NAVIGATION. 
26 1 
of any arch (as a minute) on the equator, 
k longer than the like arch of any p rallel, 
in the same proportion a; the secant of 
the latitude of that parallel is to radius. 
But since iir this projection the meridians 
are parallel, and consequently each paral- 
lel of latitude equal to the equator, it is 
plain the length of any arch (as a minute) on 
any parallel, is increased beyond its just pro- 
portion, at such rate as the secant of the lati- 
tude of that parallel is greater than radius; 
and therefore, to keep up the proportion of 
northing and southing to that of easting and 
westing, upon this chart, as it is upon the 
globe itselr, the length of a minute upon the 
meridian at any parallel must also be increas- 
ed beyond its just proportion at the same 
rate, i. t. as the secant of the latitude of that 
parallel is greater than radius. Thus to find 
the length o. a minute upon the meridian at 
the latitude ot 73 degrees, since a minute of 
a meridian i- every vvnere equal on the globe, 
and also equal to a rniimte upon the equator, 
let it be represented by unity; then making 
it as radius to the secant of 75 degrees, so is 
unity to a fourth number, which is 3.864 
nearly; and consequently, bv whatever line 
you represent one minute on the equator of 
this chart, the -length of one minute on the 
enlarged meridian at the latitude ot 75 de- 
crees, or the distance between tne paiu.ici ot 
7 5° 00'' and the parallel of75°01 / , will be equal 
to 3 of these lines, and of one of them. 
By making the same proportion, it will be 
found that the length ot a minute on the me- 
ridian of this chart at the parallel of 60°, or 
the distance between the parallel of 60° 00 
and tiiat oi 60 0 k, is equal to two of these 
lines. After the same manner, the length of 
a minute on the enlarged meridian may be 
found at any latitude; and consequently be- 
ginning at the equator, and computing the 
length°of every intermediate minute between 
that and any parallel, the sum of all these 
shall be the length of a meridian intercepted 
between the equator and that par. ilk 1 , and 
the distance of each degree and minute of 
latitude from the equator upon the meridian 
of this chart, computed in minutes of the 
equator, forms what is commonly called a 
table of meridional parts. 
If the arch BD (fig. 10.) represents the lati- 
tude of any point B, then (CD being radius) 
CE will be the secant of that latitude ; but it 
has been shown above, that radius is to secant 
of any latitude, as the length of a minute 
upon the equator is to the length of a minute 
on the meridian of this chart at that latitude ; 
therefore CD is to CE, as the length of a mi- 
nute on the equator is to the length ot a mi- 
nute upon the meridian at the latitude of 
the point 13. Consequently, if the ladius CL) 
is taken equal to the length ot a minute upon 
the equator, CE, or the secant of the latitude, 
will be equal to the length of a minute upon 
the meridian at that latitude. Iheiefore, in 
general, if the length of a minute upon the 
equator is made radius, the length of a mi- 
nute upon tlie enlarged meridian will be e very 
where equal to the secant of the arch con- 
tained between it and the equator. 
Hence it follows, since the length of every 
intermediate minute between the equator 
and any parallel is equal to the secant ot the 
latitude, (the radius being equal to a minute 
upon the equator), the sum ot ail these 
lengths, or the distance ot that paiallel on tne 
enlarged meridian from the equator, will be 
equal to (lie sum of all the secants to every 
minute contained between it and the equator. 
Consequently, the distance between any two 
parallels on the same side of the equator, is 
equal to the difference of the sums of all the 
secants contained between the equator and 
each parallel ; and the distance b tween any 
two parallels on contrary sides of the equator, 
is equal to the sum of the sums of all the se- 
cants contained between the equator and each 
parallel. 
By the tables of meridional parts given by all 
the writers on this subject, may be constructed 
the nautical chart, commonly called Mercator’s 
chart. Sec Maps. 
In fig. li, let A and E represent two places 
upon Mercator’s chart, AC the meridian of A, 
and CE the parallel of latitude passing through 
E; draw AE, and set off upon AC the length 
AB equal to the number of minutes contained 
in the difference of latitude between the two 
places, and taken from the same scale of equal 
parts the chart was made by, or from the equa- 
tor, or any graduated parallel of the chart, and 
through B draw BD parallel to CE meeting AE 
in D. Then AC will be the enlarged difference 
of latitude, AB the proper difference of lati- 
tude, CE the difference of longitude, BD the 
departure, AE the enlarged distance, and AD 
the proper distance, between the two places A 
and E also the angle BAD will he the course, 
and -\E the rhumb-line between them. 
Now, since in the triangle ACE, BD is pa- 
rallel to one of its sides CE ; it is plain the tri- 
angles ACE, ABD, will be similar, and conse- 
quently the sides proportional. Hence arise the. 
solutions of the several cases in this sailing, 
which are as follow: 
Case I. The latitudes of two places given, to 
find the meridional or enlarged difference of 
latitude between them. 
Of this case there are three varieties, viz. 
either one of the Diaces lies on the equator • or 
both on the same side of it ; or lastly, on dif- 
ferent sides. 
]. If one of the proposed places lies on the 
equator, then the meridional difference of lati- 
tude is the same with the latitude of the other 
place, taken from the table of meridional parts. 
Examp/e. Required the meridional difference 
of latitude between St. Thomas, lying on the 
equator, and St. Antonio, in the latitude of 17° 
2(/ north. We look in the tables for the meri- 
dional part answering to 17° 2(/, and find it 
to be 1036.2, the enlarged difference of latitude 
required. 
2. If the two proposed places are on the same 
side of the equator, then the meridional differ- 
ence of latitude is found by subtracting the 
meridional parts answering to the least latitude 
from those answering to the greatest, and the 
difference is that required. 
Example. Required the meridional difference 
of latitude between the Lizard in the latitude 
of 30° 00 / north, and Antigua in the latitude of 
17° 30 a north. 
From the meridional parts of 50°, 00 / 3474-5 
subtract the merid. parts of 17 30- 1066.7 
there remains - - 2407.8 
the meridional difference of latitude required- 
3. If the places lie on different sides of the 
equator, then the meridional differenc e of lati- 
tude is found by adding together the meridional 
parts answering to each latitude, and the sum is 
that required. 
Example. Required the meridional difference 
of latitude between Antigua in the latitude of 
17° 80' north, and Lima in Peru in the latitude 
of 12° 30' south. 
To the merid. parts answering to 17° 30' 1066.7 
vdd these answering to - 12 30 756.1 
the sum is 1822.8 
the meridional difference of latitude required. 
Case II. The latitudes and longitudes of two- 
places given ; to find the direct course and dis 
tance between them. 
Example. Required to find the direct course 
and distance between the Lizard in the latitude 
of 50° 00 / north, and Port Royal in Jamaica, in 
the latitude of 17° 40'; differing in longitude 
70° 46', Port Royal lying so far to tire westward 
of the Lizard. 
Preparation. 
From the latitude of the Lizard - 50° 00' 
subtract the latitude of Port Royal 17 40 
and there remains - 32 20 
equal to 1940 minutes, the proper difference of 
latitude. 
Then from the merid. parts of 50° 00' 3474.5 
subtract those of - 17 40 1057.2 
and there remains - 2397.3 
the meridional or enlarged difference of longi- 
tude. 
Geometrically. Draw the line AC, fig. 12, 
representing the meridian of the Lizard at A ; 
and set off from A, upon that line AE equal to 
1940 (from any scale of equal parts) the proper 
difference of latitude, also AC equal to 23971.3 
(from the same scale) the meridional or enlarged 
difference of latitude. Upon the point C raise 
CB perpendicular to AC, and make CB equal 
to 4246, the minutes of difference of longitude. 
Join AB, and through E draw ED parallel 
to BC : so the case is constructed ; and AD ap- 
plied to the same scale of equal parts the other 
legs were taken from, will give the direct dis- 
tance, and the angle DAD measured by the line 
of chords will- give the course. 
By Calculation.. 
For the angle of the course BAD, it will be, 
(by rectangular trigonometrv,) 
ac : cb :: r : t. bac, 
/. e. As the mend. diff. of lat. 239-7.3 3.37970 
is- to the difference of long. 4246.0 3.62798 
so is radius - - 10.00000 
to the tang, of the direct course 60° 33' 10.34828 
which, because Port Royal is southward of the 
Lizard, and the difference of longitude westerly, 
will be south 60° 33'’ west, or SW5W ^ west 
nearly. 
Then for the distance AD, it will be (by 
rectangular trigonometry), 
R • AE ;; Sec. A ; AD, 
i. e. As the radius - - 10.00000 
is the proper diff. of lat. 1940 3.28780 
so is the secant of. the course 60° 33' 10.30833 
to the distance - 394 5.6 3.59613 
consequently the direct course and distance be- 
tween the Lizard and Port Royal in Jamaica, is 
south 60° 33' 3945.6 miles- 
Case III. Course and distance sailed, riven ; 
to find difference of latitude, and difference of 
longitude. 
Example. Suppose a ship from the Lizard in 
the latitude of 50° 00' north, sails soi th 35° 4 o' 
west 156 miles : required the latitude come to, 
and how much she has altered her longitude. 
Geometrically. 1. Draw the line BK ffig.. 
13), representing the meridian of the Lizard 
at B . from B draw the line BM, making with 
BK an angle equal to 35° 40' and upon this 
line set off BM equal to 56 the given distance, 
and -from M let fall the perpendicular MKL 
upon BK. 
Then for BK the proper difference of latitude, 
it will be, (by rectangular trigonometry,) 
R ; MB ; : S. BMK ; BK, 
