i6 



NATURE 



[May 4, 1893 



with a series of topics for observation, the stars, moon, planets, 

 &c. , assuming that the readers are supplied only with an opera 

 glass or small telescope. It is to be in no sense professional, 

 "except to be accurate in statement of fact and principle with- 

 out being technical in terms." The first number can be ready 

 by September of this year if the subscribers are forthcoming. 



Optical Tests for Objectives. — In a small pamphlet en- 

 titled "Optische Untersuchung vonObjectiven," by Dr. Ludwig 

 Mach of Prague, the contents of which have appeared in the 

 Phologyaphischer Rundschau, the writer describes a very simple 

 means of obtaining photographs of objectives showing defects 

 in the glass. After first referring shortly to the methods adopted 

 by Schioder, Alvan Clark, &c., giving some excellent small pho- 

 tographs of some of the results obtained by these means, he de- 

 cribes his method of making small optical errors visible. He 

 casts, by means of an achromatic lens, an image of the sun 

 on a screen in which is a small hole. Behind this screen, at 

 some distance from it, he places the object glass to be tested, 

 together with the camera at its focus, and it is found that in all 

 places where the object glass is not perfect a system of interference 

 marks or rings is formed. Experimenting with an object glass 

 of 10 '2 cm. aperture and 143 cm. focal length, by Sir Howard 

 Grubb, the writer shows a photograph taken after this means. 



Photograph of a Bolid. — Although on fine nights many 

 telescopes carrying with them photographic plates are turned 

 towards the starry heavens for special objects, none, except a very 

 few exceptions, have had the good luck to record the passage of 

 a bright meteor. M. Lewis, at Ausonia (Connecticut) seems to 

 have been very forlunate[in this respect [Bulletin Astronomiqtte, 

 tome x., March), for on January 13 of this year, while photo- 

 graphing the comet Holmes, a very bright meteor crossed the 

 field of view. An examination of the pljte showed that the trail 

 commenced at about ih. 38m. R.A., and -f 33°40' declination, 

 terminating at oh. 8m. R. A., and -f 32° 12' declination. Under 

 the microscope he sajs that the centre of the trail is crossed by 

 a very dark axis, clearly defined, while the other part is bounded 

 by fringes of very irregular forms, indicating that fragments of 

 matter had been detached from the meteorite : signs of rotary 

 movement during its passage before the sensitised plate were also 

 visible. For orbit determinations, photographs such as these, if 

 they could be more often obtained, would be very valuable, for 

 one could then fix the different points of the irajectory with far 

 greater accuracy than is now done by the necessarily very ap- 

 proximate method of naked eye estimations. 



GEOGRAPHICAL NOTES. 



An amusing instance of newspaper science occurred in a 

 morning paper last week. A note on the salinity of the North 

 Pacific, published in this column (vol. xlvii. p. 590), was repro- 

 duced without acknowledgment, but with annotations. After the 

 quotation, " a tongue of considerably fresher water stretches 

 nearly across the ocean about 10° N." came the interpolation, 

 "caused no doubt by the dilution of the sea by the melting 

 snow and ice of the northern regions," a far-fetched hypothesis, 

 which ignores the rainy belt ol calms. A worse error was lo 

 say that the curves of equal salinity "run through Behring 

 Strait," when the original said Bering Sea. The use of a map 

 would probably have prevented the blunders. 



The Mouvement GSogj-aphique publishes a useful risumi with 

 route-maps and portraits of the officers of the various expeditions 

 of the Katanga Company from May, 1890, to April, 1893. In 

 July, 1890, the expedition of M. A. Delcommune left Europe 

 for the Congo, went by the Lomami, discovered Lake Kassali, 

 and reached Bunkeia, in Katanga on October 6, 1891. This 

 expedition spent a year in exploring the upper Lualaba and the 

 western side of Lake Tanganyika, then descended the Lukuga, 

 crossed the Congo basin in a west-by-north direction to Lus- 

 ambo, and arrived in Brussels on April 15, 1893. An expedition 

 under Le Marinel left Lusambo on December 23, 1890, reached 

 Bunkeia on April i8, 1891, and after taking possession of Kat- 

 anga, returned to Lusambo in August of the same year. On July 

 4, 1891, Captain Stairs left the east coast, and travelling by Lake 

 Tanganyika reached Bunkeia in December, but the leader died 

 on the Zambesi on his way home on June 8, 1892. In Septem- 

 ber, 1891, Captain Bia's party left Stanley Pool, ascended the 

 Sankuru, discovered Lakes Kabele and Kabire, near the 

 Lualaba, and reached Bunkeia in January, 1892. Thence in 



NO. 1227, VOL. 48] 



June they reached Lake Bangweola, and after Captain Bia's 

 death. Lieutenant Francqui led the expedition through the 

 upper regions of the Lualaba, and in January, 1893, joined 

 Delcommune at Lusambo, returning with him to Europe. The 

 discoveries made by these four expeditions are of great import- 

 ance ; they fill in much of the detail of the Congo basin hitherto 

 very lightly sketched on the maps. 



A RU,MOUR has been current that Dr. Nansen's polar expedi- 

 tion is likely to collapse at the last moment for lack of funds ; 

 but it is satisfactory to learn that this is not the case. The 

 Fram is practically ready for sea, and the party will embark in 

 the month of June, as originally intended. 



The recent advance in Arctic navigation is strikingly sh^wn 

 in the announcement by a Norwegian firm of a pleasure-trip to 

 Spitzbergen, planned for this summer, with a vessel strengthened 

 for ice-work and fitted with every comfort. 



MM. FouREAU AND Mery have during the past year carried 

 out some important journeys in the Sahara. They have suc- 

 ceeded in reaching the country of the Tuaregs, which has not 

 been visited by Europeans since the Flatters' mission was mas- 

 sacred in 1881, and they have induced the chiefs to acknowledge 

 French protection. The French officials are diligently extending 

 the cultivable area of the oases in the northern Sahara by 

 sinking artesian wells and securing artificial irrigation. 



THE USE OF HISTORY IN TEACHING 

 MATHEMATICS.' 



T H.WE ventured to make some suggestions to this Associa 

 tion as to the use of history in teaching mathematics, and 

 the restrictions and limitations under which it may be advan- 

 tageously employed. It will be perhaps the most convenient 

 course to begin with the restrictions and limitations. 



The three most important of these are: — 



(i) The history of mathematics should be strictly auxiliary 

 and subordinate to mathematical teaching. 



(2) Only those portions should be dealt with which are of real 

 assistance to the learner. 



(3) It is not to be made a subject of examination. 



Unless these conditions are observed, it is to be feared that 

 the eff^ect of the introduction of new matter for instruction 

 would be injurious rather than beneficial. The ordinary school- 

 boy or schoolgirl now takes in hand quite as many subjects as 

 he or she can satisfactorily study, and nobody wants the number 

 to be increased. 



When men look back on their school days, they constantly 

 feel some things they have always remembered and often 

 applied came to them from their masters not as part of the 

 regular course or as included in the work done for examination. 

 It is just this outside illustrative position that I propose history 

 should occupy in respect to mathematics. I want at the outset 

 to free myself from any imputation of desiring to add one grain's 

 weight to the heavy burden boys and girls have to bear in these 

 days of competitive examination. 



Coming now to the main question, which is in what ways 

 history makes mathematical study easier, clearer, or more inter- 

 esting, it may first of all be remarked that it gives us stereo- 

 scopic views instead of pictures and diagrams. A particular 

 subject may be looked at from many sides, each aspect suggest- 

 ing a different mode of treatment. Thus, although we do not 

 want to go back to the method in Whewell's Mechanical Euclid, 

 where the main truths of elementary statics were all derived 

 from the fundamental axiom that a ruler would balance if its 

 middle point were supported ; it is yet a good thing for the 

 pupil to know that such a method was successfully adopted. We 

 do not want in arithmetic to go back to the old-fashioned rules 

 of single and double false position, but the student is all the 

 better for knowing what they were, and what could be effected 

 by their means. Possibly some of us might really like to go 

 back to the proof of Euclid I. 47 in the " Vija Ganita, " depend- 

 ing only on the almost obvious truth that triangles of the same 

 shape have their sides proportional, but at all events a student 

 should know about this proof, even if he were to be warned of 

 the objections to using it. 



In some instances there is a further direct advantage in recall- 

 ing old methods that are now superseded. Though the change 



1 Abstract of a paper by Mr. G. Heppel, read before the Association for 

 the Improvement of Geometrical Teaching. 



