£ AO Pele 
423 
THURSDAY, JUNE 25,° 1914. 
MATHEMATICS AND CIVILISATION. 
Die Kultur der Gegenwart. Edited by P. Hinne- 
berg. Die Mathematischen Wissenschaften, 
unter Leitung von F. Klein. Part ili., section i. 
Fascicles i., ii. (i. H. G. Zeuthen: Die Mathe- 
matik im Alterthum u. im Mittelalter; ii., A. 
Voss: Die Beziehungen d. Mathematik sur 
Kultur d. Gegenwart, and H. E. Timerding : 
Die Verbreitung mathematischen Wissens uw. 
math, Auffassung. Berlin and Leipzig: B. G. 
Teubner, 1912-14.) Price 3 marks each. 
“HESE three monographs are agreeably 
ei different, as well as complementary; and 
even where they overlap, the variety of treatment 
is interesting. The first section is the most de- 
tailed and (comparatively) technical; its author, 
as might be expected, gives an excellent and well- 
balanced account of Greek and medieval Euro- 
pean mathematics. Something more might have 
been said about the earlier Indian inventions; 
‘only a very brief paragraph is devoted to China, 
and apparently nothing is said about Japan. 
Mr. Voss’s article is extremely interesting and 
well-arranged. He shows how mathematics have 
influenced, and been influenced by, technical crafts, 
physical theories, and philosophy; and he has the 
courage to make high, but legitimate, claims for 
a science which seems to be as unpopular in 
Germany as it is with us. He points out that 
mathematics is pre-eminently a creation of the 
spirit of man; that it is his least restricted field 
of activity; and that we are under a moral obliga- 
tion to cultivate it. It is very refreshing to find 
these truths stated with such decision and clear- 
ness; and no one who is convinced of them should 
neglect a seasonable opportunity of repeating 
them. The popular attitude towards mathematics 
is exceptionally unfair. The ordinary man does 
not despise a physician, or a judge, or a divine, 
because he himself is ignorant of medicine, or 
law, or theology; but it is very rarely that he 
regards mathematics as anything more than a set 
of rules for calculation, or mathematicians more 
than computers at best, and at worst harmless 
cranks who waste their time on puzzles, quite use- 
less to the practical man. The most exasperating 
folk of all are those who have to use mathe- 
matical formule for technical purposes, and adopt 
towards the science which serves them, while they 
do not understand it, a sort of silly, patronising 
attitude, such as that of a good-natured merchant 
to one of his junior clerks. 
To put the main argument in a form which may 
appeal to a man of common sense, we affirm, with- 
NO. 2330, VOL. 93] 
out fear of refutation, that the history of culture 
is a history of intellectual development, in which 
the main feature is a change of habits of thought ; 
instead of vague fancies, irrational dogmas, crude 
superstitions, we are gradually acquiring clear 
concepts, consistent theories, and some sort of 
ethics worthy of the name. Towards this whole- 
some change nothing whatever has contributed 
| so much as the study of pure mathematics; its 
inclusion, for instance, in a school curriculum is 
amply justified by its power of exposing intellec- 
tual dishonesty—what Smith minor calls “fudge”’ 
—to the practice of which we are all more inclined 
than we should like to admit. 
To take an illustration of what we mean. In 
the second Book of Samuel (ch. xxiv.) it is stated 
that David’s sin in numbering his people was 
punished by a heaven-sent pestilence which killed 
70,000 men. Christians having adopted the Jew- 
ish Canon as an inspired document, the prejudice 
created by this story was so great that no Chris- 
tian census was taken before 1700 A.D.; and no 
trustworthy census dates before the first year of the 
nineteenth century. Even now there are people 
who resent the census, and by making false entries 
do their best to make it untrustworthy; but there 
must be few who really think an act of simple 
enumeration sinful, and a good many who under- 
stand the value of the census for insurance 
purposes, at any rate. 
The interest of Mr. Timerding’s essay is of a 
more pedagogic kind. Among other interesting 
things we may note the references to Jacobi, his 
mode of teaching, and views about intuition (pp. 
128-30); “blackboard physics” (p. 137); and 
especially the account of recent changes in mathe- 
matical teaching in Germany. Near the end of the 
article the author makes a statement which (with 
due reservations) we are inclined to challenge. 
He maintains that in technical schools (fachliche 
Schulen) the aim of mathematical teaching is 
“entirely different”? from what it is in the general 
schools; adding, in effect, that the attention of 
technical students should not be diverted from 
such applications of mathematics as they are likely 
to have to make. We believe, on the contrary 
(and not without experience), that technical 
students (such as engineers, or accountants, or 
draughtsmen), can be interested, rather more 
easily than ordinary students, in the principles of 
mathematics, by taking them in the right way. 
This, we believe, is by beginning with definite 
numerical examples of the kind they will meet 
with in their profession, and then proceeding, by 
an inductive method, to the general formule and 
theories which solve all such problems, [In this 
way, an engineer becomes interested in electricity, 
iS) 
