April 17, 1902J 



NA TURE 



559 



may be even more potent than it is reckoned. With associa- 

 tions growing in influence, and the great facilities afforded for 

 exchange of ideas, the body of teachers is very rapidly increasing 

 in strength, and this reform in the teaching of mathematics, 

 together with many another much-needed reform, is perhaps 

 much more in the immediate future than is thought. At any 

 rate, if the bow and arrow is still the official weapon, the use of 

 the magazine rifle is being secretly taught, and we school 

 teachers look forward with no misgivings to that great fight. 

 Prof. Perry sees ahead for our people, rather we are "spoiling 

 for it," for with it w'ill come our freedom ! 



Frank L. Ward. 

 I Macdonald Place, Hartlepool, March 29. 



Rearrangement of Euclid Book I., Pt. i. 



In answer to Prof. Lodge's letter I should like to say that we 

 have for some time followed much the order he suggests. 

 Euclid's order unnaturally separates propositions which should 

 come together, e.g. L 4, S, 26, and is, therefore, a serious 

 hindrance to a clear grasp of the subject-matter as distinct from 

 mere exercise in logic. 



The following order — substantially that suggested by Prof. 

 Lodge — seems natural, and we have certainly found it work 

 very well in practice. 



(i) The propositions on angles, viz. 13, 14, 15, 27, 29, 32, 

 cor. 2, 32. At this stage logical deduction from definitions and 

 axioms is difficult and, to a boy, unconvincing. The following 

 proof of L 32 cor. 2 is convincing, at least : " If a man walks 

 right round a rectilineal figure (starting and ending at a point in 

 the middle of a side), he turns once round. Hence the exterior 

 angles, which are the angles through which he turns, are 

 together equal to 4 right angles." Similar proofs of 27 and 29 

 are equally convincing. Any attempt to analyse these proofs 

 into the axioms on which they depend seems to me at this stage 

 foolish ; it is work for a highly trained and speculative mind, 

 not for a boy, 



(2) Triangulation, I. 4, 8, 26. 



These are, I think, best presented as the outcome of ex- 

 perience passing into intuition, and as special cases of the general 

 fact that three data are necessary and sometimes sufficient to 

 determine a triangle. The special case of right-angled triangles 

 with hypotenuse and one side given should be added and proved 

 deductively from I. 5. 



The rest of Book I. consists of exercises on these fundamental 

 propositions : — Properties of a single triangle, I. 20, 5, 18, 6, 

 19; loci; quadrilaterals; areas. The order in which these 

 last three subjects are taken is immate.ial. 



A special advantage of this arrangement is that it makes it 

 easy to combine practical with theoretical work. It was, indeed, 

 from the attempt to do this, that we were led to follow this 

 order, but even in purely theoretical work it has proved a great 

 gain. 



As to the omission of "constructions" from the deductive 

 course, we agree — they are properly treated as exercises. 



As to the effect of this change on real progress we have no 

 doubt. As to examinations, we hope that they will before long 

 (l) permit freedom in the order of propositions, (2) diminish 

 bookwork and insist upon riders and practical work, as some, 

 indeed, already do. 



It seems illogical, but even in deserting Euclid's order we 

 adhere to his numbers. The constant reference to cardinal 

 propositions is a great help to thoroughness and clearness of 

 knowledge, as well as to ease of questioning and answering. 

 Probably no one will ever succeed in fixing fresh labels on to 

 the propositions, and for the present at least we find the old 

 ones useful, though they are to our boys quite arbitrary. 



W. C. Fletcher. 



Liverpool Institute. 



I QUITE agree with Prof. Alfred Lodge as to the order of 

 propositions he proposes, which is practically the order I 

 adopted in my " Foundations of Geometry." But he does not 

 in his letter refer to what seems to me the chief reason for it, 

 which is that the elementary geometry of straight lines and 

 angles should precede the geometry of plane surfaces, including 

 any propositions about areas. And to carry out this idea, the 

 fundamental propositions which Euclid gives so badly in his Xlth. 

 book (props. 1-9) ought to be taken before such propositions as 

 his I. 35 and 36. On the other hand, there are impjrtant pro- 



NO. 1694, VOL. 65I 



positions in the Xlth. book, notably prop. 10 (if this is not 

 included in the definition of parallelism) and props. 20 and 21, 

 which come properly in what Prof. Lodge calls the first part of 

 Book I. 



By the way, I may mention that it seems to me illogical to 

 prove I. 27, as Prof. Lodge does, by a simple " which is 

 impossible," and to refer I. 29 to " Playfair's axiom." Neither 

 proposition is nearer a priori truth than the other, and it is just 

 as easy to disguise the difliculty, if you wish to do so, in either 

 case. Edward T. Dixon. 



Racketts, Hythe, Hants, April 14. 



We have had the following arrangement of Euc. , 1-32, in 

 use for three years with more than two hundred pupils. 13, 14 

 (from the definitions); 15; 32, cor. 2, 32, 16, 17; 23, 8, 9; 

 4, 10. Locus of a point equidistant from two given points. 

 II, 12, 5 ; 26, 6. Locus of points equidistant from two inter- 

 secting straight lines. 



This gives fourteen propositions ; thirty-seven more complete 

 all the plane geometry of Euc. I. -VI. and XII. required in 

 mathematics or science. We have no superposition " proofs " ; 

 they merely obscure obvious truths. Parallels by superposition 

 have been found beyond the capabilities of beginners. Why not 

 alter the definition ? At present it gives the least obvious 

 property of parallels. 



A caution to the professors who are teaching us how to teach. 

 We are seeking a system of geometry suitable for boys of ten, 

 and the most logical method is not necessarily the best ; it is 

 better to separate 4, 8, 26 by examples of their use and to leave 

 the remaining case for trigonometry. Again, an ideal course 

 must be inventional, and must grow out of practical work ; 

 therefore it must introduce problems as early as possible : a 

 beginner should not be allowed to quote a construction which 

 he cannot perform. Is not the demand for a purely theoretical 

 course due to a desire to use I, 9, in proving i, 5, whilst 

 retaining Euclid's proof of I, 8 ? T. Petch. 



Leyton Technical Institute, April 14. 



In reply to the appeal of Prof. Alfred Lodge for opinions 

 with reference to his proposal to alter the sequence of Euclid's 

 propositions by introducing those relating to parallels at the 

 earliest possible stage, permit me to express what I hold to be 

 insuperable objections to his proposed innovation. 



Whatever other objections may be raised to Euclid's sequence 

 of propositions, it at any rate has this distinguishing merit, that 

 it separates the propositions (I. 1-2S) which are independent of 

 the postulate of parallels from those which are true only when 

 that postulate is admitted. To obscure this distinction, as, for 

 instance, by treating props. 16, 17 as corollaries of prop. 32 and 

 so appearing to depend on the postulate of parallels, would to 

 my mind, especially now that the non-Euclidean geometry of 

 Lobatchewsky and others is an established part of mathematical 

 science, be a distinctly retrograde step. 



Further, this innovation is not in the least necessary to secure 

 Prof. Lodge's object (with which I entirely sympathise), namely, 

 a better and more natural grouping of the propositions about 

 triangles. 



For this purpose all that is necessary is to add I. 16 to the 

 three (13, 14, 15) with which he proposes to begin. This 

 proposition may at once be proved as follows : — 



The triangle being ABC, the side B C produced to D and E 

 the mid-point of K C, turn the triangle A E B about E until 

 E A comes on E C and A on C, then E B comes to a position 

 E F in the same straight line as B E, and since B E F, BCD 

 meet in B. they cannot meet again, so that F lies on the same 

 side of B D as A [N.B., here comes in the difference between 

 plane and spherical surface geometry], and E C F or the angle 

 A is less than the exterior angle A C D. 



This proved and I. 17 as its corollary, the propositions about 

 a single triangle and those about the comparison of triangles 

 easily fall into a simple and natural sequence and grouping. 



Shanklin, April 12. Robt. B. Hayward. 



Winter Phenomena in Lakeland. 

 There being no record within my knowledge as to whether 

 holly and ivy are starch-trees or fat-trees, i.e. as to whether 

 their wood-starch disappears or otherwise in winter, a strict 

 watch was set upon the phenomena. During the months of 

 December, January and February, sections were taken at 



