702 



SCIENCE 



[N. S. Vol. XXX. No. 777 



The nations that have been considered thus 

 far emphasized only the practical side of 

 geometry, and we find with them no plan of 

 education that provided for its systematic in- 

 struction. The early Christian schools taught 

 geometry in a small way, but the practical was 

 almost entirely neglected. The medieval uni- 

 versities made provision for the applications of 

 geometry, but such work was independent of 

 Euclid. 



The Greeks were the first nation that de- 

 veloped and consistently taught a logical sys- 

 tem of geometry. Although they were inter- 

 ested in astronomy and the physical sciences 

 (which undoubtedly stimulated their study of 

 geometry), yet the practical was completely 

 divorced from the logical, as is shown in the 

 text of Euclid. The chief function of educa- 

 tion, according to the old Greek idea, was the 

 perfection of the human being, body and soul. 

 Hence gymnastics and music constituted 

 almost entirely the prograra of studies for the 

 growing boy. When the new education with 

 its philosophy and mathematics entered into 

 the Greek life, it served as the completion of 

 an edifice whose foundations had already been 

 laid. Hence the logic of geometry thrived in 

 Greece. But in the development of this sci- 

 ence, the Greeks were stimulated by a sort of 

 practical aim in attempting to solve the three 

 famous problems of antiquity: the quadrature 

 of the circle, the duplication of the cube, and 

 the trisection of any angle. In the actual 

 teaching of the subject, however, the Greeks 

 were more interested in the chain of reasoning 

 than in the subject-matter itself. 



Logical geometry next found a place in the 

 medieval universities. Under the influence of 

 monasticism and mysticism the church schools 

 were more interested in religious than in intel- 

 lectual things. Then scholasticism arose and 

 dominated European education from the 

 eleventh to the fifteenth century. It sought 

 " to bring reason to the support of faith," and 

 logical inquiry was stimulated. The univer- 

 sities began their careers under such influences, 

 and when Euclid became known to medieval 

 Europe, it found a place in the curricula of 

 these institutions, where it was taught in the 

 highest class. Undoubtedly it was looked 



upon as the instrument that completed and 

 knit together the logical faculties of the mind. 

 The universities did not neglect science. In 

 particular the " sphsera " was studied, but it 

 bore no relation to the logical study of geom- 

 etry. A thing for us to remember is that these 

 institutions followed the example set by the 

 Greeks. Geometry and the physical sciences 

 were both studied, but the former was devel- 

 oped without any reference to the latter. 



In the teaching of geometry, the different 

 European countries have held to the strictly 

 logical in varying degree. Italy has Euclidean 

 traditions, but England above all has taught 

 geometry primarily on the logical basis. 

 Euclid has there reigned supreme. Until re- 

 cent years English higher education has meant 

 the education that fits for the so-called higher 

 callings. The public schools, which prepare 

 largely for the universities, have had this same 

 conception. Any training that smacked of 

 " trade " was not considered to be a part of 

 the education of an English gentleman. The 

 result has been that the classical side has been 

 particularly emphasized and practical educa- 

 tion has been almost neglected. In recent 

 years the modern university movement in 

 England has furthered technical and industrial 

 education, and we now find secondary and 

 higher technical schools that are beginning to 

 fill this gap in the English school system. 



Thus far we have mentioned the marked 

 tendencies among certain nations and institu- 

 tions to hold either to the practical or the 

 logical in the teaching of geometry. With 

 reference to other nations, Germany and 

 Erance, for instance, have never held to the 

 rigors of Euclid, and Russia has begun only 

 in comparatively recent years to emphasize the 

 scientific teaching of geometry. Let us look 

 into the aims in some of those countries where 

 the extremes of aim have not been so disasso- 

 ciated. 



The Renaissance of the fifteenth and six- 

 teenth centuries brought no change in the 

 teaching of mathematics in the universities. 

 This result could hardly be expected from a 

 movement entirely classical in its nature. It 

 was not until the latter part of the seventeenth 

 century, under the influence of realism, that 



