518 Notices respecting New Books. 



free from inaccuracies. Thus, on page 340, § 221, the word 

 "circuitous" (which does not express " umst'dndlich") does not 

 convey the author's opinion of a method necessarily full of detail 

 and one which is required in certain special circumstances. A 

 few other errors such as "normal" for " tangential " (page 83, 

 theorem iv.), and the occasional use of "magnitude" for "quantity" 

 are the most serious defects. 



The author has made several additions containing results 

 obtained chiefly during the past year. 



The printing and general appearance of the book are excellent, 

 and Dr. Atkinson deserves the thanks of English physicists for 

 introducing to them so important an addition to the fascinating 

 subject of Magnetism. 



"Die Ausdehnungslehre, vollstandig und in strenger Form bearbeitet 

 von Hermann Grassmann (Berlin, 1862)." 



This book is the second part of the first volume of the miscella- 

 neous mathematical and physical works of their illustrious author, 

 which are in course of publication under the editorship of Dr. Er. 

 Engel, in cooperation with a strong body of five other eminent 

 mathematicians. The volumes are being brought out at Leipzig, 

 and the one before us gives a reprint of the edition brought 

 out in Grassmann's lifetime in 1862. Clifford would have 

 been delighted with this edition. In 1878 he contributed to 

 the first volume of the ' American Journal of Mathematics ' 

 an article on "Applications of Grassmann's Extensive Algebra " 

 (Mathematical Papers, pp. 266-276), which opens with the 

 statement: — "Until recently I was unacquainted with the Aus- 

 dehnungsleJire, and knew only so much of it as is contained 

 in the author's geometrical papers in Crellis Journal and in 

 1 HankeFs Lectures on Complex Numbers.' I may, perhaps, there- 

 fore be permitted to express my profound admiration of that 

 extraordinary work, and my conviction that its principles will 

 exercise a vast influence upon the future of mathematical science." 

 Prof. Tait writes': — " Hamilton and Grassmann, while their earlier 

 work had much in common, had very different objects in view. 

 Hamilton had geometrical application as his main object ; when 

 he realized the quaternion system, he felt that his object was 

 gained, and thenceforth confined himself to the development of 

 his method. Grassmann's object seems to have been, all along, of 

 a much more ambitious character, viz. to discover, if possible, a 

 system or systems in which every conceivable mode of dealing 

 with sets should be included. That he made very great advances 

 towards the attainment of this object all will allow; that his 

 method, even as completed in 1862, fully attains it is not so 

 certain" ("Quaternions" in Encycl. Brit. 9th edit., where more 

 remarks on the same writers will be found). After these quota- 

 tions as to the value of Grassmann's work, we need only state the 

 following partciulars. There are a few words, by way of preface, 

 by the editor, from which we learn that the text of the " 1862 " 

 edition (the Jirst edition was published in 1842) has been carefully 



