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XVI. Notices respecting New Books. 



Conic Sections by the Methods of Coordinate Geometry. By Charles 



Smith, M.A. New Edition, revised and enlarged. Macmillan 



& Co. : London, 1910. 

 Elements of Analytical Geometry. By Prof. Gr. A. GriBSON and 



Dr. P. Pinkerton. Macmillan & Co. : London, 1911. 

 Coordinate Geometry of Three Dimensions. By H. J. T. Bell, 



M.A., B.Sc. Macmillan & Co. : London, 1911. 

 r [^HESE admirable books are all of the nature of elementary 

 -*- treatises, intended for students beginning the study of coor- 

 dinate geometry. Mr. Smith's treatise has long held its place as a 

 thoroughly sound and well-planned text-book; and in its latest 

 form will no doubt commend itself to a still wider circle of eager 

 students. Among the additions to the new edition is a short 

 chapter on Invariants. 



The text-book written by Professor Gibson and Dr. Pinkerton 

 is somewhat more elementary than the first named, but covers a 

 wider field. The straight line, circle, and conic sections generally 

 are treated separately ; but other curves are considered. One 

 special feature of the book is the great number of excellent 

 diagrams drawn on squared paper. These cannot fail to be of 

 great instructive value to the pupil. Another excellent feature 

 is the early introduction of what Chrystal called the "freedom 

 equation " of a curve, in which as and y, instead of being expressed 

 in terms of one another, are each expressed in terms of a third 

 variable. The method is known in some books as the parametric 

 representation : but there is no doubt that Chrvstal's distinction 

 between the Ereedom and the Constraint equations is one that 

 should be early brought to the mind of the student, and the 

 phraseology is particularly happy. 



Mr. Bell, in his treatise on surfaces and lines in space, makes 

 good use of the Ereedom equations; and it is a pity that he 

 missed the opportunity of calling them by this picturesque title. 

 "Why do not geometers describe the relation among the coordinate 

 axes in terms of right-handed screw motion, instead of defining 

 the relation in terms of counter-clockwise motion ? Why not 

 clockwise? naturally asks the }^outhful critic. But there can be 

 no criticism when he is told that forward motion along the .r-axis 

 is to be combined with right-handed motion round that axis as 

 we pass from the ?/-axis to the 2-axis, and so cyclically through 

 the set. Once the student gets this simple cork-sere w idea into 

 his head he has never any difficulty in laying down his axes in 

 -space. Throughout the book the author uses the old-fashioned 

 expression, "equation to" a curve or surface, instead of the more 

 usual "equation of." These, however, are trifling matters in 

 comparison with the general excellence of the book. 



All three treatises abound in numerous examples and are models 

 of mathematical printing. 



