Septesibee 13, 1907] 



SCIENCE 



337 



tically the same form in which it was left 

 by Euclid 2,200 years ago. The great 

 Greek mathematicians, Pythagoras, Plato 

 and Aristotle were, however, primarily 

 philosophers, and the geometry they orig- 

 inated is instinct with Greek idealism. In 

 fact the era of Greek culture may be char- 

 acterized as the adolescent stage in the in- 

 tellectual development of humanity. With 

 the Greeks the worship of the ideal and the 

 beautiful rose to the height of a religious 

 cult, and the chief boast of the founders of 

 geometry was that they had raised it above 

 the common needs of humanity, and ele- 

 vated it to the dignity of pure logic. This 

 view of geometry as typical of the adoles- 

 cence of the race explains why it appeals 

 to youth and at the same time is criticized 

 as unpractical by those of greater maturity. 

 Euclidean geometry, however, should not 

 be viewed from a practical standpoint only, 

 for since the full power of maturity is only 

 attained by the proper unfolding of the 

 preceding stages of childhood and youth, 

 geometry is an important factor in develop- 

 ment, and should not give place to more 

 utilitarian subjects until it has fully served 

 its purpose as a mental stimulus. 



As geometry is a characteristic expres- 

 sion of Greek culture, so algebra sprung 

 from, and fitly symbolizes, the mystic spirit 

 of the Hindoos and their Aryan conquerors. 

 For the modern youth, therefore, the diffi- 

 culties met with in geometry are by no 

 means so great as those encountered in 

 algebra, for the Greek spirit in its two chief 

 features of freedom and individuality has 

 much in common with our own, whereas 

 that of the Hindoos is its exact antithesis. 

 Fettered by the bonds of caste, the Hindoo 

 spirit could not attain objective realization 

 and became lost in a maze of abstraction; 

 the highest good becoming a mere negation 

 of existence both physical and intellectu&l. 

 Moreover, the divinity ascribed to the 



Brahmin caste resulted in the degradation 

 of religion, and the absorption of the 

 spiritual in the merely physical. Thus the 

 morality involved in respect for life and 

 its Creator was lost, and the ideal of virtue 

 was abstraction from all activity. In short, 

 concrete reality gave way to abstraction, 

 imagination became dominant, and spirit 

 was characterized by the fanciful imagin- 

 ings of dreams. The difficulties met with 

 in algebra are therefore inherent in the 

 thought processes involved and can only be 

 lessened by establishing relations with more 

 familiar ideas by the frequent introduction 

 of concrete numerical illustrations. 



Besides explaining the nature of the diffi- 

 culties encountered, the historical method 

 also furnishes a means of estimating their 

 relative magnitude. In other words, the 

 historical method affords a criterion for 

 making a quantitative as well as a qualita- 

 tive estimate of the intellectual content of 

 the subjects considered. Thus the long 

 period of time occupied by the Egyptians 

 in reducing fractions to a working basis is 

 significant as being the prototype of the 

 serious difficulty experienced by the mod- 

 ern youth in attaining an equal proficiency 

 in the subject. The pause which frequent- 

 ly intervenes between two successive stages 

 of development is also significant, and is 

 analogous to that which occurs at intervals 

 in the growth of the child. As in the case 

 of physical growth, so here, the pause 

 marks a drop in potential due to accelerated 

 development, and its length indicates the 

 importance of the next successive advance. 

 It is in fact a sort of hysteresis due to the 

 mental inertia of the race. Political and 

 religious conditions are not sufficient to ac- 

 count for such halts in progress. Although 

 social conditions must be recognized as 

 powerful factors in aiding or arresting 

 development, j'et from the standpoint of 

 universal history such external relations 



