021 
NAVIGATION 
J By Conjlruftion. 
With the chord of 6o° defcribe the circle N.E. S.W. 
fig. 3. the centre of which reprefents the place the fliip 
failed from : draw two diameters N.S. E.W. at right an¬ 
gles to each other; the one reprefenting the meridian, 
and the other the parallel of latitude of the place failed 
from. Take each courfe from the line of rhumbs, lay it 
off on the circumference from its proper meridian, and 
number it in order 1, 2, 3, 4. Upon the firft rhumb C 1, 
lay off the firft diftance 163 miles from C to A; through 
it draw the lecond diftance AB parallel to C 2, and equal 
to no miles; through B draw BD equal to 180 miles, 
and parallel to C 3 ; and draw DE parallel to C 4, and 
equal to 68 miles. Now CE being joined, will reprefent 
the diftance made good; which applied to the fcale will 
meafure 281 miles. The arch S n, which reprelents the 
courfe, being meafufCd on the line of chords, will be 
found equal to 415°. From E draw EF perpendicular 
to CS produced ; then C F will be the difference of lati¬ 
tude, and F E the departure made good; which, applied 
to the fcale, will be found to meafure 210 and 186 miles 
refpeftively. 
Of PARALLEL SAILING. 
The figure of the earth is fpherical, and the meridians 
gradually approach each other, and meet at the poles. 
The difference of longitude between any two places is 
the angle at the pole contained between the meridians of 
thofe places ; or it is the arch of the equator intercepted 
between the meridians of the given places ; and the meri¬ 
dian diftance between two places in the fame parallel, is 
the arch thereof contained between their meridians. It 
hence follows, that the meridian diftance, anfwering to 
the fame difference of longitude, will be variable with 
the latitude of the parallel upon which it is reckoned; and 
the fame difference of longitude will not anfwer to a 
given meridional diftance when reckoned upon different 
parallels. 
Parallel failing is, therefore, the method of finding the 
diftance between two places lying in the lame parallel 
whole longitudes are known ; or, to find the difference 
of longitude anfwering to a given diftance, run in an 
eaft or weft direction. This failing is particularly ufeful 
in making low or fmall iilands. 
In order to iiluftrate the principles of parallel failing, 
let C A B P, fig. 4, reprefent a ferfion of one-fourth part 
of the earth, the arch ABP being part of a meridian ; 
CA the equatorial, and CP the polar femi-axis. Alfo, 
let B be the fituation of any given place on the earth; 
and join B C, which will be equal to CA or CP. The 
arch AB, or angle ACB, is the meafure of the latitude 
of the place B ; and the arch B P, or angle B C P, is that 
ofits complement. If BD be drawn from B perpendicu¬ 
lar to CP, it will reprefent the cofine of latitude to the 
radius B C or C A. 
Now, fince circles and fimilar portions of circles are in 
the direft ratio of their radii, therefore. 
As radius 
is to the cofine of latitude ; 
So is any given portion of the equator 
to a fimilar portion of the given parallel. 
But the difference of longitude is an arch of the equa¬ 
tor : and the diftance between any two places under the 
fame parallel is a fimilar portion of that parallel. 
Hence, R : Cofine latitude :: Diff. longitude : Diftance. 
And by inverfion, 
Cofine latitude : R :: Diftance : Diff. of longitude. 
Alfo, 
Diff. of longitude : Diftance :: R : Cof. latitude. 
Prob. Given the latitude of a parallel, and the num¬ 
ber of miles contained in a portion of the equator, to find 
the miles contained in a fimilar portion of that parallel. 
Ex. Required the number of miles contained in a de¬ 
gree of longitude in latitude 55 0 58' f 
By ConJlrnStion. 
Draw the indefinite right line AB, fig. 5. make the 
angle BAC equal to the given latitude 55 0 58', and AC 
equal to the number of miles contained in a degree of 
longitude at the equator, namely 60 ; from C draw CB 
perpendicular to AB; and AB, being ineafured on the 
line of equal parts, will be found equal to 33-5, the miles 
required. 
By Calculation. 
As radius - - 10*00000 
is to the cofine of latitude, - 55° 58' 974794 
So is miles in a deg. of Ion. at eq. 60 177815 
to the miles in a deg. in the given par. 33*58 1*52609 
Of MIDDLE-LATITUDE SAILING. 
In plane failing the earth was confidered as a plane, re¬ 
prefenting a bowling-green, having the meridians parallel 
to each other, and confequently the degrees of longitude 
equal in all places; but this cannot be true, as the earth 
is a globe or fphere ; for, as the meridians are circles on 
the terraqueous globe, meeting in the poles, it is obvious 
that any two of thofe circles muft recede more at greater 
diftances from the poles ; and at equal diftances from 
each pole, or at the equator, the diftance between the me¬ 
ridians is greateft. 
The true place of a ftiip at fea depends upon its dif¬ 
tance from the equator, and fome noted meridian ; and 
fince the meridional diftance, that is, the diftance between 
any two meridians, varies in every latitude, it is there¬ 
fore convenient that this diftance fhould be reckoned in a 
fixed latitude, and where the degrees are of the fame mag¬ 
nitude with thofe of the meridian, which can be no-where 
but on the equator, where 60 geographical miles make a 
degree. 
The circumferences of all circles are in direct propor¬ 
tion to each other, as their radii; and, fince the earth 
turns once round its axis in twenty-four hours, every 
point on its furface muft defcribe circles parallel to the 
equator; hence it follows, that the circumference of any 
parallel of latitude, in miles, is to the circumference of 
the equator, in miles, as the cofine of that latitude is to 
radius ; and, that the breadth of a degree, in any parallel 
of latitude, is to the breadth of a degree upon the equator, 
as the fine complement of that latitude is to radius. 
By the laft proportion was the following Table calcu¬ 
lated, which /hows the breadth of a degree of longitude 
in every latitude 5 and may be made to anfwer for any 
degrees or minutes by taking proportional parts. 
Table of the Miles and Parts of a Mile in a Degree of 
Longitude at every Degree of Latitude. 
D.L. 
Miles. 
D.L 
Miles. 
D.L- 
Miles. 
|D.L. 
Miles. 
D.L. 
Miles. 
D.L 
1 Miles | 
I 
59'99 
l6 
S 7'6 7 
3 1 
51-43 
46 
41*68 
61 
29*09 
76 
IT 5 1 
2 
59'9 7 
1 7 
57 '36 
32 
50-88 
47 
40*92 
62 
28*17 
77 
1 3 " 5 °j 
3 
59*92 
18 
57-06 
33 
5 °' 3 2 
48 
40*15 
63 
27-24 
78 
i2'48| 
4- 
59*86 
19 
S 6 '73 
34 
49-74 
49 
39*36 
64 
26*30 
79 
ii *45 
5 
59'77 
20 
56-38 
35 
49-15 
5 ° 
38-57 
65 
25*36 
80 
10*42 
6 
59*67 
21 
56-01 
36 
48*54 
5 i 
37*76 
66 
24-41 
81 
9-38 
7 
59 ’Sfi 
22 
55-63 
37 
47-92 
5 2 
36*94 
67 
2 3'45 
82 
8-35 
8 
5 9'44 
23 
55' 2 3 
38 
47*28 
53 
36*11 
68 
22-48 
83 
7*32 
9 
59 ' 2 fi| 
24 
54 - 8 i 
39 
46*62 
54 
35*26 
69 
21*50 
84 
6-28 
IO 
59*08 
2 5 
5 T 38 
40 
45’95 
55 
34 - 4 I 
70 
20*52 
85 
5-23 
—- 
. - 
— 
- . 
— 
— 
— 
- , 
I I 
58*89 
26 
5 3'9 3 
41 
45-28 
56 
33"55 
7 i 
J 9-54 
86 
4* 18 
12 
58-68; 27 
53 46 
42 
4 T 95 
57 
32*68 
7 2 
18-54 
87 
3" J 4 
13158*46128 
52-97 
43 
43-88 
58 
3 I '79 
73 
17-54 
88 
2'09 
14 
58-22 
29 
5 2 '47 
4-4 
43 - i 6 
59 
SO^O 
74 
1653 
89 
1*05 
15 
57'95 
30 
51*96 
45 ' 4 2 "43 
60 
30*00 
75 
I 5 - 5 2 
90 
0*00 
Hence 
