January i6, 1902] 



NATURE 



259 



same rule is needed in dealing with mean proportionals in 

 geometry. 



Cube root is harder and should be postponed until it can be 

 studied as a particular case of Horner's method of solving 

 equations approximately. 



Passing to algebra, we find that a teacher's chief difficulty is 

 the tendency of his pupils to use their symbols in a mechanical 

 and unintelligent way. X boy may be able to solve equations 

 with great readiness without having even a remote idea of the 

 connection between the number he obtains and the equation he 

 started from. And throughout his work he is inclined to regard 

 algebra as a very arbitrary affair, involving the application of a 

 number of fanciful rules to the letters of the alphabet. 



If this diagnosis is accepted, we shall be led naturally to 

 certain conclusions. It will follow that elementary work in 

 algebra should be made to a great extent arithmetical. The 

 pupil should be brought back continually to numerical illustra- 

 tions of his wcrk. The evaluations of complicated expressions 

 in a, b and c may of course become wearisome ; a better way of 

 giving this very necessary practice is by the tracing of easy 



graphs. Such an exercise as plotting the graph y = 2.x - — 



4 

 provides a series of useful arithmetical examples, which have 

 the advantage of being connected together in an interesting way. 

 Subsequently, curve-tracing gives a valuable interpretation of 

 the) solutions of equations. Experience shows that this work is 

 found to be easy and attractive. 



With the desire of concentrating the attention of the pupil 

 on the meaning rather than the form of his algebraical work, 

 we shall be led to postpone certain branches of the subject to a 

 somewhat later stage than is usual at present. Long division, 

 the rule for H.C.F. , literal equations and the like will be studied 

 at a period when the meaning of algebra has been sufficiently 

 inculcated by arithmetical svork. Then, and not till then, will 

 be the time to attend to questions of algebraic form. 



But at no early stage can we afford to forget the danger of 

 relapse into mechanical work. For this reason it is much to be 

 wished that examining bodies would agree to lay less stress 

 upon facility of manipulation in algebra. Such facility can 

 generally be attained by practice, but probably at the price of 

 diminished interest and injurious economy of thought. The 

 educational value of the subject is sacrificed to the perfecting of 

 an instrument which in most cases is not destined for use. 



To come to particulars, we think that undue weight is often 

 given to such subjects as algebraic fractions and factors. The 

 only types of factors which crop up continually are those of 

 jr'--a-, y-±2a.v-l-3-, and, generally, the quadratic function of 

 X with numerical coefficients. 



In most elementary algebra books there is a chapter on 

 theory of quadratic equations in which a good deal of atten- 

 tion is paid to symmetric functions of roots of quadratics. No 

 further use is to be made of this until the analytical theory of 

 conies is being studied. Might not the theory of quadratics be 

 deferred until it can be dealt with in connection with that of 

 equations of higher degree? 



Indices may be treated very slightly. The interpretation of 

 negative and fractional indices must of course precede any 

 attempt to introduce logarithms ; but when the extension of 

 meaning is grasped, it is not necessary to spend much more 

 time on the subject of indices ; we may push on at once to the 

 use of tables. 



It will be seen that our recommendations under the head of 

 algebra are corollaries of two or three simple guiding thoughts, 

 the object in view being — to discourage mechanical work ; the 

 means suggested — to postpone the more abstract and formal 

 topics and, broadly speaking, to arithmeticise the whole subject. 



The omission of part of what is commonly taught will enable 

 the pupil to study, concurrently with Euclid VI., a certain type 

 of diluted trigonometry which is found to be within the power 

 of every sensible boy. He will be told what is the meaning of 

 sine, cosine, and tangent of an acute angle, and will be set to 

 calculate these functions for a few angles by drawing and 

 measurement. He will then be shown where to find the 

 functions tabulated, and his subsequent work for that term will 

 consist largely in the use of instruments, tables and common 

 sense. A considerable choice of problems is available at once. 

 He may solve right-angled triangles, work sums on "heights 

 and distances," plot the graphs of functions of angles, and make 

 some progress in the general solution of triangles by dividing 



NO. 1 68 1, VOL. 65] 



the triangle into right-angled triangles. Only two trigono^ 

 metrical identities should be introduced — 



sin-fl-l-cos-9= I, and =tan fl. 



cos 9 



In short, the work should be arithmetic, and not algebra. 



Formal algebra cannot be postponed indefinitely ; perhaps 

 now will be the time to return to that neglected science. We 

 might introduce here a revision course of algebra, bringing in 

 literal equations, irrational equations, and simultaneous quad- 

 ratics illustrated by graphs, partial fractions, and binomial 

 theorem for positive integral index. Side by side with this 

 it ought to be possible to do some easy work in mechanics. 

 Graphical statics may be made very simple ; if it is taken up at 

 this stage, it might be well to begin with an experimental veri- 

 fication of the parallelogram of forces, though some teachers 

 prefer to follow the historical order and start from machines 

 and parallel forces. Dynamics is rather more abstract ; a first 

 course ought probably to be confined to the dynamics of recti- 

 linear motion. 



It is not necessary to discuss any later developments. The 

 plan we have advocated will have the advantage of bringing the 

 pupil at a comparatively early stage within view of the elements 

 of new subjects. Even if this is effected at the sacrifice of 

 some deftness in handing a, b and c, one may hope that the 

 gain in interest will be a motive power of sufficient strength to 

 carry-the student over the drudgery at a later stage. Some 

 drudgery is inevitable, if he is ultimately to make any use of 

 mathematics. But it must be borne in mind that this will not 

 be required of the great majority of boys at a public school. 



We beg to remain, gentlemen. 



Yours faithfully, 

 G. M. Bell, Winchester. R. Levett, King Edward's 



H. H. Champion, Upping- School, Birmingham. 



ham. J, W. Marshall, Charter- 



H. Crabtree, Charterhouse. house. 



F. W. DoBBS, Eton. L. Marshall, Charterhouse. 

 C. Godfrey, Winchester. C. \V. Payne, Merchant Tay- 

 H. T. Holmes, Merchant lors' School. 



Taylors' School. E. A. Price, Winchester. 



G. H. J. Hurst, Eton. D. S. Shorto, Rugby. 

 C. H. Jones, Uppingham. A. W. Siddons, Harrow. 

 H. H. Kemele, Charter- R. C. Slater, Charterhouse. 



house. H. C. Steel, \Vinchester. 



T. Kensington, Winchester. C. O. Tuckey, Charterhouse. 



E. M. Langley, Bedford F. J. Whipple, Merchant 



Modern School. Taylors' School. 



CONFERENCE OF SCIENCE TEACHERS. 

 r^NE of the most important of the many educational con- 

 ^"^ ferences which it has become customary to hold during 

 the Christmas vacation is that arranged under the auspices of 

 the Technical Education Board of the London County Council. 

 The custom of inviting teachers of science from all parts of the 

 country to attend meetings in London, to discuss the best 

 methods of imparting instruction in the branches of science 

 taught in schools and colleges, was inaugurated four years ago, 

 and each successive year has seen a substantial increase in the 

 attendance. While in 1S99 fewer than a hundred teachers, 

 inspectors and others responded to the invitation of the Tech- 

 nical Education Board, there were present at the meetings held 

 on Thursday and Friday last at the South-Western Polytechnic, 

 Chelsea, upwards of four hundred persons, among whom were 

 representatives of every stage of science teaching. 



The vice-chairman of the London Technical Education 

 Board, Mr. T. A. Organ, presided at the first meeting and, in a 

 speech welcoming the teachers present, referred to the neglect 

 of science teaching in this country in the past and contrasted 

 this with the admirable efforts made in Germany since the 

 opening of their first chemical laboratories in 1827. As indicat- 

 ing the amount of leeway we have as a nation to make up, he 

 pointed to the fact that there are nearly 10,000 more or less 

 well-trained chemists employed in German factories, and, as 

 half of them have undergone a complete course of several years' 

 training in the technical high schools, it is not surprising that 

 (Germany should be gradually securing markets in which origin- 

 ally British trade was supreme. It is unnecessary to repeat 



