difference of latitude and the difference of longitude. Reduce real
difference of latitude and difference of longitude to minutes. Take log
of the difference of longitude (Table 42) and add 10. From this log
subtract the log of difference of meridional parts. The result will be
the log tan of the True Course, which find in Table 44. On the same page
find the log sec of true course. Add to this the log of the real
difference of latitude, and if the result is more than 10, subtract 10.
This result will be the log of the distance sailed. This method should
be used only when steaming approximately a North and South course.
Note.--For detailed explanation of Tables 42 and 44 see Bowditch, pp.
271-276.
Assign for Night Reading Arts, in Bowditch:
183-184-185-186-187-188-189-194-259-260-261-262-263-264-265-266-267-268.
Also, one of the examples of Mercator sailing to be done by both the
Inspection and Logarithmic method.
SATURDAY LECTURE
GREAT CIRCLE SAILING--THE CHRONOMETER
In Tuesday's Lecture of this week, I explained how a Great Circle track
was laid down on one of the Great Circle Sailing Charts which are
prepared by the Hydrographic Office.
Supposing, however, you do not have these charts on hand. There is an
easy way to construct a great circle track yourself. Turn to Art. 194,
page 82, in Bowditch. Here is a table with an explanation as to how to
use it. Take, for instance, the same two points between which you just
drew a line on the great circle track. Find the center of this line and
the latitude of that point. At this point draw a line perpendicular to
the course to be sailed, the other end of which must intersect the
corresponding parallel of latitude given in the table. With this point
as the center of a circle, sweep an arc which will intersect the point
left and the point sought. This arc will be the great circle track to
follow.
To find the courses to be sailed, get the difference between the course
at starting and that at the middle of the circle, and find how many
quarter points are contained in it. Now divide the distance from the
starting point to the middle of the circle by the number of quarter
points. That will give the number of miles to sail on each quarter point
course. See this illustration:
[Illustration]
Difference between ENE and E = 2 pts. = 8 quarter points. Say distance
is 1600 miles measured by dividers or secured by Mercator Sailing
Method. Divide 1600 by 8 = 200. Every 200 miles you should change your
course 1/4 point East.
_The Chronometer_
The chronometer is nothing more than a very finely regulated clock. With
it we ascertain Greenwich Mean Time, i.e., the mean time at Greenwich
Observatory, England. Just what the words "Greenwich Mean Time" signify,
will be explained in more detail later on. What you should remember here
is that practically every method of finding your exact position at sea
is dependent upon knowing Greenwich Mean Time, and the only way to find
it is by means of the chronometer.
It is essential to keep the chronometer as quiet as possible. For that
reason, when you take an observation you will probably note the time by
your watch. Just before taking the observation, you will compare your
watch with the chronometer to notice the exact difference between the
two. When you take your observation, note the watch time, apply the
difference between the chronometer and watch, and the result will be the
CT.
For instance, suppose the chronometer read 3h 25m 10s, and your watch,
at the same instant, read 1h 10m 5s. C--W would be:
3h -- 25m -- 10s
-- 1 -- 10 -- 05
----------------
2h -- 15m -- 05s
Now suppose you took an observation which, according to your watch, was
at 2h 10m 05s. What would be the corresponding C T? It would be
WT 2h -- 10m -- 05s
C -- W 2 -- 15 -- 05
--------------------
CT 4h -- 25m -- 10s
If the chronometer time is less than the W T add 12 hours to the C T, so
that it will always be the larger and so that the amount to be added to
W T will always be +. For instance, CT 1h--25m--45s, WT 4h--13m--25s,
what is the C-W?
CT 13h--25m--45s
WT 4 --13 --25
----------------
C--W 9h--12m--20s
Now, suppose an observation was taken at 6h 13m 25s according to watch
time. What would be the corresponding CT?
WT 6h--13m--25s
C--W 9 --12 --20
----------------
15h--25m--45s
--12
----------------
CT 3h--25m--45s
Put in your Note-Book: CT = WT + C - W.
If, in finding C-W, C is less than W, add 12 hours to C, subtracting
same after CT is secured.
Example No. 1:
CT 3h--25m--10s
WT 1 --10 --05
----------------
C--W 2h--15m--05s
WT 2h--10m--05s
+ C--W 2 --15 --05
----------------
CT 4h--25m--10s
Example No. 2:
CT 1h--25m--45s
WT 4h--13m--25s
(+12 hrs.) CT 13h--25m--45s
WT 4 --13 --25
----------------
C-W 9h--12m--20s
WT 6h--13m--25s
+ C-W 9 --12 --20
----------------
15h--25m--45s
(-12 hrs.) 12
----------------
CT 3h--25m--45s
There is one more very important fact to know about the chronometer. It
is physically impossible to keep it absolutely accurate over a long
period of time. Instead of continually fussing with its adjustment and
hands, the daily rate of error is ascertained, and from this the exact
time for any given day. It is an invariable practice among good mariners
to _leave the chronometer alone_. When you are in port, you can find out
from a time ball or from some chronometer maker what your error is. With
this in mind, you can apply the new correction from day to day. Here is
an example (Put in your Note-Book):
On June 1st, CT 7h--20m--15s, CC 2m--40s fast. On June 16th, (same CT)
CC 1m--30s fast. What was the corresponding G.M.T. on June 10th?
June 1st 2m--40s fast
16th 1m--30s fast
----------------
1m--10s
60
--
60
10
--
15) 70s (4.6 sec. Daily Rate of error losing
June 1st-10th, 9 days times 4.6 sec. = 41 sec. losing
June 1st 2m--40s fast
June 10th 41s losing
---------
June 10th 1m--59s fast
CT 7h--20m--15s
CC -- 1 --59
------------
G.M.T. 7h--18m--16s on June 10th
If CC is fast, subtract from CT
If CC is slow, add to CT
WEEK III--CELESTIAL NAVIGATION
TUESDAY LECTURE
CELESTIAL CO-ORDINATES, EQUINOCTIAL SYSTEM, ETC.
We have already discussed the way in which the earth is divided so as to
aid us in finding our position at sea, i.e., with an equator, parallels
of latitude, meridians of longitude starting at the Greenwich meridian,
etc. We now take up the way in which the celestial sphere is
correspondingly divided and also simple explanations of some of the more
important terms used in Celestial Navigation.
As you stand on any point of the earth and look up, the heavenly bodies
appear as though they were situated upon the surface of a vast hollow
sphere, of which your eye is the center. Of course this apparent concave
vault has no existence and we cannot accurately measure the distance of
the heavenly bodies from us or from each other. We can, however, measure
the direction of some of these bodies and that information is of
tremendous value to us in helping us to fix our position.
Now we could use our eye as the center of the celestial sphere but more
accurate than that is to use the center of the earth. Suppose we do use
the center of the earth as the place from which to observe these
celestial bodies and, in imagination, transfer our eye there. Then we
will find projected on the celestial sphere not only the heavenly bodies
but the imaginary points and circles of the earth's surface. Parallels
of latitude, meridians of longitude, the equator, etc., will have the
same imaginary position on the celestial sphere that they have on the
earth. Your actual position on the earth will be projected in a point
called your zenith, i.e., the point directly overhead.
[Illustration]
From this we get the definition that the Zenith of an observer on the
earth's surface is the point in the celestial sphere directly overhead.
It would be a simple matter to fix your position if your position never
changed. But it is always changing with relation to these celestial
bodies. First, the earth is revolving on its own axis. Second, the
earth is moving in an elliptic track around the sun, and third, certain
celestial bodies themselves are moving in a track of their own. The
changes produced by the daily rotation of the earth on its axis are
different for observers at different points on the earth and, therefore,
depend upon the latitude and longitude of the observer. But the changes
arising from the earth's motion in its orbit and the motion of various
celestial bodies in their orbits, are true no matter on what point of
the earth you happen to be. These changes, therefore, in their relation
to the center of the earth, may be accurately gauged at any instant. To
this end the facts necessary for any calculation have been collected and
are available in the Nautical Almanac, which we will take up in more
detail later.
Now with these facts in mind, let us explain in simple words the meaning
of some of the terms you will have to become acquainted with in
Celestial Navigation.
In the illustration (Bowditch p. 88) the earth is supposed to be
projected upon the celestial sphere N E S W. The Zenith of the observer
is projected at Z and the pole of the earth which is above the horizon
is projected at P. The other pole is not given.
The Celestial Equator is marked here E Q W and like all other points and
lines previously mentioned, it is the projection of the Equator until it
intersects the celestial sphere. Another name for the Celestial Equator
is the Equinoctial.
All celestial meridians of longitude corresponding to longitude
meridians on the earth are perpendicular to the equinoctial and likewise
P S, the meridian of the observer, since it passes through the
observer's zenith at Z, is formed by the extension of the earth's
meridian of the observer and hence intersects the horizon at its N and S
points. This makes clear again just what is the meridian of the
observer. It is the meridian of longitude which passes through the N and
S poles and the observer's zenith. In other words, when the sun or any
other heavenly body is on your meridian, a line stretched due N and S,
intersecting the N and S poles, will pass through your zenith and the
center of the sun or other celestial body. To understand this is
important, for no sight with the sextant is of value except with
relation to your meridian.
The Declination of any point in the celestial sphere is its distance in
arc, North or South of the celestial equator, i.e., N or S of the
Equinoctial.
North declinations, i.e., declinations north of the equinoctial are
always marked, +; those south of the equinoctial, -. For instance, in
the Nautical Almanac, you will never see a declination of the sun or
other celestial body marked, N 18 deg. 28' 30". It will always be marked
+18 deg. 28' 30" and a south declination will be marked -18 deg. 28' 30".
Another fact to remember is that Declination on the celestial sphere
corresponds to latitude on the earth. If, for instance, the Sun's
declination is +18 deg. 28' 30" at noon, Greenwich, then at that instant,
i.e., noon at Greenwich, the sun will be directly overhead a point on
earth which is in latitude N 18 deg. 28' 30".
The Polar Distance of any point is its distance in arc from either pole.
It must, therefore, equal 90 deg. minus the declination, if measured from
the pole of the same name as the declination or 90 deg. plus the declination
if measured from the pole of the opposite name.
P M is the polar distance of M from P, or P B the polar distance of B
from P.
The true altitude of a celestial body is its angular height from the
true horizon.
The zenith distance of any point or celestial body is its angular
distance from the zenith of the observer.
The Ecliptic is the great circle representing the path in which the sun
appears to move in the celestial sphere. As a matter of fact, you know
that the earth moves around the sun, but as you observe the sun from
some spot on the earth, it appears to move around the earth. This
apparent track is called the Ecliptic as stated before, and in the
illustration the Ecliptic is represented by the curved line, C V T. The
plane of the Ecliptic is inclined to that of the Equinoctial at an angle
of 23 deg. 27-1/2', and this inclination is called the obliquity of the
Ecliptic.
The Equinoxes are those points at which the Ecliptic and Equinoctial
intersect, and when the sun occupies either of these two positions, the
days and nights are of equal length. The Vernal Equinox is that one
which the sun passes through or intersects in going from S to N
declination, and the Autumnal Equinox that which it passes through or
intersects in going from N to S declination. The Vernal Equinox (V in
the illustration) is also designated as the First Point of Aries which
is of use in reckoning star time and will be mentioned in more detail
later.
The Solstitial Points, or Solstices, are points of the Ecliptic at a
distance of 90 deg. from the Equinoxes, at which the sun attains its highest
declination in each hemisphere. They are called the Summer and Winter
Solstice according to the season in which the sun appears to pass these
points in its path.
To sum up: The way to find any point on the earth is to find the
distance of this point N or S of the equator (i.e., its Latitude) and
its distance E or W of the meridian at Greenwich (i.e., its longitude).
In the celestial sphere, the way to find the location of a point or
celestial body such as the sun is to find its declination (i.e.,
distance in arc N or S of the equator) and its hour angle. By hour
angle, I mean the distance in time from your meridian to the meridian of
the point or celestial body in question.
Assign for Night reading, Arts, in Bowditch:
270-271-272-273-274-275-277-278-279-280-282-283-284.
WEDNESDAY LECTURE
TIME BY THE SUN--MEAN TIME, SOLAR TIME, CONVERSION, ETC.
There is nothing more important in all Navigation than the subject of
Time. Every calculation for determining the position of your ship at sea
must take into consideration some kind of time. Put in your Note-Book:
There are three kinds of time:
1. Apparent or solar time, i.e., time by the sun.
2. Mean Time, i.e., clock time.
3. Sidereal Time, or time by the stars.
So far as this lecture is concerned, we will omit any mention of
sidereal time, i.e., time by the stars. We will devote this morning to
sun time, i.e., apparent time, and mean time.
Apparent or Solar Time is, as stated before, nothing more than sun time
or time by the sun. The hour angle of the center of the sun is the
measure of apparent or solar time. An apparent or solar day is the
interval of time it takes for the earth to revolve completely around on
its axis every 24 hours. It is apparent noon at the place where you are
when the center of the sun is directly on your meridian, i.e., on the
meridian of longitude which runs through the North and South poles and
also intersects your zenith. This is the most natural and the most
accurate measure of time for the navigator at sea and the unit of time
adopted by the mariner is the apparent solar day. Apparent noon is the
time when the latitude of your position can be most easily and most
exactly determined and on the latitude by observation just secured we
can get data which will be of great value to us for longitude sights
taken later in the day.
Now it would be very easy for the mariner if he could measure apparent
time directly so that his clock or other instrument would always tell
him just what the sun time was. It is impossible, however, to do this
because the earth does not revolve at a uniform rate of speed.
Consequently the sun is sometimes a little ahead and sometimes a little
behind any average time. You cannot manufacture a clock which will run
that way because the hours of a clock must be all of exactly the same
length and it must make noon at precisely 12 o'clock every day. Hence we
distinguish clock time from sun time by calling clock time, mean (or
average) time and sun time, apparent or solar time. From this
explanation you are ready to understand such expressions as Local Mean
Time, which, in untechnical language, signifies clock time at the place
where you are; Greenwich Mean Time which signifies clock time at
Greenwich; Local Apparent Time, which signifies sun time at the place
where you are; Greenwich Apparent Time, which signifies sun time at
Greenwich.
Now the difference between apparent time and mean time can be found for
any minute of the day by reference to the Nautical Almanac which we will
take up later in more detail. This difference is called the Equation of
Time.
There is one more fact to remember in regard to apparent and mean time.
It is the relation of the sun's hour angle to apparent time. In the
first place, what is a definition of the sun's HA? It is the angle at
the celestial pole between the meridian intersecting any given point and
the meridian intersecting the center of the sun. It is measured by the
arc of the celestial equator intersected between the meridian of any
point and the meridian intersecting the center of the sun.
[Illustration]
For instance, in the above diagram, suppose PG is the meridian at
Greenwich, and PS the meridian intersecting the sun. Then the angle at
the pole GPS, measured by the arc GS would be the Hour Angle of
Greenwich, or the Greenwich Hour Angle. And now you notice that this
angular measure is exactly the same as apparent time at Greenwich or
Greenwich Apparent Time, for Greenwich Apparent Time is nothing more
than the distance in time Greenwich, England, or the meridian at
Greenwich is from the sun, i.e., the time it takes the earth to revolve
from Greenwich to the sun; and that distance is exactly measured by the
Greenwich Hour Angle or the arc on the celestial equator, GS.
The same is correspondingly true of Local Apparent Time and the ship's
Hour Angle. Suppose, for instance, PL is the meridian intersecting the
place where your ship is. Then your ship's hour angle would be the angle
at the pole intersecting the meridian of your ship and the meridian of
the sun or LPS and measured by the arc LS. And you will note that this
distance is exactly the same as apparent time at the ship, for Apparent
Time at ship is nothing more than the distance in time which the ship is
from the sun. We can sum up all this information in a few simple rules,
which put in your Note-Book:
Mean Time = Clock Time.
G.M.T. = Greenwich Mean Time.
L.M.T. = Local Mean Time.
Apparent Time = Actual or Sun Time.
G.A.T. (G.H.A.) = Greenwich Apparent Time or Greenwich Hour Angle.
L.A.T. (S.H.A.) = Local Apparent Time or Ship's Hour Angle.
Difference between apparent and mean time or mean and apparent
time--Equation of Time.
Right under this in your Note-Book put the following diagram, which I
will explain:
[Illustration]
You will see from this diagram that civil time commences at midnight and
runs through 12 hours to noon. It then commences again and runs through
12 hours to midnight. The Civil Day, then, is from midnight to midnight,
divided into two periods of 12 hours each.
The astronomical day commences at noon of the civil day of the same
date. It comprises 24 hours, reckoned from O to 24, from noon of one day
to noon of the next. Astronomical time, either apparent or mean, is the
hour angle of the true or mean sun respectively, measured to the
westward throughout its entire daily circuit.
Since the civil day begins 12 hours before the astronomical day and ends
12 hours before it, A.M. of a new civil day is P.M. of the astronomical
day preceding. For instance, 6 hours A.M., April 15th civil time is
equivalent to 18 hours April 14th, astronomical time.
Now, all astronomical calculations in which time is a necessary fact to
be known, must be expressed in astronomical time. As chronometers have
their face marked only from 0 to 12 as in the case of an ordinary watch,
it is necessary to transpose this watch or chronometer time into
astronomical time. No transposing is necessary if the time is P.M., as
you can see from the diagram that both civil and astronomical times up
to 12 P.M. are the same. But in A.M. time, such transposing is
necessary. Put in your Note-Book:
Whenever local or chronometer time is A.M., deduct 12 hours from such
time to get the correct astronomical time:
CT 15d-- 9h--10m--30s A.M.
--12
------------------------
CT 14d--21h--10m--30s
------
L.M.T. 10d-- 4h--40m--16s A.M.
--12
------------------------
L.M.T. 9d--16h--40m--16s
Now we come to a very important application of time. You will remember
that in one of the former lectures we stated that to find our latitude,
we had to find how far North or South of the equator we were, and to
find our longitude, we had to find how far East or West of the meridian
at Greenwich we were. Never mind about latitude for the present. We can
find our longitude exactly if we know our Greenwich time and our time at
ship. For instance, in the accompanying diagram:
[Illustration]
Suppose PG is the meridian at Greenwich, then anything to the west of PG
is West longitude and anything to the East of PG is East longitude. Now
suppose GPS is the H.A. of G. or G.A.T.--i.e., the distance in time G.
is from the sun. And L P S is the H.A. of the ship or L.A.T.--i.e., the
distance in time the ship is from the sun. Then the difference between G
P S and L P S is G P L, measured by the arc L G, and that is the
difference that the ship, represented by its meridian PL, is from the
Greenwich meridian PG. In other words, that is the ship's longitude for,
as mentioned before, longitude is the distance East or West of Greenwich
that any point is, measured on the arc of the celestial equator. The
longitude is West, for you can see LPG or the arc LG is west of the
meridian PG.
Likewise if P E is the meridian of your ship, the Longitude in time is
the S.H.A. or L.A.T., E P S (the distance your ship is from the sun)
less the G.H.A. or G.A.T., G P S (the distance Greenwich is from the
sun) which is the angle G P E measured by the arc G E. And this
Longitude is East for you can see G P E, measured by G E, is east of the
Greenwich meridian, P G.
In both these cases, however, the longitude is expressed in time, i.e.,
so many hours, minutes and seconds from the Greenwich meridian and we
wish to express this distance in degrees, minutes and seconds of arc.
The earth describes a circle of 360 deg. every 24 hours. Then if you are 1
hour from Greenwich, you are 1/24 of 360 deg. or 15 deg. from Greenwich and if
you are 12 hours from Greenwich, you are 1/2 of 360 deg. or 180 deg. from
Greenwich. By keeping this in mind, you should be able to transpose time
into degrees, minutes and seconds of arc for any fraction of time. It
is, however, all worked out in Table 7 of Bowditch which turn to. (Note
to Instructor: Explain this table carefully). Put in your Note-Book:
89 deg. 24' 26" = (89 deg.) 5h--56m
(24') 1m--36s
(26") 1--44/60s
--------------------
5h--57m--37s 44/60s = 38s
4h--42m--26s
4h--40m = 70 deg.
2m--24s = 36'
2s = 30"
-----------
70 deg. 36' 30"
Also put in your Note-Book this diagram and these formulas: (For diagram
use illustration on p. 40.)
L.M.T. + West Lo. = G.M.T. L.A.T. + West Lo. = G.A.T.
L.M.T. - East Lo. = G.M.T. L.A.T. - East Lo. = G.A.T.
G.M.T. - West Lo. = L.M.T. G.A.T. - West Lo. = L.A.T.
G.M.T. + East Lo. = L.M.T. G.A.T. + East Lo. = L.A.T.
If G.M.T. or G.A.T. is greater than L.M.T. or L.A.T. respectively, Lo.
is West.
If G.M.T. or G.A.T. is less than L.M.T. or L.A.T. respectively, Lo. is
East.
Example:
In longitude 81 deg. 15' W, L.M.T. is April 15d--10h--17m--30s A.M. What is
G.M.T.?
L.M.T. 15d--10h--17m--30s A.M.
--12
------------------
L.M.T. 14d--22h--17m--30s
5 --25 W +
------------------
G.M.T. 15d-- 3h--42m--30s
--------
G.M.T. April 15d-- 3h--42m--30s
L.M.T. April 15d--10h--17m--30s A.M.
In what Lo. is ship?
G.M.T. 15d 3h--42m--30s
L.M.T. 14d 22h--17m--30s
------------------
Lo. in T 5h--25m--00s W
Lo. = 81 deg. 15'W
Assign also for Night Work reading the following articles in Bowditch:
276-278-279-226-228-286-287-288-290-291-294 (omitting everything on page
114.)
THURSDAY LECTURE
SIDEREAL TIME--RIGHT ASCENSION
Our last lecture was devoted to a discussion of sun time. Today we are
going to talk about star time, or, using the more common words, sidereal
time.
Now, just one word of review. You remember that we have learned that
astronomical time is reckoned from noon of one day to noon of the next
and hence the astronomical day corresponds to the 24 hours of a ship's
run. The hours are counted from 0 to 24, so that 10 o'clock in the
morning of October 25th is astronomically October 24th, 22 hours or 22
o'clock of October 24th.