete. The problem, say from the _ physical 
sciences including engineering, should first be presented 
concretely. It should then be stated in mathematical 
symbols. The operations performed upon the symbols 
1795, VOL. 69] 
NATURE 
that the symbol | 
501 
should be accompanied by drawings or models, the final 
result reduced to numerical form, and then interpreted in 
language. Upon every problem the student must bring to 
bear the whole range of his acquired powers and be taught 
to select the shortest method within his ability. 
In other words, all typical problems should receive a three- 
fold consideraton :—(a) its statement in words, and the 
statement in words of its solution when effected; (b) its 
graphical statement and solution, involving geometry and 
mechanical drawing with squared paper; (c) its analytic 
statement and solution, ending with a numerical result. 
The purely formal should be presented as a necessity 
arising from the so-called practical, and in order that a 
body of knowledge and technical ability may be accumulated 
which will give the student easy control over the practical 
in whatever one of its various forms experience shows that 
it may arise. 
The problems chosen should be progressive in character, 
and their mastery should amount to a complete laboratory 
course in all that part of the higher analysis in which it is 
desirable that the engineering student should be well versed. 
The course should be lecture and seminarium and _ in- 
dividual, more after the manner of the German Technische 
Hochschule. The text-book should become a book of refer- 
ence. The instructor should know clearly and be able to 
state accurately the limitations of his methods, but abstruse 
discussions of obscure points should be postponed as long 
as a due regard for logical development will allow. Time 
is wasted in removing difficulties the existence and import- 
ance of which the student has not yet recognised. 
These are some of the necessary extensions into college 
work of the reformation now urged upon the secondary 
schools, and though every one of them seems familiar 
enough when taken separately, all together their united 
application to the mathematical courses in our technical 
colleges amounts to a departure from our present traditional 
methods little short of revolutionary. 
In recent years mathematical instruction in the United 
States has greatly improved in its thought content, but it 
has responded slowly and conservatively to modern methods. 
We are still more English than German. In the work of 
training a master of the physical sciences the text-book and 
the senseless repetition of words and formulas have been 
replaced by the lecture, the laboratory and the seminarium. 
Why should not mathematics, so intimately related to them, 
follow their lead and partake in the benefits of modern 
methods carried to their legitimate and logical completion ? 
UNIVERSITY AND EDUCATIONAL 
INTELLIGENCE. 
Harvarp University has, we learn from Science, re- 
ceived a gift of 50,0001. from Mr. David Sears, of Boston, 
a graduate of the class of 1847. 
A cOMMEMORATION day will be celebrated at Glasgow 
University on April 19, when an oration will be given by 
Sir William Ramsay on Joseph Biack, who was lecturer 
on chemistry from 1756 to 1766 in the old college. 
Dr. Harotp Jacosy, adjunct professor of astronomy at 
Columbia University, New York, has been promoted to a 
professorship. Prof. Jacoby will continue as acting director 
of the Columbia University Observatory during the absence 
of Prof. Rees on account of illness. Dr. C. L. Poor, formerly 
at the Johns Hopkins University, has also been appointed 
a professor of astronomy, and will be associated with Prof- 
Jacoby at Columbia. 
Tue governing body of the South-western Polytechnic, 
Chelsea, has unanimously appointed’ Mr. Sidney Skinner, of 
Christ’s College, Cambridge, to the position of principal in 
succession to Mr. Herbert Tomlinson, F.R.S., who is re- 
tiring. Since 1888 Mr. Skinner has been attached to the 
teaching staff of the Cavendish Laboratory at Cambridge, 
and also has acted as lecturer and director of natural science 
studies at Clare College. Mr. Skinner will take up his 
duties at the polytechnic about the beginning of May 
next. 
