1. The intersections of the meridians and parallels will be
rectangular.
2. The distances north and south will be exact; and any meridian will
serve as a scale.
3. The parallels thro’ _z_ and _y_, where the line R_r_ cuts the arc
L_l_, or any small distances of places that lie in those parallels,
will be of their just quantity. At the extreme latitudes they will
exceed, and in mean latitudes, from _x_ towards _z_ or _y_, they will
fall short of it. But unless the zone is very broad, neither the excess
nor the defect will be any-where considerable.
4. The latitudes and the superficies of the map being exact, by the
construction, it follows, that the excesses and defects of distance,
now mentioned, compensate each other; and are, in general, of the least
quantity they can have in the map designed.
5. If a thread is extended on a plane, and fixed to it at its two
extremities, and afterwards the plane is formed into a pyramidal or
conical surface, it may be easily shewn, that the thread will pass
thro’ the same points of the surface as before; and that, _conversely_,
the shortest distance between two points in a conical surface is
the right line which joins them, when that surface is expanded into
a plane. Now, in the present case, the shortest distances on the
conical surface will be, if not equal, always nearly equal, to the
correspondent distances on the sphere: and therefore, all rectilinear
distances on the map, applied to the meridian as a scale, will, nearly
at least, shew the true distances of the places represented.
6. In maps, whose breadth exceeds not 10° or 15°, the rectilinear
distances may be taken for sufficiently exact. But we have chosen our
example of a greater breadth than can often be required, on purpose
to shew how high the errors can ever arise; and how they may, if it is
thought needful, be nearly estimated and corrected.
Write down, in a vacant space at the bottom of the map, a table of the
errors of equidistant parallels, as from five degrees to five degrees
of the whole latitude; and having taken the mean errors, and diminished
them in the ratio of radius to the sine of the mean inclination of
the line of distance to the meridian, you shall find the correction
required; remembering only to distinguish the distance into its parts
that lie _within_ and _without_ the sphere, and taking the difference
of the correspondent errors, in _defect_ and in _excess_.
But it was thought needless to add any examples; as, from what has been
said, the intelligent reader will readily see the use of such a table;
and chiefly as, whenever exactness is required, it will be more proper,
and indeed more expeditious, to compute the distances of places by the
following canon.
_Multiply the product of the cosines of the two given latitudes by
the square of the sine of half the difference of longitude; and to
this product add the square of the sine of half the difference of the
latitudes; the square root of the sum shall be the sine of half the arc
of a great circle between the two places given._
Thus, if we are to find the true distance from one angle of our map to
the opposite, that is, from S to Q, the operation will be as follows:
L. sin. 30° = -1.6989700
L. sin. 80° = -1.9933515
2 L. sin. 55° = -1.8267290
----------
-1.5190505 = log. of 0.330408
and 2 L. sin. 25° = -1.2518966 = log. of 0.178606
---------- --------
Log. of the sum 0.509014 is -1.7067297
Whose half is -1.8533648
the L. sin. of 45° 31´, the double of which is 91° 2´, or 5462
geographical miles.
And seeing the lines TS, TQ, reduced to minutes of a degree, are
6255.189 and 3255.189 respectively, and the angle STV is 63° 5´⅗, the
right line SQ on the map will be 5594´, exceeding its just value by
132´ or ¹⁄₄₂ of the whole.
7. The errors on the parallels increasing fast towards the north,
and the line SQ having, at last, nearly the same direction, it is not
to be wondered that the errors in our example should amount to ¹⁄₄₂.
Greater still would happen, if we measured the distance from O to Q
by a straight line joining those points: for that line, on the conic
surface, lying every-where at a greater distance from the sphere than
the points O and Q, must plainly be a very improper measure of the
distance of their correspondent points on the sphere. And therefore, to
prevent all errors of that kind, and confine the other errors in this
part of our map to narrower bounds, it will be best to terminate it
towards the pole by a straight line KI touching the parallel OQ in the
middle point K, and on the east and west by lines, as HI, parallel to
the meridian thro’ K, and meeting the tangent at the middle point of
the parallel SV in H. By this means too we shall gain more space than
we lose, while the map takes the usual rectangular form, and the spaces
GHV remain for the _title_, and other inscriptions.
VII. Another, and not the least considerable, property of our map
is, that it may, without sensible error, be used as a sea-chart; the
rumb-lines on it being logarithmic spirals to their common pole _t_,
as is partly represented in the figure: and the arithmetical solutions
thence derived will be found as accurate as is necessary in the art of
sailing.
Thus if it were required to find the course a ship is to steer between
two ports, whose longitudes and latitudes are known, we may use the
following