November 19, 1900] 



SCIENCE 



701 



drawal and date are noted on this card, which 

 is then refiled. Card 12 of the numerical in- 

 dex is withdrawn and replaced by a new one, 

 at the same time peg 12 is taken from the 

 board and placed in the side compartments, 

 the vacant peg-hole showing at a glance the 

 availability of this locker. In assignment 

 of available lockers, one need only bear in 

 mind that two pegs of the same color can not 

 be placed on the same desk, and thereby con- 

 flict will be avoided. Thus, without multi- 

 plying examples, it at once becomes apparent 

 that this system gives one a ready and simple 

 control of the laboratories. By this system, 

 classes of seven hundred are handled with 

 great facility. 



The writer wishes to express his thanks to 

 Mr. Harry Mougey, of this laboratory, for 

 several suggestions made in the construction 

 of the above board. Wm. Lloyd Evaxs 



Ohio State University, 

 September 8, 1909 



EDUCATIONAL AlilS IN THE TEACIJING OF 



ELEMENTARY GEOMETRY, JllHTORW- 



ALLY CONSIDERED^ 



The two educational aims that have stood 

 out distinctly in the history of the teaching of 

 geometry are the practical' and the logical. 

 Of course in the early development of geom- 

 etry the term teaching can not be used with 

 its modern significance. The practical side of 

 geometry was developed by the Babylonians, 

 the Egyptians and the Romans; the logical by 

 the Greeks. In the medieval universities the 

 little geometry taught was according to Euclid. 

 England has followed the same standard to the 

 present day. The other European countries, 

 for the most part, have combined both of these 

 aims, and this obtains to-day, with the empha- 



' See the author's " A History of the Teaching 

 of Elementary Geometry," Teachers College Con- 

 tributions to Education, Xo. 23, for the original 

 and secondary sources consulted. The present 

 article is not an integral part of the larger work, 

 but material from the latter is utilized in the 

 former. 



'The term practical is used with reference to 

 the applications of geometry within the field of 

 mathematics or in the related fields of science. 



sis on the logical. The same is true in the 

 United States. A third aim in the teaching 

 of geometry arose when the secondary schools 

 began to assume the character of preparatory 

 schools for the universities. The last hundred 

 years have seen this generally brought about, 

 and within the last fifty years it has been fully 

 systematized in the various countries. In 

 treating these several aims it is impossible to 

 completely separate them. 



The early Egyptians and Babylonians de- 

 veloped geometry as a means toward a prac- 

 tical end. Both nations were interested in 

 astronomy, and hence a rudimentary geometry 

 found a place with them. The Egyptians em- 

 ployed geometric principles in the building of 

 their pyramids and in surveying. They meas- 

 ured lengths and areas, they built solids of 

 regular design, they showed some skill in 

 geometric drawing in their mural decorations. 

 With all this they experienced the necessary 

 propaedeutics for a developed science, yet this 

 development never came. Whether it was the 

 lack of God-given powers or due to the con- 

 servatism of the priestly class, that sacredly 

 guarded the learning, one can only conjecture. 



The Romans, also, valued geometry for its 

 utility, employing it in architecture and in 

 surveying. But, unlike the Egyptians, they 

 had the learning of other nations to draw 

 upon. This development in architecture and 

 surveying was marked in the first century be- 

 fore and the first century after Christ. Euclid 

 had written his " Elements " approximately 

 three hundred years earlier. Archimedes had 

 already developed geometry as applied to me- 

 chanics, and Heron of Alexandria, who studied 

 and wrote on practical geometry and survey- 

 ing, lived in the early years of this " Roman " 

 period. The work of Heron influenced the 

 Roman surveyors, but Euclid found little 

 favor with the Romans. When the " Ele- 

 ments " was recognized at all, it was that it 

 might be of aid in the training of the orator, 

 which was, for the Romans, a practical aim. 

 In like manner the Hindus and Arabians 

 studied geometry primarily for its practical 

 value, although both of these nations were 

 largely dependent upon the Greeks for their 

 knowledge of geometry. 



