April 6, 1893 J 



NATURE 



547 



the more northern part. Dr. Scheiner's photographs show that 

 its extension southward is very considerable, but, owing to its 

 dimness, was not seen by Prof. Barnard. The form of this 

 nebula, a copy of which is given in this number, is inclined to 

 be spirally, although not so apparent as that of Andromeda, 

 and, curiously enough, it lacks a bright nucleus, as in the 

 latter. 



Minor Planets. — The work of discovering minor planets 

 >eems, at the present time of the year, to be in a very flourish- 

 ing condition, although rather restricted to two observers, 

 according to the current number of Astronomische NachrichUn 

 (No. 3157). Charlois with lo and Wolf with 2, bringing the 

 present notation up to 1893 ■*'. is a good number for the first 

 quarter of the year, and if this average be kept up we shall 

 soon be driven to indulge in the Greek or German alphabet, or 

 both. 



GEOGRAPHICAL NOTES. 



Mr. John Bartholomew, of Edinburgh, whose reputation 

 as one of the foremost British map-makers is world-wide, died 

 on March 30, at the age of 61. His career will be remembered 

 as an epoch in the history of the perfecting and popularising 

 of English maps. Trained in Edinburgh and afterwards 

 under the late Dr. Petermann, in London, Mr. Bartholo- 

 mew succeeded his father in a cartographical business in 

 Edinburgh, which he steadily enlarged and improved, paying 

 attention not only to excellence of mechanical production, but 

 to the improvements of methods of representation. But the 

 leading characteristic of Mr. Bartholomew's work was his 

 conscientious endeavour to produce the most accurate topo- 

 graphical delineation. The general use of maps coloured 

 orographically in this country is mainly due to the efforts of 

 the Edinburgh Geographical Institute, of which he was the 

 head. Mr. Bartholomew gradually withdrew from active work 

 on account of failing health, and his son, Mr. J. G. Bartholomew, 

 lAs taken his place in the Geographical Institute. 



Mk. Theodore Bent (see p. 519) has been able to reach 

 Aksum, where, however, he only remained for eight days, on 

 account of tribal wars. The party had to retire abruptly 

 because of a threatened fight, in which they were very nearly 

 compelled to take sides, but fortunately the report of an ad- 

 vance of Italian troops to their relief solved the difficulty, and 

 they reached the coast in sn fety. Despite the shortness of the 

 working time, some good archaeological results have been 

 obtained. 



The March number of Petermann' s Mitteilungen contains a 

 valuable paper on North-west Patagonia by Dr. J. von Siemi- 

 radzki, with a map showing the results of his surveys and 

 coloured to bring out the pastoral possibilities of the region. His 

 route in 1891-92 led up the Rio Negro and Rio Limay to Lake 

 Nahuel-Huapi and thence northward through the grassy valleys 

 and bare slopes of the Cordillera to the Upper Biobio valley, 

 whence the expedition passed to the coast of Chile. 



Thk Royal Geographical Society has given a grant to Dr. H. 

 R. Mill to defray the expenses of a careful bathymetrical survey 

 of some of the larger English lakes. The work, which will be 

 carried out next summer, would be greatly facilitated if use could 

 be had for a few days of a steam launch upon any of the lakes. 

 Windermere, Coniston Water, and Wastwater will probably be 

 sounded in the first place, as they are the most interesting from 

 the limnological point of view. 



A paper on the Geography and Social Conditions of the Iberian 

 Peninsula read at the March meeting of the Berlin Geographical 

 Society by Prof. Theobald Fischer is -published in abstract in 

 the April number of the Geographical Journal. The paradoxi- 

 cal character of the peninsula in the variety of its conditions has 

 long been known. The great central plateau with its broken 

 mountain border sloping steeply to the sea throws the bulk of 

 the populatio.i towards the coast-line. In the border zone of 

 the peninsula comprising 45 per cent, of its area, more than 66 

 per cent, of the inhabitants are settled. The only large city in 

 the central plains is Madrid ; all the rest of the plateau is 

 occupied by wheat-growers and sheep-rearers ; the mining, 

 fruit-growing and industrial interests being all confined to the 

 seaward slopes. There are few parts of Europe in which the 

 physical conditions so plainly dominate the whole character of 

 a country. 



NO. 1223. VOL. 47] 



GRAPHICAL SOLUTIONS OF PROBLEMS IN 



NAVIGATION. 

 I. J F we suppose the two angles P, S of a spherical triangle SPZ 

 *° '°g«^''«'^ less than two right angles, a plane triangle 

 bjPi^i may clearly be drawn such that Pj = P and S. = S. 

 The sides of the spherical triangle PS, PZ, SZ being respectively 

 denoted by p, c, z, those of the plane triangle may be taken in 

 the followmg ratios : — 



PjSi - tan i/>, 



PiZi = 4 tan i(t -f- 2) + i tan \{c - z), 

 SiZi = 4 tan i(<r + z). - ^ tan ^[c - z), 

 These results may be easily verified. 

 Hence S,Zi -f- PjZj = tan ^(f^-l- 0), 



*°^ PiZi - SjZi = tan \(c - z). 



From these equations we infer that Z^ is the intersection of an' 

 ellipse and hyperbola which have the same foci Pj and Sj. 

 Suppose now that the line SiPj contains, say, 100 divisions, 

 and that a system of ellipses, having Sj and ?^ as foci, with 

 major axes loi, 102, 103 .. . and a system of hyperbolas 

 whose axes are 99, 98, 97 . . . are drawn on one side of 

 SjPj ; then, by finding m^, w, from the equations 

 m^ •= 100 tan \[c + z) cot \p, 

 tn^ = 100 tan \{c - 2) cot \i>, 

 we should be able to localise the point m^m„ as coming between 

 two successive ellipses and also between two consecutive hyper- 

 bolas in the diagram. 



2. The usefulness of such a diagram lies in its application to 

 problems in navigation. For/ may be taken as the north polar 

 distance of the sun, z the complement of his altitude, and c the 

 colatitude of the place of observation. Having determined 

 m^, m„ and thus localised Zj in the diagram, the angle ZjPiSi 

 is I he hour angle which may be suitably measured. 



If we interchange/ and c in the diagram, thus making Pjand 

 Zj the foci, the point to be localised is Sj from the equations 



«i ~ 100 tan i(/> -I- 2) cot \c, 



«2 = IOC tan \(p - 2) cot ^c. 

 The difficulties attending this mode of representation will pre- 

 sent themselves in another form in §4. It is sufficient to notice 

 here that this use of the diagram has the advantage of giving 

 two useful angles— SiPjZj, the hour angle, and SiZiPj the 

 azimuth. 



3. The merit of both these modes of representation consists 

 in their being each a single diagram, applicable at any time of 

 year, though in northern latitudes more favourable to accurate 

 measures in summer than in winter. Their demerit consists in 

 the preliminary calculations of Wj, m„, or «j, «.,. This, how- 

 ever, might be minimised by supplying, along with the diagram, 

 tables of the values of m for two arguments and <p given by 



m = 100 tan i fl cot i <^ 

 The whole amount of preliminary calculation would then consist 

 in adding and subtracting/ and 2, and looking out m^ and m^. 



I shall now investigate the nature of a diagram which requires 

 no preliminary calculation. 



Returning to the spherical figure SPZ, let us suppose SP to 

 be fixed while the sides PZ, SZ vary so that Z describes a curve 

 on the sphere. The corresponding point Zj will describe a cor- 

 responding locus on the plane. For example, if L describes a 

 small circle with P as centre, the locus of Z, will be given by 

 tan c = tan;i(<r + z) + i{c - z)\ 

 PiZ, 

 I - PjZi« -I- SjZ,-' 

 Now, if we draw a perpendicular ZjN to the side l\Si we shall 

 have 



PiS,^ - SiZi^ = PiN2 - SiN2 



=; tan i/)(2PiN - tani/); 

 .-. PiZj = tan <: tan i/(cosec/ - PjN). 



This shows that the curve described by Z^ is a conic section of 

 eccentricity tan c tan ^p, with focus at Pj and directrix perpen- 

 dicular to PjCi at a distance cosec/ from P^ 



Similarly, the curves corresponding to small circles about Si 

 are conies with a common directrix and with focus at Sj, their 

 curvatures being turned the opposite way from those about Pj. 



