6/8 AN INTRODUCTION TO CYBERNETICS 



y that copies that at v. Cover the central parts of the mechanism 

 and the two machines are indistinguishable throughout an infinite 

 number of tests applied. Machines can thus show the profoundest 

 similarities in behaviour while being, from other points of view, 

 utterly dissimilar. 



Nor is this all. Well known to mathematicians are equations of 

 the type 



d^z dz 

 ^^ + ^^ + ^^~ = ^ 



by which, if a graph is given showing how w varied with time {t), 

 the changes induced in z can be found. Thus w can be regarded 

 as an "input" to the equation and z an "output". If now a, b, and 

 c are given values suitably related to L, R, S, etc., the relation between 

 w and z becomes identical with those between ii and v, and between 

 A' and >'. All three systems are isomorphic. 



The great practical value of isomorphisms is now becoming 

 apparent. Suppose the problem has arisen how the mechanical 

 system will behave under certain conditions. Given the input u, 

 the behaviour v is required. The real mechanical system may be 

 awkward for direct testing: it may be too massive, or not readily 

 accessible, or even not yet made! If, however, a mathematician is 

 available, the answer can be found quickly and easily by finding 

 the output z of the differential equation under input w. It would be 

 said, in the usual terms, that a problem in mathematical physics 

 had been solved. What should be noticed, however, is that the 

 process is essentially that of using a map — of using a convenient 

 isomorphic representation rather than the inconvenient reahty. 



It may happen that no mathematician is available but that an 

 electrician is. In that case, the same principle can be used again. 

 The electrical system is assembled, the input given to x, and the 

 answer read off at y. This is more commonly described as "building 

 an electrical model". 



Clearly no one of the three systems has priority; any can sub- 

 stitute for the others. Thus if an engineer wants to solve the differ- 

 ential equation, he may find the answer more quickly by building 

 the electrical system and reading the solutions at y. He is then 

 usually said to have "built an analogue computer". The mechanical 

 system might, in other circumstances, be found a more convenient 

 form for the computer. The big general-purpose digital computer 

 is remarkable precisely because it can be programmed to become 

 isomorphic with any dynamic system whatever. 



The use of isomorphic systems is thus common and important. 



96 



