March 27, 1902] 



NA TURE 



485 



and sure that it was really impious to roast pork without burning 

 houses, and so Pel-li and his followers weie crucified. If I 

 were to tell my friends that I am Pel-li and that there is a very 

 perfect parallelism between the two cases, they would laugh at 

 the absurdity of such a statement, even if I made it from the 

 crois. And yet, O mandarins, I say to you that you have brought 

 against me the charge of impiety because you cannot imagine 

 any other way of getting to the notions of the infinitesimal 

 calculus than the way in which you yourselves arrived at them, 

 and because I say that they are easier to take in than the 

 axioms of Euclid. 



Is it impossible, then, to imagine a different logical sequence 

 from that in which one has been brought up? The very greatest 

 difficulty which I meet with is in getting men to see that if a boy 

 has practised measurement with a scale of inches and tenths of 

 an inch he can understand decimals without a philosophical 

 explanation. This fact, obvious to me, I have not been able to 

 gel believed in by any one teacher of arithmetic who plumes 

 himself on his knowledge of the theory of teaching. Well, I 

 go further and say that as an explanation is not necessary for a 

 boy, to give him a grown-up explanation is a crime. Again, I 

 like when dealing with quite easy arithmetic to make boys 

 evaluate most complex formul;«, using all sorts of tabulated 

 functions so that they may regard algebraic signs as a sort of 

 shorthand. I cannot in a few words explain the wonderful 

 mental value of this practice. My critics not only see no value 

 in it, they look upon it with abhorrence. This is not through 

 an effort of their reasoning ; it is merely because the thing is 

 strange to them, and, like the Dudley miner, they say, " 'Ere's 

 a stranger, Bill, let's 'eave a brick at him." If my critic has any 

 kind of satisfying reason for his abhorrence I suppose it is 

 because he thinks this new custom of mine resembles some most 

 pernicious slipshod habits for which he is continually blaming 

 his pupils — getting off propositions by heart and pretending one 

 knows them ; using a rule of arithmetic or dynamics in a 

 mechanical way without understanding why ; assuming that one 

 understands a part of an investigation when one does not, and 

 in all sorts of other fraudulent ways pretending to follow a 

 logical sequence and degrading it. Now I also abhor these 

 things. But what my students do is very different and is 

 perfectly logical. They make no pretence of having proved 

 anything, they are merely familiarising themselves with the 

 shorthand of algebra, a thing that they cannot do too soon. If 

 they get to look upon this as the A B C of mathematics they will 

 not after many years of study feel proud of their mathematical 

 knowledge when all they can do is to merely use formula in a 

 text-book. Not long ago in an engineering journal the writer 

 of a letter comjilained that he had been asked to evaluate the 

 expression 



ae - o' sin [bt -^g) 



(being given the values of a, a, b and g) for several values of /. 

 He said he had passed most difficult examinations and knew 

 higher mathematics, but it was quite ridiculous that anyone 

 should expect him to know so much. His anger was extreme. 

 Now some of my s'udents hive evaluated things like this before 

 they did any formal algebra at all, but they do not dream of 

 calling it higher mathematics. Surely there is every good in 

 letting a boy become famili.ir with all sorts of formula; long 

 before we lead him through the logical sequence which deals 

 with such formula, just as we let a boy learn to use words 

 before we teach him grammar or philology. But whether I am 

 right or wrong, I do wish that my critics would try to see 

 exactly what I advocate before they throw blame. The cock- 

 shies that they fling their stones at have nothing in common 

 with any part of my scheme. 



I want it to be understood that I advocate a sequence as 

 logical as the orthodox one, or rather, I should say, ever so 



NO. i6yi, VOL. 65] 



much more logical, because in the orthodox sequence a boy is 

 really unfamiliar with the ideas to which his so-called logic is 

 applied. The usual .sequence may be logical to a philosopher, 

 but it is quite illogical to the average English boy. 



I say that what is essential is that the student .should be 

 thoroughly familiar through experiment, illustration, measure- 

 ment and every other possible method with the ideas to which 

 he applies his logic. Also that the study should be of interest 

 to him. I submit that the sequence which I recommend can 

 really be made interesting to the average English boy, whereas 

 the orthodox sequence is painfully uninteresting to him. One 

 reason for this great interest lies in its immediate application to- 

 all sorts of actual problems such as he meets with in the study 

 of natural science, and I do not care to hide the fact that there 

 is a special interest which is due to the usefulness of the results 

 of the study in the life-work that lie before him. If anybody cares 

 he may misrepresent me here to any extent. Over and over again 

 the academic person has been kind enough to sneer at my utili- 

 tarianism as if I were sacrificing the spiritual for the material, as if 

 engineering were a thing of mere formula. On this I can add 

 nothing to what I have already given in my British Association 

 address and in my other papers. But if my critics only knew 

 what wonderful regions of logical thought and high emotion are 

 connected with the practical applications of natural science, if 

 they had the respect which I confess to have for common things, 

 they also might say as Ileracleitus said of his kitchen with its 

 pots and pans, " Here also are the gods." In his typical poem 

 "Shop," Browning takes the college-common-room point 

 of view. Heavens, what a sordid narrow point of view it is ! 

 The Bloomsbury bric-a-brac shopkeeper gets all the poet's 

 scorn because he does not hate shopkeeping, because to him 

 "shop was shop only." Does anybody imagine that 

 Shakespeare could not have glorified the life of the shop- 

 man ? But this poet, with all the arrogance of his caste, says, 

 "I want to know a butcher paints, a baker rhymes for his 

 pursuit." The moment a man has leisure he must escape from 

 his trade 1 For my part, I believe that whatever a man finds 

 to do he ought to do with all his might and with all love and 

 devotion, or not attempt to do it at all. If he hates shop- 

 keeping, let him give it up to someone else to whom shop- 

 keeping is a perfect happiness. If Ruskin's influence over a 

 man has been great enough to prevent his seeing the romance, 

 the wonder of engineering, so that it is to him a mere trade by 

 which he earns his bread and butter, in heaven's name let him 

 give it up altogether and take to art criticism. For my lovely 

 mistress. Applied Science, scorns a divided worship. It is 

 disgusting to see young engineers who cannot compute, who 

 know nothing of science, whose souls are not engrossed all the 

 time with the greatness of their profession, who never think of 

 their business after office hours, who think it all a mere matter 

 of formula and tools. If they were fit for their work their lives 

 would fill with happiness, and even the power to rhyme and 

 paint and to create music might belong to them ; but woe unto 

 the nation whose shopkeepers scorn shopkeeping while they 

 paint or fiddle ; whose schoolmasters rely on cricket and a 

 housemaster to do their proper business whilst they discuss 

 Browning and the musical glasses. To make a man fit for, so 

 that he may also love, his profession, is this a function to be 

 scorned by schools and colleges ? and am I to be sneered at as 

 a utilitarian because I consider this a most important function of 

 the schoolmaster? 



Nobody has contradicted my statement that the orthodox 

 method of cramming average boys with demonstrative geometry 

 stupefies them and makes them hate mathematics all their life 

 after. May I also point out that Ihe beautiful philosophy of 

 Euclid is also degraded, just as the literature of Greece and 

 Rome is degraded, by our school methods. Is there anybody 



