large that a small arc of its circumference, say 25 or 30 miles, would
be practically a straight line.
Suppose S to be the point over which the sun is vertical and GF part of
the circumference of a circle drawn around the point. Suppose you were
at B and from an altitude of the sun, taken by sextant, you worked out
your position. You would find yourself on a little arc ABC which, for
all purposes in Navigation, is a straight line at right angles to the
true bearing of the sun from the point S. You can readily see this from
the above diagram. Suppose your observer is at H. His line is GHI, which
is again a straight line at right angles to the true bearing of the sun.
He is not certain he is at H. He may be at G or I. He knows, however, he
is somewhere on the line GHI, though where he is on that line he cannot
tell exactly. That line GHI or ABC or DEF is the line of position and
such a line is called a Sumner Line, after Capt. Thomas Sumner, who
explained the theory some 45 years ago. Put in your Note-Book:
Any person taking an altitude of a celestial body must be, for all
practical purposes, on a straight line which is at right angles to the
true bearing of the body observed.
It should be perfectly clear now that if the sun bears due North or
South of the observer, i.e., if the sun is on the observer's meridian,
the resulting line of position _must_ run due East and West. In other
words it is a parallel of latitude. And that explains why a noon
observation is the best of the day for getting your latitude accurately.
Again, if the sun bears due East or West the line of position must bear
due North and South. And that explains why a morning or afternoon
sight--about 8-9 A.M. or 3-4 P.M., if the sun bears either East or West
respectively, is the best time for determining your North and South
line, or longitude.
Now suppose you take an observation at 8 A.M. and you are not sure of
your D.R. latitude. Your 8 A.M. position when the sun was nearly due
East, will give, you an almost accurate North and South line and
longitude. Suppose that from 8 A.M. to noon you sailed NE 60 miles.
Suppose at noon you get another observation. That will give you an East
and West line, for then the sun bears true North and South. An East and
West line is your correct latitude. Now you have an 8 A.M. observation
which is nearly correct for longitude and a noon position which is
correct for latitude. How can you combine the two so as to get
accurately both your latitude and longitude? Put in your Note-Book:
Through the 8 A.M. position, draw a line on the chart at right angles to
the sun's true bearing. Suppose the sun bore true E 1/2 S. Then your
line of position would run N 1/2 E. Mark it 1st Position Line.
[Illustration]
Now draw a line running due East and West at right angles to the N-S
noon bearing of the sun and mark this line Second Position Line. Advance
your First Position Line the true course and distance sailed from 8 A.M.
to noon, and through the extremity draw a third line exactly parallel to
the first line of position. Where a third line (the First Position Line
advanced) intersects the Second Position Line, will be your position at
noon. It cannot be any other if your calculations are correct. You knew
you were somewhere on your 8 A.M. line, you know you are somewhere on
your noon line, and the only spot where you can be on both at once is
the point where they intersect. You don't necessarily have to wait until
noon to work two lines. You can do it at any time if a sufficient
interval of time between sights is allowed. The whole matter simply
resolves itself into getting your two lines of position, having them
intersect and taking the point of intersection as the position of your
ship.
There is one other way to get two lines to intersect and it is one of
the best of all for fixing your position accurately. It is by getting
lines of position by observation of two stars. If, for instance, you can
get two stars, one East and the other West of you, you can take
observations of both so closely together as to be practically
simultaneous. Then your Easterly star would give you a line like AA' and
the westerly star the line BB' and you would be at the intersection S.
[Illustration]
Assign for reading: Articles in Bowditch 321-322-323-324. Spend the rest
of the period in getting times from the N. A., getting true altitudes
from observed altitudes, working examples in Mercator sailing, etc.
WEDNESDAY LECTURE
LATITUDE BY MERIDIAN ALTITUDE
A meridian altitude is an altitude taken when the sun or other celestial
body observed bears true South or North of the observer or directly
overhead. In other words, when the celestial body is on your meridian
and you take an altitude of the body by sextant at that instant, the
altitude you get is called a meridian altitude. In the case of the sun,
such a meridian altitude is at apparent noon. Now latitude is always
secured most accurately at noon by means of your meridian altitude. The
reason for this was explained in yesterday's lecture. The general
formula for latitude by meridian altitude is (Put in your Note-Book):
Latitude by meridian altitude = Zenith Distance (ZD) +- Declination
(Dec).
Zenith distance is the distance in degrees, minutes and seconds from
your zenith to the center of the observed body. For simplicity's sake,
we will consider the sun only as the observed body. Then the zenith
distance is the distance from your zenith to the center of the sun. Now
suppose that you and the sun are both North of the equator and you are
North of the sun. If you can determine exactly how far North you are of
the sun and how far North the sun is of the equator, you will, by adding
these two measurements together, know how far North of the equator you
are, i.e., your latitude. As already explained, the declination of the
sun is its distance in degrees, minutes and seconds from the equator and
the exact amount of declination is, of course, corrected to the proper
G.M.T. Your zenith distance is the distance in the celestial sphere you
are from the sun. You know that it is 90 deg. from your zenith to the
horizon. Your zenith distance, therefore, is the difference between the
true meridian altitude of the sun, obtained by your sextant, and 90 deg..
Hence, having secured the true meridian altitude of the sun, you have
only to subtract it from 90 deg. to find your zenith distance, i.e., how far
you are from the sun. This diagram will make the whole matter clear:
[Illustration:
A
|
| B (Sun)
| /
| /
| /
| /
|<-/-
| / \ 90 deg.
|/ |
+------v---------------------- C
A = Zenith, B = Sun, C = Horizon.]
The arc ABC measures 90 deg.. That is the distance from your zenith to the
horizon. Now if BC is the true meridian altitude of the sun at noon,
90 deg.-BC or AB is your zenith distance. If BC measures by sextant 60 deg., AB
measures 90 deg.-60 deg. or 30 deg.. This 30 deg. is your Zenith Distance. Now suppose
that from the Nautical Almanac we find that the G.M.T. corresponding to
the time at which we measured the meridian altitude of the sun shows the
sun's declination to be 10 deg. N. Well, if you are 30 deg. North of the sun,
and the sun is 10 deg. North of the equator, you must be 40 deg. North of the
equator or in latitude 40 deg. N. For that is all latitude is, namely, the
distance in degrees, minutes and seconds you are due North or South of
the equator. That is the first and simplest case.
Another case is when you are somewhere in North latitude and the sun's
declination is South. Then the situation would, roughly, look like this:
[Illustration: BC = Altitude of the sun, AB = Zenith Distance and DB =
Sun's Declination.]
In this case, your distance North of the equator AD would be your zenith
distance AB minus the sun's declination DB. This diagram is not strictly
correct, for the observer's position on the earth 0 appears to be South
of the equator instead of North of the equator. That is because the
diagram is on a flat piece of paper instead of on a globe. So far as
illustrating the Zenith Distance minus the Declination, however, the
diagram is correct. The last case is where you are, say, 10 deg. N of the
sun (your zenith Distance is 10 deg.) and the sun is in 20 deg. S declination.
In that case you would have to subtract your zenith distance from the
sun's declination to get your latitude, for the sun's latitude (its
declination) is greater than yours.
Now from these three cases we deduce the following directions, which put
in your Note-Book:
Begin to measure the altitude of the sun shortly before noon. By
bringing its image down to the horizon, you can detect when its altitude
stops increasing and starts to decrease. At that instant the sun is on
your meridian, it is noon at the ship, and the angle you read from your
sextant is the meridian altitude of the sun. To work out your latitude,
name the meridian altitude S if the sun is south of you and N if north
of you.
Correct the observed altitude to a true altitude by Table 46. If the
altitude is S, the Zenith Distance is N or vice versa. (Note to
Instructor: If the sun is South of you, you are North of the sun and
vice versa.)
Correct the declination for the proper G.M.T. as shown by chronometer
(corrected). If zenith distance and declination are both North or both
South, add them and the sum will be the latitude, N or S as indicated.
If one is N, and the other S, subtract the less from the greater and the
result will be the latitude in, named N or S after the greater. Example:
At sea June 15th, observed altitude of (_) 71 deg. 15' S, IE--47', HE 25 ft.
CT 3h--34m--15s P.M. Required latitude of ship.
(_) 71 deg. 15' 00 S IE -- 47'
Corr. -- 36 24 HE + 10 36
------------------- ---------
-(-)- 70 deg. 38' 36" S Corr. -- 36' 24"
-- 90 00 00
-------------------
ZD 19 deg. 21' 24" N
Dec. 23 17 15 N (G.M.T. June 15--3h 34m 15s)
-------------------
Lat. 42 deg. 38' 39" N
-------------------
Assign for Night Work or to be worked in class room such examples as the
following:
1. June 1st, 1919. (_) 33 deg. 50' 00" S. G.M.T. 8h 55m 44s. HE 20 ft. IE +
4' 3". Required latitude in at noon.
2. April 2nd, 1919. (_) 12 deg. 44' 30" N. CT was 2d 5h 14m 39s A.M., which
was 1m 40s slow on March 1st (same CT) and 4m 29s fast on March 15th
(same CT). IE -- 2' 20". HE 22 ft. Required latitude in at noon.
Assign for Night Work reading also, the following Articles in Bowditch:
344 and 223.
THURSDAY LECTURE
AZIMUTHS OF THE SUN
This is a peculiar word to spell and pronounce but its definition is
really very simple. Put in your Note-Book:
The azimuth of a heavenly body is the angle at the zenith of the
observer formed by the observer's meridian and a line drawn to the
center of the body observed. Azimuths are named from the latitude in and
toward the E in the A.M. and from the latitude in and toward the W in
the P.M.
All this definition means is that, no matter where you are in N
latitude, for instance, if you face N, the azimuth of the sun will be
the true bearing of the sun from you. The same holds true for moon, star
or planet, but in this lecture we will say nothing of the star azimuths
for, in some other respects, they are found somewhat differently from
the sun azimuths. Put this in your Note-Book:
To find an azimuth of the sun: Note the time of taking the azimuth by
chronometer. Apply chronometer correction, if any, to get the G.M.T.
Convert G.M.T. into G.A.T. by applying the equation of time. Convert
G.A.T. into L.A.T. by applying the longitude in time. The result is
L.A.T. or S.H.A. With the correct L.A.T., latitude and declination,
enter the azimuth tables to get the sun's true bearing, i.e., its
azimuth. Example:
March 15th, 1919. CT 10h -- 4m -- 32s. D.R. latitude 40 deg. 10' N,
longitude 74 deg. W. Find the TZ.
G.M.T. 10h--04m--32s
Eq. T. --09 --10
G.A.T. 9h--55m--22s
G.A.T. 9h--55m--22s
Lo. in T. 4 --56 --00 (W--)
L.A.T. 4h--59m--22s
Latitude and Declination opp. name.
TZ = N 101 deg. 30'W
We will take up later a further use of azimuths to find the error of
your compass. Right now all you have to keep in mind is what an azimuth
is and how you apply the formulas already given you to get the
information necessary to enter the Azimuth Tables for the sun's true
bearing at any time of the astronomical day when the sun can be seen. In
consulting these tables it must be remembered that if your L.A.T. or
S.H.A. is, astronomically, 20h (A.M.), you must subtract 12 hours in
order to bring the time within the scope of these tables which are
arranged from apparent six o'clock A.M. to noon and from apparent noon
to 6 P.M. respectively.
We are taking up sun azimuths today in order to get a thorough
understanding of them before beginning a discussion of the Marc St.
Hilaire Method which we will have tomorrow. You must get clearly in your
minds just what a line of position is and how it is found. Yesterday I
tried to explain what a line of position was, i.e., a line at right
angles to the sun's or other celestial body's true bearing--in other
words, a line at right angles to the sun's or other celestial body's
azimuth. Today I tried to show you how to find your azimuth from the
azimuth tables for any hour of the day. Tomorrow we will start to use
azimuths in working out sights for lines of position by the Marc St.
Hilaire Method.
Note to Instructor: Spend the rest of the time in finding sun azimuths
in the tables by working out such examples as these:
1. April 29th, 1919. D.R. latitude 40 deg. 40' N, Longitude 74 deg. 55' 14" W.
CT 10h--14m--24s. CC 4m--30s slow. Find TZ.
2. May 15th, 1919. D.R. latitude 19 deg. 20' S, Longitude 40 deg. 15' 44" E. CT
10h--44m--55s A.M. CC 3m--10s fast. Find TZ.
Note to Instructor:
If possible, give more examples to find TZ and also some examples on
latitude by meridian altitude.
Assign for Night Work reading the following Articles in Bowditch:
371-372-373-374-375. Also, examples to find TZ.
FRIDAY LECTURE
MARC ST. HILAIRE METHOD BY A SUN SIGHT
You have learned how to get your latitude by an observation at noon. By
the Marc St. Hilaire Method, which we are to take up today, you will
learn how to get a line of position, at any hour of the day. By having
this line of position intersect your parallel of latitude, you will be
able to establish the position of your ship, both as to its latitude and
longitude.
Now you have already learned that in order to get your latitude
accurately, you must wait until the sun is on your meridian, i.e., bears
due North or South of you, and then you apply a certain formula to get
your latitude. When the sun is on or near the prime vertical (i.e., due
East or West) you might apply another set of rules, which you have not
yet learned, to get your longitude. By the Marc St. Hilaire method, the
same set of rules apply for getting a line of position at any time of
the day, no matter what the position of the observed body in the heavens
may be. Just one condition is necessary, and this condition is necessary
in all calculations of this character, i.e., an accurate measurement of
the observed body's altitude is essential.
What we do in working out the Marc St. Hilaire method, is to assume our
Dead Reckoning position to be correct. With this D. R. position as a
basis, we compute an altitude of the body observed. Now this altitude
would be correct if our D. R. position were correct and vice versa. At
the same time we measure by sextant the altitude of the celestial body
observed, say, the sun. If the computed altitude and the actual observed
altitude coincide, the D. R. position is correct. If they do not, the
computed altitude must be corrected and the D. R. position corrected to
coincide with the observed altitude. Just how this is done will be
explained in a moment. Put in your Note-Book:
_Formula for obtaining Line of Position by M. St. H. Method._
I. Three quantities must be known either from observation or from Dead
Reckoning.
1. The S. H. A., marked "t."
Note: The method for finding S. H. A. (t) differs when the sun or
star is used as follows:
(a) For the Sun:
Get G.M.T. from the corrected chronometer time. Apply
the equation of time to find the G.A.T. Apply the D.R. Lo.
(-W)
(+E) and the result is L.A.T. or S.H.A. as required.
(b) For a Star:
(Note to pupils: Leave this blank to be filled in when we
take up stars in more detail.)
2. The Latitude, marked "L."
3. The Declination of the observed body, marked "D."
II. Add together the log haversine of the S.H.A. (Table 45), the log
cosine of the Lat. (Table 44), and the log cosine of the Dec. (Table 44)
and call the sum S. S is a log haversine and must always be less than
10. If greater than 10, subtract 10 or 20 to bring it less than 10.
III. With the log haversine S enter table 45 in the adjacent parallel
column, take out the corresponding Natural Haversine, which mark N_{S}.
IV. Find the algebraic difference of the Latitude and Declination, and
from Table 45 take out the Natural Haversine of this algebraic
difference angle. Mark it N_{D+-L}
V. Add the N_{S} to the N_{D+-L}, and the result will be the Natural
Haversine of the calculated zenith distance. Formula N_{ZD} = N_{S} +
N_{D+-L}
VI. Subtract this calculated zenith distance from 90 deg. to get the
calculated altitude.
VII. Find the difference between the calculated altitude and the true
altitude and call it the altitude difference.
VIII. In your Azimuth Table, find the azimuth for the proper "t," L and
D.
IX. Lay off the altitude difference along the azimuth either away from
or toward the body observed, according as to whether the true altitude,
observed by sextant, is less or greater than the calculated altitude.
[Illustration]
X. Through the point thus reached, draw a line at right angles to the
azimuth. This line will be your Line of Position, and the point thus
reached, which may be read from the chart or obtained by use of Table 2
from the D. R. Position, is the nearest to the actual position of the
observer which you can obtain by the use of any method from one sight
only.
Example:
At sea, May 18th, 1919, A.M. (_) 29 deg. 41' 00". D.R. Latitude 41 deg. 30' N,
Longitude 33 deg. 38' 45" W. WT 7h 20m 45s A.M. C-W 2h 17m 06s CC + 4m 59s.
IE--30". HE 23 ft. Required Line of Position and most probable position
of ship.
WT 18d -- 7h -- 20m -- 45s A.M.
-- 12
------------------------
WT 17d -- 19h -- 20m -- 45s
C-W 2 -- 17 -- 06 Corr. + 9' 34"
------------------------ IE -- 30
CT 17d -- 21h -- 37m -- 51s ------------
CC + 4 -- 59 + 9' 04"
------------------------
G.M.T. 17d -- 21h -- 42m -- 50s (_) 29 deg. 41' 00"
Eq. T. + 3 -- 47 + 9 04
------------------------ ------------
G.A.T. 17d -- 21h -- 46m -- 37s -(-)- 29 deg. 50' 04"
Lo. in T 2 -- 14 -- 35 (W--)
------------------------ log hav 9.48368
L.A.T.(t) 17d -- 19h -- 32m -- 02s log cos 9.87446
Lat. 41 deg. 30' N log cos 9.97473
Dec. 19 deg. 21' 25" N --------
log hav S 9.33287
N s .21521
L - D 22 deg. 08' 35" N D +- L .03687
--------
Calc. ZD 60 deg. 16' 30" N ZD .25208
-- 90 deg. 00 00
------------- TZ found from table to be
Cal. Alt. 29 deg. 43' 30" N 90 deg. E.
-(-)- 29 deg. 50' 04"
-------------
Alt. Diff 6' 34" Toward.
_ Course. Dist. Diff. Lat. Dep. Diff. Lo._
90 deg. 6' 34" 0 6.5 8.6
D.R. Lat. 41 deg. 30' N D.R. Lo. 33 deg. 38' 45" W
Diff. Lat. -- Diff. Lo. 8 36 E
---------- -------------
Most probable fix Lat. 41 deg. 30' N Lo. 33 deg. 30' 09" W
As azimuth is N 90 deg. E, Line of Position runs due N & S (360 deg.) through
Lat. 41 deg. 30' N. Lo. 33 deg. 30' 09" W.
Assign for work in class and for Night Work examples such as the
following:
1. July 11th, 1919. (_) 45 deg. 35' 30", Lat. by D. R. 50 deg. 00' N, Lo. 40 deg. 04'
W. HE 15 ft. IE--4'. CT (corrected) 5h. 38m 00s P.M. Required Line
of Position by Marc St. Hilaire Method and most probable fix of ship.
2. May 16th, 1919, A.M. (_) 64 deg. 01' 15", D. R. Lat. 39 deg. 45' N, Lo.
60 deg. 29' W. HE 36 ft. IE + 2' 30". CT 2h 44m 19s. Required Line of
Position by Marc St. Hilaire Method and most probable fix of ship.
Etc.
SATURDAY LECTURE
EXAMPLES ON MARC ST. HILAIRE METHOD BY A SUN SIGHT
1. Nov. 1st, 1919. A.M. at ship. WT 9h 40m 15s. C--W 4h 54m 00s.
D. R. Lat. 40 deg. 50' N, Lo. 73 deg. 50' W. (_) 27 deg. 59'. HE 14 ft. Required Line
of Position by Marc St. Hilaire Method and most probable position of ship.
2. May 30th, 1919. P.M. at ship. D. R. Lat. 38º 14' 33" N, Lo. 15 deg. 38'
49' W. The mean of a series of observations of (_) was 39 deg. 05' 40 deg.. IE--01'
00". HE 27 ft. WT 3h 4m 49s. C--W 1h 39m 55s. C.C. fast, 01m 52s.
Required Line of Position by Marc St. Hilaire Method, and most probable
position of ship.
3. Oct. 21st, 1919, A.M. D. R. Lat. 40 deg. 12' 38" N, Lo. 69 deg. 48' 54" W.
The mean of a series of observations of (_) was 19 deg. 21' 20". IE + 02' 10".
HE 26 ft. WT 7h 58m 49s. C--W 4h 51m 45s. C. slow, 03m 03s. Required Line
of Position by Marc St. Hilaire Method and most probable position of ship.
4. June 1st, 1919, P.M. at ship. Lat. D. R. 35 deg. 26' 15" S, Lo. 10 deg. 19'
50" W. W.T. 3h 30m 00s. C--W 0h 20m 38s. CC 1m 16s slow. (_) 16 deg.
15' 40". IE + 2' 10". HE 26 ft. Required Line of Position and most probable
fix of ship.
5. Jan. 5th, 1919. A.M. D. R. Lat. 36 deg. 29' 38" N, Lo. 51 deg. 07' 44" W.
The mean of a series of observations of (_) was 23 deg. 17' 20". IE + 01' 50".
HE 19 ft. WT 7h 11m 37s. C--W 5h 59m 49s. C. slow 58s. Required Line of
Position and most probable fix of ship.
WEEK V--NAVIGATION
TUESDAY LECTURE
A SHORT TALK ON THE PLANETS AND STARS IDENTIFICATION OF STARS
_1. The Planets_
You should acquaint yourself with the names of the planets and their
symbols. These can be found opposite Page 1 in the Nautical Almanac. All
the planets differ greatly in size and in physical condition. Three of
them--Mercury, Venus and Mars--are somewhat like the earth in size and
in general characteristics. So far as we know, they are solid, cool
bodies similar to the earth and like the earth, surrounded by
atmospheres of cool vapors. The outer planets on the other hand, i.e.,
Jupiter, Saturn, Uranus, and Neptune, are tremendously large--many times
the size of the earth, and resemble the sun more than the earth in
their physical appearance and condition. They are globes of gases and
vapors so hot as to be practically self luminous. They probably contain
a small solid nucleus, but the greater part of them is nothing but an
immense gaseous atmosphere filled with minute liquid particles and
heated to an almost unbelievably high temperature.
Of the actual surface conditions on Venus and Mercury, little is
definitely known. Mercury is a very difficult object to observe on
account of its proximity to the sun. It is never visible at night; it
must be examined in the twilight just before sunrise or just after
sunset, or in the full daylight. In either case the glare of the sun
renders the planet indistinct, and the heat of the sun disturbs our
atmosphere so as to make accurate visibility almost impossible. The
surface of Mercury is probably rough and irregular and much like the
moon. Like the moon, too, it has practically no atmosphere. Mercury
rotates on its axis once in 88 days. Its day and year are of the same
length. Thus the planet always presents the same face toward the sun and
on that side there is perpetual day while on the other side is
night--unbroken and cold beyond all imagination.
Venus resembles the earth more nearly than any other heavenly body. Its
diameter is within 120 miles of the earth's diameter. The exasperating
fact about Venus, however, is that it is shrouded in deep banks of
clouds and vapors which make it impossible for us to secure any definite
facts about it. The atmosphere about Venus is so dense that sunlight is
reflected from the upper surface of the clouds around the planet and so
reaches our telescopes without having penetrated to the surface at all.
From time to time markings have been discovered that at first seemed
real but whether they are just clouds or tops of mountains has never
really been established.