494 



SCIENCE. 



[N. S. Vol. XXIV. No. 616. 



The chief aim in this reform is to bring 

 the mathematical instruction in closer touch 

 with applications. Related variables pre- 

 sent themselves very frequently in the sci- 

 ences, and the training in algebra should 

 prepare the student to deal with these con- 

 cepts in an intelligent manner. More stress 

 should be laid on the study of expressions 

 and less on the analytic transformations 

 and the solution of equations. Graphic 

 representations are especially helpful in 

 the study of the various values of algebraic 

 expressions, and hence these should occupy 

 a more prominent place in algebra. Fortu- 

 nately, these changes are being effected 

 not only in Germany and France, but, per- 

 haps, in a more rapid manner in England 

 and in America. 



The instruction in elementary geometry 

 is at present undergoing the most radical 

 changes, especially in England, which was 

 the last great stronghold of Euclid. The 

 fact that Euclid's 'Elements' remained a 

 standard elementary text-book on geometry 

 for two thousand years reflects less credit 

 on Euclid than disgrace on those who fol- 

 lowed him slavishly. During the last few 

 years England has abolished Euclid servi- 

 tude and her freedom is clearly exhibited 

 in the recent text-booko. The 'Manual of 

 Geometry,' by the well-known author, TV. 

 D. Eggar, is, perhaps, a somewhat extreme 

 case. In this book there is a chapter de- 

 scribing the vernier, spherometer, callipers 

 and the micrometer screw gauge. It also 

 treats briefly of the mensuration of the 

 simpler geometrical solids, trigonometric 

 ratios, and four -place logarithms. 



In France the 'Nouveaux elements de 

 geometrie' by C. Meray has, perhaps, been 

 the most effective in producing a decided 

 advance. In this work the plane and the 

 solid geometry are not separated, the con- 

 cepts parallelism and perpendicularity are 

 established by means of translations and 



rotations, and the theorems relating to the 

 congruence of triangles are developed later 

 than is customary in the classical works. 

 At a recent meeting of the French Associa- 

 tion for the Advancement of Science a 

 resolution was passed requesting the min- 

 ister of public instruction to assist in se- 

 curing a more general adoption of these 

 methods in the text-books. 



AVhile we may not be able to foretell 

 what the elementary geometry of the next 

 decade will be, yet it seems probable that 

 philosophy will largely give way to obser- 

 vation, the abstract to the concrete, the 

 rigor which appeals to the scholar to the 

 elementary methods leading to a clearer 

 understanding of our physical surround- 

 ings. In view of the fact that England 

 was the last one of the great nations to 

 abandon Euclid, it seems likely that she 

 will soon produce the best text-books in ele- 

 mentary geometry, for she is in a position 

 to profit by the experience of the nations 

 which made the transition at an earlier 

 period. 



There are two subjects whose elements 

 should constitute a part of a regular course 

 in elementary geometry, viz., analytic 

 geometry and vector analysis. These open 

 up such wide fields and are of such an ele- 

 mentary character that the average student 

 would enjoy their study much more than 

 the proof that two right angles are equal 

 or that one side of a triangle is less than 

 the sum of the other two. Moreover, both 

 vector analysis and analytic geometry fur- 

 nish simpler and more direct proofs of 

 many theorems than the ordinary methods 

 of elementary geometry and it seems poor 

 policy to teach the poorer methods first. 



The increasing use of graphic methods 

 both in algebra and in the other sciences 

 calls for a much earlier study of the ele- 

 ments of analytic geometry, and the enor- 

 mous progress of physics calls for an early 



