NA TURE 



[May 27, 1909 



attributed in England to Euclid's definitions, postu- 

 lates, axioms, and propositions. Euclid's system was 

 here regarded as the highest, and an infallible, type 

 of logical accuracy. That it is still so regarded by 

 some people is evident from the somewhat flippant 

 and jesting comments made on the circular of the 

 Board of Education in some of the daily papers. It 

 may not be hopeless to point out to the writers of such 

 comments that Euclid in at least one instance contra- 

 dicts himself. His definition of a circle, for example, 

 makes it to be, not a curve, but a surface : " a circle 

 is a plane figure bounded by one line which is called 

 the circumference." This clearly makes a circle to 

 be a surface, and, moreover, it is lacking in definite- 

 ness, because it does not say w-hether the plane- 

 bounded figure is that which is contained w-ithin the 

 circumference, or that infinite external space which 

 lies outside. Again, if a circle is a plane surface, what 

 becomes of the proposition that two circles can inter- 

 sect in only two points? Further, Euclid made the 

 mistake of supposing that every geometrical concept 

 can be defi>u'd, whereas there are some that can be 

 only described : witness his attempted definition of a 

 straight line, which merely encourages a pupil to 

 deceive himself with a vague word. 



The imperfections of Euclid are an old controversy 

 which need not be enlarged upon. His merit as a 

 logician is very great, and his logic is, on the whole, 

 a type of accurate reasoning. Those who took part 

 in reforming his system sought at once to preserve 

 his logical excellence and to improve the subject- 

 matter on which it was to be exercised. This they 

 tried to do by familiarising the beginner with the 

 main concepts of geometry in wa\'s more natural and 

 more easy than those adopted by Euclid — by an early 

 use of rule and compass, for example, which dis- 

 pensed with that somewhat complicated and ridicu- 

 lous problem which forms the very second proposi- 

 tion of Book i., " through a given point to draw a 

 right line equal to a given finite right line," a most 

 gratuitous stumbling-block to the beginner. They 

 assumed the potent Baconian principle that " ex- 

 amples give a quicker impression than arguments." 

 There is no doubt that the new system has made 

 geometry much more easy in its initial stages for the 

 young pupil, but it contains one great element of 

 danger — it may, to a great extent, replace strict logic 

 by rule of thumb, and accurate expression by slip- 

 shod language. Those who have to examine papers 

 on geometry sent by pupils from scores of different 

 schools must admit that this danger has not been 

 averted, and the reason is easily found. We are at 

 present teaching geometry on syllabuses. So long 

 as this plan is adhered to, there will be most per- 

 plexing diversities in the sequence of assumptions and 

 propositions in school teaching, not unmixed with 

 inaccuracy of expression. The present writer knows 

 from experience that it is necessary for an examiner 

 to keep before him several books on geometry when 

 dealing with the work of various schools, owing to 

 the fact that a proposition which one pupil thinks 

 it necessary to prove another assumes as an axiom. 

 Moreover, the whole of the pupils of a school are 

 sometimes found to speak of a circle as touching a 

 triangle at its three vertices. This is a matter de- 

 pendent on the individual teacher, and it cannot be 

 cured by any syllabus. 



There are, of course, several excellent text-books 

 on geometry, with little difference in the order of 

 propositions, but no one of them is universally adopted. 

 The successful reformation of the teaching of geo- 

 metry seems to require an authoritative text-book 

 whic'n will serve as a definite guide to all teachers — 

 such as that sanctioned by the Minister of Education 

 NO. 2065, VOL. 80] 



in France. In the absence of such a definite guide, 

 the present somewhat chaotic system will continue. 



The writer of this article suggested, in the columns 

 of Nature, in the early days of the reformed system, 

 that such an authoritative book should be issued 

 conjointly by the universities, but the universitv 

 authorities felt difficulties. Why should not the Board 

 of Education issue such a work? Its recent circular 

 is in itself an excellent syllabus, but the practical 

 teacher will regard it simply as one more added to 

 the bundle which he already possesses. 



There is one recommendation in the circular with 

 which it is impossible to agree : — " Axioms and postu- 

 lates should not be learnt or even mentioned " — that 

 is to say, they are to be treated as suppressed premises. 

 Now every mathematical physicist encounters occa- 

 sionally what seems to be a fundamental contradiction 

 of some proved result with other known results, and 

 it is only after it is pointed out to him that his 

 reasoning contains a suppressed premise that the 

 difficulty is removed. The neglect of the explicit re- 

 cognition of an axiom is the same in kind as the su]>- 

 pressing of an important premise. 



Two excellent sentences, containing a fundamental 

 truth, must be quoted from the circular : — " It should 

 be frankly recognised that unless the power of doing 

 riders has been developed, the study of the subject 

 is a failure. Although examining bodies may con- 

 tinue to pass candidates who merely reproduce proofs 

 they have learnt, eked out by definitions or other 

 matter, masters should not be satisfied with this ; 

 the only proof of knowledge worth hai'ing is the 

 power of applying it to new matter." (The italics 

 are ours.) This is, indeed, a great truth, the import- 

 ance of which in the teaching of applied mathematics 

 is still greater than it is in the teaching of geometry, 

 and one which every teacher should lay to heart. 

 George M. Minchin. 



PHOTOMETRIC UNITS. 



A N important announcement with regard to the 

 -'*■ photometric units maintained at the Bureau of 

 Standards, .America, the Laboratoire Central d'Elec- 

 tricito, Paris, and the National Physical Laboratory, 

 Teddington, has been issued by the Bureau of 

 Standards in its Circular, No. 15, dated April i, 1909. 

 It was at first intended to make this announcement 

 simultaneously in America, France, and Great Britain, 

 but circumstances prevented this. It is desirable, 

 however, to state authoritatively that the agreement 

 described in the subjoined memorandum has been 

 arrived at, and has the approval of the gas referees ; 

 and that the photometric standards of the National 

 Physical Laboratory are being maintained in accord- 

 ance with it. 



R. T. Glazebrook. 



Memo rand urn as to Photometric Units. 



In order to determine as accurately as possible the 

 relations between the photometric units of America, France, 

 Germany, and Great Britain, comparisons have been made 

 at different times during the past few years between the 

 unit of light maintained at the Bureau of Standards, 

 Washington ; at the Laboratoire Central d'Electricit^, 

 Paris ; at the Physikalisch-Technische Reichsanstalt, 

 Berlin ; and at the National Physical Laboratory, London. 



The unit of length at the Bureau of Standards has been 

 maintained through the medium of a series of incandescent 

 electric lamps, the values of which were originally intended 

 to be in agreement with the British unit, being made 

 100/88 times the Hefner unit. 



The unit of light at the Laboratoire Central is the 

 bougie decimale. which is the twentieth part of the standard 

 defined bv the Intornational Conference on Units of 18S4, 



