278 



SCIENCE. 



[N. S. Vol. V. No. 111. 



Hyde (51 pp.); IX. 'Vector analysis and 

 quaternions,' by Alexander Macfarlane (42pp.); 

 X. ' Probability and theory of errors, ' by Rob- 

 ert S. Woodward (40pp.); XI. 'History of mod- 

 ern mathematics,' by David Eugene Smith 

 (63 pp.). 



That this collection of comparatively brief 

 and disconnected chapters, however well they 

 may be written, could be used successfully as a 

 text-book may appear doubtful. Most of the 

 chapters are too short to serve as a satisfactory 

 text for a college course. Nevertheless, the 

 work is an exceedingly valuable one. The ad- 

 vantage to be gained by putting into the hands 

 of the student a work covering so wide a range, 

 in a form so attractive and easily accessible even 

 without the assistance of a teacher, can hardly 

 be overestimated. Both as an incentive to 

 further study and as a book of reference, the 

 volume will be of great service. 



The proper selection and apportionment of 

 subjects for such a general introduction to 

 higher mathematics is a matter of great difiB- 

 culty; on the whole, the selection has been 

 made with excellent judgment. It is certainly 

 to be regretted that the proposed chapter on 

 elliptic functions had to be omitted; the sub- 

 jects treated in chapters I., II., IV., VIII. and 

 IX. would have been missed far less. Modern 

 analytic geometry, the theory of substitutions 

 and groups with its applications, non-Euclidean 

 geometry, quantics and theoretical mechanics 

 were probably excluded as too advanced or as 

 not allowing of brief presentation. 



From the point of view of pure mathematics 

 the most interesting chapters in the book are 

 Professor Fiske's 'Functions of a Complex 

 Variable' and Professor Halsted's 'Projective 

 Geometi-y. ' The geometric mode of treatment 

 which characterizes the first third of Professor 

 Fiske's chapter will arouse the interest and self- 

 activity of the student and thus prepare him for 

 the more arduous analytical investigation of the 

 critical points of the simplest monogenic func- 

 tions which occupies the remainder of the 

 chapter. The whole is written with the great- 

 est care, and although this is the longest chap- 

 ter in the book one cannot help regretting that 

 it is not longer. In but one or two cases con- 

 ciseness seems to be carried so far as to en- 



danger clearness — for instance, in the definition 

 of uniform convergence (p. 274); but in general 

 the presentation is as clear as it is precise. 



Professor Halsted gives us a carefully worked- 

 out exposition of von Staudt's system of syn- 

 thetic geometry. The logical development, as 

 was to be expected, is admirable ; the form of 

 presentation is exceedingly concise and neat. 

 A mathematician familiar with the subject and 

 with von Staudt's terminology may read this 

 chapter with pleasure. But the beginner, for 

 whom this volume is intended, will be sorely 

 perplexed. Even if he has energy and patience 

 enough to learn the new language here spoken, 

 and comes to understand such phrases as "A 

 tetrastim with dots in a conic has each pair of 

 codots costraight with a pair of fanpoints of the 

 tetragram of tangents at the dots" (Art. 91, 

 p. 85), or "Two correlated polystims whose 

 paired dots and codots have their joins copunc- 

 tal are called 'coplanar' " (Art. 51, p. 76), of 

 what use is this to him? Few persons will 

 understand him, and he himself will be unable 

 to understand the masters who have written, 

 and are still writing, on the science of projective 

 geometry. But, even apart from this passion 

 for coining new words, it seems to the writer 

 that the rigid formality and exclusiveness of 

 the treatment here adopted tends to make a 

 naturally easy and attractive subject unneces- 

 sarily diflBcult and almost forbidding to the be- 

 ginner, and to give him a one-sided idea of 

 what is now meant by projective geometry. To 

 mention a minor point, a reference to von 

 Staudt's 'Geometric der Lage,' which, by a 

 curious oversight, is nowhere mentioned, would 

 have been in place in connection with the ' fun- 

 damental theorem ' of Art. 59 (p. 77), which 

 corresponds to von Staudt's Art. 88. The 

 printer is probably responsible for assigning 

 Pappus to the age of Plato (p. 104). 



To the student of applied mathematics the 

 chapters on 'Harmonic Functions' and on 

 ' Probability and Theory of Errors ' will prove 

 of most value. The first half of Professor 

 Woodward's chapter treats of the theory of 

 probability proper, beginning with permuta- 

 tions and leading up to Bernoulli's theorem ; 

 the latter half, on 'laws of error,' is par- 

 ticularly valuable as embodying the results 



