298 



NA TURE 



[January 30. 1902 



artificial; aiming at training the pure reason they have got out 

 of touch with facts, and for many pupils degenerated into mere 

 jugglery with symbols cast loose from thought ; hence they fail 

 to interest and influence all but a very few. Look at the 

 questions set in any of the well-known examinations — and see 

 how many of them consist of stock puzzles of more or less com- 

 plexity, invented, apparently, solely in order that successive 

 generations of boys may learn how to deal with them, score 

 marTvS by them, and then lay them aside as useless ! And, of 

 course, a large portion of our text-books and our teaching is 

 necessarily devoted to such questions. So long as the chief 

 examinations maintain their present character a general reform 

 of school mathematics is well-nigh imijossible, and partial re- 

 forms at individual schools (I have Winchester in mind as a 

 pioneer in endeavour) are very difticult. I will, however, briefly 

 and without detail indicate directions in which I think real 

 improvement can be made without introducing revolutionary 

 <;hanges. 



The great aim must be to introduce as much as possible the con- 

 crete element, for there are few boys who cannot be interested 

 keenly in what they can deal with in practical fashion by 

 <irawing or handling in any other way, and fewer still to whom 

 a bare abstract idea is not repellent. Until elementary physical 

 measurements and the mathematics appropriate for dealing with 

 them are taught together, an arrangement much to be wished for 

 both from the points of view of science and mathematics, the 

 best field for the introduction of the concrete is undoubtedly 

 geometry. 



In all the earlier stages of geometrical work, theory should 

 always be kept in touch with practice by much drawing and 

 measuring of figures ; this, I am convinced, is the best way of 

 building up exact geometrical ideas, and it has besides the great 

 advantage of being intensely interesting to boys. I do not 

 refer to "geometrical drawing" as often taught and usually 

 understood in examinations which aims merely at making certain 

 constructions (though this gives a valuable bit of training to 

 those who have too often no notion of using their hands 

 efficiently for any purpose not connected with a ball), but I 

 would have it used always concurrently with, and in illustration 

 of, demonstrative geometry. This is, I know, quite possible, 

 though I have as yet come across no text-book in which it is 

 ■done. 



In theoretical geometry the only serious divergences from 

 Kuclid's methods I would advocate are (l) the introduction from 

 the first of the idea of an angle as generated by a rotation, 

 and (2) the substitution of the arithmetical and algebraical 

 treatment of proportion for Euclid's. 



Euclid's test of proportion, which appeals so strongly to the 

 grown mathematician by its elegance and completeness, is for 

 even the very best boys very difficult to grasp, and for the 

 moderate boy a rigid insistence on it (which is practically never 

 made) would involve an absolute bar to the discussion of similar 

 figures and elementary trigonometry, matters which are quite 

 easy if the difficulty of incommensurability be kept in the back- 

 ground. I would, however, while adhering to Euclid as the 

 only possible text-book, omit, on a first reading at least, many 

 of the propositions in order to push on to those which connect 

 with, and can be illustrated by, practical work. For instance, 

 after Book I., I would have read those propo.sitions of Book III., 

 some dozen or so in number, proving the angle properties of 

 circles. 



Seeing that demonstrative geometry furnishes by far the most 

 accessible example of pure deductive logic, and for most boys 

 the only one they will ever come in contact with, I would insist 

 most strongly on its never being sacrificed to so-called " proofs " 

 by measurement which are found in some books. The training 

 of the reasoning powers is one of the highest aims of education, 

 and with this end in view constant practice in riders is of the 

 greatest value ; the teaching of Euclid's text without this is a 

 most deadly waste of time, and cannot be too strongly con- 

 demned. 



In arithmetic I think the most important reform would he 

 the general recognition of the fact that decimals are not adapted 

 for exact calculation, but are preeminently valuable in approxi- 

 mation, which is the practically useful form. 



I'rom the first, therefore, boys should be taught to work out 

 results correct to a few places only— generally not more than 

 four — and all work with recurring decimals should be omitted. 



Many of the puzzling questions set on such subjects as dis- 

 count, stocks, S:c. , have very slight relation to pr.->ctical life, 



NO. 1683, VOL. 65] 



they require much time to learn to deal with them, and should 

 be discarded in favour of work on areas and volumes of simple 

 figures. An equal amount of thought can be elicited, and 

 therefore a not less amount of culture imparted, by good 

 problems on the latter subjects, with the advantage of being 

 more in touch with practical requirements. 



In algebra I would, in the earlier stages, insist much more 

 closely than is done at present on the accurate use of symbols 

 as a shorthand language for expressing arithmetical operations, 

 deferring long "sums"' of multiplication, division, &c., until 

 much work has been done on .simple equations of the first 

 degree as aids to the solution of problems. Later I would 

 omit much of the harder manipulation with fractions and 

 abnormal index expressions which is now taught, and in place 

 of these devote much time to the development of the notion of 

 one quantity as a function of another, illustrated by plotting 

 graphs on squared paper. The theory of fractional and negative 

 indices should be taught as leading up to logarithms to base lo, 

 but I deprecate the too early use of these in calculation. 



Arithmetical trigonometry involving functions of acute angles 

 only, and with constant reference to four-figure tables and 

 accurate drawings to scale, should be taught much more gene- 

 rally than it is now. For boys in the higher forms who are but 

 poor mathematicians I have found it an interesting and stimu- 

 lating change from the weary round of arithmetic and algebra 

 they had trodden ad iiattseavt before. A short course of the 

 same work should, even in the case of good boys, be pre- 

 liminary to the algebraical treatment of trigonometry. 



I have written only of the very lowest rungs of the mathe- 

 matical ladder ; those who from professorial and engineering 

 altitudes lecture us on what we ought to teach have often no 

 notion of the mind stratum in which the greater part of our 

 life's labour is spent ; hence their advice, and their books when 

 they condescend to write for us, are loo often hopelessly above 

 the mark. That by cooperation of all interested some real 

 improvements in the curriculum may enable us to get a rung or 

 two higher all round is the earnest wish of myself and many 

 other teachers. J. W. Marshall. 



Charterhouse. 



The Distance of Nova Persei. 



Since publishing, in Nature of January 2, the suggestion 

 that the cause of the apparent expansion of the nebula sur- 

 rounding Nova Persei might be explained by the illumination 

 of meteoric matter by the light sent out on the occasion of the 

 outburst of the Nova, I have seen a paper published by Prof. 

 Kapteyn in the Aitr. A'aili. (No. 3756), in which he suggests 

 the same idea. His claim to priority in the matter is therefoie 

 clear. In my note, referred to, I give the distance of the Nova 

 3s 313 light years. In calculatmg this distance I made the 

 mistake of taking the date of the outburst as February 12 

 instead of the 22nd. This made the distance of the Nova con- 

 siderably too great. 



Let 1> denote the distance of the Nova, and r the radius of 

 the nebula, in ttiiles ; and let f be its radius in seconds of arc. 

 Then we have 



'^=—''^.-. = 206265 X - (') 



D 206265 ^ p ^ 



But if V is the velocity of light in miles per second, and if T 

 be the time in days elapsed from the outbreak of the star to the 

 date of the photograph, then 



>- = 24x6ox6o. V . T (2) 



Substituting this in (i) we find 



T 

 D = 24 X 60 X 60 X V X 206265 X . 



Also if L be the distance travelled over by light in 

 365 i days, i.e. a light year, then 



L = 24 \ 60 \ 60 X \' X 365 j .... 

 Dividing (3) by (4) we find 



0^206265 T 



L 365-25 9 



year of 



. .(4) 



L..[275.S4]x'^.L 



(S) 



the figures in brackets being the logarithm of ~- ^. 



36525 



