Industrial Research 



273 



has no place in the consulting field, and that these 10 

 must be exceptional men, it does not seem unreasonable 

 to ask where they may be found. 



Most mathematicians now in industry were trained 

 as physicists or as electrical or mechanical engineers 

 and gravitated into their present work because of a 

 strong interest in mathematics. Few came from the 

 mathematical departments of universities. As scien- 

 tists they are university trained, but as matliematicians 

 they are self-educated. 



Their training has not been ideal. Industrial mathe- 

 matics is being carried on by graduates of engineering 

 or physics not so much because of the value of that 

 training as because of the wealoiess of mathematical 

 education in America. The properly trained industrial 

 mathematician shoidd have, beyond the usual coui-ses of 

 college grade, a good working background of algebra 

 (matrices, tensor theory, etc.), some geometry, particu- 

 larly the analytic sort, and as much analysis as he can 

 absorb (function theory, theory of differential and 

 integral equations, orthogonal functions, calculus of 

 variations, etc.). These should have been taught with 

 an attitude sympathetic to their apphcations and rein- 

 forced by theoretical courses in sound, heat, hght, and 

 electricity, and by heavy emphasis upon mechanics, 

 elasticity, hydrodynamics, thermodynamics, and elec- 

 tromagnetic field theory. He should understand what 

 rigor is, so that he will not unwittingly indulge in un- 

 sound argument, but he should also gain experience in 

 such useful but sometimes treacherous practices as the 

 use of divergent series or the modification of terms in 

 differential equations. He should have enough basic 

 physics and chemistry of the experimental sort to give 

 him a reahstic outlook on the power as well as the 

 perils of experimental technique. By the time he has 

 acquired this training he will usually also have acquired 

 a Ph. D. degree, but the degree itself is not now, and is 

 not lilvely to become, the almost indispensable prere- 

 quisite to employment that it is in university Ufe. 



There is nowhere in America a school where this 

 training can be acquired. No school has attempted to 

 build a faculty of mathematics with such training in 

 mind. Hence industry has had to make such shift as 

 might be with ersatz mathematicians cidled from de- 

 partments of physics and engineering. To make matters 

 worse, a student with strong theoretical interests who 

 enrolls in physics these days is almost certain to spend 

 most of his time on modern mathematical physics, 

 which insists almost as little upon fidelity to experience 

 and experiment as does "pure" mathematics, from which 

 it differs more essentially in matters of language and 

 rigor than of general philosophic attitude. At the 

 moment, therefore, engineering schools must be looked 

 upon as the most hopeful sources of industrial 

 mathematicians. 



Historically it is easy to explain how this situation 

 came about. Fifty years ago America was so backward 

 in the field of mathematics that there was not even a 

 national association of mathematicians. A quarter of 

 a century later it was just coming of age in mathematics 

 and was properly, if not indeed necessarily, devoting its 

 entire attention to improving the quaUty of instruction 

 in the "pure" field. The first faint indications that 

 industrial mathematics might some day become a 

 career had indeed begun to appear, but they were not 

 impressive enough to attract the attention of imiversity 

 executives. 



Today wo lead the world in pure mathematics, and 

 perhaps also in that other field of mathematics which 

 has somehow come to be known as modern physics. 

 We have strong centers of actuarial and statistical train- 

 ing. But in the field of apphed mathematics, wliich is 

 the particular subject of this report, we stand no further 

 forward than at the turn of the century, and far behind 

 most European countries. 



A quarter of a century ago it would have been difficult 

 to find suitable teachers. Just now it could be done, 

 primarily because a number of European scholars of the 

 right type have been forced to come here and a few 

 others have developed spontaneously within our own 

 borders. There are perhaps half a dozen of them, but 

 they are so scattered, sometimes in such unpropitious 

 places, as to have httle influence on the development of 

 industrial personnel. 



It is unfortunate that no university with strong 

 engineering and science departments has seen fit to 

 bring this group together and establish a center of 

 training in industrial mathematics. We have estimated 

 a demand of about 10 exceptional graduates per year. 

 If that estimate is even remotely related to the facts, 

 such a department would have a most important job 

 to do. 



Mathematics in Industry 



Subjects Used 



As Dr. H. M. Evjen, research physicist of the geo- 

 physical section of the Shell Oil Company, remarks: 



Higher mathematics, of course, means simply those branches 

 of the science which have not as yet found a wide field of appli- 

 cation and hence have not as yet, so to speak, emerged from 

 obscurity. It is, therefore, a temporal and subjective term. 



If this is accepted as a definition of higher mathe- 

 matics — and it is a valid one for the pure science as well 

 as for its apphcations — it follows automatically that 

 industry relies principally upon the lower branches. 

 What it uses much ceases by the very muchness of its 

 use to be high. The theory of linear differential 

 equations, for example, is a subject by which the 

 average well-trained engineer of 1890 would have been 



