258 



NA TURE 



[January i6, 1902 



{Canis vulgaris) from Savoy, presented by M. Leon Mon- 

 taigne; a White-crested Tiger-Bittern (Tigriosoiiia leiicohfhum) 

 from West Africa, presented by Mrs. F. M. Hand ; nine 

 Piieasant-tailed Jacanas I^IIyJrophasinntis chirtirgtts) from India, 

 presented by Mr. I'rank Finn; a Horned Capuchin {Celms 

 apella) from South America, a Feline Douroucouli {Nyctipilhecus 

 vociferatts) fromJSouth Brazil, four Crowned Partridges (Kolluliis 

 cristaliis) from Malacca, deposited ; a White-tailed Gnu (Con- 

 noihaelus gnu, J ), born in the Gardens. 



OUR ASTRONOMICAL COLUMN. 



Diameter of Jui'Iter. — In continuation of his series of 

 determinations of planetary diameters with the 26-inch refractor 

 at Washington, Prof. T. J. J. See gives the reduced measures 

 of Jupiter in Asironomisihe Nacluichten, Bd. 157, No. 3757. 

 The observations were made during daylight, using the colour 

 screen over the eye-piece for eliminating the secondary fringes, 

 &c. For the final evaluation of the diameter sixty-eight measures 

 are employed, extending over the period 1901 September 6- 

 October I ; from these he gives : — 



Equatorial diameter of Jupiter = 37" '646 + o"%l4 

 = 141.950 ± 53 km. 

 Prof. See thinks this very closely approximates to the absolute 

 value of the diameter, and by comparing it with the value 

 obtained at night, when the planet is seen as a very brilliant 

 object on a dark background, he obtains a measure of the irradia- 

 tion. The night value is 38"'40, which gives for the irradia- 

 tion : — 



I = o"755 + o"-040 

 = 2847 + 150 km. 

 As these values are so different, the suggestion is made of the 

 advisability of adopting two sets of planetary diameters, one re- 

 presenting the apparent size of the planet as seen at night, to be 

 used in physical observations and ephemerides, work on satel- 

 lites, &c., the other representing the true dimensions of the 

 spheroid independent of its illumination by the sun, to be 

 employed in the theory of the planet's figure, constitution, &c. 



The resulting absolute dimensions of the Jovian spheroid 

 referred to the distance 5 '20 are : — 



Equatorial diameter = 37 646 = 141,950 km. 

 Polar diameter ... =35-222= 132,810 km. 



Oblateness = I:I5'S3. 



Assumed mass ._ = 1:1047 '35 (Newcomb). 

 Density = i -35 (water = i). 



"The Heavens at a Glance," 1902. — This handy little 

 publication for the present year is issued in a slightly modified 

 form. The author has repeatedly had inquiries respecting the 

 inclusion of one or more star maps, and the present edition is 

 furnished with two, one showing the northern stars, the other 

 the southern objects visible from Great Britain. Another 

 additional feature is the small map of the moon, showing the 

 principal lunar formations. 



All the more important phenomena are given for the year, 

 and a series of summaries of the particulars relating to variable 

 and coloured stars, nebulx-, &c. 



Vakiabi.e Star Catalogue. — In the Astronomical Journal, 

 vol. xxii. No. 514, the committee appointed by the Council of 

 the Astronomische Gesellschaft publish a further catalogue giving 

 the elements of stars which have been certainly recognised as 

 variable since the publication of Chandler's third catalogue 

 (Asirotioinical Journal, vol. xvi., pp. 145-172). The present 

 list gives the definitive designations for 191 variables, and also 

 for the three Novie in Perseus, Sagittarius and Aquila. 



Catalogue OF 100 New Double Stars. — Bulletin 'Ho. 12 

 from the Lick Observatory comprises the fourth catalogue of 

 new double stars having distances under 5", discovered by W. J. 

 Ilussey with the 36inch telescope at Mount Hamilton. (The 

 first three catalogues appeared in the Astronomical fournal, 

 Nos. 480, 4S5, 494.) 



The search is being conducted in a systematic manner, and 

 it is hoped that the work when more advanced will afford 

 data for an investigation into the distribution of close double 

 stars in various parts of the sky, and of their numbers with 

 respect to magnitude. 



NO. 1 68 I, VOL. 65] 



THE TEACHING OF MATHEMATICS IN 

 PUBLIC SCHOOLS. 



'T'HE following letter has been sent to the Committee ap- 

 pointed by the British Association to report upon the 

 teaching of elementary mathematics. 



Gentlemen, — M the invitation of one of your own body, 

 we venture to address to you some remarks on the problems 

 with which you are dealing, from the point of view of teachers 

 in public schools. 



As regards geometry, we are of opinion that the most 

 practical direction for reform is towards a wide extension of 

 accurate drawing and measuring in the geometry lesson. 

 This work is found to be easy and to interest boys ; while many 

 teachers believe that it leads to a logical habit of mind more 

 gently and naturally than does the sudden introduction of a 

 rigid deductive system. 



It is clear that room must be found for this work by some 

 unloading elsewhere. It may be felt convenient to retain 

 Euclid ; but perhaps the amount to be memorised might be 

 curtailed by omitting all propositions except such as may serve 

 for landmarks. We can well dispense with many propositions 

 in the first book. The second book, or whatever part of it we 

 may think essential, should be postponed till it is needed for 

 III, 35. The third book is easy and interesting ; but Euclid 

 proves several propositions whose truth is obvious to all but the 

 most stupid and the most intellectual. These propositions 

 should be passed over. The fourth book is a collection of 

 pleasant problems for geometrical drawing ; and, in many cases, 

 the proofs are tedious and uninstructive. Xo one teaches 

 Book V. A serious question to be settled is — how are we to 

 introduce proportion ? Euclid's treatment is perhaps perfect. 

 But it is clear that a simple arithmetical or algebraical explana- 

 tion covers everything but the case of incommensurables. Now 

 this case of incommensurables, though in truth the general case, 

 is tacitly passed over in every other field of elementary work. 

 Much of the theory of similar figures is clear to intuition. The 

 subject provides a multitude of easy exercises in arithmetic and 

 geometrical drawing ; we run the risk of making it diflicult of 

 access by guarding the approaches with this formidable theory 

 of proportion. We wish to suggest that Euclid's theory of 

 proportion is properly part of higher mathematics, and that it 

 shall not in future form part of a course of elementary geometry. 

 To sum up our position with regard to the teaching of geometry, 

 we are of opinion — ■ 



(i) That the subject should be made arithmetical and practical 

 by the constant use of instruments for drawing and measuring. 



(2) That a substantial course of such experimental work 

 should precede any attack upon Euclid's text. 



(3) That a considerable number of Euclid's propositions 

 should be omitted ; and in particular 



(4) That the second book ought to be treated slightly, and 

 postponed till III, 35, is reached. 



(5) That Euclid's treatment of proportion is unsuitable for 

 elementary work. 



Arithmetic might well be simplified by the abolition of a good 

 many rules which are given in text-books. Elaborate exercises 

 in vulgar fractions are dull and of doubtful utility ; the same 

 amount of lime given to the use of decimals would be better 

 spent. The contracted methods of multiplying and dividing 

 with decimals are probably taught in most schools ; when these 

 rules are understood, there is little left to do but to apply them. 

 Four-figure logarithms should be explained and used as soon as 

 possible ; a surprising amount of practice is needed before the 

 pupil uses tables with confidence. 



It is generally admitted that we have a duty to perform 

 towards the metric system ; this is best discharged by providing 

 all boys with a centimetre scale and giving them exercise 

 in verifying geometrical propositions by measurement. Perhaps 

 we may look forward to a time when an elementary mathe- 

 matical course will include at leaist a term's work of such easy 

 experiments in weighing and measuring as are now carried on 

 in many schools under the name of physics. 



Probably it is right to teach square root as an arithmetical 

 rule. It is unsatisfactory to deal with surds unless they can be 

 evaluated, and the process of working out a square root to five 

 places provides a telling introduction to a discourse on incom- 

 mensurables ; furthermore, it is very convenient to be able to 

 assume a knowledge of square root in teaching graphs. The 



