6i6 



NATURE 



[October 24, 1895 



area has been observed upon stimulation of the central 

 end of the sciatic ner\-e during chloral ' and pyridin - 

 poisoning, showing the influence exerted by the condition 

 of the centre at the time of peripheral stimulation. 



Of the second, the so-called " practical " part of the 

 book, we have little to say. From what wc ha\ e read, 

 we regard Dr. Gillies' practice as no sounder than his 

 theories. The reprint with which he provides us of 

 Dr. Davies' original communication on blistering in acute 

 rheumatism, and the controversy thereon, is the most 

 interesting part of the book. We should like to know 

 who it is who believes that the " scrum" is '■'■ ahundunily 

 charged with lactic acid " in acute rheumatism ; and, 

 supposing it was, how much one is likely to get from 

 the serum, say, of half a dozen blisters ? (p. 88.) To 

 sum up our remarks, we do not consider the book of 

 value either to physicians or physiologists. The facts 

 it contains are not new, and the theories do not justify 

 their existence, since they fail to fulfil the conditions 

 which should be demanded of all hypotheses, viz. to 

 indicate lines of research which shall offer a reasonable 

 hope of increasing our knowledge. One merit which it 

 possesses, is that it may draw attention to some valuable 

 pieces of work which might perhaps otherwise have been 

 disregarded. F- W. T. 



A NEW DEPARTURE IN GEOMETRY. 

 Die Gruniigt-bilde der ebencn Geometric. By Dr. V. 

 Ebcrhard, Professor at the University of Kiinigsbergi.P. 

 Bd. I. 8vo. xIviii. + 302 pp. Five plates. (Leipzig: 

 Teubner, 1895.) 



THE hislor)' of Analytical Geomctr\' affords a curious 

 subject of study to the thoughtful mathematician. 

 It would seem that equations between coordinates were 

 first used to express spatial relations discovered by 

 intuitional processes, and the equations were combined 

 algebraically to discover other implied spatial relations. 

 For this purpose it was necessary to interpret in 

 geometrical terms equations arrived at by algebraic 

 processes from geometrical data, and the facility thus 

 acquired led men to seek for similar interpretations of 

 equations set down without reference to geometrical 

 conditions. Hence it happens that modern developments 

 of Analytical Geometry appear rather to present 

 algebraic facts in geometrical language than to deduce 

 results that can be apprehended by intuition from data 

 of intuition. Such a notion as that of a cubic surface, 

 for instance, would seem to be essentially analytical, and 

 although it has been proved possible to arrange a 

 geometrical construction for an algebraic curve whose 

 equation is given, yet the construction arrived at is so 

 artificial that intuition fails to grasp by its aid the 

 necessary form of the curve. Looking at the subject in 

 this way, it seems hardly too much to say that the algebra 

 which was designed to be the ser\ant of the geometer 

 has become his master. 



.Some such reflections as these form the starting-point 

 of Dr. Eberhard's work. The volume under notice is to 

 be the first of a series, and in his long preface ' he sets 



If 

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tract, ■*• ■()> w^f 



'togy, xvii. p. 37a. 

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 ri umt Zicic dcT Kaumlclirc" 



forth his aim and method. Here, after tracing the origin 

 in experience of simple geometrical notions such as those 

 of the straight line and the plane, he divides cur\-es and 

 surfaces into two classes, the regular (gesetzmassig) and 

 the fortuitous (zufallig), and proceeds to inquire after 

 intuitional criteria available for distinguishing between 

 them. He defines a regular locus as one in which a 

 relation that can be apprehended by intuition connects 

 a variable point of the locus with a finite number of 

 points fixed in it. The kind of relation which he admits 

 as capable of being apprehended by intuition is 

 essentially topographical. This will be elucidated by 

 considering the example he gives. Let a system of 

 points be taken, and let planes be drawn through them 

 three by three. These planes will in general intersect in 

 other points besides those of the original system. Let 

 planes be now drawn through the points of the extended 

 system three by three. These planes will again intersect in 

 some new points, and the process can be continued. Let 

 the process be arrested at any stage, and suppose a set 

 of four points of the extended system lie in one plane. If 

 one of the points of the original system were slightly dis- 

 placed these four points would generally cease to lie 

 in one plane, but if the particular point of the original 

 system were displaced on a certain surface, the four points 

 would remain in a plane. This property constitutes a 

 definition of the surface available for intuitional geometry. 

 It will be seen from the example that the method rests 

 upon the topographical relations of systems of points. 



The description of these relations for a given system 

 can be carried out systematically, and the process con» 

 sists in the use of two related notions. The first is the 

 notion of " characteristics," and the second is the notion 

 of the "index " of a point in a plane system. If three 

 points out of four are taken in a definite order, the triangle 

 formed by them is described in the positive or negative 

 sense by an observer on the same side of their plane as 

 the fourth point. The sense of description of the triangle 

 formed by three points in a definite order for an obser\ cr 

 on a definite side of their plane is the cliaracteristic of 

 the three. The index of a point in a plane system is the 

 order in which a line turning about that point meets the 

 other points of the system. A statement of the indices 

 simplifies the problem of stating the characteristics. 



The bulk of the present volume is taken up with 

 theorems concerning the characteristics and index- 

 systems of groups of points in a plane, and they arc fully 

 exemplified in the cases of groups of four, five, and six 

 points. In an investigation of so novel a character we 

 find, as we might expect, original methods of working 

 and difficult arguments. The w.int of figures in illustra- 

 tion of the earlier chapters, and some of the notations 

 employed, combine with the nature of the subject to- 

 render the hook difficult to read. 



The endeavour to make the geometry of curves and 

 surfaces of high degrees more intuitive is laudable, a new 

 classification of loci founded on geometric rather than 

 algebraic principles is also ;i worthy object of research, 

 and the idea of grounding such a classification in topo- 

 graphical circumstances is ingenious ; but a final judgment 

 as to Dr. Eberhard's success in these directions can only 

 be pronounced after his complete work has been given to | 

 the worid. A. E. H. L. 



NO. 1356, VOL. 52] 



