536 SCIENCE PROGRESS 



works. Both books are indispensable to the student, and he is fortunate in having 

 two such books to read. 



But there is one superiority of the German authority to which reference may 

 be made without any invidiousness. Nicht-euklidische Geometric is one of a series 

 in which the various branches of mathematics are treated by well-known 

 authorities ; Non-Euclidean Geometry is an isolated treasure found surrounded by 

 Problem papers in arithmetic for preparatory schools, Arithmetic, and Statics. 

 What a difference ! The lack of such a series in English as the Sammlung 

 Schubert is a subject for national mathematical sorrow. As far as can be judged 

 the Sammlung is a private venture, unsupported by Government subsidy or 

 university patronage, and yet its enterprising publisher, G. J. Goschen, has already 

 issued sixty or seventy books and has twenty more on the stocks. It is to the 

 honour of Messrs. George Bell that they have published a mathematical book from 

 which they cannot expect to derive a large profit. But if they do not gain by it 

 financially, they have the credit of issuing from their house a sound, useful treatise 

 upon an important subject. Considering the enormous number of high-priced 

 mathematical text-books used in schools and the profits which certainly accrue to 

 some publishers from this source, it is regrettable that apart from the great 

 University Presses of Oxford and Cambridge so little is done for the advance of 

 mathematical knowledge by the publication of treatises from which profit cannot 



be extracted. 



C. 



Plane and Solid Geometry. By W. B. Ford and C. Ammerman. [Pp. 

 ix + 321 + xxxiii.] (New York : The Macmillan Company, 1913. Price 

 5$. 6d. net.) 



Of the making of elementary books on geometry there is no end, and the study 

 of them is a weariness of the flesh. So the reviewer frames his lament, but it 

 would be unfair to make such a sweeping verdict without justifying the attitude 

 of mind which forces the unwilling utterance. The review of this book will 

 therefore take a somewhat general trend, and the remarks in it will apply not 

 only to the book before us, but to others of the same brood, whose paternity, 

 though unmentioned, is no secret. 



Some fifteen years ago an agitation was raised in this country, primarily by 

 engineers, against the method in which geometry was then taught. The writer 

 of this review willingly acknowledges the wisdom and the justice of a great deal 

 that was said by those who led the attack. The methods then employed were 

 open to charges which could not be refuted. The text-book was of very great 

 antiquity, and designed to meet the needs of pupils of mature intellect and of 

 a different civilisation from our own ; the boys and girls of tender years who 

 tried to learn geometry from it often did not realise what was being taught them, 

 and in many, perhaps most, cases studied the text only. Sufficient to say that 

 the reformers, as usual, triumphed, and the system of teaching the subject was 

 reorganised. The panacea which found favour, to judge by the literature issued, 

 was a utilitarian treatment of the subject. Though the ancient Greek philosopher 

 declared that there was no royal road to geometry, the modern civil engineer 

 decided to make one. Planes were called tables, lines were christened strings, while 

 for points even such curious objects as human heads were substituted. All was 

 done under the spell of the word practical. Every stage in the development 

 of the subject at which difficulties were encountered was zealously attacked by 

 a host of minor Euclids, until in the modern geometry no landmarks remained 



