Plane trigonometry with practical applications, by Leonard E. Dickson.

Ch. VI]

NAVIGATION: DEAD RECKONING

71

50. The Mercator chart. A clear understanding of the leading
method of making a map of the earth's surface is of great importance not only in navigation but also in geography and other
earth sciences. The student will not fail to appreciate the practical nature of this topic.
The earth's surface is mapped on the interior of a rectangle
in such a way that the meridians are represented by parallel
straight lines perpendicular to the straight line representing the
equator, while the parallels of latitude are represented by straight
lines parallel to the line representing the equator. Since the
earth's meridians converge at the poles and yet have been plotted
as parallel lines, there has been an opening out of these meridians,
i.e., a stretching of east and west lengths. But we desire that any
small figure on the map shall be of the same shape as the corresponding figure on the earth. Hence there must be simultaneously
a stretching of north and south lengths. For a very short such
vertical length in latitude L, the stretching factor is sec L, when
the earth is regarded as a sphere. For, by Art. 48, diff. long. =
dep. X see L. On a Mercator chart we agreed that the east and
west length called departure should be stretched until it becomes
equal to the corresponding diff. long. mapped unstretched on the
line representing the equator. Hence dep. has been stretched in
the ratio sec L, and we agreed to use the same stretching factor
for small vertical lines. This argument is valid only for a very
short arc, say one minute of arc. Given a longer arc extending
from the equator vertically to lat. 5~ N, we divide it into 300 arcs
each equal to a minute and hence obtain the stretched arc containing sec 1' + sec 2' +... + sec 300' minutes. It is too
laborious to compute such sums without the aid of integral calculus, which leads to a formula convenient for computation.1
1 For latitude L the number of nautical miles in the stretched latitude is
log tan (45~+ 'L) - r(e2 sin L+ Ie4 sin' L +...).4343
where r is the equatorial radius and e is the eccentricity of the ellipse whose rotation produces the earth's surface.