128 



NA TURE 



\yMne 6, 1889 



on mathematical physics, with the result that many 

 students receive very insufficient practice in acquiring and 

 developing physical conceptions. 



To this cause, also, must be attributed the fact that 

 proficiency in mathematical physics is a much rarer thing 

 than proficiency in pure mathematics, the former being 

 made, to such a great extent, subservient to the latter. 

 Indeed it was not, I think, until the publication of 

 Thomson and Tait's " Natural Philosophy " that the 

 ancient problem-grinding system, which hindered the 

 progress of exact Physics, received a severe blow, and 

 Cambridge, under the influence of the ideas and methods 

 of Thomson and Clerk Maxwell, produced a plentiful 

 supply of physicists. 



But the good that was effected by Thomson and Tait is 

 confined almost entirely to advanced students ; in the 

 hands of those of a lower standard are still to be found 

 text-books of the old sort, teaching Mathematics under 

 the guise of Physics, presenting nothing but the dry 

 husks of the latter, and, by inaccurate language, laying an 

 early foundation of erroneous notions. Some people of 

 the older school — not a few — express, indeed, a strong 

 impatience with us if we protest against the use of slipshod 

 and inaccurate language in our text-books. They tell us 

 that we are doing the matter of nomenclature to death, 

 and that, after all, we need not be very particular about 

 the choice of exact terms. This is a most unfortunate 

 attitude ; it is one, however, which this Association has 

 done, and is doing, something to counteract. Surely it 

 must be admitted that if the conceptions of Physics are 

 presented to the beginner in erroneous language, there is 

 a danger that in many instances these conceptions will 

 never be properly acquired. And is not accurate language 

 as cheap as inaccurate .'' 



1 know of several text-books still in the hands of 

 schoolboys and others in which, in whole pages of 

 answers to problems on the motion of a particle, velocity 

 is always spoken of as so many feet, and acceleratioji also 

 as so many feet— no reference being made to time^ and 

 the fact that acceleration has a double reference to time 

 being never mentioned. I assume it as evident that the 

 progress of such students is seriously hindered and 

 delayed by such teaching. 



If the writers of such text-books as those to which I refer 

 have really clear and correct notions themselves, they 

 ought to take pains to present them accurately to beginners 

 who have to learn from them. Not to do so is to ignore 

 the fact that a great deal of the difficulty of Jearning any 

 branch of science is removed if only the student is started 

 upon it with clear and correct notions as to the various 

 entities with which the science deals, and with its fun- 

 damental principles expressed in accurate language. 



Let us take, as another example of exceedingly errone- 

 ous teaching, the following exercise — one of a great many, 

 all similarly expressed — from a text-book still in the hands 

 of beginners : " The time occupied by a body in describing 

 uniformly a complete revolution in a circle, whose radius 

 is II feet, is 16 seconds, calculate the centrifugal force 

 which acts upon it. Ans. r696 feet per second." 



In this we have two bewildering errors instilled into 

 the student's mind, viz. \\i2.\. force is measured in feet per 

 second, and that when a body is revolving in a curve, it is 

 acted upon by an outward normal force — the old time- 

 honoured fallacy about centrifugal force. It is almost 

 miraculous that any accurate scientific knowledge is 

 acquired in spite of such fallacious teaching as this. 



I wish to emphasize this teaching about centrifugal force ^ 

 because, no matter how often it is condemned and shown 

 to be erroneous, it stillflourishes in our scientifictext-books. 

 In another elementary work which is extensively used, we 

 find the nature of centrifugal force more fully explained in 

 the following manner. " In order to keep a body moving 

 in a circle, there must be a force acting upon it which will 

 produce a constant acceleration towards the centre, equal 



to ^ . Hence, if W be the weight of the body, P the 

 r 



pressure tending towards the centre, P — W ." This is, 



gr 



of course, all right ; but then follows the mysterious 

 statement, " a pressure equal and opposite to P is some- 

 times spoken of as the centrifugal force." 



1 call this a mysterious statement, because it does 

 not tell us on what this pressure equal and opposite to P 

 acts, or whether it really acts on anything at all or not. 

 We know, of course, that this reversed force is the reaction 

 of the moving particle on the agent or surrounding 

 medium; but how is a beginner to know this? Will he 

 not naturally fall into the error of supposing that it acts 

 on the moving body itself? Or, if he possessed a little 

 intelligence, would he not ask, " Why do you require me to 

 think of a force equal and opposite to one which acts on 

 the body ? I have nothing to do, when considering the 

 motion of the body, with any forces except those which 

 act on the body itself ; and why do you not ask me to 

 consider a force equal and opposite to some of the other 

 forces acting on the body — a force equal and opposite to 

 its weight, for instance? It would be just as sensible to 

 do so, since this reversed weight is the body's reaction 

 against the Earth." 



The only true reply to such a student is to confess that 

 it is quite a mistake to introduce the conception at all; 

 but instead of this, we find that the discussion proceeds 

 to encourage the erroneous notion that centrifugal force 

 acts on the moving body by taking the case of a carriage 

 describing a curve of small radius. 



Now, what IS it that has given rise to this fallacy about 

 centrifugal force? Is it the fact, so often adduced, that, 

 if we tie a stone to one end of a string and, holding the 

 other end in the hand, whirl it round, we feel an outward 

 force pulling the hand ? But then, if we imagine the 

 stone to be attached to an elastic string, one end of 

 which is tied to the hand, while the stone is projected 

 vertically upwards, the hand would experience an upward 

 pull ; and are we thence to conclude that the stone is 

 continually acted upon by an upxvard force ? 



Possibly the whole fallacy is traceable to D'Alembert, 

 who gave us a sort of dynamical memoria tcchnica, usually 

 expressed in the words, " forces equal and opposite to the 

 effective moving forces of a material system are in 

 equilibrium with the external forces," — a very imperfect 

 and misleading statement for the principle obviously 

 contained in Newton's first and second axioms, viz., 

 " the mass-accelerations of the particles of any material 

 system have at each instant the same total component 

 along any line, and the same total moment round any axis 

 as the external forces acting on the system." 



D'Alembert's fictitious reversal is unnatural and un- 

 necessary ; and, in introducing fictitious forces and 

 reducing the state of the moving material system to one 

 of equilibrium, he ignored the actual state of affairs. He 

 should have faced the state of motion as it exists, and 

 recognized the fact that the production and maintaining 

 of a given state of motion requires the action of a definite 

 system of forces assisting the motion, instead of con- 

 centrating his attention on something which would stop 

 the motion. It is this determination to fix the mind on a 

 state of equilibrium, and to ignore the actual motional 

 state that is, I think, responsible for the fallacy of 

 j centrifugal force. 



I [Other examples of erroneous teaching with regard to 

 centrifugal force, the confusion oiwork with horse power, 

 &c., were then given.] 



We must remember, also, that a teacher who has often 

 to correct the erroneous language of the text-book which 

 he employs with his class is by no means certain of in- 

 spiring confidence in himself and convincing his pupils 

 that the teaching of the author is at fault ; for, young 

 students, very unhke Prince Bismarck, have a profound 



