A TRANSVERSE POLYCONIC PROJECTION FOR GENERAL 
MAPS OF CANADA 
The ordinary polyconic projection, invented by Professor Ferdinand 
Hassler, first Superintendent of the United States Coast and Geodetic 
Survey, owes its popularity in United States and Canada, to the facility of 
its construction from elaborate tables published by that institution, the 
United States Geological Survey, the Smithsonian Institution, and others. 
But it has the grave drawback of introducing on maps, increasing errors 
in meridional distances as meridians recede from the central meridian. Its 
use is recommended for maps covering any distance north and south, but 
limited east and west to such extent that the distortion is at a minimum 
consistent with the purposes of the maps. 
The projection is, therefore, unsuited for a map of the whole of the 
Dominion of Canada whose great east-west dimension is predominating 
and whose site is in high latitudes. However, it seems that a polyconic 
projection in a transverse position, as suggested by Charles H. Deetz 
(U.S.C. and G. Survey Spec. Pub. No. 47), with the poles assumed — ^for 
construction only — to be transferred to the equator, and the middle meridian 
of the country taken as auxiliary equator, may be used with advantage 
for a general map of Canada. On account of the shape of the country, the 
greatest distortion and largest errors, inherent to the projection, would 
fall outside its boundaries; the general properties of the polyconic system 
would be preserved, and, by the fact that angles subtending the developed 
contact bases of the tangent cones would be smaller than in the ordinary 
projection, the consequent curve deflexions affecting the length of arcs of 
parallels of latitude, would also be smaller, and tend to lessen those errors. 
The following computation of a transverse polyconic projection is, there- 
fore, undertaken in the hope that it may serve a useful purpose, and interest 
the general map draughtsman to whom this paper is dedicated. Elementary 
knowledge only of plane and spherical trigonometry is required for the trans- 
formation of rectangular co-ordinates and no mathematical discussion of 
this projection will be entered into. 
The problem may be worked out on the hypothesis of a spherical 
earth whose radius would be such that the surface of the map-area would 
most nearly coincide with the corresponding ellipsoidal surface of the earth. 
The radius of such a sphere is equal to the geometrical mean of the meridian 
radius of curvature, and the radius of curvature of the normal section of 
the earth ellipsoid at the latitude of the centre of the area considered: 
^ pm pn ^ 
But this applies to relatively small areas only, as errors with contrary 
signs are introduced in meridional and parallel distances. On a map of 
the whole of Canada, unless the scale be very small, such errors are inad- 
missible. 
1 Germain, A.; Trait6 dea Projections dea Cartes G6ographiques. 
75342—2 
