September 19, 19 12] 



NATURE 



95 



T 



THE PLACE OF MATHEMATICS IN 

 ENGINEERING PRACTICE.^ 



HE foundations of modern engineering have been 

 laid on mattieinatics and physicai science ; tlie 

 practice of engineering is now governed by scientific 

 methods applied to the analysis of experience and the 

 results of experimental research. Engineering has 

 been defined as "the art of directing the great 

 sources of power in nature for the use and conveni- 

 ence of man." An adequate acquaintance with the 

 laws of nature, and obedience to those laws, are 

 essential to the full utilisation of these sources of 

 power. It is now universally recognised that the 

 educated engineer must possess a good knowledge 

 of the sciences which bear upon his professional 

 duties, in combination with thorough practical train- 

 ing and experience in actual engineering work. 

 Neither side of his education can be neglected without 

 hampering him seriously, especially when he has to 

 go beyond precedent and face new pioblems. Of 

 these sciences, the mathematical is undoubtedly of 

 the greatest importance to engineers. The range and 

 character of mathematical knowledge which can be 

 considered adequate are gradually being agreed upon 

 as experience is enlarged ; and present ideas are em- 

 bodied in the courses of study prescribed in the 

 calendars of schools of engineering. Absolute identity 

 in the course of study and the standards laid down 

 for degrees in engineering has not been attained, but 

 the approach thereto has already been considerable, 

 and the movement will undoubtedly continue in the 

 same direction. 



The preponderance of opinion amongst- engineers 

 now favours the teaching to students of engineering 

 of science generally, and of mathematics in particular, 

 being undertaken by recognised authorities in the 

 several branches, and on lines which shall ensure 

 greater breadth of view and fuller capability for dealing 

 with new problems arising in their professional work 

 than can be secured by means of special courses of 

 instruction arranged for students of engineering as a 

 class apart. Whatever branch of engineering a man 

 may select for his individual practice, he must need 

 a fundamental knowledge of mathematics, and in 

 some branches, in order to do his work well, he will 

 require to add considerably to the mathematical know- 

 ledge which is sufficient for a degree. 



As time passes the mathematician and the practising 

 engineer have come to understand one another better, 

 and to be mutually helpful. While engineers as a 

 class cannot claim to have made many important or 

 original contributions to mathematical science, some 

 men trained as engineers have done notable work of 

 a mathematical character. The names of Rankine, 

 William Froude, and John Hopkinson among British 

 engineers also hold an honoured place in mathe- 

 matics. Mathematicians of eminence have spent 

 their lives in the tuition of engineers, and in that way 

 have greatly influenced the practice of engineering ; 

 but while they have necessarily become familiar with 

 the problems of engineering as a consequence of their 

 connection therewith, they have not accomplished 

 much actual engineering work, and none of it has 

 been of first importance. Speaking broadly, there is 

 an abiding distinction between mathematicians . and 

 engineers. Mathematicians regard engineering 

 chiefly from the scientific point of view, and are 

 primarily concerned with the bearing of mathematics 

 on engineering practice, the construction of theories, 

 and the framing of useful rules. Engineers, even 

 when well equipped with mathematical knowledge, are 

 primarily devoted to the design and construction of 



1 Lecture drlivered at Cambrklje before the Fifth Internation.il Congress 

 of Mathematicians by Sir William H. While, K.C.B., F.R.S. 



NO. 2238, VOL. 90] 



ellicient and durable works, their main object being 

 to secure the best possible association of efficiency 

 and economy, and so to achieve practical and com- 

 mercial success. There is evidently room for both 

 classes, and their collaboration in modern times has 

 produced wonderful results. 



The proper use of mathematics in engineering prac- 

 tice is now generally agreed to include the following 

 steps : first comes the development of a mathe- 

 matical theory based pn assumptions which are 

 thought to embody and to represent conditions dis- 

 closed by past practice and observation. Frequently 

 these theoretical investigations give rise to valuable 

 suggestions for further observation or experimental 

 investigations. Mathematical analysis must be 

 applied to the results of observation and experiment ; 

 and, as a result, amendments or extensions are made 

 of the original mathematical theory. Useful rules are 

 also devised, in many instances, which serve for guid- 

 ance in the future practice of engineers. Formerly 

 it was thought by men of science that purely mathe- 

 matical investigation and reasoning would do all that 

 was required for the guidance of engineering prac- 

 tice ; now it is admitted that such investigations will 

 not suffice, and that the chief services which can be 

 rendered to engineering by mathematicians will con- 

 sist in the suggestion of the best directions and 

 methods for e.xperimental research, the conduct of 

 observations on the behaviour of existing works, the 

 establishment of general principles based on analysis 

 of experience, and the framing of practical rules em- 

 bodying scientific principles. 



The contrast between present and past methods can 

 be illustrated by comparing investigations made 

 during the eighteenth century into the behaviour of 

 ships amongst weaves by Daniel Bernoulli, who won 

 the prize offered by the Royal French .Academy of 

 Science in 1757, and work done by William Froude 

 a century later in connection with the same subjects. 

 Bernoulli was the greater mathematician, but had 

 only a small knowledge of the sea and of ships. 

 His memoir was a mathematical treatise; his prac- 

 tical rules, although deduced from mathematical 

 investigations which were themselves correct, depended 

 upon certain fundamental assumptions which did not 

 correctly represent either the phenomena of wave- 

 motion or the causes producing and limiting the roll- 

 ing oscillations of ships. Bernoulli realised and dwelt 

 upon the need for further experiment and observation 

 and showed remarkable insight into what was needed ; 

 but the fact remains that he neither made such ex- 

 periments himself nor was able to induce others to 

 make them. As a consequence, his practical rules for 

 the guidance of naval architects were incorrect and 

 would have produced mischievous results if they had 

 been applied in practice. 



William Froude was a trained engineer who had a 

 good knowledge of mathematics and a mathematical 

 mind. His acquaintance with the sea and ships was 

 considerable, his skill as an experimentalist was 

 remarkable,' and he was fortunate enough to secure 

 the support of the .Admiraltv through the Construc- 

 tive Department. He thus 'obtained the services of 

 the officers of the Royal Navy in making a long series 

 of accurate and detailed observations of the charac- 

 teristic features of ocean waves as well as the rolling 

 of ships amongst waves or in still water. In this 

 way, starting with the formulation of a mathe- 

 matical theory of wave-motion, and of a theory for 

 unresisted rolling in still water and amongst waves, 

 Froude added corrections based on experimental re- 

 search, and succeeded eventually in devising methods 

 by means of which naval architects can make_ close 

 approximations to the probable behaviour of ships of 

 new design when exposed to the action of waves, 



