98 



X. I 77 T RE 



[November 30, 1905 



ingly he made many investigations upon special 

 quartii curves, both plane and twisted; he considered 

 the meaning of a focus of a curve traced on a sphere, 

 showing that the method of inversion changed the 

 ordinary foci of such a curve into foci of the inverse' 

 curve; he published many investigations upon a class 

 of loci to which at the time considerable attention 

 was paid, anallagmatic curves and surfaces. Other 

 papers treat of the cyclide of Dupin, the curvi "l 

 intersection of two quadric surfaces, and a family of 

 curves called b\ Laguerre Cassinians : and it is to 

 be noted that while writing upon a particular curve 

 he would at limes include theorems of wider appli- 

 cation. 



A passing mention may be accorded to an interest- 

 ing statement of the addition theorem of hyperelliptic 

 functions closely resembling that derived from 

 Poncelet's polygons in elliptic functions; to researches 

 upon Steiner's Roman surface and its reciprocal, 

 known as Cayley's cubic surface, and other appli- 

 cations of the theory of forms to geometry. Finally, 

 at the end of the book we meet with a series of papers 

 in which Laguerre's discovery of geometry of direc- 

 tion is developed. The idea from which this sprang 

 is elementary enough; a straight line or a circle may 

 be traced out by a moving point in two opposite 

 senses, and therefore is regarded by Laguerre as com- 

 posed uf two " half-lines " or two cycles. The notion 

 of tangency is modified when a curve is described 

 in a definite direction, so that a cycle is regarded as 

 possessing one tangent only parallel to a given half- 

 line. Following up this thought, Laguerre is led to 

 divide all curves into curves of direction, which can 

 be divided analytically into two trajectories traced 

 out in different senses, and curves which have not 

 this propertj ; he finds the form of the tangential 

 equation of the most general curve of direction. Bv 

 help of a hiehly ingenious " transformation by re- 

 ciprocal half-lines," it is shown how certain problems 

 may be greatly simplified; the problem of drawing 

 a circle to touch three given circles, for example, is 

 reduced to that of drawing a circle through three- 

 points. The theory is extended also to spherical 

 geometry. 



Laguerre's life-work in geometry forms a volume 

 which no mathematician can study without being 

 profoundly impressed by the ingenuity of the author 

 and his skill in handling every method which he 

 employs; papers such as his, models of clear polished 

 style, are read with keen intellectual enjoyment. 

 Yet when the book is laid down and we reflect 

 on the work as a whole, there comes a regretful con- 

 viction that what has been accomplished is very far 

 from all that could have been hoped for from the 

 powers of the author and his brilliant first achieve- 

 ment. Delicate in health and of retired life, 

 Laguerre's isolation from the march of scientific 

 thought is betrayed in his writings. General notions 

 appealed to him solely by their applicability to par- 

 ticular problems, and he therefore chose to bestow 

 the utmost care upon a number of short discussions 

 pi special topics. Let the reader, if In- would 

 appreciate what is best in these collected writings of 



NO. T 8 8 3 , VOL. 73] 



Laguerre, realise when he takes up the book that 

 its author was one of those who are content to apply 

 to small things powers capable of far higher work, 

 and he will find matter to arouse his interest and 

 admiration in every paper reprinted in the volume. 



PHILOSOPHICAL STUDIES. 



(1) Goethe's Philosophie aus seinen Werken. Edited 

 with an introduction, by Max Heynacher. Pp 

 viii + 428. (Leipzig: Diirr'sche Buchhandlung 

 1005.) Price 3.60 marks. 



(2) Immanuel Kant, Physische Geographie. Second 

 edition. Edited by Paul Gedan. Pp. xxx + 386. 

 (Leipzig: Diirr'sche Buchhandlung, 1905.) Price 

 2.50 marks. 



(3) Dialoge iiber naturliehe Religion, nbcr Selbstmorc 

 mid Unsterblichkeit der Seele. By David Ilium 

 Translated into German and edited by Dr. F 

 Paulsen. Third edition. Pp. 165. (Leipzig . 

 Diirr'sche Buchhandlung, 1905.) Price 1.50 marks 



(4) Immanuel Kant's Kleinere Schriften our Logik 

 mid Metaphysik. Second edition. Edited by Karl 

 \ oil uicler. In four parts. Pp. xxxii+169, xl + 

 172, xx+175, xxxi+i7b. (Leipzig: Diirr'sche 

 Buchhandlung, 1905.) Price 5.20 marks. 



(5) (,'. 11'. F. Hegel, Encyclopiidie dcr philosophischen 

 Wissenschaften im Grundrisse. Second edition. 

 Edited bj Georg Lasson. Pp. Ixxvi + 522. 

 (Leipzig: Diirr'sche Buchhandlung, 1905.) Price 

 3.60 marks. 



(t) /"* OETHE'S work was so many-sided, and 

 VJ withal so voluminous, that it is a real 

 service to educated thought to have presented, as 

 here, a volume of extracts, in moderate compass, con- 

 taining in his own words an account of the great 

 writer's philosophic and scientific views, and of the 

 influences exerted on him by different systems. 

 Herder, Spinoza and Kant all obviously attracted 

 him at various times, and his name must find a place 

 in any account of the theory of colour or of com- 

 parative anatomy -to name only two of the scientific 

 subjects in which he was interested. With these and 

 kindred matters the editor deals in a well-informed 

 introduction. He knows the literature well, his 

 Eckermann, the Goethe Jahrbuch, and Goethe's 

 poetry. Goethe's title to be regarded as a fore- 

 runner of Darwin is duly emphasised. 



(2) That Kant should thus have lined the wings 

 of his spirit in the droits of the sensible world will 

 astonish the average reader, for this work con- 

 descends to minute details regarding the animal, 

 vegetable, and mineral kingdoms, the characteristics 

 of different races of men, and the like. Even one of 

 the earlier parts, dealing with mathematical pre- 

 liminaries, is not at all speculative in its nature, and 

 only one or two paragraphs in the introduction, which 

 point out that geography deals with facts in space 

 as history with events in lime, remind us of the 

 Critique of Pure Reason; but the services of Kant to 

 geography are not negligible, and have been attested 

 by Hehnhollz. 



The present edition contains a full statement of 



