G.— ENGINEERING 173 



However that may be, engineering as I see it still calls for this nineteenth- 

 century gift of visualisation, and if now mathematical analysts see fit to 

 eschew visualisation, that is no concern of ours except as meaning that we 

 must go our own way. I feel profoundly certain that in the engineering 

 student who intends research a gift of visualisation must be fostered 

 deliberately : he must develop intuitions not only in geometry plane and 

 solid, but of membranes, gases, elastic solids, incompressible fluids. It 

 is a gift very difl^erent from a gift for observation, because a solid may be 

 visualised clearly which is unlike any solid that he has ever seen. So in 

 hydrodynamics the fluid that he visualises has no colour, scent, taste, 

 viscosity, compressibility, surface-tension : it is a fluid in his own brain, 

 and it is unlike real fluids in this at least, that its presence there does no 

 harm. 



11. I ought not to spend more time on this heading of my thesis, yet 

 one point I would try to make because it has been very much in my 

 thoughts during the past three years. So far from our being always 

 dependent on professional mathematicians, I suspect that the time is 

 coming when we shall have methods of our own for doing most of what, 

 hitherto, we have looked to them to do for us. Those methods will not 

 be exact in the mathematical sense, but I think they will be none the 

 worse for that, even philosophically speaking. For there is, as it seems to 

 me, something wrong philosophically in an approach which envisages 

 even the possibility of an exact solution to any actual problem. In practice 

 data are subject to a margin of error, no less than the quantities required ; 

 yet in theoretical work (perhaps as a bad result of the examination system) 

 we almost invariably start as though the data had absolute certainty. 



On two occasions in the past three years this Section has borne with 

 patience my exposition of ' Relaxation Methods ' — an attempt to construct 

 a ' mathematics with a fringe '. Grateful for that indulgence, I will not 

 weary you this morning with a recital of problems which have been 

 attacked with success up to date. Some you will hear of later, when short 

 papers are given by my research students in accordance with a scheme 

 which Section G is trying as an experiment this year. I will only say that 

 I have been astonished as well as gratified by the way in which problems 

 regarded as difficult have yielded to the new attack. 



12. You are thinking, perhaps, that I lay too much stress on theory 

 and on calculation, that I have been talking only of the ' high-brow ' 

 sort of study that would have been the preserve of physicists before the 

 war. But that is how I visualise the trend of university research, con- 

 sidering how generous is the provision which now exists for more ad hoc 

 and expensive studies. Inevitably, as I believe, there will be some shift 

 of the focus of our interest, — schools of engineering will find problems 

 diflFerent from those which engaged their energies a generation ago. 

 It is not that those problems have lost importance or been solved, but that 

 better facilities now exist elsewhere, and can be made available. When a 

 problem can be turned over to trained men who will work on it full time, 

 common sense suggests that it is uneconomic both of brains and money 

 to pursue it at universities in hard-won spells of leisure from the duties 

 of teaching and administration. 



Moreover, though paradoxical is it not the fact that engineers, usually 



