184 



SCIENCE. 



[N. S. Vol. XXI. Xo. 527. 



suiaed relations are adopted which lead to 

 Euclidean geometry. In this respect the au- 

 thor is appealing to the attention of element- 

 ary schools, where no geometry other than 

 the practical geometry of our world has a 

 right to be taught. 



The first chapter deals with the first group 

 of assumptions, the assumptions of associa- 

 tion. Thus, the first assumption is that two 

 dinlinct points determine a straight line. This 

 associates two things called points with a thing 

 called a straight line, and is not a definition 

 of the straight line. The definition of a 

 straight line as the shortest distance between 

 two points involves at once an unnamed as- 

 sumption, the conception of distance, which is 

 a product of our physical senses, whereas the 

 rational develojiment of geometry seeks the 

 assumptions which imderlie and are the 

 foundations of our physical senses. In the 

 higher court of pure reason, the testimony of 

 our physical senses has been ruled out, not as 

 utterly incompetent, but as not conforming to 

 the legal requirements of the court. However, 

 there is no objection to shortness in names, 

 and a straight line is contracted into a 

 straight, a segment of a straight line, to a 

 sect, etc. 



In the second chapter we find the second 

 group of assumptions, the assumptions of be- 

 tweenness, which develop this idea and the 

 related idea of the arrangement of points. In 

 the next chapter we have a third group, the 

 assumptions, of congruence. This chapter 

 covers very nearly the ordinary ground, with 

 respect to the congruence of angles and tri- 

 angles, and all the theory of perpendiculars 

 and parallels which does not depend upon 

 Euclid's famous postulate. This postulate 

 and its consequences are considered in chapter 

 IV. 



All the school propositions of both plane 

 and solid geometry are eventually developed, 

 although there is some displacement in the 

 order of propositions, due to the method of 

 development. Numerous exercises are ap- 

 pended at the end of chapters, which are num- 

 bered consecutively from 1 to YOO. 



T'ndoubtedly the enforcement upon logic of 

 a blindness to all sense perceptions introduces 



some difficulties which the ordinary geometries 

 seem to avoid, but as in the case of our con- 

 ception of a blind justice, this has its com- 

 pensation in the greater weight of her de- 

 cisions. It seems as if the present text-book 

 ought not to be above the heads of the average 

 elementary students, and that it should serve 

 to develop logical i)0wer as well as practical 

 geometrical ideas. Doubtless, some progress- 

 ive teachers will be found who will venture 

 to give it a trial, and thus put it to the tests 

 of experience. At least, the work will appear 

 as a wholesome contrast to many elementary 

 geometries which have been constructed on 

 any fanciful plan of plausible logic, mainly 

 with an eye to the chance of profit. 



Arthur S. Hathaway. 

 Rose Polytechnic Institute. 



A Treatise on the British Freshwater Algo". 



By G. S. West, M.A., A.R.C.S., F.L.S. 



Cambridge, The University Press. 1904. 



Certainly there is no book upon any phase 

 of cryptogamic botany for which there has 

 been so much need, and for which the demand, 

 in recent years, has been so great, as one deal- 

 ing comprehensively with the fresh-water 

 alga". It is nearly twenty years since any 

 work of the kind has appeared in English, 

 and whatever may have been said in favor of 

 the works of Cooke and WoUe when they were 

 l)ublished, there can be no question about their 

 having been out of date for a long time. 

 Indeed, the tremendous strides made in algol- 

 ogy during the last ten years has made it diffi- 

 cult for any one but the specialists to keep 

 informed regarding the physiology, phylogeny 

 and morphology of this group, to say nothing 

 of the new genera and species. Of the fresh- 

 water algJE alone, ajjproximately one fourth of 

 the genera now recognized have been described 

 since the appearance of Engler and Prantl's 

 classification in 1890. Consequently, nearly 

 all of the important literature upon the alga; 

 has been in periodicals and separates, often 

 difficult to obtain, the result being that the 

 general student of botany has, of necessity, 

 been many years behind in his ideas regard- 

 ing this most imi)ortant and interesting group 

 of plants. 



