62 



SCIENCE. 



[Vol. XII. No 2i 



tant and interesting. The making of a general map of the United 

 States offers peculiar difficulties on account of the enormous size of 

 the territory they occupy, which is so large that the curvature of 

 the earth's surface has a very appreciable influence upon the map. 

 This will be better understood by the fact that a circle circum- 

 scribed about the country cuts off a section of the globe of a little 

 more than 42° aperture, its centre being situated north of Omaha. 

 It is impossible to make a single map of the United States which 

 in all its parts is an exact likeness of the country as it exists on 

 the globe : therefore it is the task of the map-maker to find that 

 method of projecting the country upon a flat piece of paper that 

 results in alterations which are as slight as possible, and that agree 

 best with the special object of the map. 



We will consider in how far the existing maps satisfy this de- 

 mand. On the accompanying map the polyconic projection of the 

 Coast Survey which is in general use is shown in red. The lines 

 of latitude and longitude are obtained in the following way. The 

 portion of the globe situated between two parallels that are very 



States with one another. But, besides this, the form of the single 

 countries is distorted. We have indicated by arrows that direction 

 in which the increase in the lengths of lines is a maximum. In the 

 direction vertical to these arrows the decrease in the lengths of 

 lines is a maximum. The distortion of each part of the map de- 

 pends upon the relation of these two measures, and may be ex- 

 pressed by an angle. We observe on our map that this distortion 

 exceeds 3° in the outlying parts of the country. 



After having thus found the character of alterations in this map, 

 we will proceed to consider whether it meets the demands that 

 must be made upon such a map. The principal objection is, that 

 the surfaces are very much enlarged in the outlymg districts, which 

 are at the same time among the most thickly settled. It is true 

 that measurements may be made on sectional maps which have 

 various central parallels ; but when it is the object of the map t& 

 present at a single glance the relative extent of certain phenomena, 

 of whatever character they may be, climatological, geological, or in- 

 dustrial, it is of prime importance that the surfaces in various parts 



SKETCH-MAP OF THE UNITED STATES SHOWING THE DIFFERENCE BETWEEN THE DISTORTIONS IN THE POLYCONIC 



AND CONIC PROJECTIONS. 



close together may be considered as part of a cone the apex of 

 which lies on the axis of the earth. By developing this cone, the 

 zone assumes the shape of a portion of a circular ring. One zone 

 after another is thus developed. As the side of the cone varies 

 according to the latitude of the zone, the rings do not exactly join 

 each other ; but this is remedied by enlarging the lengths of the 

 meridians so as to fill up the gap. It will easily be understood that 

 by this process those parts of the map lying far away from the cen- 

 tral meridian will become distorted and too large. We will first 

 study the enlargement of surfaces. On our map the lines on which 

 the enlargement amounts to i, 2, 3, 4, 5, and 6 per cent are indi- 

 cated by heavy lines. Thus it will be seen, that while, in measur- 

 ing a surface in the central points of the United States, we commit 

 an error no greater than i per cent, the latter amounts to more than 

 6 per cent in the New England States and in California. If we 

 have a large-scale map of the United States showing the distribution 

 of forests, 10.6 square inches in New England will represent the 

 same surface as 10 square inches in Minnesota.. Thus we see that 

 we cannot compare the surfaces of various parts of the United 



of the map should be equivalent to each other. Maps which have 

 this property are called ' authalic ' or ' equivalent ' maps, a,nd we 

 should say that it is the fundamental point that each map used for 

 census purposes should be authalic. 



But we might ask, if we agree that this should be the case, will 

 not the alterations of angles become enormous, and thus our ideas 

 of distances become altogether wrong.' In order to satisfy this 

 question, we must consider what means we have to preserve the 

 same relation of surfaces all over the map, and at the same time to 

 make the alteration of angles — and therefore that of distances 

 also — as small as possible. We may first ask that this alteration be 

 smaller than on the ordinary polyconic projection, and then we will 

 have a map exceeding in value the former. But we present here a 

 map that has not only slighter alterations of angles than the poly- 

 conic projection, but the surfaces of which are also preserved all over 

 the map. While in the polyconic projection the alteration of angles 

 very nearly reaches 4°, it does not exceed in our map 1° 16'. 

 While the surfaces on the former projection are enlarged more 

 than 6 per cent in the outlying regions, they are not at all altered 



