Chapter 4
4. Use this method only when steaming approximately an East and West course.
For an example of this method, see Bowditch, p. 77, example 3.
THURSDAY LECTURE
EXAMPLES ON PLANE AND TRAVERSE SAILING (_Continued_)
1. Departure taken from Cape Horn. Lat. 55 deg. 58' 41" S, Lo. 67 deg. 16' 15" W, bearing by compass SSW 20 knots. Ship heading SW x S, Deviation 4 deg. E, steamed the following courses:
+----------+------------+-----------+---------- C. Cos. | Wind | Leeway | Deviation | Distance ---------+----------+------------+-----------+---------- SW x S | SE | 1 pt. | 4 deg. E | 40 WNW | N | 2 pts. | 5 deg. E | 25 S 40 deg. E | NE | 2 pts. | 4 deg. W | 20 ---------+----------+------------+-----------+----------
_Remarks_
Variation 18 deg. E throughout. Current set NW magnetic 30 mi. for the day. Required Latitude and Longitude in and course and distance made good.
2. Departure taken from St. Agnes Lighthouse, Scilly Islands, Lat. 49 deg. 53' S, Lo. 6 deg. 20' W, bearing by compass E x S, distance 18 knots, Deviation 10 deg. W, Variation 23 deg. W. Ship headed N steamed on the following courses:
+------+-------+-------+------+------------------------------ C. Cos.| Wind |Leeway |Devia- |Dis- | Remarks | | |tion |tance | -------+------+-------+-------+------+------------------------------ N | | .. | 10 deg. W | 60 |Variation 23 deg. W. Current set S 1/2 E| W | 3 pts.| 10 deg. E | 40 |SE mg 1-1/2 miles for 24 hrs. NNE | NNW | 2 pts.| 8 deg. W | 45 |Req. Lat. and Lo. in and | | | | |course and distance made | | | | |good, -------+------+-------+-------+------+------------------------------
Assign for Night Work the following articles in Bowditch: 179-180-181-182. Also additional problems in Dead Reckoning.
FRIDAY LECTURE
MERCATOR SAILING
This is a method to find the true course and distance between two points. The method can be used in two ways, i.e., by the use of Tables 2 and 3 (called the inspection method) and by the use of logarithms. The first method is the quicker and will do for short distances. The second method, however, is more accurate in all cases, and particularly where the distances are great. The inspection method is as follows (Put in your Note-Book):
Find the algebraic difference between the meridional parts corresponding to the Lat. in and Lat. sought by Table 3. Call this Meridional difference of Latitude. Find the algebraic difference between Longitude in and Longitude sought and call this difference of Longitude. With the Meridional difference of Latitude and the difference of Longitude, find the course by searching in Table 2 for the page where they stand opposite each other in the latitude and departure columns. Now find the real difference of latitude. Under the course just found and opposite the _real_ difference of Latitude, will be found the distance sailed in the distance column. Example:
What is the course and distance from Lat. 40 deg. 28' N, Lo. 73 deg. 50' W, to Lat. 39 deg. 51' N, Lo. 72 deg. 45' W?
Lat. in 40 deg. 28' N Meridional pts. 2644.2 Lat. sought 39 51 N Meridional pts. 2596.0 --------- ------ 0 deg. 37' Mer. diff. Lat. 48.2
Lo. in 73 deg. 50' W Lo. sought 72 45 W --------- 1 deg. 05' = 65'
On page 604 Bowditch you will find 48.7 and 64.7 opposite each other, and as 48.7 is in the Lat. column only when you read from the bottom, the course is S 53 deg. E. The real difference of Lat. under this course is opposite 62 in the distance column. Hence the distance to be sailed is 62 miles.
If distances are too great, divide meridional difference of Lat., real difference of Latitude and difference of Longitude by 10 or any other number to bring them within the scope of the distances in Table 2. When distance to be sailed is found, it must be multiplied by the same number. For instance, if the difference of Lat., difference of Lo., etc., are divided by 10 to bring them in the scope of Table 2, and with these figures 219 is the distance found, the real distance would be 10 times 219 or 2190.
Now let us work out the same problem by logarithms. This will acquaint us with two new Tables, i.e., Tables 42 and 44. Put this in your Note-Book:
Lat. in 40 deg. 28' N Mer. pts. 2644.2 Lo. in 73 deg. 50' Lat. sought 39 51 Mer. pts. 2596.0 Lo. sought 72 45 --------- ------ ------- Real diff. 0 deg. 37' 48.2 1 deg. 05' 60 -- 60 5 -- (Table 42) log (+ 10) 65 = 11.81291 Log 48.2 = 1.68305 -------- Log tan TC (Table 44) 10.12986 TC = S 53 deg. 26' E
Log sec TC (53 deg. 26') = 10.22493 Log real diff. Lat. = 1.56820 + -------- 11.79313 - 10. -------- 1.79313
Distance (Table 42) = 62.11 miles
Find algebraically the real difference of latitude, meridional 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:
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.
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.