184 



SCIENCE. 



[N. S. Vol. XL No. 266. 



methods by which the undoubted advantages 

 of the polyconic projection can be preserved and 

 its disadvantages greatly reduced, to which 

 Blundau cannot be an entire stranger. One 

 way would be not to adhere strictly to one cen- 

 tral meridian, but in the case of an oblique map 

 to shift the apices of the tangent cones in such a 

 manner that the central meridians pass as nearly 

 as may be through the middle of the map ; the 

 meridians would then assume a spiral shape. I 

 have used this method on several occasions, but 

 Hammer is the first one, I believe, who has 

 called public attention to it. If the map should 

 have a predominating east and west dimension, 

 the developing cones may be applied in a trans- 

 verse position ; some great circle passing cen- 

 trally through the map might be treated as a 

 central meridian and the poles might be trans- 

 ferred to the equator. In the accompanying 

 sketch I have constructed a projection of the 

 United States on this principle ; the 95° long, is 

 substituted for the equator and the great circle, 

 which in lat. 39°, is perpendicular to the 

 meridian of 95°, is taken as central meridian. 

 The distortion, which in an ordinary polyconic 

 projection, accumulates near the right (east) 

 and left (west) borders is here transferred to 

 the vicinity of the upper (north) and lower 

 (south) borders.* It may not be amiss to 

 mention that for certain purposes Blundau 

 has recommended the employment of abnormal 

 conic projections, in which case the axis of ihe 

 cone does not coincide with the axis of the 

 earth and gives as illustration a map of Africa, 

 in which the point in which the equator in- 

 tersects the western coast of the continent is 

 chosen as apex of the cone. On this projection 

 the elements of distortion show very favorable 



* In the polyconic projeetion the lines of equal 

 linear (and areal) distortion are parallel to the central 

 meridian, and the distortion for modern distances 

 increases as the square of the distance from this line. 



[Distortion : 



"■"■(s^)" 



where A = distance from 



central meridian in degrees of arch of great circle.] 

 Since the distance across the United States from north 

 to south, is only about three-fifths of that from east to 

 west, it follows that by the above manipulation the 

 maximum of distortion is reduced from 6J per cent, 

 to about 2J per cent. 



conditions, but it has the serious defect of 

 leaving a blank space by the complete de- 

 velopment of the cone on a plane, and since 

 Hammer has shown in Petermann's Mitt, of 

 1894, that just as favorable conditions may be 

 attained by an equivalent azimuthal projection, 

 the application of abnormal conic projections 

 does not appear to deserve much encourage- 

 ment. The polyhedral projection has a trap- 

 ezoidal shape. It is now generally adopted in 

 Europe for the single sheets of serial publica- 

 tions of government surveys on a large scale 

 (between 1/20000 and 1/100000) ; it is similarly 

 used by the U. S. Geological Survey, and has 

 been proposed by Penck for the prospective map 

 of the world on the one millionth scale. It is the 

 shape which any part of the earth's surface, en- 

 closed by two parallels and two meridians will 

 assume in many kinds of projections now in 

 use, provided the size of the section is small 

 enough (not more than 15 or 30 minutes or 

 one degree of latitude and longitude) to allow 

 the substitution of the chord for the arc 

 of the parallels. Consequently this so-called 

 polyhedral projection, properly speaking, is 

 no projection at all ; the separate sheets may 

 be joined in different ways, such as will conform 

 to either a polyconic or to a simple tangent 

 conical projection. 



Cylindric Projections. — The transition from 

 conic to cylindric projections takes place when 

 the constant factor (re) of the azimuthal angle 

 becomes 0. In this case the meridians become 

 straight parallel lines and the contact occurs at 

 the equator. This great circle as well as the 

 parallels appear also as straight lines, intersect- 

 ing the meridians at right angles. These con- 

 ditions are common to all cylindric projections, 

 and the only difference between the several 

 varieties consists in the particular function of the 

 latitude j/ =/(<?) which is adopted as measure 

 for the distance of the parallels from the equa- 

 tor. If the meridional arcs are given in their 

 true dimensions {y = arc f) we have the square 

 projection which should not be used for more 

 than 15° from the equator. For broader zones 

 an ' intersecting' cylinder should be substituted 

 (corresponding to the intersecting cone of the 

 De Lisle's projection) which will transform the 

 square into a rectangular projection. The cylin- 



