i_.i'rHEiiA'ricAL.] perspective. But, as Lagrange has remarked, one may regard geographical maps from a more general point of view as representations of the surface of the globe, for which purpose we have but to draw meridians and parallels according to any given law; )- tlieii any place we have to fix must take that position with reference to these lines 1, that it has on the sphere with reference to the circles of latitude and longitude. Let the law which connects latitude and longi- tude, gb and to, with the rectangular co- ordinates x and 3/ in the representation be such that clx = mdgb + mic», and cly = m’¢Igb + n’dw. In fig. 6 let the lines -5‘ Fig. 6. intersecting in the parallelogram PQRS be the repre- sentations of the meridians 772, sq and parallels rs, pg in- tersecting in the indefinitely small rectangle pqrs on the surface of the sphere. The coordinates of P being at and 3/, while those of 7) are gb and w the coordinates of the other points will stand thus- q . . . 45 w+a'w 7‘ . . . ¢> -l-d¢> to S . . . ¢> +d¢> w-l-dw Q . . ac +’)ldw 1/+'n'¢lw It . . ac +md¢> 3/+m’il¢> S . . :1: +md¢>+ndw 3/+m’d¢>+n’dw. Thus we easily see that PR=(m‘3+m'2)%dqS; and PQ = (12.2 + n'3)l¢lw; also the area of the paiallelogram PQRS is equal to (m’n-12m’)dgbdw. If 90°=¥=iIx are the angles of the parallelogram, then mm + m’n’ tan == - ‘P m’n—'mn' If the lines of latitude and of longitude intersect at right angles, then mu + m'n' = 0. Since the length of 1)?‘ is = dgb, its representation PR is too great in the proportion of (m'-’+m’2)! : 1; and pq being in length cosgbdw, its repre- sentation PQ is too great in the ratio of (n.2+n'2)l: : cosgb. Hence the condition that the rectangle PQRS is similar to the rectangle pqrs is (m‘-’ + m"3) cosgx/g = 7;"-’+ n'2, together with out +m'n' = 0 ; or, which is the same, the condition of similarity is expressed by — 7i’:m cos ¢> ; n=m' cos ¢>. Since the area of the rectangle pqrs is COSgb(]<[)dco, the exaggeration of area in the representation will be expressed by 172'». — mu’: cos gb. Thus when the nature of the lines representing the circles of latitude and longitude is defined we can at once calculate the error or exaggeration of scale at any part of the map, whether measured in the direction of a meridian or of a parallel ; and also the misrepresenta- tion of angles. The lines representing in a map the meridians and par- allels on the sphere are constructed either on the principles of true perspective or by artificial systems of developments. The perspective drawings areindeed included as a particu- lar case of development in which, with reference to a certain point selected as the centre of the portion of spherical sur- face to be represented, all the other points are represented in their true azimuths,—tlie rectilinear distances from the centre of the drawing being a certain function of the cor- responding true distances on the spherical surface. For simplicity we shall first apply this method to the projection or development of parallels and meridians when the pole is the_centre. According to what has been said above, the meridians are now straight lines diverging from the pole, dividing the 360° into equal angles ; and the parallels are represented by circles having the pole as centre, the radius of the parallel whose co-latitude is n being p, a certain func- tion of u. The particular function selected determines the nature of the development. GEOGRAPHY 201 Let Ppq, Prs (fig. 7) be two contiguous meridians crossed by parallels rp, sq, and Op’q’, Or’s’ the straight lines representing these meridians. If the angle at P is (in, this also is the value of the angle at 0. Let the co-latitude Pp=u, Pq=u+du; Op'=p, Oq’=p+dp, the circular arcs 12'7", q’s’ representing the P parallels pr, qs. If the radius of the sphere be unity, " Y 2/q’=dp ; 21'?’ =p§li4. .5- pq =du ; or =siii ud,u. Put 0 1), (lp p : 0"‘ j ' 0": , , q du ’ sin it F, then p’q'==apq and }*'7’ -=o’pr. That is to .5‘: say, a, 6' may be regarded as the relative Fig_ 7_ scales, at co-latitude u, of the represen- tation, a applying to meridional measurements, a’ to measurements perpendicular to the meridian. A small square situated in co- latitude we, having one side in the direction of the meridian—the length of its side being i—is represented by a rectangle whose sides are {U and ia-’; its area consequently is ‘Foo’. If it were possible to make a perfect representation, then we should have 0' = 1, 0' =1 throughout. This, however, is impossible. Ve may make o-=1 throughout by taking p = n. This is known as the Equidistant Projection, a very simple and effective method of representation. Or we may make 0' = 1 throughout. This gives p = sin u, a perspective projection, namely, the 0-rtlzo_q1'aplu'c. Or we may require that areas be strictly represented in the de- velopment. This will be effected by making ca’: 1, or - pcIp=sin min, the integral of which is p= 2 singer, which is the Equivalent Projection of Lambert, sometimes referred to as L01-_qna’s 1’rry'ection. In this system there is misrepre- sentatioii of form, but no misrepresentation of areas. Or we may require a projection in which all small parts are to be represented in their true forms. For instance, a small square on the spherical surface is to be represented as a small square in the development. This condition will be . . , dp du attained by making o-=0, or 7‘='Sin u which is, c being an arbitrary constant, p=ctan§-u. This, again, is a perspective projection, namely, the Stereo- grapkic. In this, though all small parts of the surface are represented in their correct shapes, yet, the scale varying from one part of the map to another, the whole is not a similar representation of the original. The scale, o'= §c sec‘-’;-u, at any point, applies to all directions round that point. These two last projections are, as it were, at the extremes of the scale; each, perfect in its own way, is in other re- speets very objectionable. We may avoid both extremes by the following considerations. Although we cannot make o-=1 and a'=1, so as to have a perfect picture of the spherical surface, yet considering 0-1 and o-’—1 as the local errors of the representation, we may make (o- - 1)‘-’ +' (o-'—1)'-’ a minimum over the whole surface to be repre- sented. To effect this we must multiply this expression by the element of surface to which it applies, viz., sinududp, and then integrate from the centre to the (circular) limits of the map. Let ,8 be the spherical radius of the segment to be represented, then the total misrepresentation is to be taken as B I %<a—1>:< which is to be made a minimum. Putting p——_-n+3/, and giving to 3/ only a variation subject to the condition 35/: 0 when 21: O, the equations of solution—using the ordinary notation of the calculus of variatioiis—-are . till‘) _ _ I - =0 , PB—0, P beintr the value of Zpsinu when u= B. This gives ll 0 X. -— 26 . the integral of '2 _P — 1 z siiiudu,
sin n