September 15, 1904] 



NA TURE 



479 



solulion and theory of quadratic equations. The style 

 is attractive, and well suited to beginners. The 

 answers themselves occupy more than fifty pages, and 

 the volume can, if desired, be obtained without these 

 at a reduction in cost. 



The movement of reform in the teaching of 

 elementary mathematics has affected trigonometry, 

 and an outcome is seen in the nevi- text-book by Messrs. 

 Borchardt and Perrott, in which the authors have 

 aimed at presenting the subject in a manner suited to 

 ihe new conditions. Four-figure mathematical tables 

 are provided for general use. Graphs are introduced 

 from time to time. The first endeavour of the authors 

 is to give the student a good working knowledge of 

 the elements of trigonometry, with facility in practical 

 computation. The opening chapters are thus confined 

 to acute angles. Easy problems are solved on heights 

 and distances, but more attention might well have been 

 given to the solution of right-angled triangles under 

 all sorts of conditions, as this is fundamental. In 

 establishing the general formula for angles of any 

 magnitude, the authors have not perceived that a satis- 

 factory account can only be given by introducing the 

 conception of a vector, with the projections of rotation 

 vectors and vector polygons. Consequently, the heart 

 of the matter is missed, and the foundation for future 

 development is not completely laid. The usual formulae 

 for triangles, and for compound, multiple, and sub- 

 multiple angles are established, the work consisting 

 largely of a mass of trigonometrical transformations. 

 It is not evident why practical applications should be 

 confined to problems in surveying. There are other 

 rich sources to draw from. Thus, an investigation of 

 simple harmonic motion could be made to throw a flood 

 of light on the significance of trigonometrical formulae, 

 awakening a living interest in dead symbols. This is 

 the first half of the complete text-book. The part 

 which is to follow will contain chapters on De Moivre's 

 theorem, the exponential theorem, trigonometrical 

 series, &c. The book is a fair attempt to teach 

 elementary trigonometry in a more rational manner, 

 but falls far short of the ideal text-book on the subject. 



The remarks just made apply generally to the volume 

 by Messrs. Lachlan and Fletcher, except that the whole 

 of the subject-matter, including De Moivre's theorem, 

 &c., is compressed into one small volume, and in order 

 to be suitable for beginners would require considerably 

 to be amplified and supplemented by the teacher. 



The " Preliminary Practical Mathematics " by 

 Messrs. Starling and Clarke is the result of experience 

 gained with technical students, who find, when enter- 

 ing the laboratory or workshop, that their mathe- 

 matical knowledge is not suited to the requirements, 

 and who have not time to enter on an extended mathe- 

 matical course to supply the deficiency. If the subject 

 were properly taught to boys at school, a book like 

 the present would not be required. 



The ground covered in Mr. Kerr's " Constructive 

 Geometry " is substantially that of the first three books 

 of Euclid's Elements, but the treatment of the subject 

 is in accordance with modern ideas, and is very suit- 

 able indeed for beginners. Starting with a few con- 

 crete objects and using simple drawing appliances 

 NO. 1820. VOL. 70J 



for quantitative experimental work, the pupil " is 

 helped to build up ideas about lines, points, triangles, 

 circles, &;c., in precisely the same way as that followed 

 in dealing with the elements of physics and chemistry." 

 Deductive reasoning is increasingly employed as the 

 pupil advances, and we think the author is justified 

 in hoping that " there will be as net result an extensive 

 fund of available knowledge on which more advanced 

 work can rest securely, and also such habits of inquiry 

 and thought as will give a stimulus to further study." 



The " New School Arithmetic " by Messrs. Pendle- 

 bury and Robinson is based on the sixteenth edition 

 of the Arithmetic by the former, the first edition of 

 which was published eighteen years ago. During this 

 time commercial conditions have changed, scientific 

 requirements have advanced, and a reform in the teach- 

 ing of elementary mathematics has been inaugurated. 

 The authors have aimed at producing a text-book 

 which shall be fully abreast of the times. Thus metric 

 weights and measures are given and used along with 

 British, and these, being introduced at an early stage, 

 afford excellent concrete examples in illustration of 

 vulgar and decimal fractions. The new style of multi- 

 plication is used exclusively, and thus at the proper 

 time approximate methods become natural and easy. 

 Squared paper and other graphical illustrations are 

 used with good effect, and algebraical symbols are 

 introduced on appropriate occasions. The present 

 volume deals with money, weights and measures, 

 and examples thereon, with vulgar and decimal frac- 

 tions, and with the decimalisation of money. The 

 examples are very numerous and well graduated. The 

 style is simple and clear, and altogether this excellent 

 text-book deserves a wide circulation. 



OIJR BOOKSHELF. 

 Die Vorgeschichte des Menschen. By G. Schwalbe. 



Pp. 52+1 plate. (Brunswick: Vieweg und Sohn, 



1904.) 

 The author of this work is already well known by his 

 writings on Pithecanthropus erectus, the Neanderthal 

 skull, and that of Egisheim. The basis of the 

 pamphlet now before us is a lecture delivered by the 

 author at the meeting of the Society of German 

 Naturalists and Physicians, held at Cassel in 1903, but 

 two valuable appendices have been added to the 

 original lecture. The line of argument runs in the 

 main on palaeontological and anatomical evidence, 

 though the existence of man in pre-Glacial times is 

 regarded as an established fact. The writer claims 

 for the Neanderthal man a specific distinction from the 

 " homo sapiens " of Linnaeus, and would term him 

 " homo primigenius." He traces the relations of this 

 early representative of the human race not only with 

 the Pithecanthropus erectus, but with the Drvopithecus 

 and some of the more anthropoid forms of living apes, 

 and in the illustrative plate gives diagrams of the 

 forms of the different skulls. The agreement of the 

 human remains from Spv, in Belgium, with those 

 from the Neanderthal is accepted, and those from the 

 Krapina cave, in Croatia, though varying in the 

 brachycephalic direction, are regarded as belonging to 

 the " homo primigenius." Mr. Schwalbe seems even 

 inclined to accept evidence of the existence of man in 

 Tertiary times. Whether his conclusions can in all 

 cases be adopted without hesitation or not, his argu- 

 ments are worthy of careful consideration, and the 



