November 8, 1900] 



NATURE 



45 



forth to the as yet ignorant world in all their magnitude there 

 will be some astonishment. But it is our duty to keep up our 

 high standards. We must change what is bad when we Know 

 it tCT be bad, and not let bad things ' continue to exist, parasitic 

 growths, maintained because on the whole we are strong and 

 healthy. You will perhaps think that this is a very serious exordium 

 when I tell you that I have introduced it all on account of the 

 state of mathematics in our profession. I feel a sort of degrada- 

 tion every time that I hear a successful, clever old member of 

 this Institution sneering at mathematics. There is a plaqsibility 

 about his statements ; he himself has been very successful in life 

 without much help from mathematics ; but indeed his sneer is 

 doing a great deal of harm to the younger members who admire 

 his success, who forget that he has succeeded in spite of, and 

 not because of, his neglect of mathematics. 



Our knowledge of electrical phenomena must be quantitative 

 to be of practical use ; we must be able to calculate. Mathe- 

 matics is the science of calculation, and we must therefore be 

 able to employ, and we all do necessarily employ, less or 

 more mathematics every hour of our professional lives. The 

 draper and the grocer and the housekeeper merely need 

 arithmetic. Everybody now knows some arithmetic. Every- 

 body can add and subtract and multiply and divide, and 

 keep accounts in some simple sort of way. This is due to the 

 fact that arithmetic is no longer taught in the old Greek method 

 with its twenty-seven independent characters (for our ten figures), 

 the study of which required a lifetime, so that only old men 

 could do multiplication, and they not only needed many hours 

 to do one easy bit of multiplication, but declared that if the art 

 were not practised every day it could not be remembered. 

 Reading and writing and ciphering are now taught to everybody. 

 It used to be that only learned men and philosophers could 

 read, write and compute. You will remember the charge that 

 was brought against one of Shakespeare's characters, who was 

 said to possess mere bookish theory without practical knowledge. 

 "And what was he?" "Forsooth a great arithmetician." 

 Nowadays, when everybody can compute, we should say of the 

 possessor of mere bookish theory, "Forsooth he knows the 

 calculus. " 



For in mediaeval times things were taught in such a way that 

 only a few men had a chance of knowing how to read, write and 

 cipher. We have been compelled to change all that ; the 

 pedagogue has, by compulsion, given up his mediaeval methods 

 of teaching in these things, although in all other matters he 

 retains them. But a time has come when we see that ciphering 

 is not enough mathematics for us to be familiar with, we need a 

 little algebra, we need co-ordinate geometry, we need the 

 differential and integral calculus. The pedagogue tells us that 

 we must follow the orthodox course of study, which takes many 

 years ; and some of us, many of us, who have followed the 

 orthodox method, find that we have spent so much time and 

 mental power upon it and its thousands of unnecessary tricks and 

 contrivances and philosophy, that we can take in no more ideas. 

 We cannot utilise our mathematics on engineering problems 

 because we are too old and tired and blasd to comprehend these 

 problems. Nevertheless we are the only people who know 

 mathematics, and so we publish volumes of unmeaning and use- 

 less disquisitions on problems that we do not understand. Or 

 we know just enough mathematics to be able to show our ignor- 

 ance to experts, but quite enough to impress engineers with our 

 knowledge ; and we know just enough about engineering problems 

 to show our ignorance to engineers, but quite enough to impress 

 mathematicians, and what we publish is merely as the crackling 

 of thorns under a pot. 



As for the man who does understand electrical problems, 

 he remembers that there was a something called a study of 

 mathematics at his school, that he did pass certain examina- 

 tions with much difficulty and tribulation, that the subject had 

 no real meaning to him even when he was supposed to know 

 it, and he now hates the sight of anything that looks like mathe- 

 matics. 



I tell you, gentlemen, that there is only one remedy for this 

 sort of thing. Just as the antiquated method of studying 

 arithmetic has been given up, so the antiquated method of 



^ Such as our wretched system of weights and measures. Oh, young 

 America and Austri lia, is it wise to waste a year of every child's life, and 

 years of the life of every business man, merely because we do it in England ? 

 You get many of your pedagogues from us, and of course they say that 

 without cwts.. qrs., lbs., and Latin declensions and Euclid, the mind cannot 

 be trained. Do you believe them, or are you with open eyes making a 

 great sentimental sacrifice? 



Studying other parts of mathematics must be given up. The 

 practical engineer needs to use squared paper. What is the 

 use of telling him that he has taken an unauthorised way to the 

 study of co-ordinate geometry, that he cannot approach it 

 except through Euclid and modern geometry and geometrical 

 conies and algebra and trigonometry? He says the youngest 

 child can be made to understand diagrams on squared paper. 



So again the idea underlying the calculus is one that every 

 child, every boy, every man possesses and uses every day of his 

 life, and there are useful methods of the calculus that might 

 be taught quite quickly to boys, and which it would be a plea- 

 sure to boys and men to use continually in all sorts of practical 

 problems ; but of course the subject of the differential and inte- 

 gral calculus is one that must come at the end of a long course 

 of what is to the average boy utterly uninteresting and unmean- 

 ing mathematics. Indeed, the average boy never reaches the 

 subject, whose very names, differential and integral calculus, 

 are enough to drive him frantic. 



Yes, the schoolmasters say that we must follow the mediaeval 

 rules of the game, and all sorts of fine things are said about 

 them ; but as a matter of fact we only need to bring a little 

 common sense to bear upon schoolmasters. At present most of 

 us stick to our arithmetic as a safe and well-tried friend. We 

 compute after the manner of the draper and grocer and house- 

 keeper. In finding out what is the best size of conductor, or 

 armature winding or core, or iron and winding of a field magnet, 

 we calculate by mere arithmetic for one size and then for 

 another ; perhaps we have weeks of arithmetical computation 

 before we find the right size of thing to use, and we cannot 

 frame general rules. And some foolish person who knows a 

 little mathematics, works at the problem (as we ought to be able 

 to do, but are not), and he frames a general rule and we laugh 

 at it, and sneer at mathematics because he has probably left 

 out of account the most important consideration. We know 

 that the result is wrong, but we cannot say why it is wrong. 



Then there are some far-reaching, labour-saving ideas that we 

 simply cannot get into our heads at all ; we cannot comprehend 

 them. Am I sinning against the rule as to good comradeship 

 which exists here if I say that some of us are ignorant of the 

 most fundamental facts regulating economy in arranging sizes of 

 conductors ? Suppose we find the total cost of installing a 

 conductor of a certain length, using one square inch section of 

 copper. We do the same thing for other sizes, and we plot total 

 cost and weight of mere copper on squared paper. I do not 

 care what system we adopt if it is the same system for all sizes, 

 and if we buy our materials from the same manufacturers and 

 use the same kind of labour, our points will lie very nearly in a 

 straight line on the squared paper. Hence increased cost will 

 be proportioned to increased weight of copper, and, indeed, 

 increased total cost will be like the mere increase in the cost of 

 copper, taking a slightly higher price of copper per ton. Some 

 of us, ignorant of the elementary mathematics involved in the 

 problem, think that the mistake has been made of assuming that 

 the cost of an installed conductor is merely the cost of the copper 

 in it, and, of course, he must feel that it is too absurd a mistake 

 not to be laughed over. With an elementary knowledge of 

 mathematics his mistake would be impossible, and without such 

 a knowledge the clever electrical engineer is constantly dis- 

 covering mares' nests in the investigations which he criticises. 



I know of long misleading accounts of the results of good 

 experimental observations which might have been described in 

 a few clear words by the aid of elementary mathematics. I know 

 men who spend on a particular problem ten times the amount of 

 worrying thought that would enable them to master the easy 

 mathematics that includes all such problems. Quite recently 

 one of our most eminent members declared to me that he had 

 not really grasped the reason for small economy at a power 

 station when there is a small load factor until he studied the 

 common-sense mathematical form which has been given in a re- 

 cent publication. And yet he is a man who has heard much, 

 and read much, and talked much on this subject. 



Every electrical engineer has a correct idea of how a trans- 

 former acts, or how the E.M.F. in one of the coils of an arma- 

 ture of a direct current or other generator, or, let us say, a rotary 

 transformer, changes during a revolution, and how the E.M.F.'s 

 of all the coils are combined to produce currents in the external 

 circuits. But through how much mathematical tribulation must 

 most of us have passed from our state of ignorance to our present 

 state of knowledge ! It is no wonder that we are disinclined 

 to the study of a new phenomenon which seems as if it 



NO. 1619, VOL. 63] 



