44 



SCIENCE 



[N. S. Vol. XXX. No. 758 



serts that greater rigor should be intro- 

 duced, quoting Hilbert as saying that it is 

 a.- error to believe that rigor in the proof 

 is tlie enemy of simplicity. 



With the new powers of insight given by 

 the non-Euclidean geometry, and the intro- 

 duction of Lobachevski 's new principle in 

 geometry, it was found that even Euclid 

 made implicit assumptions. Thus, to make 

 an angle congruent to a given angle in- 

 volves a continuity assumption, while to 

 prove other propositions requires a new set 

 of assumptions which Halsted calls "be- 

 tweenness assumptions," viz., of any three 

 points of a straight line, there is always 

 one, and only one, which lies between the 

 other two. 



The Euclidean method of superposition 

 is also characterized as a worthless device ; 

 for if triangles are spatial but not material, 

 there is a logical contradiction in the no- 

 tion of moving them, while if they are 

 material, they can not be perfectly rigid, 

 and when superposed they are certain to be 

 slightly deformed from the shape they had 

 before. 



Furthermore, so-called hypothetical con- 

 structions found in most text-books are 

 criticized as illogical. Thus, certain propo- 

 sitions may require the construction of a 

 regular heptagon or the trisection of an 

 angle, although such constructions are im- 

 possible by elementary geometry. Thus, in 

 many constructions, existential propositions 

 are assumed. Helmholtz says of this: "In 

 drawing any subsidiary line for the pur- 

 pose of demonstration, the well-trained 

 ■geometer asks always if it is possible to 

 draw such a line. ' ' 



This leads to the importance of not 

 placing too great reliance upon diagrams. 

 Bertram Eussell says of Euclid I., that the 

 first proposition assimies that the circles 

 used in the construction intersect, an as- 

 sumption not noticed by Euclid because of 



the dangerous habit of using figures. Hil- 

 bert believes in making frequent use of 

 figures, but never depending upon them. 

 The operations undertaken on a figure must 

 always retain a purely logical validity. 

 Halsted says that the beginners' course 

 should consist largely in becoming familiar 

 with figures, while in rational geometry 

 that treatment of a proposition is best 

 which connects it most closely with a visual- 

 ization of the figures. 



In rigorously founding a science, he be- 

 lieves that we should begin by setting up 

 a system of assumptions containing an ex- 

 act and complete description of the rela- 

 tions between the elementary concepts of 

 this science. These axioms are at the same 

 time the definitions of these elementary 

 concepts. No statement within the science 

 should be admitted as exact unless it can 

 be derived from these assumptions by a 

 definite number of logical deductions. 



This criticism of the Perry movement 

 and the laboratory method of instruction 

 has recently been summarized by Professor 

 Halsted as follows ■^'^ 



We knew that the so-called laboratory method 

 for mathematics, the " measuring " method, was 

 rotten at the core, since mathematics is not an 

 experimental science, since no theorem of arith- 

 metic, algebra or geometry can be proved by 

 measurement; but, even granting the impossible, 

 granting the super-human power of precise meas- 

 urement, we could not thereby ever prove our 

 space Euclidean, ever prove it the space taught 

 in all our text-books. 



THE PRACTICAL VIEW OP MATHEMATICS AS 

 THE EXTENSION OP EXPERIENCE 



One of the most sane and sensible views 

 of the teaching of elementary mathematics 

 yet presented is due to Professor Simon 

 Newcomb.^' 



After recognizing the great difficulties 



^ " Even Perfect Measuring Impotent," Halsted, 

 Science, October 25, 1907. 



^ Newcomb, Educational Review, Vol. 4. 



