Septembeb 13, 1907] 



SCIENCE 



339 



matics, and if used for this purpose in 

 the lower grades of instruction will prove 

 a valuable adjunct to the methods ordi- 

 narily followed. So far from being unique 

 in its inception and aims, however, it is 

 merely a corollary of the historical method, 

 and can only be used to advantage when it 

 is recognized as such. 



Another corollary of the historical meth- 

 od is what is known as the "spiral method" 

 of instruction. This consists iu taking the 

 pupil several times over the same ground, 

 but each time reaching a higher level and 

 attaining a more general point of view. 

 The method is founded mainly on experi- 

 ence, but its theoretical basis is evidently 

 historical. 



The specific application of the historical 

 method to mathematical pedagogy consists 

 primarily in obtaining the proper historical 

 perspective. From this aspect its principal 

 use is in arranging the details of a curricu- 

 lum, and a few suggestions follow relative 

 to its application for this purpose. 



Perhaps the most obvious suggestion is 

 that subjects which developed simultane- 

 ously should form parallel courses instead 

 of being taught serially, as is now common 

 in all mathematical instruction. For in- 

 stance, algebra and geometry originated 

 simultaneously and served as a mutual 

 stimulus to growth and development. It 

 is evident, therefore, that it is possible to 

 teach these subjects in the same academic 

 grade, and that they can undoubtedly be 

 made mutually helpful by so doing. This 

 opinion is verified by the fact that this 

 method has been used for some time in the 

 higher schools of Prussia with results which 

 indicate a decided advantage for such cor- 

 relation of subjects.^ 



Following out the historical idea, the 

 curriculum should be based on a thorough 

 •J. W. A. Young, "The Teaching of Mathe- 

 matics in the Higher Schools of Prussia." 



grounding in the principles of number, the 

 amount of time devoted to the several sub- 

 jects being proportionate to their relative 

 difficulty as indicated by their historical 

 rate of development. This should be fol- 

 lowed by a course in elementary algebra, 

 taught as a generalization of arithmetical 

 ideas, and accompanied by a parallel course 

 in elementary geometry. The course in 

 elementary algebra would naturally consist 

 in a logical development of the six funda- 

 mental processes, including logarithms. 

 At present the latter usually follows quad- 

 ratic equations and the binomial theorem, 

 whereas historically it precedes both. The 

 natural sequence is, in fact, to teach multi- 

 plication as an abbreviation of addition, 

 thus leading to the theory of exponents, 

 and then passing to logarithms as an ab- 

 breviation of multiplication. Historically 

 the subject of logarithms arose in this con- 

 nection, having been invented by Napier 

 about 1614 for the purpose of facilitating 

 the long numerical calculations fashionable 

 in his day.. 



Conforming to the natural lines of de- 

 markation, these elementary courses would 

 be succeeded by advanced courses in 

 algebra and solid geometry, the former be- 

 ginning with simple equations and em- 

 phasizing chiefly the theory of equations. 

 At present the natural sequence is not 

 followed in teaching algebra, at least three 

 subjects, namely, proportion, logarithms 

 and series being out of proper historical 

 perspective. Proportion, or the old- 

 fashioned "rule of three," was developed 

 by the Hindoos for the solution of 

 numerical equations by the "rule of 

 false assumption, ' ' and as it is now obsolete 

 for this purpose, does not properly belong 

 in algebra, and should be reserved for 

 arithmetic and geometry, where it properly 

 has a place. The proper setting for loga- 

 rithms has already been mentioned. As 



