TWENTY-SEVEyTH ANNUAL MEETING. 91 



a cii-cle with O as a center and ] a- h'- as a radius. Now it is easy to produce a 

 straight line by P'. To perform this it is only necessary to move the point P on 

 a circle passing through O. This can be done by connecting the center C of the 

 circle with the point P by a seventh bar C P. 



As to mathematical means of illustrations in general, and their utility, 

 it may be well to mention some points of Felix Klein's lectures on mathe- 

 matics, on the occasion of the World's Fair, in Chicago. (Felix Klein, of 

 Gottingen, at present one of the most eminent mathematicians, and well 

 known to American universities, delivered those lectures before the members 

 of the Congress of Mathematics, at Northwestern University, Evanston, 111.) 



Among mathematicians in general three main categories may be dis- 

 tinguished, and perhaps the names logicians, formalists and intuitionists 

 may serve to characterize them. (1) The word logician is here only intended 

 to indicate that the main strength of the men belonging to this class lies in 

 their logical and critical power, in their ability to give strict definitions, and 

 to derive rigid deductions therefrom. (2) The formalists among the mathe- 

 maticians excel mainly in the skilful formal treatment of a given question, 

 in devising for it an "algorithm." (3) To the intuitionists, finally, belong 

 those who lay particular stress on geometrical intuition, not in pure geometry 

 only, but in all branches of mathematics. What Benjamin Pearce has called 

 "geometrizing a matliematical question" seems to express the same idea. 



Klein ranks himself to the logicians and intuitionists, and possesses the 

 power and ability of both. The main feature of this combination lies in the 

 "refined intuition." Refined intuition characterizes the most successful 

 mathematical schools of the present, and does not properly mean intuition as 

 it is required for an artisan or a draughtsman. The latter, or the naive in- 

 tuition, is not exact, while the refined intuition is not properly intuition at 

 all, but arises through the logical development from axioms considered as 

 perfectly exact. Not all mathematicians have this point of view, what 

 might be explained by the fact that the degree of exactness of the intuition 

 of space is different in different individuals, — perhaps in different races. 

 Klein points out, as if a strong naive space-intuition were an attribute pre- 

 eminently of the Teutonic race, while the critical, purely logical sense is more 

 fully developed in the Latin and Hebrew races. 



But in general the heuristic value of the applied sciences as an aid to 

 discover new truths in mathematics is at present more recognized than ever 

 before. Besides the great importance in technics on other applied sciences 

 of mathematics, the heuristic method is the most successful in pedagogics. 

 An education which develops in the student a strong intuition produces, as 

 the history of pedagogics shows it, the best results. It was one of the first 

 principles of Pestalozzi, and is useful as well in the higher as in the common 

 schools. The incomparable success of the lectures of the great mathematician 

 Steiner was due to this method. Steiner himself had such an unusual in- 

 tuition of space that it was easy for him to make the most difficult construc- 

 tions of descriptive geometry by imagination, and to see clearly before his 

 powerful mind complicated geometrical configurations. That a strong intu- 

 ition is of a general value may be stated by the same Steiner. Whenever he 

 visited an art exhibition artists and experts were surprised by his quick and 

 correct criticism. It took him only a glance on a person to not forget her 

 image any more. 



People generally believe that intuition, like fine arts, must be born with 

 a person. This may be true in a certain sense, and only to a certain limit. 



