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THEORETICAL NAVIGATION
NAUTICAL ASTRONOMY.
BY
LEWI'S CLARK.
Lieut .-Commander, U. S. N.
NEW YORK:
> D. VAN NOSTRAND, PUBLISHER,
23 Muesat and 27 Warren Street.
18T2.
\J
Entered according to Act of Congress, in the year 1872, by
D. VAN NOSTRAND,
in the Office of the Librarian of Congress at Washington.
INTRODUCTION
The following pages have been prepared for use at the U. S.
Naval Academy.
Napier's and Bowditch's Rules have been used in deducing
the formulae, which are generally those used in Bowd. Nav.
Beferences to Trigonometry are to the treatise of Prof. Chau-
venet.
Not seeing any good reason for making distinctive " Sailings"
while still considering the earth's surface as a plane, the author
has taken the liberty of placing them together under the head of
" Common Sailing."
For the method of deducing the equation of "Mercator's
Sailing " the thanks of the author are due to Prof. J. M. Bice,
of the Naval Academy.
CHAPTER I.
DEFINITIONS AND NOTATION.
1. Meridians are great circles of the sphere, passing through
both poles.
2. Suppose a ship to sail so that the line of her keel makes a
constant angle with each successive meridian ; this line is called
the ship's track or loxodromic curve. In old nautical works, the
rhumb line.
3. The constant angle made by this line with each meridian is
called the true course. In the following problems the word
course will be understood to mean true course, and will be
denoted by C.
4. The compass needle, undisturbed by local causes, points to
the magnetic pole, and great circles passing through this pole
are called magnetic meridians. The angle which the loxodromic
curve makes with the magnetic meridians is called the magnetic
or compass course. Compass course must be reduced to true
course previous to the solution of nautical problems in which
course is considered.
5. The portion of the loxodromic curve considered in any
problem, is called the distance. It is necessarily the number of
miles passed over by the vessel on the course which belongs
with it.
6. Latitude is angular distance north or south of the equator,
measured in degrees, minutes, etc., of a great circle, denoted
hjL.
7. Difference of latitude, denoted by I, is the portion of a
meridian included between two parallels of latitude.
6
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
8. Longitude is the angular distance between any meridian
and a fixed or prime meridian. The prime meridian is usually
that of Greenwich. It may be considered as angle at the pole,
of which the corresponding portion of the equator is a measure.
It is denoted by X.
9. Difference of longitude is angle at the pole, or the cor-
responding arc of the equator between any two meridians, rep-
resented by D.
10. Departure is the angular distance between any two merid-
ians measured on any parallel of latitude. As parallels of lati-
tude vary in size, the units (degrees, etc.) become smaller. If,
however, we have departure determined in angular units of its
own circle, the corresponding difference of longitude would be
the same. Departure is, however, found in the linear value of
units of a great circle of the sphere. In order, then, to determine
the corresponding difference of longitude, it will be necessary to
know first the relation between the units
of any parallel of latitude and the corre-
sponding units of the equator.
11. To find these relations, we have in
Fig.l
ED = D
A B =p the departure in Lat. L.
D and p are similar arcs of circles, and
therefore are to each other as their radii.
Fig. 1.
P =
Dr
1R
CA0 = A0JD=L -jf= cos L
which substituted in above gives
p = D cos L or
JD =jj sec L.
which give the required relations.
COMMON SAILING.
Having therefore the departure expressed in units of the equa-
tor (in nautical miles), we find the corresponding difference of
longitude by multiplying it by the secant of the latitude in which
the departure is situated.
COMMON SAILING.
12. For such small distances as an ordinary day's run at sea,
it is customary to consider the small portion of the earth's sur-
face passed over as a plane. The difference of latitude and de-
parture corresponding to the course and p IG 2
distance sailed are determined by the solu-
tion of a plane right angled triangle.
In Fig. 2
the difference of latitude I = d cos C
the departure p = d sin C
p = Z tan G
This is sufficientlv accurate for small
distances.
These equations are employed in what is called by navigators
41 working dead reckoning." Their computation is facilitated by
the use of Tables I. and II. Bowd., which are tables for the so-
lution of any plane right triangle, calling the distance hypothenuse,
difference of latitude side adjacent, and departure side opposite.
When several courses are sailed, the triangle is solved separately
for each value of C, and the algebraical sum of V, I", V", etc., p',
p", p'", etc., are taken for the whole difference of latitude and the
whole departure.
13. The equations above are strictly true when
d I = d d cos G.
d p = d d sin G.
d p = d I tan C.
The smaller I, d, and p are taken, therefore, the nearer correct
will be the result.
The departure p, formed from the sum of several partial
8 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
departures, is, of course, for different latitudes. It is customary
to assume it upon the middle parallel. That is, the middle lati-
tude is found (in the figure) between each extremity of I and the
departure assumed upon it. The difference of longitude is found
from
I) = p seo. L.
L being this middle latitude,
L == L' + \ I.
L == L'i- \ I.
The difference of latitude found being applied to the latitude
left, with proper sign gives latitude in.
The difference of longitude applied to longitude left, with
proper sign will give longitude in.
14. Several problems arise in Common Sailing which are solved
on the supposition that the triangle is a plane right triangle.
They are solved generally by inspection of Tables I. and II. They
may be solved by logarithms, using some form of the preceding
equations.
The two following are selected as examples :
15. Problem 1. To rind current.
The difference between the latitudes as found by observation
and by " dead reckoning," is taken, and also the difference be-
tween the longitudes as determined in same manner. The
observed position is considered as the correct position, and any
difference in the two positions may be due to current.
The difference of longitude is changed to departure by
p== D cos L.
The course or direction of the set of the current is then deter-
mined by
and its amount or distance by
tan C =
p
J
sby
d = .
P
G
d=-
I
cos
G
COMMON SAILING. 9
16. Problem 2. To find the course and distance "made
good."
The difference between the latitude left and that arrived at
(by observation) is taken.
The difference between the longitudes is changed, as in pre-
ceding problem, to departure. The same equations are then
solved as before, C being in this case the course made good, and
d the distance made good.
This problem, as we shall see, is more correctly solved by
Mercator's Sailing.
CHAPTER II
MERCATOR'S SAILING.
1. We have, in Common Sailing, considered a small portion of
the earth's surface as a plane. This is sufficiently correct for
small distances as an ordinary day's run. A more rigorous solu-
tion of problems appertaining to the loxodromic curve is neces-
sary.
Fig. 3.
In Fig. 3, E is a portion of the loxodromic curve. E Ea
parallel of latitude passing through origin, E y a great circle of
the sphere through the same point, p equals P E the co-Lat. of
E. P E 0=^ C, the course.
Decompose s along p and <£ 3 and we have
Cot C^~-
d have a common tangent at the point E, and as d X and
3 (p denote angular velocities, they are to each other as the cor-
responding radii.
. •. d(f> = d X cos L, or d = d . X sin p.
d p _ dp
and cot C= r ^-=
d sin p d X
P*
\d.p
sin J p cos \ p
p 2
(a.)
Dividing numerator and denominator by cos J p and integra-
ting first member.
f Pi Pi
Cot C ^-X 2 )=- wtjpd.hp = r j tan j
/pi Pi
(b.)
But
- log tan | p = log cot \ p = log cot \ (90© - £) = log tan (45°+
| Z) which substituted in (6) gives
Cot C (X x -X,) = log tan (45° + \L\ (c.)
the limits A and A changing for those of p x and^) 2 .
j - , rn tan (45°+ | A)
& tan ( 45°+ J A)
If i 2 = 0, or the point E be at the equator
X,-X 2 = D = tan Clog tan (45° + J £).
In this the logarithm isNaperian, and D is in angular measure.
To change to the common system of logarithms, we multiply by
the reciprocal of the modulus, — = 2.302585093 ; and to change
m
D for the globe, multiply by the radius of the earth in minutes
= 3437.74677, and we have
D = 7915.70447 log tan (45° + J L) tan C.
12
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
5>. The relation existing between D and C in this expression is
F IG . 4 # that of an angle and side oppo-
site in a plane right triangle and
may be represented as in Fig. 4
in which
D is the difference of longi-
tude. C is the course, and the
side F = 7915.70447 log tan.
(45° + J L) . F is called the
Augmented Latitude, and will be
represented by 31.
In the expression
M = 7915.70447 log tan (45° -f \ L)
different values of L may be assumed, and the corresponding
augmented latitude computed and tabulated. Table III. Bowd.
is such a table, computed for each minute of L from 0° to 84°.
3. From the foregoing, we see that any portion of the loxo-
dromic curve, or ship's track, may be represented by a straight
line, as E F, in Fig. 4. Charts constructed on these principles
are called Mercator's Charts. By means of a Table of Aug-
mented Latitudes they are easy to construct, and possess the
advantage of showing the ship's track by a straight line, and
the course being represented by the angle which this line makes
with any meridian.
4. Problem 1. — A ship sails from a latitude 1/ until she
arrives at a latitude L", upon a given course C, find the
difference of longitude D.
; For the difference of longitude from where the loxodromic
curve intersects the equator, to its intersection with the meridian
of L', we have
D' = 31' tan C,
and to the second latitude
D" = 31" tan C,
D = D" - D' = {31" -M) tan C.
Find, from a table of augmented latitudes, or by computation,
M" and M' corresponding to L" and L respectively, and take
mercator's sailing. 13
their difference. This is called the augmented difference of lati-
tude. Representing it by m, we have
D = m tan G
5. Problem 2. — Given the latitudes and longitudes of two
places ; find the course, distance, and departure. — (Bowd., p.
79, Case!)
L' and L" being given, find M' and M" by computation, or by
Table III., Bowd.
l = L" - L', m = M'-M D = X"-X\
by Mercator's Sailing
Tan (7= —
m
and from Common Sailing
d = I sec G
p = I tan G
6. In Common Sailing we find the difference of longitude by
taking the departure upon the middle parallel of latitude. The
proper parallel is one situated nearer the pole. Strictly the
departure should be taken upon
L m = \ (L' + L") + aL,
To find A L (see Tables, Bowd., p. 76, and Stanley, p. 338.)
In Common Sailing we have,
(a.)
CosX w =|-
From Common Sailing
2> =
I tan G,
and from Mercator's
D =
m tan G
which substituted in (a) gives
Cos L m =
I
m
.-. l-2sin 2 !i: m =
m
Sin J L m =
/m-l
* on
(b.)
(e.)
14 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
For different values of L' + L" we may find / = L"—L'
The middle latitude used in Common Sailing, is
L m being found by (c) we have
CHAPTER III.
GREAT CIRCLE SAILING.
1. The shortest distance between any two points on the globe,
is the arc of a great circle joining them. In running long dis-
tances, it is best to follow the arc of a great circle. Strictly
speaking, this would be impossible, as the course would have to
be changed each instant. It is customary to determine certain
points of the great circle, and run from point to point on a loxo-
dromic curve. Circumstances of wind and weather must govern
the navigator in choosing his route. Most of the convenient
great circle routes have already been computed. A knowledge
of Great Circle Sailing is necessary, however, in order to know
which tack to put the ship upon in case of adverse winds.
2. Problem 1. — The latitudes and longitudes of two
places being given, to' find distance and course from one or
both of them.
In Fig. 5, we have given
P A = 9Q°-L'
P B = 90°-L"
A P B = A = A"-A'
P A B = C, the course from A
P B A = C, the course from B
Letting fall the perpendicular B
B K and representing P K by 0, we have
By Napier's Eules
AK= 90°-(i' + 9)
Cos A = tan tan L"
Tan = cos A cot L"
16
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
By Bowd. Kules, or by Napier's Kules and eliminating the per-
pendicular,
Cot C = cos (L' -f- 0) cot X cosec (j> (b.)
and in triangle BAR,
Cot d = cos C tan {L'-\- _(90°-Z').
The course is determined in degrees and minutes, and is meas-
ured from the meridian of L'. Attention must be paid to the
signs of and X. The distance is also found in degrees and
minutes of the great circle. Reduce to minutes for distance in
nautical miles.
6. Having found the latitudes and longitudes of as many
points of the great circle as are desired, plot them on chart, and
by hand trace through these points the curve ; owing to the
principles of construction of a Mercator's chart, this will be an
irregular curve except when coincident with the equator or a
meridian.
CHAPTER IV.
TIME.
1. Time is the hour angle of some heavenly body whose
apparent diurnal motion is taken as a measure.
The instant when any point of the celestial sphere is on the
meridian of the observer is called transit,
2. Sidereal time is the hour angle of the first point of Aries
(y). The instant of its transit is sidereal noon, h.
Eight ascension is the angular distance of a heavenly body
from the first point of Aries reckoned towards the east. Hence
when any heavenly body is on the meridian of a place its B. A.=
the sidereal time.
As the earth revolves 360° in order to bring any meridian two
successive times under (y), we can find the space passed over in
one hour by dividing 360 by 24, equals 15°. Hence when the
H. A. of y is 15° the sidereal time is 1 h. The interval between
two successive transits of y is the sidereal day. Evidently the
interval between two successive transits of any fixed point over
the same meridian would be equal in length to a sidereal day.
3. Apparent time is the hour angle of the true sun.
The true sun has motion in R. A., and therefore is not a fixed
point in the celestial sphere. Its motion is not uniform in the
ecliptic, and this of itself would tend to make apparent solar
days irregular in length. Besides, as the plane of the ecliptic is
inclined to the plane of the equator, the true sun's apparent
daily path is not perpendicular to the plane of the meridian ; in
other words, the true sun approaches the meridian at a constantly
varying angle ; this also tends to cause irregularity of apparent
time. Instruments cannot be constructed to keep apparent time,
TIME. 19
and astronomers have resorted to the following device in order
to obtain a uniform 'time.
4. Mean time is the hour angle of a mean sun (supposed)
which has for its annual path, the celestial equator. A first
mean sun is supposed to move in the ecliptic at a uniform rate,
so as to return to perigee and apogee with the true sun. This
obviates the first difficulty mentioned. The changes in longi-
tude of this mean sun are equal in equal times, but equal changes
in longitude do not give equal changes in E. A. So a second
mean sun (sometimes called simply the mean sun) is supposed to
move in the equator at the same rate that the first moves in the
ecliptic, and to return to the vernal equinox with it. The time
therefore denoted by this second mean sun, although not equal
to sidereal time, is perfectly uniform in its increase. The daily
difference will evidently be equal to the daily increase in the
right ascension of this mean sun = 3 m. 56 s. The instant of
transit of the true sun over the meridian of the observer is
called apparent noon. The instant of transit of mean sun is
mean noon.
5. The equation of time is the difference between apparent
and mean time. It is also the difference of the hour angles of
true and mean suns. It is also the difference between the right
ascensions of the true and mean suns. From what has preceded,
we know that the first mean sun's longitude, or, as it is some-
times called the true sun's mean longitude, is equal always to the
right ascension of the mean sun. Hence the equation of time is
equal to the difference between the true suns right ascension and its
mean longitude.
6. Astronomical time commences at noon or Ohrs., and is
reckoned to the westward 24 hrs. An astronomical day (apparent
or mean) is the interval between two successive transits of the
sun (apparent or mean).
7. Civil time commences at midnight 12 hrs. before the
commencement of astronomical time, and is divided into two
periods of 12 hrs. each, marked A. M. and p. m.
20 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
7. To convert civil into astronomical time.
Remember that the civil day of same date commences 12hrs-
before the astronomical day.
9. Time at different meridians.
It is evident that as any time at one meridian is the H. A. of
the heavenly body or point whose motion is considered, to find
the corresponding time at any other meridian it is only necessary
to add or subtract the angle between the two meridians. In
Nautical Astronomy it is generally necessary to convert the given
local time to the corresponding Greenwich time, in order to in-
polate quantities from the Nautical Almanac, which are com-
puted for the meridian of Greenwich.
10. Having given the local time of any meridian, to find
the corresponding Greenwich time.
To the local time add the longitude if west, and subtract if
east ; the result will be the corresponding Greenwich time of the
same kind as the given local time. Conversely, the differ-
ence between the time at two meridians (of the same kind) will
be the difference of longitude expressed in time. Remembering
that lh. = 15°, this may readily be converted to arc.
11. To convert apparent time, at a given meridian, into the
mean time, or mean into apparent.
If M = the mean time
A = the corrresponding time
E = the equation of time
we have from Art. 5
M - A = E
M=A + E
A = M- E
JE'may be + according as the apparent is greater or less than
the mean time. E is found on Page II. of the American Nautical
Almanac for Greenwich Mean Noon, and is to be interpolated to
the instant of the given Greenwich mean time. Where the given
Greenwich time is apparent time, then E must be taken from
Page I.
TIME. 21
12. To change sidereal into solar time it will be first ne-
cessary to know the relative value of their units.
In consequence of the earth's annual revolution about the
sun, there will be one less transit of the sun across any meridian
than there will be of any fixed point outside of the earth's orbit,
during the period of this revolution.
There are in one year
3662.4222 sidereal days.
365.24222 solar days.
whence we have
1 sid. day = lll'lf^l sol. day = 0.99726957 sol. day,
or 24 hrs. sid. time =23 hrs. 56 m. 4.091 s. solar time. And
1 sol. day = y^owoo sid - d& y — 1.00273791 sid. day.
or 24 h. solar time =24 h. 3 m. 56.555 s. sid. time.
From these relative values Tables II. and III. of the American
Nautical Almanac are computed. The first is for converting an
interval of sidereal time to the corresponding interval of mean
time. The second for changing an interval of mean time into
the corresponding interval of sidereal time.
It is evident that, in Table II., the corrections are nothing
more than the changes in right ascension of the mean sun during
the given intervals of sidereal time. It is this change in right
ascension which causes the different values of the units. In
Table III., the corrections are the changes in right ascension of
the mean sun in the given intervals of mean time.
13. To convert an interval of sidereal time into the cor-
responding interval of mean time.
Enter Table II. with the sidereal interval, as an argument.
Find the change in R. A. of the mean sun and subtract this change
from the given sidereal interval.
14. To convert an interval of mean time into the corre-
sponding interval of sidereal time.
Enter Table III. with the mean time interval, as an argument.
22
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
Find the change of B. A. of the mean sun, and add this change
to the given mean time 'interval.
15. "We have now found a means of changing an interval of one
kind into an interval of another. It is frequently necessary to
find the corresponding time, having given another time. To do
this it will be necessary to be able to find the B. A. of the mean
sun at any instant. The B. A. of the mean sun is given in N. A.
for the instant of Greenwich mean noon (Page II. of the month),
marked " sid. time, or B. A. of mean sun." This being given for
the instant of Greenwich, mean noon must be interpolated to the
instant of Greenwich mean time. Hence we have
16. Given the local mean time at any meridian, to find
the corresponding sidereal time.
Convert the local mean time into Greenwich mean time, by
applying the longitude in time. Enter Table III. of the N. A.,
and find the change in B. A. of the mean sun for the elapsed
Greenwich time ; add this to the B. A. given on Page II. of the
month for the 'preceding Greenwich noon, and result will be the
correct B. A. of mean sun at the instant of time given.
Fig. 7.
Then, in Fig. 7,
y P S = R A mean sun
A P S = HA mean sun or LMT
and APy=HA of y or L ST
B.eneeAPy = APS+yPS,oi
The sidereal time is equal to
the mean time plus the R. A. of
mean sun.
17. Given the local sidereal time, at any meridian, to
find the corresponding mean time.
Convert the local sidereal time into Greenwich sidereal time,
by applying the longitude.
Enter Table II. of the N. A., and find the change in B. A. of
the mean sun for the elapsed Greenwich sidereal time. From
Page III. of the month, take the " mean time of preceding side-
real noon " (which is evidently 24 hrs. minus B. A. of mean sun
TIME.
at that instant). Subtract from this the correction obtained from
Table II., and the result is the correct negative R. A. of the mean
sun at the given sidereal instant.
Then, in Fig. (7.),
APy = given L. S. T.
y P S = JR. A. of mean sun, and
AP S= the L. 31. T.
Hence APS=APy-yPS.
We have obtained y P S, however, negatively, and A P S = A
P y + the corrected negative R. A. of mean sun.
The mean time is equal to the sidereal time, minus the B. A. of
mean sun, or plus the negative B. A.
18. Given the apparent time at any meridian, to find
the corresponding sidereal time.
Change apparent to mean time (Art. 11.), and proceed as in
Art. 16, or
Apply longitude to local apparent time, giving Greenwich ap-
parent time.
Find R. A. of true sun on page I., N. A., and correct by means
of given hourly difference to the instant of Greenwich apparent
time.
Then in the Fig. (7.)
y P S = R. A. of true sun
A P S= given L. A. T. and
APy=APS+yPS,ov
The sidereal time is equal to apparent time plus the B. A. of the true
sun,
19. Given the sidereal time at any meridian, to find the
corresponding apparent time.
Proceed as in Art. 17, then change the mean time to apparent
by Art. 11.
20. Given the hour angle of a star, at any meridian, to
find the local mean time.
Find in the N. A. the R. A. of the star. To this apply the H.
24 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
A. of star plus, when west of the meridian, and minus when east.
The result is local sidereal time.
Then proceed as in Art. 17.
21. To find the hour angle of a star at a given meridian
and mean time.
Find the corresponding sidereal time by Art. 16. To this
apply the star's E. A. ; the difference is star's H. A. -f- when the
sidereal time is greater than the K. A., — when K. A. is greater
than sidereal time.
22. Given the hour angle of the moon at any meridian
to find the local mean time.
Apply the H. A. of moon to the longitude of the place, which
gives the longitude of place which has the moon on its meridian.
The N. A., page IV., of the month gives the time of moon's me-
ridian passage at Greenwich, or the angle between the moon and
sun. The hourly difference multiplied by difference in time
(lougitude), and result added to the Greenwich time of passage
when longitude is west, subtracted when east, gives the local
time of meridian passage, or the corrected angle between the
sun and moon. We now have the time at the place which has
the moon on its meridian. Applying H. A. of moon gives the
time at the given meridian.
In Fig. 8
A PM= HAoi moon
A P G + AP M=^ Long, of
meridian P. M. from Green-
wich.
Having found M P S as
stated, AP S = AP M -\-
MP S.
If the Greenwich time be
given and longitude AP G required. Find R A. of moon from
N. A. and correct for Greenwich time, and proceed as in case of
star. Art. 20.
TIME. 25
23. To find the hour angle of the moon at any meridian
and time.
Proceed as in case of star. Art. 21.
24. Given the hour angle of a planet at any meridian, to
find the local mean time.
The N. A. gives the time of meridian passage of each of the
planets over the Greenwich meridian, and the local mean time
may be found as in first of Art. 22.
If Greenwich time be given and not the longitude, proceed as
in second part of Art. 22.
25. To find the hour angle of the sun at a given meridian
and time.
The hour angle of the sun is the L. A. T. Proceed as in Art.
11, for changing mean to apparent time. If the apparent time
be more than 12 hrs., subtracting it from 24 gives the negative
H. A.
26. To find the time of meridian passage of any celestial
body, the longitude of the place or Greenwich time being
given.
It is only necessary to find from the N. A. the E. A. of the
body for the Greenwich time. This E. A. is the sidereal time of
transit, change this sidereal time to corresponding mean time
by Art. 17.
27. Eeference has been made to the American Nautical Al-
manac, and rules given for taking out some required quantities.
There are other quantities frequently required in Nautical
Astronomy, such as
Declination of sun, moon, and planets ; Equation of time,
Semi-diameter, Horizontal Parallax of moon, etc.
In general it is necessary to take out the required quantities
for the nearest Greenwich time to the given time, and interpo-
late in either direction to the given instant of Greenwich time.
Hourly differences are given to facilitate this work. As, how-
ever, the hourly differences themselves change quite materially
in some cases, it may be found necessary to use second differences.
26 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
Formulae have been given to meet each particular case. The
author has found that in general thej are of no practical assist-
ance to the student, and even, in some cases, confusing. One
thing may, however, be advantageously impressed upon the
student, and that is, that almost invariably it is necessary to first
obtain the Greenwich time before consulting the Almanac. At sea
this is found from the chronometer, and on shore either by
chronometer, or by applying to the local time of the place the
longitude. When the Greenwich time is apparent time, quantities
pertaining to the true sun must be interpolated from Page I. of the
month. When the time is mean time, then from Page II. Quantities
pertaining to other bodies are invariably given for the Greenwich
mean time, excepting the negative R. A. of mean sun, which is
given for the instant of Greenwich sidereal noon.
NOTATION FOR FOLLOWING CHAPTERS.
L = latitude
d — declination
t = hour angle
p = polar distance = 90° - d
z = true zenith distance = 90° — h
z = apparent zenith distance = 90°— h'
h = true altitude
h' = apparent altitude
Z = azimuth
A = amplitude = 90° -Z
q = position angle, or angle at the body.
CHAPTER V .
LATITUDE.
1. Latitude is the angular distance of a place on the surface of
the earth, north or south of the equator.
As the celestial equator is in the same plane as the equator,
and celestial meridians in the same planes with corresponding
terrestrial meridians, it is evident that the zenith of an observer
is the same angular distance from the celestial equator that his
place is from the terrestrial equator. Distance north or south
from the celestial equator is called declination. Hence the
declination of an observer's zenith is equal to his Latitude.
2. To find the latitude from the altitude of any heavenly-
body on the meridian, the Greenwich time of the obser-
vation being known.
The observed altitude must be changed to true altitude, by
applying errors of instrument, semi-diameter (if limb of body be
observed), dip. parallax, and refraction. This is necessary in
all observations, and, hereafter, when altitude is mentioned, it is
to be considered as true altitude.
In Fig. 9, let Z H N Q be a projection on the plane of the
meridian of observer.
E Q its intersection with plane of equator.
I H its intersection with plane of true horizon.
P P' the prolongation of the axis of the earth.
For the body X on the meridian, we have
EZ=L=ZX + JEX=Z+d = 90°-h + d
for the body X'
EZ = L=ZX' -EX' = Z - d = dO°-h-d.
3D
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
These are the two cases where d is north and south, or -f-
ancl — , and less than the latitude,
For the body X",
L = d-Z=d- (90° -/i)
for the body X",
3. Practical Navigators, in order that they may find the lati-
tude instantly upon observation of the sun upon the meridian,
make use of the following forms :
1st. When latitude and dec. are of same name, we have
L = 90° - h + d and h = h' + corr.
£ = 90°-(/i'-f-corr) +d
L= (90°+ d - corr)- A 7
The portion within parentheses can be computed before the
observation. All that remains to be done is to subtract observed
altitudes, which may be done mentally.
2c?. When lat. and dec. are of different names
L = 90°-h — d
L = 90° -(/i' + corr) - d
L= (90° -d- corr) - h'
LATITUDE.
31
In same way the portion within parentheses may be computed
previous to the observation.
4. To find the latitude from an observed altitude of any-
heavenly body, at any time, the Greenwich time of the
observation being given.
The declination of the body is found from the Greenwich
time. The altitude corrected
and hour angle of the body
found, we then have,
ZM=Z=dO° -h and
MP Z=t = houx angle given,
to find
PZ=90° - L
Let fall the perpendicular M
X, and Z=90°- (0 -f 0') and
if 0"=9O° — 0, <{>"= the decli-
nation of foot of perpendicular,
and as perpendicular may fall
without the triangle
L = $" + '
Cos t = tan d cot 0"
Tan 0" = tan d sec t
Sin d : sin h = sin 0" : cos 0'
sin 6" sin h
Cos 0'
sin d
(2.)
which afford the solution. ' when the perpendicular falls with-
in the triangle is negative.
5. When the body is on the prime vertical, the perpendicular
will fall near Z and 0' = nearly. "When, therefore, the body
is near the prime vertical, 0' becomes very small, and cannot be
determined accurately by its cosine.
0" is marked N or S like the declination, and is in same quad-
rant as t, as the sign of its tangent in (1) is dependent upon that
of sec. t. When t > 6h, 0" > 90°.
When the body has no declination, the perpendicular falls at
32 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
0, and " = 0, L = '. When d is nearly 0, (1) approaches
the undeterminate form. There are two values of L in the equa-
tion,
Zr = 0" -{- 0', but unless 0'
be very small, the one may be selected which coincides most
nearly to the supposed latitude. When 0' is less than 12 hrs.
use 7 — place tables.
6. To find the effect of an error in the altitude we have
Cq S 0' = sin 0" sin ft (2 .
r sin d v '
Differentiating
■ a.' j jl» sin 0)" cos ft 7 7
— sin a = r = — a ft.
sin a
7 ,. sin 0" cos ft 7 , / v
d 0' = - V^ <2 ft (a. )
sin a sin 0'
From (2)
cos 0' sin 0"
sin ft sin d
which substituted in (a) gives
d 0' =— cot ft cot 0' tZ ft.
In triangle Z M x of figure we have
cos Z= tan 0' tan ft
hence
sec Z= - cot 0' cot ft
d c/)==d h . sec ^,
substituting small finite differences.
A 0' = A ft sec Z, nearly.
d 0' is the error of 0' due to an error of ft.
The correction to 0' for error of ft would be
A 0' = - A ft sec ^
When the body is on the meridian Z=0, and numerically
A 0' = A ft.
The nearer Z is to 90° the greater will be A 0'.
LATITUDE. 33
7. To find the effect of an error in the time, or hour angle.
We have
Sin h — sin L sin d -\- cos L cos d cos t
= cos L d L sin d — sin L d L cos d cos £ - cos P cos d sin £ d t
j T cos Pcos d sin £ d £ * \
cos 1/ sin =*— (si „ £_+T >)&'h (c.)
The difference of equations (a) is
h-h' = (T' 2 - T 2 ) &'h = 2 T Q t A' /i,
hence
_ , (ft - fr ) _ j(h-h' )
2 M'/i — i M'/i
substituting this in (c), we have
Hence the mean of the two altitudes, plus the square of one-
half the interval between the observations multiplied by the
change of altitude in one minute from noon (Table XXXII.,
Bowd.), plus the square of one-fourth the difference of altitude,
divided by the first correction, is equal to the meridian altitude.
The meridian altitude obtained may be proceeded with as
usual.
10. To find the latitude from several altitudes taken near
the meridian, the apparent times of observation being
known.
See Bowd., page 202.
This method is commonly called the method of circum-meridian
altitudes.
Let h' t h", h ! ", etc., be the several altitudes (observed)
t\ t", t'" s etc., the corresponding hour angles.
We have for each reduction to the meridian from Art. 9,
A x h = i! % A' h .'. h = h' -f A x h
A 2 h = t m A' h :. h = h" -f A, ft
etc., etc.
LATITUDE. 37
or
1
Aii
*i + A 2
h-\- AJi
■7
n
t'
2 + t
" 2 + p :
■ A' h.
n °— n
*, = *' + *"+»- - +
n '
Hence the meridian altitude is equal to the mean of all the alti-
tudes, plus the mean of the squares of the hour angles multiplied
by the change of altitude in one minute from noon.
Table XXXIII., Bowd., contains the squares of hour angles
up to 13m., and Table XXXII. the change in altitude in one
minute from noon. When the heavenly body passes through or
near the zenith, the change of altitude is too rapid for the
assumption.
h =h"+A'hT 2
h« = K'"+A'h(T+xy
Subtracting the half sum of first and third equations from second,
we deduce
A> x * = h"-\{h'-\-ir) (a.)
The difference of first and third gives
A' h T= K
l(h'-h'")
which substituted in second equation, gives
Substituting in this the value of A / h x 1 from (a)
which affords solution by giving h 0i the meridian altitude.
38 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
12. To find the latitude by the rate of change of altitude on
prime vertical. (Prestel's Method.)
sin h = sin L sin d -|-cos L cos d cos t
cos h d h =— cos L cos dsintdt
■j 7 cos L cos d sin t d £ , N
&h=- , ■ (a.)
cos A '
From the astronomical triangle we have
cos d ; cos 7i = sin Z [ sin t
. „ cos tf sin t
.'.sinZ=
cos A
which substituted in (a) gives
d h= — cos .L sin Z dt.
Multiplying the second member by 15 to reduce to arc, changing
sign for correction and transposing, we have
15 cosi> sinZ
If now T' and T are respectively the hour angles of the alti-
tudes h and A', we have for a small interval of time and small
change of altitude
h'-h
T'-T=t=
15 cos L sin Z
cosL = -— — cosec Z (b.)
lb t
and when body is on prime vertical Z=90° and
h'-h
cos L =
15*
To use this observe two altitudes and note the times carefully.
A very good approximate latitude may be obtained when the
body is within 2 Q or 3° of the prime vertical, (b) may be used
when Z is approximately known.
LATITUDE.
39
13. To find the latitude by an altitude of the Pole Star,
the longitude of the place and local mean time being given.
In figure 11 let fall the perpen-
dicular 31 x, then in triangle M x P
cos t = cos p tan .
tan = tan p cos t
and as ; sin h = cos 0: (sin L-\- -j- d>) -
1 ' COS
but as p and are small, and their cosines nearly equal to 1, we
have
Sin h = sin (L -f- )
L = h-<1>
When £ is more than 6 hrs. and less than 18 hrs., cos t will be
negative, and will be negative in (a), and numerically,
L = h-\-
t is the hour angle of the star. The local time must be changed
to sidereal time, and if S = sid. time, then
= p cos (#-*'s K. A.)
If we consider p and star's R. A. to be constant, may be
computed and tabulated for different values of S.
Owing to the changes of E. A. and dec. of Pole Star, such
a table would require correction each year. It is better in prac-
tice to compute . Bowd. gives a table, page 206, for , but at
the present time the table is incorrect.
40
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
15. To find the latitude by two altitudes with the elapsed
time between them, supposing the declination to be the
same at both observations, and the Greenwich time approx-
imately known.
In the Fi- 12, let 31 and 31' be
the two positions of the body.
h = 90° - Z M, first altitude.
h' = 90° — Z M', second altitude.
d = the declination common to
each of the triangles.
t = 31 P M', the difference of the
hour angle ; the elapsed apparent
time in case of the sun ; the elapsed
sid. time in case of a star.
In the case of observations of a
star the elapsed mean time noted by a watch or chronometer can
be changed to a sid. time interval. In the case of the moon, the
elapsed mean time can be corrected for the change in R. A. of
moon during the interval.
If T == hour angle of body at 31, and
T' = " " " 31'
t = T'-T
We have the above given.
Let The the middle point of M 31'
Let A = MT=M' T=%MM'
B == 90° -P T, the declination of T }
H= 90° -Z T, the altitude of T.
q = P T Z, the position angle of T.
By assumption P T 31 and P T 3f / are equal right triangles,
and "angles P T 31, P T 31' = 90°, hence q = 90°- Z T 31 = Z
T 31' — 90°.
In the triangle P T 31, by Nap. Rules, we have
Sin A = cos d sin It), .
A \ {a)
Sin B = sin d sec
Tan B
by which A and B may be found.
tan d sec J t.
(b.)
LATITUDE. 41
In the triangles Z M T, Z M' Thy Spher. Trig. (4)
Sin h = sin II cos A -f- cos H sin A sin q ) , .
Sin/i'=sin II cos A - cos if sin A sin q )
The half difference and half sum of these
Sin \ (h—h') cos J (/i -|- h') = cos 5" sin ^4 sin q
Cos J (/i - h') sin | (/i + ft') = sin H cos ^.
from which
«. jt cos \ (h — h') sin | (h -f- ft')
cos J.
(*)
c . sin A (ft— ft') cos £ (ft -4- ft') , >.
Sin 5 = aA. ^ 2^ — ! L ( e .)
cosizsm^l
which gives 90° - Z I 7 and the angle P T Z.
We now have in the triangle P T Z
Z T=90°-H
q = P T Z
B= 90° -PT
given, to find P Z = 90 -L.
Let fall the perpendicular Z and represent it by C.
Let T = Z.
In triangle Z T by Nap. Eules
(/)
cos q= tan Z tan H )
tan Z= cot H cos 5 j
sin H= cos iT cos (7 ) , ,
cos (7= sin ^"sec ^ J ' '
which determines ^ and (7.
p 0= 90°-(-B±Z).-Zwhen perp. falls without the triangle
PT Z.
In triangle P Z
sin L =cos (7 sin (5 + Z) (ft.)
In order to simplify the solution of the whole work, it will be
necessary to find the values of C and Z, if possible, by using first
data. To do this, we have in triangle Z T :
42 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
sin G = cos H sin q
rr sin G
cos H =
sm q
substituting this in (e) we have
s i n c = sin l (h-h ') cos & (ft-fft 1 ) (*• )
sin J.
and substituting first of (#) in (d) gives
cos A cos
The values of C and ^ thus obtained may be substituted in
(h) and latitude found.
To condense the formula, and taking reciprocals of equations
(a) and (6) we have
cosec A= sec d cos J £
cosec B = cosec d cos A
sin (7 = sin J (/i — /i ) cos i (h -\-h') cosec J. [> (&)
sec^= sec J (h-h') cosec J (/i-j-/i') cos A cos #
sin L = cos C sin (i? 4r ^)
It is unnecessary to take out A and C from the tables, as the
log. cos A may be taken corresponding to log. cosec A, and log.
cos C corresponding to log. sin G.
The equations given above, give the form of Bowd. First
method. They can be further simplified by finding B by its tan.
in (b) and we may use
tan B == tan d sec \ t
sin G = sin \ (h-h') cos J (h-\-k) sec d cosec \t
sec Z = sec J {h — h') cosec J (/i -|- h') sin e£ cosec 5 cos G
sin iv = cos G sin (B + Z)
(0
^4, 5, (7, ^ and L, are each numerically less than 90°. A is in
1st quadrant.
q is -|- when 1st alt. is the greater, — when the smaller. It
really makes no difference about C, if we keep it less than 90°,
as only its cosine is used. B has the same name as the declina-
tion.
LATITUDE. 43
16. We have seen that Z may be plus or minus according as
the perpendicular Z falls without or within the triangle. By
reference to the figure it will be seen that the perpendicular can
fall without the triangle only when the continuation W 21 crosses
the meridian between P and Z.
Hence the rule : mark Z north or south according as the
zenith and elevated pole (N or S) are on the same side of the
great circle, forming the two positions of the body. (See Bowd.
p. 181.)
In the figure P Z M> P Z IT and Zis -f- or has same name
as the latitude.
Hence, when the greater azimuth corresponds to the greater
altitude, Z has the same name as the latitude.
By projecting a figure with perpendicular Z falling without
the triangle P T Z. we would see that the greater azimuth cor-
responds to the lesser altitude, and we have : When the greater
azimuth corresponds to the lesser altitude, Z has a different name
from the latitude.
As Z is determined by its secant it cannot be accurately deter-
mined when very small. This will be the case when the altitudes
are very great ; when M and 31 are near the prime vertical ; or,
in general, when the differences of the azimuths of M and J/ 7 are
very small or nearly equal to 180°.
In the case of the sun this will be when the latitude and decli-
nation are nearly equal. This method cannot, therefore, be used
with accuracy, when the sun crosses meridian near the zenith.
17. To find the latitude (circumstances as in last problem)
•with an assumed latitude. (Douwe's method. Bowd. 2d
method )
In figure of last example
Let L' = the assumed latitude.
T = i {T'+T) = ZP Tthe middle hour angle.
%t = ^ (T'—T) = half difference of hour angles.
From the second of (I), Art. 15, we have
sin c = sini (h-h') cos j (h+h l ) (a. )
cos d sin J t
44
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
and in triangle P Z
sin T =
sin G
cos L
and (see Art. 8)
cos z = sin h -\- 2 cos d cos L' sin 2 £ T
cos z = sin h'-\- 2 cos d cos L'sin 2 \ T
In place of 2 sin 2 ^ T, 2 sin 2 J I 7 ' we may use versin T, and
versin T'.
The latitude obtained by " Sailings," may be used, and should
the latitude obtained differ largely from the assumed latitude,
the work may be gone over again with the new latitude.
For computing (a) the first part of Tab. XXLLL, Bowd., may
be used conveniently.
18. To find the latitude from two altitudes of different
bodies, or of same body when the change of declination is
considerable, the Greenwich times being known.
Reduce the observed altitudes to true altitudes ; the difference
between the correct chronometer times must be taken, and when
different bodies have been observed this interval changed to sid.
interval. From this data compute T and T' the hour angles of
the two bodies.
Fig. 13. Given in Fig. 13 :
N d = 90°-PM
d' = 90°-P M'
h = 90°-Z3I
h' = 90° -ZM'
T=Z P M
T =Z P M'
t = T'-T=M P M'
Let fall the perpendicular
M 0, and represent declina-
tion of by D>
P = 90°- D'
and we have
LATITUDE. 45
cos t — tan d cot D'
tan D' = tan d sec t (a.)
M' = d—D' and, representing M M' by 5
sin d; cos 5 =sin P'; cos (d'—D')
„ sin dcos (cf— P') (6.)
cos P = : — ^- ' x '
sin D
letting P = M' M, and P' its supplement M M[ P
cos P' : sin (d' — D') = cot £ : cot P
cos P' : sin (D'—d') = cot £ ; cot P'
, „, cot t sin (D'—d') (c.)
cos P'
In the triangle ZM' M calling the angle ZM' if, Q\ we have
sin i Q'= / cos j (-g + fr + &) sin j (P + /V-/Q
^ cos h' sm P (d.)
and if q'= position angle P M Z
q'=P'-Q< (e.)
In the triangle P 3P Z, letting fall the perpendicular Z n, and
representing 31' n by N', we have
cos q' = tan A' tan JSf'
tan JV'= cot h' cos 5'
Pn = 90° — (rf'+'nO
and in the two triangles we have
sin h' ! sin P = cos N' '. sin (d' -\-N')
. T sin h' sin (d'-\- N') (a.)
sm P = ^— ! Vi/ y
cos iV'
In (5) if the perpendicular JkT falls within the triangle, M'
would be =D — d numerically.
The radical in (d) may have the positive or negative sign, and
hence we may have two values of q'= P' + Q'.
In the figure
P M' M—Z M' M.
q' will equal P M' M -\- Z M' M when the greater azimuth
corresponds to the lowest altitude. The ambiguity may there-
46
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
fore be removed by noting the azimuths at each observation.
The other unknown quantities may be determined by their
proper sign by restricting t to positive values less than 12 hours.
19. The hour angle T in the triangle P 31' Z may be found,
and thence the longitude if the. times have been noted by a chro-
nometer regulated to Greenwich time.
We have in the triangle Z 31' n and P n Z
M' n = N'
P n = 90° ~(N'+d')
cos q' = tan N' tan h'
tan N' — cot h' cos q'
sin N' : cos (iV'+ d') = cot q : cot T'
cot T = CQt g' cos ( N '+ d ')
sinN'
If L has been already found, we have also
cos h' '. cos L = sin T' : sin 5'
sin T
sin 9' cos /i'
cos L
Sin T' and sin q are positive when M' is west of meridian ; nega-
tive when it is east.
20. In Arts. 18 and 19, we have employed the angles at M' in
p IG j^ the triangle P 31' Z. If in the
accompanying Fig. 14, we had
employed the angle 31, and con-
sidered t positive in the direc-
tion opposite the diurnal rota-
tion, remembering that q is less
than 180° east of meridian, and
greater than 180° west of the
meridian, we should have
tan D = tan d' sec t
„ sin d' cos (d — D)
cos B = : — ^- i
sin D
, n Cot t sin (D—d)
cot P = 1
cos D
LATITUDE. 47
sin i q = / cos i (ff 4- ft + fr) sin i (ff-L.ft-.fr)
* cos /i sin i?
GP M;
the difference would still be the longitude east, or —
In order then to obtain the longitude at sea, it is neces-
sary to determine the hour angle of some heavenly body at the
same instant, at the meridian of the place and at Greenwich.
The local hour angles of heavenly bodies are found by computa-
tion. The Greenwich hour angles are found indirectly by means
of the chronometer.
2. The chronometer is a time measurer. A chronometer is
called a Greenwich chronometer when it is regulated to Greenwich
mean time. When we say regulated to Greenwich mean time, we
mean that the reading of the chronometer, plus or minus a known
correction, is the Greenwich mean time. In order to find this
correction, we must know the error of the chronometer on some
given day, and its daily rate.
LONGITUDE. 49
The error of a chronometer is the amount that the chronometer
is slow or fast at a given time.
The rate of a chronometer is the amount that it gains or loses
daily.
It is evident that if we have then the error of a chronometer
on some given date, and wish to find it on some other date, we
must multiply the rate by the interval in days (and if necessary,
decimal parts of days) and apply the result to the given error,
according as the chronometer is gaining or losing, and also
according as the date on which error is required is previous to
or after the date on which the error is given.
3. To find the rate of a chronometer, it is only necessary
to know or find its error on different days ; the difference in
errors divided by the elapsed number of days, giving the rate.
4. The chronometer correction is the quantity which must
be applied to the face of the chronometer to obtain the correct
time. If the chronometer is slow the correction is -)-, if fast,
correction is —
5. To find the correction for a G-reenwich chronometer
by equal altitudes of the sun.
In the case of a fixed star, the mean between the time of two
equal altitudes is the time of transit. This may be compared
with the computed time of transit and error of timepiece
deduced.
In the case of the sun, owing to the
change in declination, equal altitudes of
if and M do not give equal hour angles.
The first observation is, however, taken
at M, when the body has a certain de-
clination, and the angle M P Z is not
y M changed by the change of the declina-
tion from M to the meridian. In the
two triangles, Z P M, Z P M 'to find the
error in t due to change in d,
We have
sin /i=sin L sin d -\- cos L cos d cos t
= sin L cosddd — cos £ sin d cos tdd— cos L cos d sin t d t
50 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY,
sin L cos d J d — cos L sin d cos t d t
&t
d; =
15 cos L cos d sin t
tan Ldd tan 6? d d
15 sin £ 15 tan t
which gives the error in t due to change of d.
We may put, as the change of declination A c£is small,
A . tan L & d tan d a d
15 sin t ~ 15 tan £
If A' d be the hourly difference of d given in the Ephemeris,
and t the hour angle be expressed in hours
. tan Lt a' d tan d t a' d (a.)
15 sin t 15 tan £
This gives an approximate expression for the error of t.
The correction to t would be
_ taniU'd tan d t a' d (b.)
15 sin t ~*~ 15 tan £
If now the sun be observed at M and M and the times noted
by Greenwich chronometer, the middle chronometer time is the
mean of the noted times. If the elapsed time is 2 t, the middle
chronometer time would be
T-j-t
or
T'-t
This middle chronometer would be in error of time of transit
by A t found above, and we should have for chronometer time of
apparent noon
T -f- 1 + A t, or
T - t + A t, or
If the first observation had been P if it would be necessary
to find the chronometer time of apparent midnight. By a
LONGITUDE. 51
similar process to the above, paying attention to the signs, we
would have
tan Lt^d tan dt a' d (c.)
15 sin t ' 15 tan t
6. If in (b) we put
A = — — — : — - and B =-.>-, ,
15 sm t lo tan c
we will have
a t = Aa' d tan X -(- 5 A' d tan d
L and c? are -f- when north, — when south.
A c/ and A' d are -f- when the change of the sun's declination is
towards the north,— when towards the south.
A is — since t is <12 hours.
B is -|- when t is < 6 hours — when t > 6 hours.
J[ and 5 may be computed for different values of f, and their
logarithms tabulated. Such tables are given in " Chauvenet's
method of finding the error and rate of a chronometer." The
argument is 2 t, or the elapsed time.
In the equation for the lower branch of meridian the sign of A
is changed as in (c).
2 t should be properly the elapsed apparent time. The interval
is so small that this is generally neglected and the elapsed mean
time used. It may be also corrected for the supposed or known
rate of the chronometer. A" d is taken from the Nautical Almanac
for the instant of apparent noon or apparent midnight.
In observing equal altitudes, use equal intervals of 10' or 2(K
It is not necessary that the altitudes be correct, but only that
they should be the same on each side of the meridian. Use there-
fore, the same instruments at both observations, and be especially
careful to use the same end of the roof of artificial horizon.
T A- T
7. — X — -f~ A t gives the chronometer time of apparent noon
or of apparent midnight.
By applying the equation of time we have the chronometer time
of mean noon, the difference between which and the longitude is
52
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
the chronometer correction. If the correction of the chronom-
eter to local mean time is required, we have only to omit the
application of the longitude.
8. To find the correction to a Greenwich chronometer
by a single altitude of any heavenly body.
As before, the observation must be taken at some place whose
latitude and longitude are well determined.
We will have, therefore, in the astronomical triangle, the case
when three sides are given to find the angle t, the formulae for
which are (Spher. Trig. 164, 165, 166).
sm
i A
2 JL
COS
/sin (s— 6) sin(s
* sin 6 sin c
/sin ssin (s — a)
o)
tan J A = yf
sin b sin c
/sin (s
b) sin (x— c)
sin s sin (s— a)
We have given in Fig. 17
Fig. 17.
N
L, to find
PM = p = 90°— d
ZM = z= 90°— h
P Z=coZ =90°
ZPM=t
The chronometer times of the alti-
tudes are taken and their mean plus
the supposed chronometer correction,
gives us the Greenwich time, with suf-
ficient accuracy for determining the
declination of the body. The mean of
the altitudes is taken and used as a
single altitude. For finding t by its
sine, we have, using the sides of the triangle directly,
to
sm
tu/ ^l (P + g
co L) sin J (z -\- co L — p)
cos L sin p
It has been found more convenient to use the following values
of the sides, viz. :
90° — L, 90°— A, and p which gives
sin 1 1 = M* 2- (L
p -f- /i) sin j ( .£ -[~ P — ^)
cos L sm p
LONGITUDE. 53
or, if we put
s'=h(L + P + h)
sinii= / cos s' sin (s'—h) ( b )
* cos L sin p
Which is Bowd. formula, p. 209.
To determine t by its cos, using direct values of the sides, we
would have
cos i t== / sin ± (co L -{-p + z) sin \ (co L + p - z)
* cos L sinp
or, if
s" = \ (co L +p + z)
cos i t= / sin s" sin (s" — z ) ( c -)
^ cos Z> sin p
To determine t by its tangent, using direct values of the sides, we
would have
tan J t = / sin(s w --coZr)Bin(s , "-j3)
^ sin s"' sin (s"' — z )
in which
s"' = |(coi+z + p)
t is —when the body is east of meridian.
In case the sun is observed, if P. M., t is the L. A. T ; if A. M., it
is 12 hrs. - L. A. T.
Bowd. Tab. XXVII. contains the direct value of t in P. M.
column, and also 12 hrs. — t in A. M. column. In case any
other heavenly body is observed, t is its hour angle, + when west
and — when east of meridian. The L. M. T. may be found, and
thence the Greenwich time by the method in the Chapter on
time.
In the case of the sun, the L. A. T. is changed to mean time,
and by applying the longitude, to Greenwich mean time. The
mean chronometer time is compared with this to find the chronom-
eter correction.
When t < 6 hrs. J t < 45° and is better determined by b, as the
sines of angles less than 45° vary more rapidly than the cosines.
(See Chauv. Trig. Art. 112.)
When great precision is required, t is better determined by d,
b and c are the most convenient formulae for finding L. M. T. at
54
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
sea. Many Navigators determine the errors and rates of their
chronometers by single altitudes. It is advisable then to use (d).
In taking observations for single altitudes take half the obser-
vations with each end of the roof. The times may be noted by
a watch compared with the chronometer. If the interval between
the comparison and observations is long, or the rate of the watch
considerable, the watch times must be corrected for this change.
9. The two methods given are the only convenient methods
which the Navigator can use with the instruments at his dis-
posal for finding the correction for his chronometer. There are
many ports where time-balls are dropped at the same instant each
day for the convenience of the shipping in the harbor. Unless,
however, they are dropped by electricity from some respectable
observatory they are not to be depended upon.
10. As before stated, the methods of finding longitude at sea
depend upon finding difference of time. The Greenwich chro-
nometer, carefully regulated, furnishes the Navigator with the
Greenwich time. The local time is found by observation of
some heavenly body. The most common method is by (b) and
(c) in Art. 8. Other methods are given. The latitude is found by
applying the run of the ship to the latitude found at noon or by
some other observation. The declination is taken from the Nau-
tical Almanac for the Greenwich mean time, as shown by chro-
nometer.
U. To find the hour angle (and thence the losal time) of a
heavenly "body just visible in the horizon.
Let M be the body
P M = p = 90° - d
P N=L
In the triangle M P N (Fig. 18),
right angled at N, we have
cos MP N= tan P N cot P M
cos M P N= — cos t = tan L tan d
cos t = - tan L tan d.
LONGITUDE.
55
12. To find the hour angle of a heavenly "body when on or
nearest to the prime vertical.
In the case of the body at n, d> L, the body is nearest to the
prime vertical when Z n is tangent to its diurnal circle, and
P n Z = 90°. We then have
cos t = tan p tan L = cot d tan L.
If d < L and of same name, as for a body whose path is m
m\ the body will be on prime vertical at m and P Z m — 90°, and
cos t === cot L tan d.
If d and X are of different names, as in the case of the body
whose diurnal path is o o', the nearest visible point to the prime
vertical is in the horizon, and the solution is effected by the
equation
cot t = — tan L tan d, of Art. 11.
13. As A. M. and p. m. sights are enjoined in the directions
of the Navy Department, it would be well if Navigators used the
same altitudes for both observations. The corrections to the
observed altitudes would be the same, and generally the longi-
tudes determined would be at nearly equal intervals from noon.
56
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
They could each be reduced to the noon longitude, and their
mean taken.
14. To find the longitude at sea by the intersection of
circles of position.
At any instant of time the sun is in the zenith of some place
whose latitude is equal to the declination of the sun at that
instant, and whose longitude is equal to the Greenwich apparent
time.
In Fig. 20, P G is the meridian of Greenwich. P is the
meridian of the place which has the sun S in its zenith. S is
the latitude of this place and is equal to the declination of the
sun. G P S is the Greenwich apparent time, and is equal to the
longitude of S.
If now any number of observers Z, Z', Z", etc., situated on the
circle, observe the sun at the same instant of Greenwich appar-
ent time GPS, their zenith distances Z S, Z' S, Z" S are equal.
Such a circle is called a circle of equal altitudes.
Their Greenwich times being equal, they would each obtain
from the N. A. the same declination S 0. Each observer would
have in the astronomical triangles S P Z, S P Z', S P Z", etc.,
LONGITUDE. 57
the side P 8 = p = 90° — d common, and the sides 8 Z, 8 Z\
etc., = 90° - h equal. The hour angle of the sun &t Z is Z P 8,
at Z\ Z' P 8, and at Z", Z" P 8. The difference in the values
of these hour angles must be due to the different values of the
third sides, P Z, P Z\ P Z", etc. These sides are 90° - L,
90° - L, 90° - L", etc. Hence the different values of the
latitudes cause the different values of the hour angles, and
thence the different values of the longitudes G P Z, G P Z',
G P Z", etc.
An infinite number of circles of equal altitude may be drawn
about 8 possessing the same properties as those described. If,
therefore, with the sun as a centre, a circle be drawn upon a
globe, all points upon this circle will have the same altitudes of
the sun at the same instant.
As, therefore, the Greenwich time and altitudes are constant
for any particular circle, an observer at Z, by using his own
altitude and Greenwich time, and assuming the latitudes of Z, Z\
Z", etc., can determine the corresponding longitudes G P Z,
GP Z,GP Z', etc.
Suppose an observer at Z' , his latitude unknown, with his
zenith distance Z' 8, polar distance P 8, and the assumed lati-
tudes of Z and Z" should determine their corresponding longi-
tudes.
These assumed latitudes and determined longitudes may be
plotted upon a globe, and the arc Z Z " of the circle of equal
altitudes drawn through them. The observer has a line Z Z"
upon which his position Z' is known to be. Such a line is called
a line of position.
The direction of this line at any point is the direction of the
tangent to the curve at this point. The direction of this tangent
will be at right angles to the bearing of the sun at that point.
Hence by a line of position we may determine the azimuth of the
sun.
If now the observer wait until the sun has changed its bearing
n°, and with the new values of the altitude and declination of the
sun, and same values of the latitude, compute again a portion of
the new circle of equal altitudes, as he is also on the second line
of position, he must be on the intersection of the two. If this
58 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
be plotted as before upon the globe, the intersection will be at Z\
the latitude and longitude of which may be taken from the
globe. To plot these curves accurately would require a larger
globe than would be convenient. They may, however, be plotted
upon a Mercator's chart.
By reference to the principles of construction of the Mercator's
chart, it will be seen that only the loxodromic curve plots as a
straight line. The circle of equal altitudes would plot as an
irregular figure, its greatest diameter coinciding with the arc of
the meridian N 0. The whole figure could not indeed be plotted
upon ordinary charts, unless the zenith distances Z S, Z S\ etc.,
were very small.
It is customary to plot only the small portion of the curve
lying between the assumed latitudes, as that is all that is
required. For small differences of latitude this would be prac-
tically a straight line. If the difference of latitude be great, or
the chart a large scale one, latitudes between Z and Z" may be
assumed, the corresponding longitudes found and plotted, and
the curve traced by hand through them.
In the practical use of this problem at sea, it is customary to
assume latitudes 10' or 20' on each side of the supposed one, and
determine the corresponding longitudes.
In general, assume the latitudes to cover any supposed error
of the latitude.
In the foregoing, the discussion has been confined to the sun.
The body S may be any other heavenly body which can be con-
veniently observed.
15. If, between the observations, the observer should change
his position, as is generally the case at sea, the first observation
may be corrected to the position of the second by correcting the
altitude, or, more conveniently, by moving the first line of posi-
tion to the place of the second observation.
If the first observation be taken at Z (Fig. 21), and the ship
run to Z, Z / or Z is the correction to the altitude, or zenith
distance Z S, to find the zenith distance Z S at the same
instant.
LONGITUDE. .
59
If the distance be small, Z 0' may be considered as a right
line having the direction of the tangent at Z, and
Z' 0'=Z = A z = ZZ' smZZ' 0'=Z Z'sinZ' Z 0'
N Z Z'=C = course
NZ = 180° — Z
A z = - A h = Z Z' cos [G - (180°-^)]
A h= -ZZ' cos (C-Z)
If the first observation be at Z' in the same way we will have
A h = ZZ' cos (C — Z)
Fig. 21.1
The difference between C and Z is taken, and Z reckoned from
the north point 180°. Then, if the difference between C and Z
is < 90°, A A is additive ; if > 90°, A h is subtractive. (See
Bowd. Eule, p. 183.) The equation for A h may be solved by
the Traverse Table. Find (C-Z) at top or bottom of page, and
the distance sailed in distance column, opposite in difference of
latitude column, is the correction in minutes and tenths to be
added to altitude when difference is less than 90°. If the dif-
ference is greater than 90°,find 180° — {C-Z), as before, and cor-
rection is subtractive.
To move the line of position, lay off on the chart the distance
ZZ in the direction of the course sailed between the observa-
60 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
tions, through the extremity draw the line Z parallel to Z 0'.
This evidently accomplishes the same result as correcting the
altitudes. It possesses the advantage of being simple, and when
the chart has the magnetic compass plotted upon it, the compass
course can be laid off between the observations.
The method of correcting the altitude must be used, however,
in the case of Double Altitudes for Latitude.
We have seen that the line of position is at right angles to the
bearing of the sun. If the sun is on the prime vertical at both
observations, the lines of position will run north and south, and
there will be no intersection.
If L = d nearly, the lines of position will not change their
direction sufficiently to depend upon their intersection.
When the body is near the prime vertical, errors in the lati-
tudes have the least effect upon the corresponding longitudes.
When the body is near the meridian, errors in the longitude
have their least effect upon corresponding latitudes.
Latitudes may be assumed, and the corresponding local mean
times found, or the longitudes may be assumed, and the corre-
sponding latitudes determined by Art. 4, Chap, on Latitude.
16. If there is an uncertainty in the altitude, draw on each
side of the line of position lines parallel to it, and distant from
it, the amount of the supposed uncertainty, and the position will
be somewhere within this belt.
In the same manner, if there is an uncertainty in the Green-
wich time, parallels may be drawn upon each side of the line of
position equal to this uncertainty.
17. Near the coast, when charts aro on a sufficient scale, there
is no difficulty in determining the position with a considerable
degree of accuracy. At long distances from the coast line, our
charts are generally upon too small a scale to admit of an accu-
rate plotting of the lines. This may be remedied best by project-
ing upon a piece of paper a sectional chart which shall cover the
difference of latitude and longitude.
The latitude may be found by computation, as follows :
LONGITUDE.
61
Let / r l 2 the longitudes of A and
B in latitude L.
// l{ the longitudes of A'
and B' in latitude 2/
L = latitude of C. J
From the similarity of the
triangles A B C and A' B' C
{h'-l:) + (h
U k'
u. - w
The Navigator will find the method of Art. 17 preferable to
this. It does not require great nicety in the construction of the
chart. The latitude and longitude of the intersection may be
transferred from this chart to the one in use.
Fig. 23.
18. To find the longitude by means of observed lunar dis-
tances. (See Vol. II., No. 4, of the Ast. Journal. Chauv.
Method.)
The observation is supposed to give
the apparent distance and apparent
altitudes of the two objects ; but if
the latter cannot be observed, they
must, in order to apply the present
method, be previously computed by
known rules. Taking at once the
most general case, namely, that in which the object observed
with the moon also has parallax, let us take " the sun." The
formulae will require no change for a planet, and for a star no
change beyond making the parallax zero.
Let, then in Fig. 23, Z being the zenith of the observer,
d = S' H' = the apparent distances of moon's and sun's
centres.
h = M ' H= the moon's apparent altitude.
H= S' H' = the sun's apparent altitude.
62 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
d 1} h lf Hi, the distance and altitudes referred to that point of the
earth's axis which lies in the vertical of the observer, which point
we shall distinguish as the point 0.
We shall then have
and if
cos d x — sin h x sin H x cos d - sin ft sin H
cos h x cos H x cos ft cos H
sin ft sin H x cos h x cos H x
m = . . — ~ n = 1 - 1
sin ft sin M cos ft cos izi
then
cos d — cos d x = (1 — ?i ) cos rZ — (??i — n) sin ft sin H (1.)
Let
&d = d x - d, Ah = h x - ft, A H= H - #;,
then
cos d — cos d x = 2 sin J A d sin (d -f- \ A t?)
cos (ft - f A ft) cos (H - AH)
cos ft cos i?
i A 5- sin (H- \ AH.,
w =
/. _2 sin J A ft sin (A-f-iA ft)W/i_i 2 sin i
cos iT
2 sin i A ft sin (ft -f i A h) _ 2 sin i A H sin {H - % A H)
~ cos ft cos H
, 4 sin ^ A ft sin i A 77 sin (ft -\-jAh) sin g- \ A J?)
COS ft COS 5"
also observing the relations
sin lfi x cos ft = \ [sin (2 ft -f- A ft) -j- sin A ftj
cos ft x sin ft = 4 [sin (2 ft -f- A ft) — sin A ft]
sin H x cos H = J [sin (2 jS" - A H) — sin A H]
cos i7 x sin #= i [sin (2 H - A H) -f sin A 5"]
we find
sin ft, cos ft sin i£ cos H — cos ft, sin ft cos H x sin ^
sin ft cos ft sin Hcos H
sin A ft sin (2 H - A H) - sin A .3" sin (2 ft -f A ft)
2 sin ft cos ft sin H cos -9"
LONGITUDE. 63
if then we put
A 2 sin \ A h sin (h -j- \ A /i) cos cZ
cos 7i
i> = sin A 7l sin 2 ( H ~ A g )
1 2 cos h cos .2"
r _ _ 2 sin i A Rsin (E- \ A E) cos fZ
1 "~ " " 2 cos E
D __ sin gsin (2 h -f A 7?)
1 2 cos /i cos H
the equation (1) becomes
2 sin -i A d sin (7Z + J A d) = A-+ A + 6\ + A — A d sec tf. (2.)
This rigorous formula may be adapted for practical use in
several ways requiring auxiliary tables. I proceed to give the
transformation which appears to require the fewest and simplest
tables.
If the terms of (2) are reduced to seconds, we shall have
A d sin (d + J Ad) = A 1 + B 1 +C 1 + B 1 - A x C x sin 1" sec d. (3.)
in which
Ay — _ L . sin (h -\- J A h) cos r/
A=-
COS /i
A/i sin (2 # - A #)
cos /i 2 cos 17
C\= — ^?=.. sin (H-} 2 A E) cos rZ
A5=
cos 5"
A/i sin (2 /i+ A h)
cos ii 2 cos h
Let
p = moon's horizontal parallax reduced to the point 0.
r = moon's refraction.
P, R, the same quantities for the sun, then
A h — p cos (h — r) — r
A E= R - P cos (J?— i2).
The neglect of i? in the term P cos (i? — R) produces an error
altogether inappreciable in practice ; but the error produced by
64 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
omitting r in the term p cos (h-r) may amount to 1", and we shall
therefore take
cos (h - r) = cos h -f- sin r sin h
A h = p cos h — r -J- p sin r sin ft
= (pcos-r)(l+-^ n J^\
V p cos A — r /
If we develop the last term, and put
h = r tan h,
we shall have, designating the term by K,
T r. p sin r sin h , . ., „ /\ . /j \
K / =^ ; = k sin 1" 1 4- — — I
p cos /i — r \ ' p sin ^p/
in which jp may be taken at its* mean value ; and since h and h
decrease together, it will be found that K is nearly constant, its
maximum being .000296, and its minimum .000285. A wider
range will be admitted if we allow for the variations of the ba-
rometer and thermometer, and of p; but without here entering
into more details, it will suffice to state that the error of the
value
K = .00029
is always less than .00006 so long as h > 5°, and the formula
Ah= (p cos h - r) (1 -f K)
gives A h within /7 .05 at a mean state of air, and within 0".2 in
all cases.
Let now
R
E l
cos h s cos H
The quantities r 1 and R l will be given by a " Eefraction Table
for Lunars," which with the argument apparent altitude will give
the refraction divided by the cosine of the altitude, and will be
arranged precisely like the ordinary tables of refractions. The
corrections for the barometer and thermometer may be arranged
as usual in nautical tables, with the arguments height of barometer
(or thermometer) and apparent altitude ; or, which is preferable,
with the refraction itself instead of the altitude, for with the latter
arrangement the same table will serve to give the correction
LONGITUDE. 65
either of r or of r 1 . These quantities then being substituted, the
corrections of the apparent altitudes become
A h = (p — r l ) (1 -f- JS") cos h
A h= {B 1 - P) cos E
and the terms of (3) become
A x = {p - r) (1 + K) sin {h + \ A h) cos d
d == - (i? 1 - P) s\n(H - \A H) cos d
D = (B l — P) sin (2 /t + z/ 7Q
1 2 cos A
The term ^ 0! sin 1" sec c£ is very small, its maximum being
only about 1". It is easy to obtain an approximate expression
for it, and to combine it with the term A 1 ; for in so small a
term we may take
G x = — R ' sin H cos d = — h' cos d
where h'=B tan H; and without sensible error in most cases
we may take h ' sin 1" = K, so that
G 1 sin 1" sec d = — K
and
^ - A x G x sin 1" sec d={p — r') (1 -|- 5T) 2 sin (/i-f |/l 7i) cos . «M*+^*)
sin h
G _ sin ( g ~l z? - g )
sin ZZ I (A. )
Z) _ sin (2 h+A h)
sin 2 /i
-4' = (p —r') A sin h cot d
B / = — (p — r) B sin .ETcosec <2
0' = - (-B'-P) C7 sin E cot rf
D' = (B' — P) D sin A cosec d
then our formula (3) becomes
A d Bin (d + j/ld) =A , +B . c ,
sin d iii
Developing the first number, it becomes
4 d (l_\ 2 sin t ^ ^ cog ( d + I ^ <*) \
\ sin d /
so that if we put
_ _ A d 2 sin 1" cos (d -\- \ A d)
sin and n m The
similar correction of the sun's or a planet's parallax is insensible
in practice.
If, then, in the computation of (A), (B), and (C), we employ
for p the value p = tt 1 we obtain d L . To reduce finally to the
centre of the earth, we have
(6.)
cos d' = sin A sin 6 -j- cos A cos 6 cos a )
cos d' = sin A sin d t -\- cos ^ cos d x cos a )
from which combined with (4) we find
p cos . v J
sm d l tan d x
in which
y (1 — e' 2 sin 2 0)
and we may employ without sensible error the value of N corre-
sponding to 0=45°, or log N=l. 8170, the compression being ^.
The computation of this correction would be rendered at once
70 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
simple and accurate in practice, if the • ephemeris contained the
log of
, T sin A sin d , T
sin a' tan a
(which is equivalent to a logarithm introduced by Bessel into his
ephemeris for the same purpose), for we should then have
cV — dj, = N' sin (p (7.)
4tii. Corrections for the Contraction of the Moon's and
Sun's semidiameters by Refraction. — The apparent distance
of the centres of the moon and sun t has been supposed above to
have been found in the usual manner from the observed distance
of the limbs, by adding the apparent semidiameters ; or when
the moon has been observed with a planet or star, by adding or
subtracting the moon's semidiameter alone, according as her
nearest or farthest limb has been observed. At low altitudes
the elliptical figure of the disc must be taken into consideration ;
for the refraction being different at points of the limb which
have different altitudes, the result is an apparent contraction of
every semidiameter, the vertical ones being the most, and those
perpendicular to the vertical the least contracted. It becomes
necessary to obtain a general expression for the contraction of
that semidiameter which lies in the direction of the distance, and
makes an angle q with the vertical circle. If we put
s = horizontal semidiameter of the moon -|- the augmentation,
s = the apparent vertical semidiameter
s' = " inclined "
A s = contraction of vertical " = s — s
A s / = " inclined :< = s — s'
A r = difference of refractions at the centre of the moon and the.
observed point on the Jimb,
we have nearly
A s = A r cos q.
Bat the apparent altitude of the centre being h, A s is the dif-
ference of refractions at the apparent altitudes h and h -f- s , while
A r is the difference of refractions at h and h -f- s' cos q,
LONGITUDE. 71
whence
A s ! A r = s : s' cos q
A r = J—, z7 § cos q = A s cos q (nearly)
J's = A s cos 2 5 (8.)
a known formula which agrees very nearly with the hypothesis
that the figure of the disc is an ellipse. It is evident, however,
that the lower half of the disc is more flattened than the upper
half ; but if As be taken as the mean of the contractions of the
upper and lower vertical semidiameters, the preceding formula
will be in error only 0".4 at the altitude 10 D , and 1".2 at 5° ; the
maximum values of As at those altitudes being respectively
10 7/ and 30 '. The changes of the thermometer and barometer
may also sensibly affect the value of As at low altitudes, but only
by 4" in the improbable case of the highest barometer and lowest-
thermometer, and h = 5°. It will hardly be necessary to attend
to this small error in practice ; nevertheless, it can readily be
done without any further reference to the refraction tables, for
the computer will already have before him r\ the mean value of
r', and Ar\ the sum of the corrections of r\ for barometer and
thermometer ; so that he may find at once the proportional cor-
rection of As' , which is
r '
Now the angle q is given by the formula
sin H — sin h cos d
cos q = ,
cos h sin a
and we have from the formula (A)
B' _ sin H A' sin h cos d
B {p — ?") cos h cos h sin d ' A (p — r') cos h cos h sin d >
(B' . A\ 1
CQSg = (iT+x) (p-Ocob* -
If we assume A = 1, B = 1, we shall have
A' + B'
cos q = - 4 Y
\P — r ) cos ' l
{p-r')*coH 2 h (E.)
72 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
which is easily put into tables. A table with the arguments h
and p — r may give the value of
As tt
\jp — r'y cos -h
and a second table with the arguments A' -f- B' and " the num-
ber from the first table" may give A s'.
In order to ascertain the degree of accuracy of the formula
(E) i we observe that the errors in k cos q produced by taking
A = 1, B = 1, are
e=(A- 1*EL* *'=(!_ J) sin ^
tan d ' cos h sind'
the errors in cos 2 q are
2 e cos 5 and 2 c cos g
and the errors in A s ' are therefore
2 A .% (^4—1) tan h cos q t 2 A s (1 — B) sin H cos q
tan rf cos h sin d
The greatest values of e x and e\ at different altitudes, are
shown as follows, taking cos q = 0, H = 90°, in order to repre-
sent the extreme cases :
h e x tan d e'\ sin d
o
5
0.45
0.02
10
0.16
0.00
15
.08
.00
30
.02
.00
50
.00
.00
It appears, therefore, that the error of the formula (E), like
that of (8), becomes sensible only at] those low altitudes where
extreme precision is unattainable on account of the uncertainty
of the refraction. We may therefore safely employ it as suf-
ficiently accurate for all cases.
When the sun is observed with the moon, a similar correction
must be applied to his semidiameter. If
LONGITUDE. 73
Q = angle at the sun,
S = true semidiameter of the sun,
S = apparent vertical semidiameter of the sun,
S' = " inclined
A S — contraction of vertical semidiameter =8 — S 0i
A 8' = " inclined " === 8 - #',
then as above
A 8' = A S cos 1
n s in h — sin H cos (7 / O D' \ 1_
C0S V— eoaJBBmd ~VC D ){E - P)
and assuming
C == 1, D = 1,
we have
a + iy
cos R
cos Q
(jR' - P) cos IT
(i2' — i^) cos i/
which is even more accurate than (i7), and is put into tables in
the same manner.
The corrections A s' and A 8' should strictly be applied to the
semidiameter, and should appear in the value of a employed in
the computation A d; but since the values of A' , B', C , and
D' are required in finding A &' and A 8', we have to employ a
value of d which may in extreme cases be in error by about
30". This produces a small error in each of the terms A' , B',
C' f D', which could in practice be eliminated only by repeating
the computation with the corrected value of d. But this repeti-
tion is unnecessary, as the error in A d is rarely more than 0."5 ;
and it will suffer to apply A s' and A 8' directly to d x .
In order, however, to show generally the effect upon A d of
small errors in d, let us differentiate the equation (C), regarding
the term x (of the second order) as constant, and taking A = 1,
B = 1, 0=1, D = 1 (which also amounts to considering
terms of the second order as constant). We find
a a J — C 1 — r ) ( s * n ^ — s * n & G0S d} s ^ n 1" ^ ^
sin "d
74 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
_ {R — P) (sin H— sin h cos d) sin 1" <5 rf
"~ am z d~
(5z/ d= — [(p — r) cos (/cos J?— (Til' — P) cosq cosh]
sin V 6 d
sin d
(9.)
This formula shows at once that the maximum of 6 A d occurs
when the two bodies are in the same vertical circle, the moon
being the higher body, for this condition gives cos q = — 1, cos
q = 1, so that the two terms obtain the same sign.
The following table shows the maximum effect upon A d of the
error of 1" in d, computed by formula (10), for the several values
of h and H; the least value of h — H {= d) being 20°.
H
h
25°
35°
45°
55°
65°
75°
85°
90°
o
/
//
//
n
a
//
//
lt
5
3.6
2.4
1.9
1.5
1.3
1.2
1.1
1.1
15
3.2
2.2
1.7
1.4
1.2
1.1
1.0
25
2.9
2.0
1.5
1.3
1.1
1.0
35
2.6
1.8
1.4
1.1
1.0
15
2.2
1.5
1.2
1.0
55
m
1.8
1.2
1.0
65
..
••
1.3
0.9
and at the same time p — r' and R' — P have their greatest
values. In this position, we have d = h — H> and the formula
for the maximum of 6 A d is therefore
sin 1'' 6 d
d A d = - [{p - V) cos H+ {& -P) cosh] - m{h _ Hy
This table of extreme errors shows clearly enough that the
error arising from the neglect of A s' and A S' in the value of d
employed in computing A d, is too small to require any departure
from the process already indicated. For the Navigator must
bear in mind that ail observations at very low altitudes are sub-
ject to two principal sources of error : — 1st, the uncertainty of
the refraction, which no process of calculation can eliminate ;
LONGITUDE. 75
and 2d, the imperfect definition of the limb of the moon or sun
in the vicinity of the horizon. If a method of computation in-
volves only errors which in every case are less than these un-
avoidable errors, it satisfies the essential condition of a good
method.
CHAPTER VII.
THE COMPASS.
1. A magnetized needle or bar of steel balanced and allowed
to turn freely on a pivot, will take a position in a particular
direction, which is called the magnetic meridian.
The direction in which the north end points is the magnetic
north. It varies or declines from the true north differently at
different places on the earth ; and even at the same place at dif-
ferent times. Delicate observations show a small diurnal fluctua-
tion of a few minutes, also a progressive change or one of very
long period, — on the Atlantic coast of the United States, of 2' to
5' westerly in one year.
2. If a circular card marked with the horizon points be
attached to such a needle, its several points will deviate from
the corresponding points of the horizon, all by the same amount
and in the same direction.
Let P be any place,
N S its true meridian,
N ! S' its magnetic meridian,
N P' N' is the variation.
In Fig. 24 it is east, N' S' being to
the right of N S.
THE COMPASS.
77
3. The magnetic declination, or variation of the compass, at any
place, is the angle which the magnetic meridian of that place
makes with the true meridian. It is east, if the magnetic meri-
dian is to the right ; west, if the magnetic meridian is to the left
of the true meridian.
Fig. 25.
In Fig. 25 it is west, JV 1 S' being to
the left of NS.
The point of view, or position from
which the observer is supposed to look,
being at P.
4. Let
be any object, terrestrial or celestial.
P 0, its horizontal direction from P.
A = NP 0, the true azimuth or bearing of from the N point
of the horizon.
A' = N' P 0, the magnetic bearing of from the magnetic
north.
D =N P N', the variation.
If towards the right be regarded as the positive direction of
these angles, and towards the left as the negative direction, we
have from both figures, and with in any position,
NP N' = NP - N' P O, or,
D =* A — A',
positive, or to the right, for Fig. 1 ;
negative, or to the left, for Fig. 2.
78 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
If A and A' denote bearings or angular distances from any
other points than the true north and magnetic north, for instance
the east or the west, we evidently still have —
D = A - A';
or, translated into common language :
The magnetic declination, or variation, at any place is equaL
to the difference of the true and the magnetic bearings of any
object ; it is east if the number or point which expresses the
true bearing is to the right of the number or point which ex-
presses the magnetic bearing ; but west if to the left.*
5. From equation (1) we also have
A = A'-\-JD;
or, the variation must be applied to a compass bearing (or
course) to the right hand if east, to the left hand if west, in order
to find the true bearing (or course).
6. The same equation also gives
A' = A - D;
or, the variation must be applied to a true bearing (or course)
to the left hand if east, to the right hand if west, in order to find
the compass bearing (or course).
7. To find the variation, it is necessary to determine both the
true bearing and the magnetic bearing of some object ; at the
same instant if the object be in motion.
8. The true bearing or azimuth of a celestial object may be
found —
First. — From an observation of its altitude (Prob. 1). This
may be used to the best advantage when the azimuth changes
most slowly with the altitude, t. e., when a given change or sup-
posed error of the altitude produces the least change of azimuth.
The most favorable position of any object is when its azimuth is
* The numbers or points are supposed to be read from tbe same compass card,,
the observer looking at them from the centre.
THE COMPASS. 79
nearest 90° ; the unavailable position is on the meridian. High
altitudes and great declinations, especially if of a different name
from the latitude of the place, are to be avoided.
Second. — When it is in the horizon, or its apparent altitude
above the sea horizon is 33'+ the dip, (Prob. ) its amplitude,
or bearing from the east or west point of the horizon, is readily
determined by the solution of a spherical right triangle ; or when
the declination is less than 23° 28', by Tab. VII. (Bowd.).
Third. — From the local time.
The most favorable time for a circumpolar star is that of its
greatest elongation from the meridian ; for other objects, when
they are on or near the prime vertical. A more exact knowl-
edge of the time is requisite, when the observation is made near
the time of meridian passage, especially at very high altitudes. _
Fourth. — From the measurement with a theodolite, or other
azimuth instrument, of the azimuth angle between the two posi-
tions of the body at the same altitude east and west of the
meridian.
9. The true bearing of a terrestrial object at any point may
be found, from the measurement —
First. — With a theodolite, or other azimuth instrument, of the
horizontal angle ; or,
Second. — With a sextant, of the angular distance between the
terrestrial object and some celestial object, whose azimuth at
the same instant is found either from its altitude or the local
time*
It is not necessary to have two observers, or that the obser-
vations of altitudes and horizontal (or oblique) angles should be
simultaneous. One observer may measure an altitude, then the
horizontal (or oblique) angle, then another altitude, noting the
time. On the supposition that the altitudes increase or decrease
uniformly we have, as the interval of time between the observa-
* Sometimes called " an astronomical bearing'/'
80 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
tious for altitude, is to the interval between the first observation
for altitude and the observation for hor. (or ob.) angle, so is the
difference of altitude to the reduction of the first altitude.
Measurements of the horizontal (or oblique) angle may be
made before and after the observation of altitude, and interpo-
lated in the same way.
10. When precision is requisite, it is necessary to keep in
mind —
First. — That a change of the point of observation of .001 of the
distance of the terrestrial object may change its bearing more
than 3'.4.
Second. — That the higher the altitude of an object, the more
requisite is the careful adjustment of the instrument used in the
measurement of the horizontal angle.
Third. — That greater care is requisite in the measurement of
the direct angular distance, the greater the inclination to the
horizon of the oblique plane which passes through the two
objects ; the apparent altitude, or angle of elevation, of the terres-
trial object above the eye of the observer must also be deter-
mined.
Fourth, — That in measuring terrestrial angles with a sextant or
circle of reflection, the axis about which the index moves is the
proper centre of the instrument, and the reading should be
increased by the parallactic angle, which is inversely as the dis-
tance of the object seen direct.
For a distance of 500 feet it is about 1' in the common sex-
tant. But it is combined with the index correction, if the obser-
vation for the latter be made with an object at the same or nearly
the same distance.
These are important considerations in accurate surveys, and
in making with precision meridian lines.
Ordinarily the sun is the most convenient celestial object.
For use in connection with a compass, precision in the true
bearing to the nearest 5 ' is generally sufficient.
THE COMPASS. 81
11. The magnetic bearing is observed directly with a compass.
The two chief forms of this instrument are th.e*surveyors com-
pass, in which the graduated circle revolves with the line of
sight, while the reading points, which are the extremities of the
needle, remain fixed ;
And the mariner's compass, and in its more refined form, the
azimuth compass, in which the graduated circle attached to the
needle remains fixed, while the pointer revolves with the line of
sight.
With the best surveyor's compasses a precision of 5', or with
the best azimuth compasses a precision of 10 ' , is rarely attainable.
12. To obtain even this degree of precision, it is necessary —
First — To correct for the index-error of the instrument. This
correction is the same for all bearings ; and may be found for
each compass (and compass-card) by bearings of a number of
objects in different directions, whose true magnetic bearing has
been determined by more delicate instruments. Once carefully
found, it nay be marked as a constant correction.
If it is neglected, the bearings observed are " compass bear-
ings," and the variation found is the variation of that particular
compass ; in distinction from the true magnetic bearings and the
true magnetic declination.
Second. — To correct for eccentricity, or for the pivot not being
in the centre of the graduated circle.
With the surveyor's compass this error is eliminated by oppo-
site readings of the graduated circle.
Azimuth compasses are not sufficiently delicate for the refine-
ments of this correction. But the maximum error may be found
by measuring horizontal angles of about 90°, which have been
measured by a more reliable instrument.
Third. — To attend to the balancing of the needle or compass
card. Sealing wax dropped on that part of the card which re-
quires depression is sometimes used.
As the north end of the needle dips or is depressed in north
magnetic latitude, and the south end in south magnetic latitude,
readjustment is generally necessary after a considerable change
of latitude.
82 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
Fou rlh. — That the sight vane or vanes and their axis of rota-
tion should be parallel, also perpendicular to the graduated cir-
cle, if there be one on the compass box.
Observations on a plumb line, or other well defined vertical
line, made on the land, furnish a test of these adjustments.
Fifth— ^-That at the instant of observation the sight vanes should
be vertical.
This is the more important the greater the elevation of the ob-
ject.
Azimuth compasses are furnished with a mirror attached to
the sight vane, so that objects of considerable elevation may be
observed by reflection. This mirror should be perpendicular to
the plane passing through the eye-vane and the thread of the
sight-vane, to which the mirror is attached. This may be tested
by observations on a well-defined vertical line on shore.
13. For ordinary sea purposes a precision of 30', or even 1°, is
sufficient. But even this requires some attention to the several
points of the last article.
It is desirable that all compasses on board ship should be
tested — those for steering as well as those for more delicate use,
and their errors noted or adjusted, if of sufficient importance.
The error arising from the motion of the ship is less sensible
if the plane of the gimbals coincide with that of the card (when
the instrument is at rest), and pass through the point of the
pivot. Generally, however, the pivot is placed above the gimbals
and the card, since it is necessarily above the centre of gravity
of the needle and its attachments.
14. Magnetic needles, when not suspended, should be put away
in pairs, parallel, and with the north pole of one against the
south pole of the other, and separated, either in different boxes
or by a piece of cork or soft wood.
15. Small pieces of iron in the vicinity of a compass may pro-
duce a sensible deflection of the needle. Ships have often wan-
dered far from their intended course from a few nails or a knife
or other small iron article being carelessly placed in a binnacle.
If two compasses are near each other the north pole of one
THE COMPASS. 83
needle repels the north and attracts the south pole of the other.
They will then be deflected, and both in the same direction (and
equally if equal magnets), unless their direction from each other
is N.E.S. or W. (magnetic). In some intermediate direction,
near four points from the meridian, the deflection will be the
greatest.
Hence the comparison of two compasses placed side by side
is an imperfect test of their agreement or accuracy. When two
binnacles are used they should be at least 4| feet apart. The
disagreement of the compasses placed in them is, however, not
wholly due to their influence upon each other, but to other
sources of disturbance.
16. Electricity will disturb the needle. If the glass cover be
rubbed with dry silk, a delicate compass may be rendered for the
time useless. A strong electric current may weaken the magnet-
ism of a needle, or even reverse its poles. Lightning may pro-
duce such a change.
17. On shore, in particular locations, very marked deviations
of the needle are observed.
In ships, particularly those of iron, and in a less degree those
which have iron as a part of their cargo or armament, there are
peculiar causes of disturbance. The observations of Professor
Airy show that a part of the iron is permanently magnetic, or
nearly so, changing only very slowly, and that another portion is
magnetic by induction, and varies with its position with refer-
ence to the meridian and in different magnetic latitudes.
A ship may be regarded as two assemblages of magnets, one
permanent, the other variable ; and each acting upon the com-
pass in any particular position as a single magnet, whose force
is the resultant of the combined forces of all its parts. The dis-
turbance will be different in different parts of the ship. Obser-
vations have been instituted on board of some iron ships for
determining the position where the compass is least disturbed.
The standard compass on board some ships is placed between
the binnacles, and elevated so as to command a view of the
horizon, to affect less the steering compass, and to be farther
above the level of the disturbing magnets.
84
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
Fig. 26.
18. The deviation of the compass produced by these local
causes varies with the direction of the ship's head.
The resultant of the permanent magnet may be resolved into
two forces : one tending to draw * the N. pole of the needle
towards the ship's head, and having a maximum effect when the
ship heads E. or W. by compass ; the other tending to draw *
it towards the starboard side, and having a maximum effect when
the head is towards the N. or S. The variable magnet is regarded
by Prof. Airy as having its maximum effect when the ship heads
N.E., S.E., S.W., or N.W., by compass ; but this may not always
be the case.
19. To find the local deviation for different directions of the
ship's head, it is necessary as the ship turns round — either by
being swung round intentionally, or at sea in a calm, or with
light baffling winds, or at anchor by the tides — to observe the
bearing of some well-defined object as the head comes successively
to each point of the compass. The direc-
tion of the ship's head should be carefully
noted at the time of taking each bearing.
It is well to note it by the binnacle compass
as well as by that employed in the observa-
tions, f The compass must occupy the
same position during the whole series of ob-
servations, as the local deviation determined
is for that position only.
20. If the object be terrestrial and so dis-
tant, that the swinging of the ship produces
no sensible change in its actual direction
as seen from the position of the compass,
no other observations are necessary.
To ascertain what this distance must be
in a given case,
Let be the object, CC the extreme positions
of the compass, as the ship swings round the
point A.
* Or to repel it, in which, case the effect is regarded as negative,
f The heading by other compasses in different parts of the ship may also be
observed simultaneously.
THE COMPASS. 85
Put d = A C ;
D = A, the distance of the object ;
= A G, the parallactic angle.
We have
or, since is very small (in minutes),
= — A— = 3138- 4-
D sin 1 ; i>
If d is expressed in feet and D in sea miles,
0= | = 0' .5648 -4-
6087 D sin V D
whence
0' X -5648 d
Examples.
(1) d = 300 ft,, Z> = 6 miles ; then = 28'.
(2) d = 500 ft., and it is desirable that shall not exceed 30';
.t A u i il O' X -5648 X 500 aA •-,
then D must not be less than — ^ — /x o^ 94 sea miles.
If the bearings have been taken as the ship headed at the in-
tended points, that is at equal intervals round the compass, the
mean of the whole series will be the true compass bearing ; the
difference of this mean from each observed bearing will be the
local deviation for the corresponding direction of the ship's
head, and should be marked east, if this mean bearing is to the
right of the observed ; west if the mean bearing is to the left of
the observed. A table of the local deviations may be formed by
writing in one column the direction of the ship's head, and in
another the corresponding deviations.
Or, the " ship's head by compass " may be laid off on a straight
line at proper intervals as abscissas, and the corresponding
deviations as ordinates, and a curve drawn through the several
points thus determined.
Or the differences of some conveniently assumed bearing from
the observed bearings, may be laid off as ordinates, and a line
86 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
drawn parallel to the axis, and so as to divide the curve symme-
trically. The distance of the curve from this line at the several
points will be the local deviation.
The scale for the ordinates may be greater than that for the
abscissas.
This graphical method is more conveDient when the bearings
have not been taken as the ship headed at the intended points,
but at unequal intervals ; or if any have been omitted.
21. If the object be near, an observer may be stationed at it
who will make observations at the same times that the bearings
are taken on board the ship ; the instants being indicated by
some preconcerted signals made on board.
First. — With a theodolite carefully adjusted, and with its hori-
zontal limb clamped, he may direct the telescope towards the
position of the ship's compass, and read the instrument ; or
Second. — With a sextant he may measure the horizontal angles
between the ship's compass and some well-defined object, taking
into account, when necessary, the angles of elevation* of the two ;*
or
Third. — With a plane-table he may draw on paper lines in the
direction of the ship's compass, and measure the angles which
th*ey make with some lines drawn at pleasure ; or
Fourth. — With a good compass he may take reciprocal bear-
ings.
By any of these instruments the changes in the direction of
the ship's compass from the object (and as well, of the object
from the compass) are directly measured. These observations,
then, furnish the means of reducing the bearings observed on
board to what they would have been if made at a fixed position,
or upon an object whose direction was not varied.
* Let A and A' be the two angles of elevation, then the horizontal or azimuth
angle will be an angle of a spherical triangle, of which the two adjacent sides are
(90° — A) and (90° — A'), and the opposite side is the observed angular distance of
the two objects.
THE COMPASS. 87
Such fixed position, or rather its direction from the shore ob-
ject, is entirely arbitrary. That the reductions may be small,
and all applied in the same direction, and conveniently com-
puted, let the assumed zero line of direction be that for which the
shore instrument would read the smallest number of degrees
noted.*
The several readings or angles measured by the shore instru-
ments, diminished by this assumed number of degrees, are
respectively the parallactic reductions to be applied to the cor-
responding bearings observed with the compass on board the
ship. They are to be applied to the right when the zero line is
to the right of the actual line of direction ; to the left, when the
zero line is to the left of the actual line of direction.
This precept is easily demonstrated :
Ftg. 27.
S
Let O be the object.
C the position of the ship's compass.
C the position to which the bearings are to he reduced.
* If the readings are on different sides of a zero-point, the line of direction for
that zero-point is most convenient ; or the readings on one side may be increased
by 183° or 363°.
THEORETICAL NAVIGATION AND NAUTICAL ASTEONOMY.
Fig. 38.
S
CO is the line whose bearing is observed.
C 0, parallel to C , is the line whose bearing is required.
The reduction is the angle C0 = C OC.
In Fig. 27, 0C o is to the right of OC, and the reduction is to be
applied to the right.
In Fig. 28, C is to the left of C, and the reduction is to be
applied to the left.
This is evidently true, whatever may be the direction of the
meridian line NS.
The bearings observed on board the ship having been thus
reduced, they may be used as if they had been made on a very
distant object, and the local deviations computed and tabulated,
or plotted, as in Art. 20.*
* If a good chart of the harbor on a scale sufficiently large is available, the posi-
tion of the ship's compass at each observation may be found either by cross bear-
ings on two distant objects, or by measuring with a sextant the horizontal angles
between them ; and plotted upon the chart. The magnetic bearings of the shore
object may then be measured on the chart : the differences of these from the cor-
responding compass bearings will be the deviations.
In some harbors, poles or other well defined marks are placed so as to range with
a distant object on particular magnetic bearings, as each 5° or 1 point. With such
facilities, the observer on board has only to note the range, or between what two
ranges, and where between, in order to find the magnetic bearings with which to
compare his compass bearings.
THE COMPASS. 89
22. If a good compass is used at the shore station, and
its position may be regarded as free from any peculiar local
disturbance, the bearings observed with it may be assumed as
the true magnetic bearings of the ship's compass ; and the dif-
ferences of the opposites of these from the compass bearings ob-
served on board, may be taken as the deviations, and tabulated
or plotted.
The deviation is east, if the bearing by the shore compass is to the
right of the corresponding bearing by the ships compass ; west, if
the bearing by the shore compass is to the left of the corresponding
bearing by the ships compass.
This method is generally preferred, especially for iron ships,
where the local disturbance is large. It assumes that the shore
compass gives the magnetic bearing more truly than the mean
resultant of the observations made on board.
23. At sea the observations may be made on the sun, and to
better advantage when it is near the horizon. Its true azimuth
at each instant of observing its bearing by the compass must be
found either —
First, By means of simultaneous altitudes ; or,
Second, By noting the local apparent times ; or,
Third, By altitudes at equal intervals of 10 m., 20 m., or 30 m.,
and the computed azimuths interpolated for the time of the com-
pass observation.
The differences of the true azimuths from the compass bear-
ings will be the declination combined with the deviation.
The mean of a series of observations made at equal intervals
round the compass will be the decimation of the compass used ;
the differences from that mean, the deviations for the several
directions of the ship's head. A graphical process may be used
similar to that described in Art. 20, and advantageously when
the series of observations is not symmetrical, or any of them
have been omitted.
24. To allow time for the needle to settle, and for several bear-
90
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
ings to be taken, it is desirable to keep the ship's head steady
for a few minutes at each of the points selected.
Unless the deviations are great, observations on each of the
sixteen principal points are sufficient. Observations carefully
made on eight points may give a result sufficiently accurate for
the ordinary purposes of navigation. Very careful observations
on the four following points (by compass), N.E., S.E., S.W., and
N.W., have sometimes been used in default of others.
25. Prob. 1. — To find from an observed altitude the true
azimuth of a heavenly body at any place, the Greenwich
time of observation being known.
We have as in the figure (29) the
folio wing given :
P 31= p = 90° - d
P Z = 90° - L
Z31=z = 90 Q - h
to find the angle P Z 31.
From Spher. Trig. 164, 165, 166,
we have
sin h A= ,/ sin (s ~ b ) sin (8- c)
sin b sin c
v
cos
i^__ / sin s sin (s — a)
* sin b sin c
tani^= / sin ( s ~ b) sm ( s - c )
* sin s sin (s — a)
Using the formula for the sine, and these values of the sides,
viz. :
Co L, 90°- h, and 90° - d
we will have
siniZ= / cos \ {Co L + h + d) sin \ {Go L + h -d)
* cos L cos h
THE COMPASS. 91
or, if we put
Sf=\{Co L+h + d).
sin i Z = / cos & sin (ff - d) O )
* cos £ cos h
Using the formula for the cosine and the following values for
the sides,
90° - L, 90° - h and p
we have
cos i Z= ^
cos \ (L -\- h -\- p) cos I (L ■
j-h-p)
cos L cos h
or if we put
S" = \(L + h+p)
cos \
Z //cos S'' cos (S" — p)
* cos L cos h
(b.)
Using the formula for the tangent, with the
following
values
of the sides, viz. :
Co L, p and z,
and putting
S» = i(CoL+p + z)
we have
tan i Z= / sin &" — Go L ) sin (^ f " ~ z )
* sin S'" sin (S'" — p)
When
^is less than 90° use (a).
When
Z is greater than 90° use (b).
If greater accuracy is desired than is generally necessary at
sea, use (c).
The formula for the cos \ Z is generally used in case of the
sun in connection with A. M. and P. M. time sights. The data
required is the same as that for determining the hour angle. Z
is the true bearing or azimuth of the body, reckoned from the
north point of the horizon in north latitude, and from the south
point in south latitude. If reckoned as positive toward the east,
it must be negative toward the west.
92
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
It is generally best to use the supplement of Z when it is
greater than 90°, as the readings of azimuth compasses are from
0° to 90°. If, when the altitudes are observed, the bearings of
the heavenly body be taken by an azimuth compass, by compar-
ing the magnetic and true bearings we may obtain the variation
and deviation of the compass combined. It is marked E when the
true bearing is to the right of the magnetic bearing, otherwise TV.
26. Problem 2. — To find the amplitude and azimuth of a
heavenly "body -when in the horizon, the Greenwich time
being given.
In Fig. 30, the body M be-
ing in the true horizon, W M
is its amplitude, NM its azi-
muth.
In the triangle P N M t right
angle at N, we have.
cos P 31= cos M N cos P N
cos p = sin d = cos Z cos L.
If a = amplitude = 90° - Z
cos Z = sin a = sin d sec L
27. Problem 3.— To find the altitude and azimuth of a
heavenly body at a given place and time.
In Fig. 31 we have given
P Z=90° - L
Z P M = t, the hour angle of body M.
P M==d0° -d, to find
ZM= 90° — h, and
P Z M = Zthe azimuth.
cos t = cot " tan d
tan 0" = tan d sec t (a.)
0' = — L
THE COMPASS.
. , cos (0 — L) sm d
sin h = ^ -^
sm 0"
cos + #' ~ h') sin j (D - H'+h')
^ cos H' cos h
for cosine
cos i Z= / CQS i ( H> + U + D) cos j{H' + h-D)
* cos H' cos h'
If is in the true horizon, or its measured altitude equals the
dip, the right triangle MHO' gives
cos z = cos H / = cos D sec H?
2, thus determined after sextant measurement, may be applied
as before to the computed azimuth of M t to obtain the azimuth
of the terrestrial object.
CHAPTER VIII.
REFRACTION.— DIP.— PARALLAX, AND SEMIDIAMETER.
1. When a raj of light passes obliquely from one medium to
another of different density, it is bent or refracted from a recti-
linear course. The ray before it enters the second medium is
called the incident ray, afterwards the refracted ray. The differ-
ence between the directions of these two rays is the refraction.
The angle which the incident ray makes with a normal to the
surface of the refracting medium, when the incident ray meets
it, is called the angle of incidence. The angle which the refracted
ray makes with the normal is the angle of refraction. The differ-
ence between these two angles is therefore the refraction.
Fig. 33.
M
In the figure (33), if S A is an incident ray upon the surface
B B' of a refracting medium, A G the refracted ray, and M N A
normal to the surface at A, S A 31 is the angle of incidence, C
A N or S' A M is the angle of refraction, and 8 A S' the re-
fraction. An observer situated anywhere along the line A C will
receive the ray as if it had come directly to his eye without re-
REFRACTION. 97
fraction from S'. S' A C is called the apparent direction of the
ray.
2. It is shown in works upon optics, that refraction take place
according to the following general laws :
1st. When a ray of light falls upon a surface of any form,
which separates two media of different densities, the incident ray,
refracted ray, and normal to that surface at the point of inci-
dence, are in one plane.
2d. When a ray passes from a rarer to a denser medium, it is
refracted towards the normal ; and when a ray passes from a
denser to a rarer medium it is refracted from the normal.
3d. When the densities of the two media are constant, there
is a constant ratio between the sine of the angle of incidence
and the sine of the angle of refraction. If a ray passes from a
vacuum into a given medium, the number expressing this con-
stant ratio is called the index of refraction for that medium.
This index is always an improper fraction, being equal to the
sine of the angle of incidence divided by the sine of the angle
of refraction.
4dh. When a ray passes from one medium into another, the
sines of the angles of incidence and refraction are reciprocally
proportional to the indices of refraction of the two media.
3. Astronomical Refraction. — The rajs of light from a heaven-
ly body in coming to the observer must pass through our atmos-
phere. If the space between the star and the upper limit of the
atmosphere be regarded as a vacuum, or as filled with a medium
which exerts no sensible effect upon the direction of a ray of
light, the path of the ray until it reaches the atmosphere, will
be a straight line ; but upon entering the atmosphere will be
refracted toivards the normal to the surface of the atmosphere
at the point of incidence. The atmosphere not being of uniform
density, the ray is continually passing from a rarer to a denser
medium, so that its path becomes a curve concave towards the
earth.
98
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
The apparent direction of the ray will be that of a tangent to
the curve at the point where it reaches the eye. The difference
in direction of this tangent and the ray before it reaches the
atmosphere is called the astronomical refraction.
The ray (Fig. 34) from the star S entering the earth's atmos-
phere at B is bent into the curve A B.
Fig. 34.
The observer at A sees it in the direction of the tangent A S'.
From the first law given, the vertical plane of the observer which
contains the tangent A S' must also contain the normal E C and
the incident ray B S. Hence refraction increases the altitude
of a heavenly body without changing its azimuth.
The angle Z A S is the apparent zenith distance of the heavenly
body. The angle E B S is, the angle of refraction, and Z A S,
the apparent zenith distance, is the angle of refraction. If we
represent the refraction by r, we have
r = EB S - ED 8'
and from the third law
sin E B S
m,
smZAS'
a constant ratio for a given condition of the atmosphere and a
given position of A.
REFRACTION. 99
4. To find the refraction r.
In the figure, let
z = Z A S', the apparent zenith distance,
r = E B S - E I) S', the refraction,
u = Z C E,
Then
EDS'=ADC=ZAS' — Z CE = z- u
EB S=Z -u-\-r
sin E B S sin (z — u 4- r)
= : ! £- = m
sin Z A S' sin Z
sin [z — u — r)] = m sin z.
sin [z — (it — r) ] -|- sin z m -|- 1
sin [z — (u — r)] — sin 2; m - 1
which by (109) Plane Trigonometry becomes
tan \ [z — (w — r) -|- z] m -f- 1
tan ^ [z — (u - r) — z] m — 1
which reduces to
tan [z — J (u — r)~\ m -\- 1
tan ^ — (u — r) m — 1
hence
tan i (u — r) = ^-^ — tan [z — 4 (w< — r)] * '
In this u and r are both unknown, but are both small angles,
being when the zenith distance is 0, and increasing with the
zenith distance. Assuming that they vary proportionately, and
that
u
* = q
r
and substituting in (a) we have
tan i (q - 1) r = 1^- tan J [z - J fa - 1) r]
as J (g — 1) r is very small we may put
tan J (q — 1) r = | (g — 1) r sin 1"
100 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
and have
"whence
Putting
i (q - 1) r sin 1" = —25 tan [z - J (? -1) r]
1-f-m
r = . ~ m tan [z - \ (q - 1) r]
? - 1 sm 1" l-\-m z
2 1 - m
and
we have
(5 — 1) sin 1" 1-f-m
P = i(2-1)
r = u tan (2 — p ?*)
which is known as Bradley's Formula.
If at two given zenith distances z' and z" the refractions r' and
r" are formed by observations in a mean state of the atmosphere,
then we have the two equations
r' = n tan (2' — p r ),
r" = n tan (z n — ^) r") ;
and the two unknown quantities n and p may be found.
By comparing observations in this way at various zenith dis-
tances, the values of n and p are found to be very nearly the
same; so that the assumption made is found to be nearly
correct.
The values of n and p used in the computation of Table XII.
(Bowd.) are
n= 57".036 andp = 3
These values correspond to the height of
the barometer, b = 29.6 inches,
the thermometer, t = 50° Fahr.
5. Refraction in different conditions of the atmosphere is
nearly proportional to the density of the air ; and this density,
the temperature being constant in proportional to its elasticity ;
that is, to the height of the barometer. Then, if
A is the noted height of the barometer,
r, the refraction of Tab. XII.
A r, the barometer correction
REFRACTION. 101
- A r b
r — 29.6
! A &
r + Jir== -29T6 r
A b
A r
(i->>
i
A p — 29.6
z? r = r
29.6
The correction for the barometer in Table XXXVI. (Bowd.) is
computed from the formulae.
The elastic force being constant, the densityjncreases by ¥
part for each degree of depression of the thermometer (Fahr.)
Hence, if
A / r = the correction for the thermometer,
t = the noted temperature
A'r= 50 °~ t (r+A'r)
400 v ;
400" A' r = (50° - t{ (r + A' r) ,
= 50° r+ 50° A' r - tr — Art
350-° A r' -f A' r t = 50° r — * r
50° -*
J' r
350° -M
by which the correction for thermometer, Tab. XXXYI. (Bowd.),
is computed.
7. To find the radius of curvature of the path of a ray in
the earth's atmosphere.
By the radius of curvature, is meant the radius of a circle
which most nearly coincides with the curve.
If in Fig. 35 we consider the curvature to be uniform from B
to A, the problem is reduced to finding the radius of this arc.
Let C' be the centre of the arc A B,
R' = C A y the radius of curvature,
R = G A, the radius of the earth.
102
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY,
S B' and S A' are tangents to the curve at the points B and A
respectively. The angle between the radii C'A and C' B is equal
to the angle made by these tangents with each other, which is
the refraction r. As A B is a very small arc, we may put
A B — R sin r
&n,d nearly
A D = A B = R' sin r
Fig. 35.
In the
triangle ADC
R / sin r
R '
sin
w
~ sin (z
-«)
whence
sm (u - r)
sin tt
sm r
and as u
and r are small
R' —
R
sin z
r
But by preceding work
— =q andp
r
=*(«-
1) = 3
whence
3 = 7,
u — 7 r
DIP.
103
so that
When
IB
sin z
0, or the star is at the zenith,
When
z = 90°, or the star is in the horizon,
B'=7 B
This is for a mean condition of the atmosphere for which the
values of p and q were obtained. The curve is greatly varied for
extraordinary states of the atmosphere.
We have seen that infraction increases the apparent altitude
of a heavenly body. As a correction, therefore, to an observed
altitude, to obtain true altitudes, it is always subtractive.
DIP.
8. A plane, tangent to the earth's surface, is called the true
horizon. If an observer be elevated above the plane, the visual
ray will be tangent at some other point on the earth's surface.
If it were not for the effect of refraction, the angle between the
visual ray and the true horizon would be a correction to be
applied to an observed altitude to obtain true altitudes. The
effect of refraction is to determine this angle.
Fig. 36.
In Fig. 36, the most h
distant point visible from
A is H'' where the visual
ray A H' is tangent to
the earth's surface. The
apparent direction of H"
iaAH'. EA .ff" is called
dip of the horizon. It in-
creases in apparent alti-
tude, and as a correction
is subtractive.
104 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
9. To find the dip of the horizon.
Let G be the centre of the earth
C the centre of the arc H" A
H" G C are in the same straight line, since the arcs
H" A and H" B are tangent to each other at H"
G A and C A are'perpendicular respectively to A JET and A H",
hence
HAH'=GAG' = AH, the dip.
If 7i = the height B A afeove the sea level,
CA = B+h
C A — 7 B, the radins of curvature of the arc H" A
GG' = 6 B.
In the triangle G A G\ by Plane Trigonometry, we have
sin i A H
= y(6 B - J h) (i h)
7 B(B + h)
h is so small that it may be omitted when additive to B, and we
have
As
i
2
A if is a
AH=
small angle
\ A JTsin 1" =
_ 2 /3/1 _
sin 1" V 7 ^
'3 ft
7i2
/
2
3/i
IB
, /
3
sin 1
IB
Vh
sm ± ■ k 7 ^ sin 1' " 7ii
- — T „ J -A=r is a constant and may be computed. Its value
sm 1 V 7 B
will depend upon the value of R used. Bowditch uses the value
in Yince's Astronomy. The logarithm of the constant used by
him is 1.7712711.
log A H= 1.7712711 + i log h
h is expressed in feet, and A H found in seconds.
10. To find the distance of an object of known height just
visible in the horizon.
In figure of previous Art.
h = B A. the height of A
d = H" A, the distance of A.
As this arc is small, we shall have
d = H" G' A sin 1" X OA = 7 B X H" C' A sin 1" (a.)
PARALLAX.
105
In the triangle C C A, we have
or nearly
sin \ H" C A = J hh{X +hW*
J JB" C'^sinl" = J_ h
84 R
H" G' A sin l"=y_J}L
which substituted in (a) gives
d=7B
21 E
V
/ *
21 i*
■■Vl/ZRh
If c?, A and i2 are expressed in feet, in geographical miles
1
d =
6087
V 7/3 i* A
Table X., Bowd., is computed for d in statute miles. It would
be more useful to the Navigator if it were in geographical miles.
PARALLAX.
11. Change in direction due to change of position is called
Parallax. In astronomical observations, the observer is on the
surface of the earth. It is convenient to reduce them to the
earth's centre. The change in direction of a heavenly body, as
viewed from the earth's surface and from its centre, is called
geocentric parallax. Geocentric parallax may be denned as the
angle at the body subtended by that radius of the earth which
passes through the observer's position.
In Fig. 37, the geocentric par-
allax of the body S will be
S=ZAS-ZC S
This is regarding the earth as
a sphere, which is sufficiently
accurate for all nautical prob-
lems except the complete reduc-
tion of lunar distances, when the
spheroidal form of the earth
must be taken into consideration.
Fig. 37.
106 THEORETICAL NAVIGATION AND NAUTICAL ASTEONOMY.
12. To find the parallax of a body in the horizon H.
Let n = the parallax, called in this case the horizontal parallax,
d, the distance of the body from the centre of the earth,
then
R
Sin 7T = —
a
13. To find parallax of a heavenly body for a given alti-
tude.
In the triangle C S A, letting p = the parallax, we have
R sin z
sm»=
d
Substituting in this the value of the horizontal parallax, gives
sin p = sin tt sin z
or nearly, as n and p are small angles,
p = it sin z
p = 7T COS h
The horizontal parallax n is given in the Nautical Almanac
i'or the sun, moon, and planets. From the figure it is evidently
the semidiameter of the earth as viewed from the body. As the
equatorial semidiameter of the earth is larger than any other, so
will be the equatorial horizontal parallax. This the Nautical
Almanac gives for the moon. For refined observations this will
Jiave to be reduced for the latitude of the observer.
Tables X., A., and XIV. are computed by the above formulae.
Table XIX., Bowd., contains a quantity to be subtracted from
59' 42", the remainder being the combined corrections of parallax
and refraction for the moon's altitude.
APPARENT SEMIDIAMETERS.
14. The apparent semidiameter of a body is the angle sub-
tended by its radius at the place of the observer. Observations
of the sun and moon with sextant are made by bringing either
the upper or lower limb in contact with the sea horizon, or (in
using the artificial horizon) by bringing two opposite limbs of
direct and reflected limbs together. The altitude of the centre
APPARENT SEMIDIAMETEES.
107
of the body being required, the angular semidiameter of the
heavenly body must be applied plus or minus, according to the
limb observed.
15. To find the apparent semidiameter of a heavenly
body.
Fig. 38.
In Fig. 38,
Let if be the body,
d == G M, its distance from
earth's centre,
d' = A M, its distance from A
S = MC B, its apparent semi-
diameter as viewed from C
S' = MA B' } its apparent semi-
diameter as viewed from .4
B = G A, the earth's radius
r = M B = M B', the linear
radius of the body.
For finding 8, the right triangle C B M gives
sin b = — -
(a.)
Were the body 31 in the horizon of A, its distance from A and
C would be sensibly the same, so the angle S is called the hori-
zontal semidiameter.
From Art. 12, we have for the horizontal parallax
B 7 B
sm tt = — or d=
a sm 7r
which substituted in (a) gives
sin S = — - sin n
or
s=^«
108 THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
v
-p- is constant for any particular body, and representing it by
ra, we have
log S = log m -\- log n
The Nautical Almanac gives the semidiameters of the sun,
moon and planets.
16. To find S', the apparent semidiameter as seen from A, the
right triangle A B' M gives
sin 8- = - r - (»•)
d'
In the triangle G MA
sin MAG d
sin M G A = ~d
If
h=90° - ZAM, the apparent,
A'== 90° - Z G M, the true altitude of M.
cos h d
cos A' d'
d'=d eosh '
cos /l
which substituted in (b) gives
r cos h
sin #'
6? cos h'
T
substituting for -y its value from (a) we have
0l • cr COS h
sin a = sm 8 —
cos /i
CY . „ cos A
COS /l'
gives an approximate value for S' t when S and A are known.
As h < h', cos ^ > cos h' and consequently S ! > S, or the semi-
diameter increases with the altitude of the body. This excess
is called the augmentation, and is only sensible in the case of
the moon.
17. To find the augmentation of the moon's semidiameter.
COS li
APPARENT SEMIDIAMETERS. 109
which by Plane Trigonometry (108) becomes
A s = ^ 2 sin i (h' -f h) sin \ (h' - h)
cos h'
ti — h = p, the parallax, and being small
2 sin \ (h' — h) = 2 sin \ p = p sin 1" = 7r cos h sin 1"
and as A S is small, we may take ^ Qi + 7i) = h,
and cos A for cos li' ; and then
A S = S 7T sin 1" sin h
and as
For the moon
A S = — - 7r 2 sin 1" sin /i.
It
: 0.2729: then
i2
J S== .000001323 7T 2 sin 7i.
Using the mean value of tt = 5T 20".
^ 5 =15". 65 sin/i.
Tab. XY. (Bowd.) is computed'from a formula nearly like this.
CHAPTER IX.
SEXTANT.— ARTIFICIAL HORIZON.
1. The optical principle of the construction of the sextant is
the following : " If a raj of light suffers two successive reflec-
tions in the same plane by two plane mirrors, the angle between
the first and last directions of the raj is equal to twice the angle
of the two mirrors."
Fig. 39.
In Fig. 39, let M and m be the two mirrors. The direct and
reflected rajs are always found in the same plane — called the
plane of reflection. In order that the last direction of the ray
after suffering two reflections shall be in the same plane as the
first direction, the plane of reflection must be perpendicular to
both mirrors. In the diagram the plane of the paper is the
plane of reflection. The shaded lines M and m are the inter-
sections of this plane with the mirrors. Let 8 31 he the direct
raj falling upon the mirror M (lying in the direction M I). Let
M m be the direction of the ray after the first reflection, and m
E its direction after the second reflection. Draw M B parallel
SEXTANT. Ill
to m E, M P perpendicular to M C, and M p perpendicular to
the mirror m. The angle S M B is the angle between the rays
S 31 and m E. The angle P Mp, being obviously equal to MC m,
is the angle between the mirrors. We have, then, to prove
SMB=2PMp.
If m, draw the perpendicular mn, Mmn = p Mm, is the angle
of incidence of the ray Mm on the mirror m ; nm E ' = p M B
is the angle of reflection of the same ray. The angle of incidence
and the angle of reflection being equal, we have
pMm = pMB = PMp-\- P MB
On the same principle we have
P Mm = P MS = S MB + P MB
Taking the difference of these two equations we have
P 31 p = SMB - P Mp
hence
SMB = 2P Mp = 2 31 Cm.
2. This principle is applied in the sextant as follows : The
mirror Mi& attached to a bar MI, called the index bar, which
revolves upon a pivot at M in the centre of a graduated arc N.
The mirror 31 is firmly fixed at right angles to the plane of this
arc. The mirror 31 is called the index glass ; the mirror m the hori-
zon glass. Place the index bar in the position 31 so that the
two glasses are parallel. In this position an incident ray from
an object B will be reflected first to n o and then in the direc-
tion m E. The first and last directions of the ray will be paral-
lel. If, then, the object is so distant that two rays from it, B 31
and B' m, falling upon the two mirrors are sensibly parallel, the
the observer at E will receive the direct and reflected ray at the
same time, or will see two images of the same object in coinci-
dence,. Commence the graduation of the limb at 0, marking it
zero. Move the index bar to the position M 7, so that a ray
from the object /Sis reflected in the direction m E ; the observer
E sees the object B and S in coincidence, and the angle S 31 B
112
THEORETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
between the two objects is equal to twice the angle through
which the index glass has been moved.
As the centre of rotation is at M, this angle will be twice the
angle M I.
If, now, the arc N be graduated, and the marking of the
graduation doubled, we can read at once the angle SMB. The
angles are read to a nicety by means of a vernier, on the index
bar at I.
THE VEKNIER.
3. Let M N, Fig. 40, be a portion of the limit of a circle, C B
the arm which revolves with the index glass about the centre of
the circle. At the end of this arm, construct a b graduated into
a number of divisions which occupy the space of n 1 of the limb.
The first line a is the zero of the vernier, and the reading is to
be determined from the position of this zero on the limb M N.
THE VEENIEK. 113
If we put
x = the value of a division of the limb.
y = the value of a division of the vernier, we will have
(n — I) x = n y
hence
n - 1
y = ■ x
J n
and
1
x — y = x
n
The difference x — y is called the 7eas£ count of the vernier,
which is, therefore, th of a division of the limb. If the zero
n
of the vernier falls between the two graduations P and P -f 1,
the whole reading is P plus the fraction from P to a. To meas-
ure this fraction, m, observe that if the ?7ith division of the ver-
nier is in coincidence with a division of the limb, the fraction is
m X ( x — y) or — x - l n the figure the vernier is divided into
ten equal parts, equal to nine divisions of the limb, and if the 4th
division is in coincidence, the whole reading is P 4- — — x ; and if
4
x = 10', then the whole reading is P -f- — — - . 10 = P -j- 4'. Sup-
pose that P is the division of the limb marked 35° 40', then the
reading is 35° 44'. The least count in this case is 1'. The frac-
tion is obtained in practice bj the numbers placed above (or
)elow) the divisions on the vernier.
Sextants generally read to 10" ; in other words the least count
10 '. From the above it will be seen that for this 60 divisions
>f the vernier equal 59 divisions of the limb. Verniers are
line times constructed (seldom for sextants) with the divisions
>n the vernier greater than those upon the limb. The only dif-
ference will be that the reading of the vernier will be in a direc-
ion opposite to that of the reading of the limb.
For the adjustments of the sextant see Chauvenet's Astronomy,
>p. 95 to 99, inclusive, or Bowd., pp. 133 -136. Circles of reflec-
tion and octants are similar in construction to the sextant.
114 THEOKETICAL NAVIGATION AND NAUTICAL ASTRONOMY.
4. The artificial horizon is a small basin partially filled with
mercury, over which is placed a roof consisting of two plates of
glass fitted in a frame at right angles to each other. The roof is
to protect the surface of the mercury from wind and dust. The
best form have a wooden basin fitting inside of a metallic one.
A small funnel screws into a hole at one end of the wooden basin ;
a channel underneath conveys the mercury to the centre of the
basin. The funnel acts as a strainer, retaining a greaber portion
of the oxide. If the mercury be amalgamated with tin, all impuri-
ties will float upon the surface, and may be removed by passing
lightly over the surface the edge of a piece of paper.
If, in Fig. 41, B B' be the horizontal
surface of the mercury, S A a ray of
light from a heavenly body incident
upon the surface at A, it will appear to
an observer at E in the direction S' E.
The angular depression B A S' below
the horizontal plane is equal to S A B,
the altitude above this plane. If, then,
S E is a direct ray from the heavenly
body parallel to S A, and the observer
at E with a sextant makes the direct
image S and the reflected image S'
coincide, the reading of the sextant
will be S E S' = S A S' = 2 SAB.
The surface of the mercury being in the plane of the true
horizon, the altitude obtained has only to be corrected for
parallax and refraction, and in case the limit of a body has been
observed, for semidiameter. The index correction of the sex-
tant, as is obvious, must be applied to the reading of the sextant.
Parallax and refraction to the altitude of the body, and semi-
diameter to the altitude or diameter to the reading of the sextant.
The glasses in the roof should be made of plate glass with paral-
lel faces. To eliminate any error that may arise from a pris-
matic form of the glasses, observe one half of a set of altitudes
with one end of the roof towards the observer, and one half with
the other end towards the observer. In the case of equal alti-
tudes, keep the same end towards the observer.
Naval B
AVAL JDOOKS.
A
TREATISE ON ORDNANCE AND NAVAL GUNNERY,
Compiled and arranged as a Text-Book for the U. S. Naval Acad-
emy, by Commander Edward Simpson, U. S. N. Fourth edition,
revised and enlarged, i vol. , 8vo. Plates and cuts. Cloth. $5.
"As the compiler has charge of the instruction in Naval Gunnery at the Naval Academy,
hi* work, in the compilation of which he has consulted a large numher of eminent authorities,
is piobably well suited for the purpose designed by it— namely, the circulation of infor-
mation which many officers, owing to constant service afloat, may not have been able to col-
lect. In simple and plain language it gives instruction as to cannon, gun-carriages, sun-
pov*der. projectiles, fuses, locks and primers ; the theory of pointing guns, rifles, the practice
of gunnery, and a great variety of other similar matters, interesting to fighting men on sea and
land.''— Washington Daily Globe.
G
UNNERY CATECHISM. As applied to the service of Naval Ord~
nance. Adapted to the latest Official Regulations, and approved
by the Bureau of Ordnance, Navy Department. By J. D. Brandt,
formerly of the U. S. Navy. Revised edition. 1 vol., iSmo.
Cloth. $1.50.
" Bureau of Ordnance— Navy DEPARTafENT, |
Washington City, July 30, lSt>4. )
"Mr. J. D. Brandt,—
" Sir:— Your ' Cathechism op Gunnery, as applied to the service of Naval Ordnance,' having
been submitted to the examination of ordnance officers, and favorably recommended by thein,
Is approved by this Bureau. I am, Sir, your obedient servant,
"H. A. WISE, Chief of Bureau. 1 '
ORDNANCE INSTRUCTIONS FOR THE UNITED STATES
NAVY. Part I. Relating to the Preparation of Vessels of War
for Battle, and to the Duties of Officers and others when at Quarters.
Part II. The Equipment and Manoeuvre of Boats, and Exercise of
Howitzers. Part III. Ordnance and Ordnance Stores. Published
by order of the Navy Department. I vol., 8vo. Clcth. With
plates. $5.
THE NAVAL HOWITZER ASHORE. By Foxhall A. Parke*,
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Approved by the Navy Department.
r pHE NAVAL HOWITZER AFLOAT. By Foxhvll A. Parker,
1 Captain U. S. Navy. 1 vol., 8vo. With plates. Cloth. $4.00.
Approved by the Navy Department.
20 D Van JSFostrancCs Publications.
RUNNER Y INSTRUCTIONS. Simplified for the Volunteer Officers
v-J" of the U. S. Navy, with hints to Executive and other Officers. By
Lieutenant Edward Barrett, U. S. N. , Instructor of Gunnery,
Navy Yard, Brooklyn, i vol., i2mo. Cloth. $1.25.
'It is a thorough work, treating plainly on its subject, and contains also some valuable hintt
Lo executive officers. No officer in the volunteer navy should be without a copy."— Boston
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CALCULATED TABLES OF RANGES FOR NAVY AND
ARMY GUNS. With a Method of finding the Distance of an
Object at Sea. By Lieutenant W. P. Buckxer, U. S. N. i vol.,
8vo. Cloth. $1.50.
NAVAL LIGHT ARTILLERY. Instructions for Naval Light Ar-
tillery, afloat and ashore, prepared and arranged for the U. S.
Naval Academy, by Lieutenant W. H. Parker, U. S. N. Third
edition, revised by Lieut. S. B. Luce, U. S. N., Assistant Instructor
of Gunnery and Tactics at the United States Naval Academy. 1
vol., 8vo. Cloth. With 22 plates. $3.
ELEMENTARY INSTRUCTION IN NAVAL ORDNANCE AND
GUNNERY. By James H. Ward, Commander U. S. Navy,
Author of "Naval Tactics," and " Steam for the Million." New
Edition, revised and enlarged. 8vo. Cloth. $2.
" It conveys an amount of information in the same space to be found nowhere else, and given
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iV. Y. Evening Post.
MANUAL OF NAVAL TACTICS ; Together with a Brief Critical
Analysis of the principal Modern Naval Battles. By James H.
Ward, Commander U. S. N. With an Appendix, being an extract
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8vo. Cloth. $3.
NAVIGATION AND NAUTICAL ASTRONOMY. Prepared for
the use of the U. S. Naval Academy. By Prof. J. H. C. Coffin,
Fourth edition, enlarged. 1 vol., 12 mo, Cloth. $3.50.
SQUADRON TACTICS UNDER. STEAM. By Foxhall A.
Parker, Captain U. S. Navy. Published by authority of the Navy
Department. 1 vol., 8vo. With numerous plates. Cloth. $5.
JEFFERS. Treatise on Nautical Surveying. By Capt. W. N. Jeffers,
U. S. N. Illustrated. 8vo. Cloth. $5.
OSBON'S HAND-BOOK OF THE UNITED STATES NAVY.
Being a compilation of all the principal events in the history of
every vessel of the United States Navy, from April, 1S61, to May,
1864. Compiled and arranged by B. S. Osbon. i vol., 12 mo
Cloth. $2. so.
Naval Boohs. 21
HISTORY OF THE UNITED STATES NAVAL ACADEMY.
With Biographical Sketches, and the names of all the Superin-
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some of the earliest votes by Congress, of Thanks, Medals, and
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i vol., i2mo. Cloth. Plates. $i.
NAVAL DUTIES AND DISCIPLINE : With the Policy and Prin-
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life. Everything is st ited as tersely as possible, and this is one of the advantages of the book,
considering that the experience and professional knowledge of twenty-five years' service,
are crowded somewhere into its pages."— Army and Navy Journal.
MANUAL OF THE BOAT EXERCISE at the U. S. Naval Acad-
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Naval Tactics. iSmo. Flexible Cloth. 75c.
MANUAL OF INTERNAL RULES AND REGULATIONS
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hands of every midshipman, and officers of high rank in the navy would often find it a useful
x>mpanion."— Boston Journal.
UCE'S SEAMANSHIP : Compiled from various authorities, and
Illustrated with numerous Original and Selected Designs. For
the use of the United States Naval Academy. By S. B. Luce, Lieu-
tenant-Commander U. S. N. In two parts. Fourth edition, revised
and improved. 1 vol., crown octavo. Half Roan. $7.50.
LESSONS AND PRACTICAL NOTES ON STEAM. The Steam-
Engine, Propellers, &c, &c, for Young Marine Engineers, Stu-
dents, and others. By the late W. R. King, U. S. N. Revised by
Chief-Engineer J. W. King, U. S. Navy. Twelfth edition, enlarged
Svo. Cloth. $2.
L
22 I). Van 2sustraia£$ Publications.
S
TEAM FOR THE MILLION. A Popular Treatise on Steam and
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H. Ward, Commander U. S. Navy. New and revised edition, l
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I^HE STEAM-ENGINE INDICATOR, and the Improved Mano-
meter Steam and Vacuum Gauges : Their Utility and Application.
By Paul Stillman. New edition, i vol., 12 mo. Flexible cloth.
$1.
C CREW PROPULSION. Notes on Screw Propulsion, its Rise and
*>3 History. By Capt. W. H. Walker, U. S. Navy. 1 vol., 8vo.
Cloth. 75 cents.
POOR'S METHOD OF COMPARING THE LINES AND
DRAUGHTING VESSELS PROPELLED BY SAIL OR
STEAM, including a Chapter on Laying off on the Mould-Loft
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illustrations. Cloth. $5.
HARWOOD'S LAW AND PRACTICE OF UNITED STATES
NAVAL COURTS-MARTIAL. By A. A. Harwood, U. S. N.
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binding. $4.
pLEET TACTICS UNDER STEAM. By Foxkall A. Parker
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CTUART'S NAVAL DRY DOCKS OF THE UNITED STATES.
^ By Gen'l C. B. Stuart. Illustrated with twenty-four fine engravings
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HTREATISE ON THE MARINE BOILERS OF THE UNITED
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DEAD RECKONING; Or, Day's Work. Bv Edward Barrett,
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SUBMARINE WARFARE, DEFENSIVE AND OFFENSIVE. Com-
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its employment in War, and results of its use. Descriptions of th/s
various forms of Torpedoes, Submarine Batteries and Torpedo Boa's
actually used in War. Methods of ignition by Machinery, Cc:
Fuzes, and Electricity, and a full account of experiments made tc
determine the Explosive Force of Gunpowder under Water. Also a
discussion of the offensive Torpedo system, its effect upon Iron-Clad
Ship systems, and influence upon Future Naval Wars. By Lieut.-
Commander John S. Barnes, U. S. N. With illustrations. 1 vol,
8vo. Clo. $5.00.
Scientific Books.
| 'RANCIS' (J. B.) Hydraulic Experiments. Lowell Hydraulic Ex-
i- periments — being a Selection from Experiments on Hydraulic
Motors, on the Flow of Water over Weirs, and in Open Canals ot
Uniform Rectangular Section, made at Lowell, Mass. By J. B.
Francis, Civil Engineer. Second edition, revised and enlarged, in-
cluding many New Experiments on Gauging Water in Open Canals,,
and on the Flow through Submerged Orifices and Diverging Tubes.
With 23 copperplates, beautifully engraved, and about 100 new
pages of text. 1 vol., 4to. Cloth. $15.
Most of the practical riles given in the hooks on hydraulics have heen determined from ex
periments made in other countries, with insufficient apparatus, and on such a minute scale, that
In applying them to the large operations arising in practice in this country, the engineer cannot
but doubt their reliable applicability. The parties controlling the great water-power furnished
by the Merrimack River at Lowell, Massachusetts, felt this so keenly, that they have deemed it
necessary, at great expense, to determine anew some of the most important rules for gauging
the flow of large streams of water, and for this purpose have caused to be made, with great care,
several series of experiments on a large scale, a selection from which are minutely detailed in
this volume.
The work is divided into two parts— Part I., on hydraulic motors, includes ninety-two exper.
ments on an improved. Fourneyron Turbine Water- Wheel, of about two hundred horse-power,
with rules and tables for the construction of similar motors :— Thirteen experiments on a inodei
of a centre-vent water-wheel of the most simple design, and thirty-nine experiments on a centre
vent water-wheel of about two hundred and thirty horse-power.
Part II. includes seventy-four experiments made for the purpose of determining the form ot
the formula for computing the flow of water over weirs ; nine experiments on the effect of back-
water on the flow over weirs ; eighty-eight experiments made for the purpose of determining
the formula for computing the flow over weirs of regular or standard forms, with several tables
of comparisons of the new formula with the results obtained by lormer experimenters ; five ex-
periments on the flow over a dam in v/hich the crest was of the same form as that built by the
Essex Company across the Merrimack River at Lawrence, Massachusetts ; twenty-one experi-
ments on the effect of observing the depths of water on a weir at different distances from tha
weir; an extensive series of experiments made for the purpose of determining rules for gaug-
ing streams of water in open canals, with tf Mes for facilitating the same ; and one hundred and
»ne experiments on the discharge of water through submerged orifices and diverging tubes, the
jphole being mlly illustrated by twenty-three double plates engraved on copper.
In 1S55 the proprietors of the Locks and Canals on Merrimack River, at whose expense most
of the experiments were made, being willing that the public should share the benefits of the
scientific opeiations promoted by them, consented to the publication of the first edition of this
work, which contained a selection of the most important hydraulic experiments made at Lowell
up to that time. In this second edition the principal hydraulic experiments made there, subse-
luent to 1S55, have been added, including the important series above mentioned, for determin-
ing rules for the gauging the flow of water in open canals, and the interesting series on the flo*
through a submwged Venturi's tube, in which a larger flow was obtained than any we find re-
corded.
FRANCIS (J. B.) On the Strength of Cast-iron Pillars, with Tables
for the use of Engineers, Architects, and Builders. By James B d
Francis, Livil Engineer. 1 vol., 8vo. Cloth. $2.
H
D. Van Nostrantfs Publications.
OLLEY'S RAILWAY PRACTICE. American and European
Railway Practice, in the Economical Generation of Steam, in-
cluding the materials and construction of Coal-burning Boilers,
Combustion, the Variable Blast, Vaporization, Circulation, Super-
heating, Supplying and Heating Feed-water, &c, and the adaptation
of Wood and Coke-burning Engines to Coal-burning ; and in Per-
manent Way, including Road-bed, Sleepers, Rails, Joint Fastenings,
Street Railways, &c, &c. By Alexander L. Holley, B. P. With
77 lithographed plates, i vol., folio. Cloth. $12.
44 This is an elaborate treatise by one of our ablest civil engineers, on the construction and nw
of locomotives, with a few chapters on the building of Railroads. * * * All these subjects
are treated by the author, who is a first-class railroad engineer, in both an intelligent and intelli-
gible manner. The facts and ideas are well arranged, and presented in a clear and simple etyle,
accompanied by beautiful engravings, and we presume the work will be regarded as indispens-
able by all who are interested in a knowledge of the construction of railroads and rolling stock,
or the working of locomotives."— Scientific American.
HENRICI (OLAUS). Skeleton Structures, especially in their Appli-
cation to the Building of Steel and Iron Bridges. By Olaus
Henrici. With folding plates and diagrams. 1 vol., 8vo. Cloth.
$3.
WHILDEN (J. K.) On the Strength of Materials used in En-
gineering Construction. By J. K. Whilden. i vol., i2mo.
Cloth. $2.
44 We find in this work tables of the tensile strength of timber, metals, stones, wire, rope,
hempen cable, strength of thin cylinders of cast-iron ; modulus of elasticity, strength of thick
cylinders, as cannon, &c, effects of reheating, &c, resistance of timber, metals, and stone to
crushing; experiments on brick-work; strength of pillars; collapse of tube; experiments on
punching and shearing ; the transverse strength of materials ; beams of uniform strength ; table
of coefficients of timber, stone, and iron ; relative strength of weight in cast-iron, transverge
strength of alloys ; experiments on wrought and cast-iron beams : lattice girders, trussed cast-
iron girders ; deflection of beams ; torsional strength and torsional elasticity."— American Ar-
tisan.
C AMPIN (F.) On the Construction of Iron Roofs. A Theoretical
and Practical Treatise. By Francis Campin. With wood-cuts and
plates of Roofs lately executed. Large 8vo. Cloth. $3.
BROOKLYN WATER-WORKS AND SEWERS. Containing a
Descriptive Account of the Construction of the Works, and also
Reports on the Brooklyn, Hartford, Belleville, and Cambridge
Pumping Engines. Prepared and printed by order of xe Board of
Water Commissioners. With illustrations. 1 vol., folio. Cloth.
$15.
ROEBLING (J. A.) Long and Short Span Railway Bridges. By
John A. Roebling, C. E. Illustrated with large copperplate en-
gravings of plans and views. Imperial folio, cloth. §25.
CLARKE (T. C. ) Description of the Iron Railway Bridge across
the Mississippi River at Q.uincy, Illinois. By Thomas Curt?k
• Clarke, Chief Engineer. Illustrated with numerous lithographed
plans. 1 vol., 4to. Cloth. $7.50.
Scientific Boohs.
WILLIAMSON (R. S.) On the Use of the Barometer on Survey*
and Reconnaissances. Part I. Meteorology in its Connection
with Hypsometry. Part II. Barometric Hypsometry. By R. S.
Willia.vsox, Bvt. Lieut. -Col. U. S. A., Major Corps of Engineers.
With Illustrative Tables and Engravings. Paper No. 15, Professional
Papers, Corps of Engineers. 1 vol., 4to. Cloth. $15.
" San Francisco, Cal., Feb. 27, 1887.
'Gen. A. A. Humphreys, Chief of Engineers, XJ. S. Army:
" General— I have the honor to submit to you, in the following pages, the results of my in-
vestigations in meteorology and hypsometry, made with the view of ascertaining how far tha
barometer can be used as a reliable instrument for determining altitudes on extended lines oi
•urvey and reconnaissances. These investigations have occupied the leisure permitted me from
my professional duties during the last ten years, and I hope the results will be deemed of suffix
jcient value to have a place assigned them among the printed professional papers of the United
States Corps of Engineers. Very respectfully, your obedient servant,
"R. S. WILLIAMSON,
"Bvt. Lt.-Col. TJ. S. A., Major Corps of TJ. S. Engineers. rt
TUNNER (P.) A Treatise on Roll-Turning for the Manufacture ot
Iron. By Peter Tunner. Translated and adapted. By John B.
Pearse, of the Pennsvlvania Steel Works. With numerous engrav-
ings and wood-cuts. 1 vol., 8vc, with 1 vol. folio of plates. Cloth. $10
SHAFFNER (T. P.) Telegraph Manual. A Complete History and
Description of the Semaphoric, Electric, and Magnetic Telegraphs
of Europe, Asia, and Africa, with 625 illustrations. By Tal. P.
Shaffxer, of Kentucky. New edition. 1 vol., 8vo. Cloth. 850pp.
$6.50.
MINIFIE (WM.) Mechanical Drawing. A Text-Book of Geomet-
rical Drawing for the use of Mechanics and Schools, in which
the Definitions and Rules of Geometry are familiarly explained ; the
Practical Problems are arranged, from the most simple to the more
complex, and in their description technicalities are avoided as much
as possible. With illustrations for Drawing Plans, Sections, and
Elevations of Buildings and Machinery ; an Introduction to Isomet-
rical Drawing, and an Essay on Linear Perspective and Shadows.
Illustrated with over 200 diagrams engraved on steel. By Wm
Mixifis, Architect. Seventh edition. With an Appendix on the
Theory and Application "of Colors. 1 vol., Svo. Cloth. $4.
*• It is the be-^t work on Drawing that we have ever seen, and is especially a text-book of Geo-
metrical Drawing for the use of Mechanics and Schools. No young Mechanic, such as a Ma-
chinist. Engineer, Cabinet-Maker, Millwright, or Carpenter should be without it."— Scientific
American.
" One of the most comprehensive works of the kind ever published, and cannot but possesi
great value to builders. The style is at once elegant and substantial. 1 '— Pennsylvania Ir>qui*w.
" Whatever is said is rendered perfectly intelligible by remarkably well-executed diagrams on
■ieel. leaving nothing for mere vague supposition ; and the addition of an introduction to iso-
metrical drawing, linear perspective, and the projection of shadows, winding up with a useful
aidex to technical terms." — Glasgow Mechanics' Journal.
%W° The British P^vernment has arthorized the use of this book in their schools of art at
Somerset House, London, and throughout the kingdom.
MINIFIE (WM.) Geometrical Drawing. Abridged from the octavo
edition, for the use of Schools. Illustrated with 48 steel plates.
Xew edition, enlarged. 1 vol., i2mo, cloth. $2.
" It is well adapted at a text-book of drawing tc be used in our High Schools and Academiei
Wfeere th .8 useful branch of the fine arts has beer hitherto too much neglected."— Boston Journa*
D. Van NostrancPs Publications.
PEIRCE'S SYSTEM OF ANALYTIC MECHANICS Physical
and Celestial Mechanics, by Benjamin Peirce, Perkins Professol
of Astronomy and Mathematics in Harvard University, and Con-
sulting Astronomer of the American Ephemeris and Nautical Al-
manac. Developed in four systems of Analytic Mechanics, Celestial
Mechanics, Potential Physics, and Analytic Morphology, i vol.,
4to. Cloth. $10.
C^ ILLMORE. Practical Treatise on Limes, Hydraulic Cements, and
J Mortars. Papers on Practical Engineering, U. S. Engineer De-
partment, No. 9, containing Reports of numerous experiments con-
ducted in New York City, during the years 1858 to 1861, inclusive.
By Q. A. Gillmore, Brig. -General U. S. Volunteers, and Major U.
S. Corps of Engineers. With numerous illustrations. One volume,
octavo. Cloth. $4.
ROGERS (H. D.) Geology of Pennsylvania. A complete Scien-
tific Treatise on the Coal Formations. By Henry D. Rogers,
Geologist. 3 vols., 4to., plates and maps. Boards. $30.00.
BURGH (N. P.) Modern Marine Engineering, applied to Paddle
and Screw Propulsion. Consisting of 36 colored plates, 259
Practical Woodcut Illustrations, and 403 pages of Descriptive Matter,
the whole being an exposition of the present practice of the follow
ing firms : Messrs. J. Penn & Sons ; Messrs. Maudslay, Sons, &
Field ; Messrs. James Watt & Co. ; Messrs. J. & G. Rennie ; Messrs.
R. Napier & Sons ; Messrs. J. & W. Dudgeon ; Messrs. Ravenhill
& Hodgson ; Messrs. Humphreys & Tenant ; Mr. J. T. Spencer,
and Messrs. Forrester & Co. By N. P. Burgh, Engineer. In one
thick vol., 4to. Cloth. $25.00. Half morocco. $30.00.
ING. Lessons and Practical Notes on Steam, the Steam-Engine,
Propellers, &c. , &c, for Young Marine Engineers, Students,
and others. By the late W. R. King, U. S. N. Revised by Chief-
Engineer J. W. King, U. S. Navy. Twelfth edition, enlarged. 8vo.
Cloth. $2.
ARD. Steam for the Million. A Popular Treatise on Steam and
its Application to the Useful Arts, especially to Navigation. By
J. H. Ward, Commander U. S. Navy. New and revised edition.
1 vol., 8vo. Cloth. $1.
WALKER. Screw Propulsion. Notes on Screw Propulsion, its
Rise and History. By Capt. W. H. Walker, U. S. Navy. 1
vol., 8vo. Cloth. 75 cents.
THE STEAM-ENGINE INDICATOR, and the Improved Mano-
meter Steam and Vacuum Gauges ; Their Utility and Application.
By Paul Stillman. New edition. 1 vol., i2mo. Flexible cloth.
$1.
SHERWOOD. Engineering Precedents for Steam Machinery. Ar-
ranged in the most practical and useful manner for Engineers. By
B. F. Isherwood, Civil Engineer U. S. Navy. With illustration*
Two volumes in one. 8vo. Cloth. $2.50.
K
W
I
Scientific Books.
POOR'S METHOD OF COMPARING THE LINES AND
DRAUGHTING VESSELS PROPELLED BY SAIL OR
STEAM, including a Chapter on Laying off on the Mould-Loft
Floor. By Samuel M. Pook, Naval Constructor. i vol., 8vo.
With illustrations. Cloth. $5.
SWEET (S. H.) Special Report on Coal ; showing its Distribution,
Classification and Cost delivered over different routes to various
points in the State of New York, and the principal cities on the
Atlantic Coast. By S. H. Sweet. With maps. 1 vol., 8vo. Cloth.
$3-
ALEXANDER (J. H.) Universal Dictionary of Weights and Meas-
ures, Ancient and Modern, reduced to the standards of the United
States of America. By J. H. Alexander. New edition. 1 vol.,
8vo. Cloth. $3.50.
" As a standard work of reference this book should be in every library ; it is oue which we
have long wanted, and it will save us much trouble and research."— Scientific American*,
CRAIG (B. F. ) Weights and Measures. An Accouni; of the Deci-
mal System, with Tables of Conversion for Commercial and Scien-
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