92 KANSAS ACADEMY OF SCIENCE. 



Education and training, however, can produce a good deal of what is called 

 ability and art in common life. I shall give an example which may suggest 

 a test for the quality of the naive intuition of a person. Most of the people 

 that look at a tree think it to be a very simple object of art. If they would 

 shut their eyes at the moment of this intuition, and try to imagine exactly 

 the tree which they just were going to see, they would find out how strong 

 their intuition was. Even representants of the purely analytical school 

 would be surprised after this test. Thus, it is the aim of the leading mathe- 

 maticians of the present to proceed from the naive intuition to the refined 

 intuition, i. e., to combine the critical power of logic with the usefulness of 

 intuition. 



When I was in Zurich, an eminent mathematician, who is now in Germany, 

 lectured on the theory of functions; but his treatment of the subject was 

 purely analytical. One day a student showed him a short geometrical demon- 

 stration of one of his theorems. The professor looked at it, and said: "This 

 seems to be a very elegant prove, indeed, but I am sorry to say that I cannot 

 understand it — I am not a geometrician." However valuable those lectures 

 have been, such a one-sided standpoint cannot be taken any more. One- 

 sidedness must disappear. 



The question now arises as to how produce a stronger intuition. It has 

 been answered many times with more or less success, and by the most promi- 

 nent educators. According to their statements, the only way is a thorough 

 course in what is called mathematical graphics. 



From what has been said before it will be understood that we define 

 mathematical models in the most general sense, i. e., as geometrical figures 

 and configurations, and mathematical models in a narrower sense as geo- 

 metrical and mechanical models. All these objects serve the one educa- 

 tional purpose to refine the intuition, and thereby the efficiency of applied 

 sciences, or technics. Now I will proceed to what might appear first from 

 the title of the paper, namely, the geometrical models or geometrical means 

 of illustration. 



In the accounts of the investigations of Euler, Newton and Cramer we find 

 numerous figures. The interest for the construction of models was first 

 produced in France, where, under the influence and the example of Monge, 

 a great number of thread models of surfaces of the second order, conoids 

 and helicoids were being constructed. A further progress was made by 

 Bardin (1855). He had made cast models of stone-cuts, gearings, and many 

 other things. His collection was greatly enlarged by Muret. These schemes 

 were not appreciated by French mathematicians, while Cayley and Henrici 

 (1876) exhibited in London mathematical models besides other scientific 

 apparatus of the universities of Cambridge and London. In Germany the 

 construction of models received an impetus, but after the assimilation of 

 projective by descriptive geometry. Plucker drew already in 1835 the shapes 

 of the curves of the third order, and made in 1868 the first larger collection 

 of models, consisting of models of complex surfaces of the fourth order. 

 Klein added to it some more models of the same character. A special surface 

 of the fourth order, the wave surface of crystals with two axes, was con- 

 structed by Magnus in Berlin, and Soleil in Paris, in 1840. Fiedler, in Zurich, 

 is the constructor of the first model of a surface of the third order with its 

 27 straight lines. He constructed it in 1868, when he was a professor in 

 Prague, and always used to have it suspended at the ceiling of his study or 

 parlor. Now this model is a historical curiosity of the mathematical collec- 



