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KANSAS ACADEMY OF SCIEXCE. 



MATHEMATICAL MODELS. 



By ARNOLD EMCH, of the State University, Lawrence. 



By an ordinary consideration of geometrical forms it might appear that 

 a mathematical model could have but a theoretical interest, and non-mathe- 

 maticians are mostly liable to the opinion that solid representations in 

 space be rather curiosities than practical schemes. This opinion was pre- 

 vailing even among mathematicians until some 20 years ago, when besides 

 of the analytical results particular stress was also laid upon the conception of 

 the geometrical forms. It was perceived that real progress in mathematics 

 could only be made by aid of geometrical illustrations, and since that time 

 the geometrical standpoint — geometry taken in the most general sense — is 

 dominant in mathematical investigation. At present it is not sufficient to 

 know and to discuss the analytical equation of a geometrical form; its real 

 shape must be studied also. Thus, if a plane configuration, or a representa- 

 tion by descriptive geometry is too complicated, or not conspicuous enough, 

 a model of the form is constructed. This enables the student or the investi- 

 gator to see the essential features of the form, and suggests to him new ideas. 

 Moreover, it is obvious that by the constant reference to real forms many 

 problems of technics come into account which by a purely analytical 

 method, with the exclusion of every configuration, never would be taken up. 

 In this manner the very important connection between science and technics 

 can be maintained, and there is no danger that mathematics will branch off 

 too much from its technical application. 



A simple illustration for this method is, for instance, Peoncellier's diagram, 

 which transforms a circular movement into a movement of a straight line. 

 Peoncellier published a solution of this problem in the "Nouvelles Annales de 

 Mathematic," in 1864. The main part of the apparatus for the realization 

 of this movement on a straight line is called "inversor," because it pro- 

 duces the relation of the inversion. 



The "inversor" consists of two bars of equal length which are connected 

 in the point O (see figure), and between which a system of other bars in shape 

 of a rhombus is put in. At all connection points of the bars are links, so 

 that the whole system is moveable. If the point O is kept fixed, and if one 

 point of the rhombus, P, moves at pleasure, the other point, P', will move such 

 that it is inverse to P, or OP X OP' ^ const. For, by using the designations of 

 the figures, the relations exist : 



a sin e = b sin .r, or 



O = a'- sin'^ e — b- sin- x 

 O P = a cos e — b cos x 

 O P' = a cos e-\-h cos x 



O P X t) P' ^ a'- cos- € — b'- cos' x 

 adding O = a'- sin- e — b- sin- x 

 O P X O P' = a- — b- = const. 



Thus, the inversor realizes the transformation by reciprocal radii in regard to 



