220 
NATURE 
[JANUARY 7, 1897 
Again, the history of mathematics is of direct benefit 
to the teacher himself in more ways than one. The 
progress of scientific discovery rarely, if ever, proceeds 
along strictly logical lines ; and of this fact mathematics 
affords a conspicuous example. The science took its rise 
from concrete problems of calculation and measurement : 
for a long time it remained partly empirical ; and it is 
only in quite recent times that its axioms and postulates 
have been submitted to rigorous examination. The 
significance of this for the teacher lies in the fact that the 
intellectual growth of the race repeats itself to a large 
extent in the individual ; thus in teaching the rudiments 
of algebra and geometry it is by far the best plan to 
begin with concrete problems and examples, avoiding 
abstract formule, and introducing the appropriate 
notation gradually, as the necessity arises. 
Another wholesome result of this historical knowledge 
is the lesson of patience which it conveys. A teacher who 
is vexed and disappointed by the slowness of his class 
may console himself by reflecting upon the length of 
time which has been required by mature intellects to 
obtain a true conception of the nature of fractions, of 
negative, irrational, and complex quantities, and so on. 
The controversies to which these subjects gave rise, 
even among professed mathematicians, are very instruc- 
tive in throwing light upon the psychological difficulties 
which every honest student of mathematics must en- 
counter, and which the aid of a sympathetic teacher 
helps him to overcome. 
From all these different points of view the teacher 
will find Prof. Cajori’s book very helpful and suggestive. 
It is easy to read, without being superficial ; it is com- 
posed with a due sense of proportion ; and the limita- 
tion of its scope enables the author to enter into sufficient 
detail without making his work too large. Thus, for 
example, the account of early printed books on arithmetic 
and algebra is rendered extremely vivid by the insertion 
of numerous specimens of notation ; this, more than any- 
thing else, enables the reader to appreciate not only the 
immense advantage of modern notation, but also the 
extraordinary power of men like Fermat and Tartaglia, 
who were able, in spite of most imperfect and incon- 
venient apparatus, to make discoveries of first-rate 
importance. 
Although, perhaps, the sections devoted to the Middle 
Ages and the Renaissance are the most interesting and 
profitable, the early pages on number-systems and 
numerals, and on the Rhind papyrus, are likely to be 
the most novel to the general reader. For the mathe- 
matician it still remains to account for such resolutions 
as fs = ads + os + 35+ aby a list of which occurs in 
the papyrus referred to. It is difficult to see how they 
were obtained, or what practical purpose they were in- 
tended to serve ; apparently the standard way of repre- 
senting a fraction was to express it as the sum of aliquot 
parts with different denominators. It is not merely a 
question of aliquot parts. Ahnes and his contemporaries 
must surely have known that 4; = ;4; + ;/s, although even 
about this it may not be safe to dogmatise. 
The most difficult part of a work of this kind is the 
discussion of modern developments. Thanks to a 
number of unselfish scholars, the stages of ancient and 
mediaeval discovery have been traced out with great care 
NO. 1419, VOL. 55] 
and precision ; and although much, no doubt, may still 
be done, the landmarks are familiar, and the data, after 
all, are limited. Buta history of modern mathematics, 
beginning, say, with the present century, has yet to be 
written ; and here the amount of material is so vast, and 
its ramifications so numerous, that to discuss it properly 
seems to require a trained band of specialists. In deal- 
ing with recent times, Prof. Cajori has wisely confined 
himself to a few points of special interest, such as the 
progress of trigonometry, the researches on the value 
and nature of 7, the rise of projective geometry, and the 
discussion of the merits of Euclid’s “ Elements” as an 
elementary text-book. He is against the retention of 
Euclid, but confesses, at the same time, that in his 
opinion the ideal text-book has not yet appeared. He 
very properly protests against attempts to prove the 
postulate about parallels ; and scores a fair hit by pointing 
out the inconsistency of the conservatives, who swear by 
Euclid and then ignore his fifth book. 
The “ Hints on Teaching,” referred to in the title, con- 
sist of remarks suggested by the historical context, and 
do not form an integral part of the work. They are the 
observations of a practical teacher, and commend them- 
selves by their reasonableness and common-sense. Thus, 
on the one hand, Prof. Cajori recommends that the 
concrete should precede the abstract, that proofs of the 
obvious should be avoided, and that in some cases logical 
rigour should be temporarily sacrificed; on the other, 
he insists on the necessity for intelligent teaching, 
such as will train the mind to independent thought and 
observation. 
For the reader who wishes to pursue the subject further, 
there are a number of references. It may be suggested 
that it would be an improvement if the proportion of 
primary references were larger than it is. Thus, to take 
an example at random, on page 231 there is an account 
of Girard’s discussion of imaginary roots of equations, 
and a footnote gives a reference to Cantor II. 718. This 
is all very well as an acknowledgment on the author’s 
part, but the reader would prefer a reference to the page 
in Girard where the original passage is to be found. 
Secondary references of this kind involve waste of 
energy. 
One point to which the author might well have drawn 
attention is the variety of projective properties of conics 
to be found in Apollonius. It is certain that Pascal read 
the “ Conics”; and itis by no means unlikely that modern 
projective geometry springs from the study of the works 
of the great geometer of Perga. 
A small but rather irritating matter is the inconsistency 
shown in the transcription of Arabic names. Why, for 
instance, should we have Al Battdni in two words and 
Albirini in one? And, again, why use the German tran- 
scription dsch when the English 7 or the international ¥ 
is ready to hand? Thus, on page r1o, Abu Ga‘far al- 
Hazin appears as Abii Dscha ‘far Alchazin, and our old 
friend ‘Omar Khayyém (‘Omar al-Hajj4m) masquerades 
as ‘Omar Alchaijami. Let us have either a popular English 
approximation, or a scientific transliteration ; a German 
popular version is simply hideous. 
We niay conclude with a word of praise for the 
biographical part of the work. So far as the limits of the 
book allow, the author tells us about the mathematicians 
