356 



THE POPULAE EDUCATOR. 



line c B, produced towards B, while it touches the sphere in a 

 small eircle or parallel of latitude passing round the sphere 

 through the point F, in which the straight line D E touches 

 the arc A B. We will assume this small circle, which is indicated 

 in the diagram by the dotted line F G, 

 to represent the 55th parallel of north 

 latitude, and divide the whole arc A B 

 into eighteen arcs of 5 degrees each, in 

 the points marked 5, 10, 15, etc. Now, 

 supposing straight lines to be drawn 

 from the centre c, through the points in 

 the arc marked 45 and 65, cutting D E 

 in. the points H, K, it is manifest that the 

 straight line H K, the projection on the 

 surface of the cone of the arc between 45 

 and 65 on the surface of the sphere, does 

 not differ materially from the arc itself in 

 length, and that if this process wero 

 adopted for a small portion of the sur- 

 face of the sphere on either side of the 

 arc between 45 and 65, instead of the 

 arc itself only, the result on the surface of 

 the cone, when the cone was unrolled and 

 spread out flat on a table, would represent 

 accurately enough for all practical pur- 

 poses on a flat surface the portion of the 

 sphere which is thus projected on the cone. The measure- 

 ment on the surface of the cone between places situated on or 

 very near the 55th parallel, would be precisely the same 

 as their distances 

 from each other on 

 the sphere, while the 

 distances between 

 places on the cone 

 near the parallels 

 passing through 45 

 and 65 would be a 

 little in excess of 

 their distances from 

 each other on the 

 surfaceof thesphere. 

 If, however, instead 

 of touching the 

 sphere, we suppose 

 the circumscribing 

 cons to pass through 

 it, through two pa- 

 rallels of latitude, as 

 M L, the section of 

 the side of a cone 

 which cuts the 

 sphere in the paral- 

 lels of 45 and 65, it 

 is manifest that we 

 ensure a greater 

 degree of accuracy 

 in delineating the 

 features of that por- 

 tion of the sphere 

 that are to be de- 

 picted on the cone, 

 especially when a 

 larger extent of the 

 sphere has to be pro- 

 jected on the cone, 

 as we have two paral- 

 lels, namely, those of 

 45 and 65, along 

 which measurements 

 on the cone are iden- 

 tical with measure- |t 

 ments on the sphere. 



In projecting the arc B A on the surface of the circum- 

 scribing cone, it is manifest that the distances between every 

 two parallels at intervals of 5 degrees would be accurately 

 determined by drawing straight lines through the centre 

 to the points of division, and producing them till they meet the 

 Bide of the cone ; but as the points so obtained would exhibit 



unequal divisions for the measure of degrees on the meridian 

 (as the learner may see more clearly if he will take the trouble 

 to construct Fig. 14 on a large scale), in making, for example, 

 a map of any portion of the surface of the sphere between the 

 35th and 75th parallels, it is merely 

 necessary to set off equal spaces on the 

 line of the central meridian, as the 

 spaces 95 to M, marked at a, h, c, d, 

 and e, each of which is equal to a fourth 

 of the chord from 45 to 65. This is 

 done to obtain equal distances between 

 the parallels of latitude on the conical 

 projection as on the sphere. It may be 

 said, indeed, that the great advantages 

 presented by the conical projection are 

 the preservation of equal distances be- 

 tween the parallels of latitude and rec- 

 tangular intersections of the parallels 

 and meridians, as on the globe ; the 

 opposite diagonals measured across any 

 space contained by two parallels and two 

 meridians being in all cases equal to one 

 another. 



As the learner, if he have read care- 

 fully what has been said above, will now 

 thoroughly understand why the conical 

 projection is far better suited than any other for developing por- 

 tions of a sphere on a flat surface, we will proceed with instruc- 

 tions for making a projection for a map of Europe, after saying 

 , . . ..... that the conical pro- 



jection is the easiest 

 that a learner can 

 construct, as it con- 

 sists of nothing more 

 than straight lines 

 and concentric arcs 

 of circles, which can 

 be readily drawn by 

 means of a ruler and 

 pair of compasses. 



In Fig. 1 7 is given 

 a conical projection 

 for a map of Europe, 

 which the learner 

 should construct on 

 a larger scale by the 

 process about to be 

 described, on stout 

 cartridge paper, 

 pasted or pinned to 

 a drawing board, 

 taking care that the 

 board is large enough 

 to include the centre 

 from which the arcs 

 representing the pa- 

 rallels of latitude 

 are to be described. 

 First draw the base 

 line, c D, as shown 

 in the figure, with a 

 fine pencil ; bisect 

 it in E, and through 

 E draw the straight 

 line, A B, at right 

 angles to c D. The 

 straight lines A B, 

 c D should be drawn 

 as far as the paper 

 will admit. An in- 

 spection of a map 

 of Europe will show 



that the whole of this continent is included within the 35th and 

 75th parallels of north latitude on the south and north, and 

 bisected by the meridian of 20 or the 20th degree of longitude 

 east from Greenwich. The straight line A B may therefore be 

 taken to represent longitude 20 east from Greenwich in our pro- 

 jection. Take any space to represent five degrees, but bf. careful 



