MATHEl[ATICAL.] GEOGRAPHY 207 determined so as to give, upon the whole, the least amount I sphere contained by two consecutive meridians the differ- of exaggeration for the entire map. This idea of a cutting ' cone seems to have originated1 with the celebrated Gerard Mercator, who in 1554: made a map of Europe on this prin- ciple, selecting for the parallels of intersection those of 40° and 60°. The same system was adopted in 1745 by De- lisle for the construction of a map of Russia. Euler in the Ada .-lc.ru]. Imp. Petrop., 1778, has discussed this pro- jection and determined the conditions under which the errors at the northern extremity, at the centre, 0 and at the southern extremity of a map so con- structed shall be severally equal. Let c, c’ be P P the co—latitudes of the extreme northern and southern parallels, -y, 7’ those of two intermediate (3 m parallels, which are to be truly represented in the G projection. Let OC’, Um’ (fig. 23) be two consecu- tive meridians, as represented in the developed cone; the difference of longitude being (1), let the G. " angle at 0 be km. The degrees along the'meri- C’ m’ dian being represented by their proper lengths, CC’=c’ —c ; and P corresponding to the pole, let Fis- 23- OP=2, then OC=2+c; and so for G, G’, C’. The true lengths of G’n’ and G22, namely, (1) sin y’ and wsin y, are equal to the represented lengths, namely, /aw (2+y’) and In.) (2+y) respectively, wl1e11ce -y and y’ are known when It and 2 are known. Comparing now the represented with the true lengths of parallel at the extremities and at the centre, if e be the common error that is to be allowed, then c:Izw(:+c) — to sin c, c: —Izw(:+ .'3c+§c’)+w sin §(c+c’), 6:71-w(’: + c’) — to sin c’. The difference of the first and third gives 7:, and then sub- tracting the second from the mean of the first and third, we get
- + .‘/_I'+c')=_}(c’— c) cot ‘;(c’—c) tan .l_,(c’+c).
Thus 2 being known, the common centre of the circles re- presenting the parallels is given. The value of /I, is given by the equation /z(c’ — c) =sin c’ — sin c, and y and -y’ can be easily computed. But there is no necessity for doing this as we may construct the angles at O, which represent- ing a difference of longitude to are in reality equal to law. For instance, to construct a map of Asia on this system, having divided the central meridian into equal spaces for degrees, 2 must be calculated. Here we have c= 20°, c’= 80°, whence 2 + 50° =15’ tan 50° cot 15° = 66°'7. Hence in this case the centre of the circles is 16°'7 beyond the north pole ; also /L = ‘G138, so that a difference of longitude of 5° is represented at O by an angle of 3° 4’ 9". The de- grees of longitude in the parallel of 70° are in this map re- presented too large in the ratio of 1°] 50 :1; those in the mid—latitude of 40° are too small in the ratio of 0933 : 1; and those i11 10° latitude are too large in the ratio of 1'05 to 1. G'.rmss’s Projection may be considered as another variation of the conical system of development. Meridians are represented by lines drawn through a point, and a difference of longitude to is represented by an angle /to), as in the preceding case. The parallels of latitude are circular arcs, all having as centre the point of divergence of the meridian lines, and the law of their formation is such that the representations of all small parts of the surface shall be precisely similar to the parts so represented. Let u be the co—1atitude of a parallel, and p, a function of 2:, the radius of the circle representing this parallel. Consider the infinitely small space on the 1 See page 178 of Traité dcs Projections des Caries Géographzyzzcs, by A. Germain, Paris, an admirable and exhaustive essay. See also the work entitled Coup cl’o2z'l hz'sto7'z'que sur la P7'0jcctz'on dos C’m'tes dc Gé0_r/raphie, by M. d’Avezae, Paris, 1863. I ;;’g’ =dp, 32'9" = p/LC]/1., the angle at O ence of longitude of which is dp, and two consecutive parallels whose co—latitudes are 2: and u+du. The sides of this rectangle (fig. 24) are pq=d2z, pr=sinm[,,, whereas in the representation 7)’q’?"s’, P r , '1 being:/nip. Now, as the represen- tation is to be similar to the original, 8 1/41’: (lp ___}J_q: the girl’ Iz.pd,u.l pr sinuclp. ’ 0 )3; q _ 3;_ c zo , _ ’ whence P —}b5f—i,,,, , and integrating, ,,, I *5” p=l.'(tan l v 1«‘ig_ 24_ where the constant /2, is to be determined according to the requirements of each individual case. This investigation was first made in 1772 by the German mathematician J. H. Lambert,‘~’ but in 1825 it was again brought forward by Gauss in an essay written in answer to a prize question pro posed by the Royal Society of Sciences at Copenhagen. A translation of this essay is to be found in the P/rilosop/zical Jildj/CLZZ’7l€ for 1828 (seepage 112), where Lambert’s projec- tion comes out as a particular solution of the general pro- blem. Again, in a general investigation of the problem of “similar representation,” Sir John Herschel, in the 30th volume of the Journal of the I303/al G'eo_qrapIzical Society (1860), deduced as a particular case this same projection. A large 111ap of Russia was constructed and published on this system by the Geographical Society of St Petersburg in 1862. The relative scale in this development is—— (lp ilk (tan; )7’ r7°zo=E ' sin u where (L is the radius of the sphere. It is a minimum when u= cos‘1 /1. This minimum should occur in the vicinity of the central parallel of the map; if no be the co- latitude of this parallel, we may put P: k<tan g)COS 11., I Or if we agree that the scale of the representation shall be the same at the extreme co—latitudes c, c’, then _ log sin c’ — log sin c "log tan §c’— log tan ,l,c' To construct a map of North America extending from 10° latitude to 70°, we may take /l.= and it‘ such as shall make the diflbrence of radii of the extreme parallels: 60, namely It = 101-315. The scales of the representation at the northern and southern limits"are 1'll6 and 1096 respect- ively. The radii of the parallels are these- 70° . . . 32'801 30° . 72328 60°. . . 43‘356 20° . . . 82255 50°. . . 53177 10° . . . 92'801 40°. . . 02728 0° . . . 10-P315 Having drawn a line representing the central meridian, and selected a point on it as the centre of the concentric circles, let arcs be described with the above radii as parallels. For meridians, in this system a difi'erence of longitude of 10° is represented by an angle of two-thirds that amount, or 6° 40’. The chord of this angle on the parallel of 10°, whose radius is 92801, is easily foundto be 10792. N ow stepping this quantity with a pair of com- passes along the parallel, we have merely to draw lines through each of the points so found and the common centre of circles. The points of division of the parallel may be checked by taking the chord of 20°, or rather of 13° 20’, 3 Bcit7'('i_qe zum Gebrauchc dcr Illa!/zeinatik zmd deren Anwcndung,
vol. iii. p. 55, Berlin, 1772.