332 Notes on Map Projections. 
Mercator’s principle enables him to run directly from one point 
to another. For the polar portion of the earth in which this 
projection totally fails, a central projection can be used to some 
distance. A projection on Mercator’s principle might be made 
relative to the prime meridian instead of the equator, its prime 
verticals serving as the equidistant parallels, (as in Cassini’s) and 
the circles parallel to the prime meridian being projected by the 
e of increasing degrees. This would require the investiga- 
tion of the formule for conversion of coérdinates in this case. 
The parallels and meridians of the earth might then be con- 
structed by points. Another mode would be to make a radial 
and concentric circular projection around the pole, in which the 
length of the latitude degree should be deduced from the same 
condition as in Mercator’s method, the divergence of meridians 
being duly considered. The amount of distortion in Mercator’s 
projection wholly unfits it for land maps; and the variation of its 
ale in different parts would give rise to endless inconvenience 
were it applied to any other purpose than that of nautical off- 
shore charts, in which direction is so much more important than 
area or distance. 
BONNE’S OR THE MODIFIED FLAMSTEAD’S PROJECTION. 
This method of projection is that which has been almost uni- 
versally applied to the detailed topographical maps based on the 
trigonometrical surveys of the several States of Europe. It was 
originated by Bonne, was thoroughly investigated by Henry and 
wissant in connexion with the map of France, and tables for 
France were computed by Plessis. 
In constructing a map on this projection, a central meridian 
and a central parallel are first assumed. A cone tangent along 
the central parallel is assumed, the central meridian is develope 
on that element of this cone which is tangent to it, and the cone 
is then developed on a tangent plane. The parallel falls into an 
arc with its centre at the vertex, and the meridian into a grad- 
uated right line. Concentric circles are conceived to be traced 
through points of this meridian taken at elementary distances 
along its length. The zones of the sphere lying between the 
parallels through these points are next conceived to be developed 
each between its corresponding arcs. Thus, all the parallel zones 
of the sphere are rolled out on a plane in their true relations t0 
each other and to the central meridian, each having in projection 
the same width, length, and relation to its neighboring zones, 4 
on the spheroidal surface. As there are no openings between 
consecutive developed elements, the total area must in this case; 
and in all like developments of surfaces of revolution, remain 
wholly unaltered by the development. Each meridian of the 
projection is so traced as to cut each parallel in the same point 1 
which it intersected it on the sphere. 
