REVIEWS 677 



by a tendency that students may have towards a hasty satisfaction with results, 

 such as existence-theorems, that are complete solutions only theoretically. Such 

 theorems are nearly quite unconnected with the practical needs that are felt 

 in every application of mathematical reasoning for numerical and graphical 

 calculations. Of the present tracts, all, with the exception of Nos. 3 and 6, 

 are concise guides to such practical applications. The tracts are very well 

 printed on good paper ; and the covers are fairly thick paper and of a pleasant 

 dark green colour. It would be a good thing if the example of the more 

 modern numbers of Ostwald's Klassiker could be followed in that the covers 

 were made still more stiff. The fault is one which is shared by the Cambridge 

 Tracts, and must, to a certain extent, interfere with the usefulness of an excellent 

 series. 



There are, broadly speaking, two main directions in which the teaching of 

 science should be reformed. In the first place, we may require, on what seem 

 to be conclusive grounds, that the subjects should be made more interesting and 

 more readily grasped by the student, by a very free use of historical and critical 

 methods. In the second place, a tendency has grown up of late years, even in 

 pure mathematics, to make practical work of modelling, drawing, visualising, and 

 calculating play a great part in education. All intelligent teachers will welcome 

 efforts in both directions. Both directions are followed in these tracts, although 

 the historical method is much less emphasised in them than the' practical. 

 Indeed, even some space is devoted to describing a thorough systematisation of 

 the practical laboratory conveniences for drawing and calculating. Yet several 

 valuable historical notes and aids to historical study will be found in them. The 

 originality in these tracts is to be sought rather in the choice of subjects and 

 method of treatment than results. 



It is important to develop the student's ability for visualisation so that he can 

 form easily a mental picture of three dimensional systems, and for this purpose the 

 best thing is a course in descriptive geometry — a subject which has been unduly 

 neglected in Great Britain. In No. 1, in which the influence of Monge is 

 naturally very pronounced, there is a discussion of the purpose and methods of 

 descriptive geometry ; of the straight line, plane, surfaces, and curves of double 

 curvature, in orthogonal projection ; and of perspective. An admirable account 

 of the evolution of descriptive geometry is given (p. 13) ; and the final chapter 

 is devoted to photogrammetry— the method of obtaining from ordinary photographs 

 correct metrical details about three dimensions. 



No. 5 is an account of various methods— numerical and graphical— of 

 solution of spherical triangles. The subject is important in applications to 

 astronomy and navigation, and develops the power of dealing with questions in 

 the geometry of position. 



No. 2 is devoted to the practice of interpolation and numerical integra- 

 tion, and contains chapters on some theorems in the calculus of finite 

 differences ; formulas of interpolation ; the construction and use of tables ; 

 numerical and integration. There are a large number of examples for the student 

 to work out. 



No. 4 is also mainly concerned with practical applications, and the theory 

 of Fourier's series is left on one side. Numerical and graphical methods in 

 harmonic analysis are also dealt with ; and a new feature in text-books is 

 " periodogram analysis," in which we have to find the incommensurable periods 

 of periodic terms whose sum may represent a given not purely periodic 

 graph. 



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