38 Prof. L. Boltzrnann on the 



we recognize as existing ; whether substances, or their energy, 

 or, in general, their properties, so that we may perhaps at 

 last define away even our own existence. 



Bnt let this pass ; the necessity exists for the most complete 

 utilization of our different powers of conception ; and since 

 it is by aid of the eye that the greatest mass of facts can be 

 grasped simultaneously, it becomes desirable to make the re- 

 sults of our calculation perceptible, and that not merely by 

 the imagination, but visible to the eye and at the same time 

 palpable to the touch by means of gypsum and of cardboard. 



How little was done in this direction in my student days ! 

 Mathematical instruments were almost wholly unknown, and 

 physical experiments were often made in such a manner that 

 they could only be seen by the Lecturer himself. And as, 

 further, owing to shortness of sight I was unable to see 

 writing on the blackboard, my imagination was constantly 

 kept on the stretch. I had almost said to my good fortune. 

 Yet this latter statement would be in opposition to the object of 

 the present Catalogue, which can only be to praise the infinite 

 equipment of models in the mathematics of the present day ; 

 and it would, moreover; be quite incorrect. For even if my 

 growing organically powers of conception had gained, it could 

 only have been at the expense of the range of my acquired 

 knowledge. At that time the theory of surfaces of the second 

 order was still the summit of geometrical knowledge, and an 

 egg, a napkin ring, or a saddle was sufficient. What a host 

 of shapes, singularities, and of forms growing organically out 

 of each other, must not the geometrician of the present day 

 impress on his memory ! and how greatly is he not helped 

 by plaster casts, models with fixed and movable strings, links, 

 and all kinds of joints ! 



Not only so, but those machines make more and more way 

 which serve not for mere illustration, but save the trouble of 

 making actual calculations, from the ordinary four rules of 

 arithmetic to the most complicated integrations. 



As a matter of course both kinds of apparatus are most ex- 

 tensively used by physicists, who, in any case, are continually 

 accustomed to the manipulation of all kinds of apparatus. 

 Optical wave- surfaces, thermodynamical surfaces in gypsum, 

 wave-machines of all kinds, apparatus for illustrating the laws 

 of the refraction of light and other laws of nature, are 

 examples of models of the first kind. 



In the construction of instruments of the second kind some 

 have even gone so far as to attempt the evaluation of the 

 integrals of differential equations which hold equally for a 

 phenomenon difficult to observe (like the friction of gases) 



