Thursday, February 19, 2009

Celestial Navigation Notes

This was designed to help you prepare to pass the Coast Guard exam on celestial navigation oceans, or if you are just interested in learning celestial navigation. You do not need a sextant to work these problems or the ones on the exam. Learning to use the sextant is a personal skill that is acquired by practice. The Coast Guard test consists 10 multiple choice questions with a passing score of 90%.

Corrections to Sextant, Time, Sunrise, Sunset, Twilight, Moonrise, Moonset, Finding GHA and Declination, Assumed Position and Local Hour Angle, Computed Altitude and Azimuth, Amplitudes, Interpolation, Altitude Intercept, Using Position Plotting Sheets, Plotting Lines of Position (Sun, Moon, Stars) LAN, Running Fixes.

You will need Pub. 229 (15-30) and a reprint of the 1981 Nautical Almanac, plotting sheets, and a starfinder, dividers, parallel rules, triangles, and a calculator. The Coast Guard exam questions may cover:

Zone Time of Local Apparent Noon (LAN)
Local apparent noon (LAN) is a sight taken on the Sun at the instant it transits an observer's meridian. At that time, the Sun is at its highest altitude. The purpose of observing the Sun at LAN is that it allows you to establish your latitude. The purpose of determining watch time of LAN is to enable you to arrive on the bridge within a few minutes of the time you should take your sight. Since zone time of LAN is based upon your dead reckoning (DR) longitude, the exact moment it transits over your meridian will seldom be precisely calculated, but you should be able to figure it very closely. The Greenwich Mean Time of meridian passage, or GMT Mer. Pass, is found on the lower right daily pages of "The Nautical Almanac" just below the sunset table. This is also the time that the sun will be over any standard meridian, no matter what zone you maybe in. Be sure that you use the correct page for the appropriate date. To compute the time of LAN, use a step by step process using the work form shown below. In the example that follows, we will calculate the 1st estimated time of LAN.

Example: Your ships 0700 DR position on 24 January 1981 is Lat. 22°25.0 N, 46°10.0 W. Your vessel is course 110°T at a speed of 12.0 knots. What is the zone time of local apparent noon? Here are the steps to take to compute the first estimate of LAN.
Step 1: Enter the appropriate date.

Step 2: Go to ''The Nautical Almanac" and extract the time of meridian passage for 24 January April 1981.

Step 3: Set up a universal plotting sheet. Plot your 0700 DR position and DR ahead 1212.

5 hours 12 minutes at 12 knots = 62.4 miles

Step 4: Enter your DR longitude for the time of meridian passage you extracted from the almanac (1212).

Step 5: Enter the nearest standard meridian.

Step 6: Calculate the difference in arc between the standard meridian and your DR longitude.

Step 7: Convert the arc in step 5 into time using the "Conversion of Arc to Time" from the Nautical Almanac.

Step 8: Now apply the time correction to the time of meridian passage from the almanac. Because we are west of our standard Step meridian, we must add the difference. By applying this correction you will have the time of LAN at the meridian for your DR position. This is called the first estimate of LAN. Note: if you clocks are set to daylight savings time (DST), add one hour to your zone time or use the standard meridian that your clocks are set to.

Date 24 Jan. 1981
Meridian Passage 1212
DR Longitude 45°11.5 W
Standard Meridian 45°
difference of longitude (arc) 0° 11.5
difference of longitude (time) + 0m 46s
ZT LAN (1st) Estimate) 12h 12m 46s = 1213

The Sextant Use and Adjustment
A sextant is not difficult to use but it does take practice to get a sight quickly and accurately, especially aboard a bouncing vessel. The instrument is held vertically in the right hand and the sighting is made through the telescope. The horizon is observed in the horizon glass while the celestial object is found in the mirror and positioned such that it is in line with the horizon. In the case of the Sun or Moon, the edge of the disk is placed on the horizon. If the lower edge is used, the sight is referred to as a lower limb sight. An upper limb sight is less often used with the Sun but is often necessary with the Moon since the lower edge may not actually be a circular one, depending on the phase.

Once the body is lined up properly, the sextant is "rocked" or pivoted as if the top of the index arm were attached to the rod of a pendulum and the arc were at the bottom with the swinging action. This is done to insure that the sextant is held vertically when the sight is taken. As the rocking is done, the celestial body will seem to trace an arc with respect to the horizon. The sextant is vertical or plumb when the body is at the bottom of the arc. The sight is then marked, the observer says "mark" to his timekeeper or observes the time himself.

The angular height of the celestial body is read on the arc and on the micrometer drum. The arc displays the degrees whereas the drum displays the minutes and tenths of minutes (or in some cases minutes and seconds). An arrow on the index arm points to the degrees on the arc. The degree is chosen that rests just to the right of the arrow. If the arrow pointing to the micro­meter drum lies between two minutes, an estimation is made as to how many tenths of the way between it is or sometimes a vernier is available on the index arm for that purpose.

There are several techniques of getting the celestial body in the field of view, an important step in sextant use that I skipped over quickly a couple of paragraphs ago. In sighting the Sun, assuming reasonably good sea conditions, the observer can get the horizon under the Sun in the glass and then move the index arm back and forth, homing in on the glare surrounding the Sun until the Sun's disk is seen. Filters will be needed in front of the index mirror to protect the eye from the Sun's brightness. Also, filters may be necessary in front of the horizon glass if the Sun's sparkle on the water is too bright. A second method is useful for non-glaring objects such as the planets, stars and daytime Moon. Hold the sextant upside down in the left hand and sight through the glass toward the celestial object. Then move the index arm until the horizon appears in the mirror. The advantage in this method is that it is easier to find the celestial object by direct observing and leave the easily found horizon line for the moving mirrors. Once the object is reasonably well lined up with the horizon, the sextant is turned right side up and the final adjustments with the micrometer drum are made.

If some mathematical calculations are made ahead of time, the rough altitude of the celestial body can be figured allowing a third method to be used. This involves presetting the sextant to the prefigured altitude and then scanning the horizon with the horizon glass until the celestial body comes into the field of view of the mirror. The rough azimuth of the body can also be prefigured so that the area of scanning can be limited. For this method, the sextant would be held right side up the whole time. Some practical hints on using the sextant are in order especially if the instrument represents a considerable investment and happens to be the only sextant aboard. A lanyard attached to the sextant and to the observer saves accidental dropping of the instrument, either to be damaged on the deck or to be lost to Davy Jones Locker. Wrapping oneself around the shrouds when taking a sight over the rail saves the navigator from the same fates.

Sighting when the ship gets to the top of a wave is important to insure that the real sea horizon is used rather than the closer top of a nearby wave. The real sea horizon can vary in distance depending on the height of the observer's eye but corrections for this can be made. Once the sextant is obtained, adjustments to the mirrors may be necessary to reduce the index correction to a minimal amount. One or two adjusting screws are located on each mirror for this purpose. Each mirror should be perpendicu­lar to the sextant frame and when the sextant is set at zero the two mirrors should be parallel to each other. Three tests are involved. The first test is for perpendicularity of the index mirror. Hold the sextant on its side (with handle down) and with the index arm set to 35°. Place your eye close to the sextant near the index mirror so that you can see the sextant arc in the mirror (reflected) and also just to the right of the mirror (direct). If these two images are not in a straight or continuous line, the mirror is not perpendicular to the frame. Adjusting the screws will bring the images in line.

The second test is for perpendicularity of the horizon glass. Actually, the "glass" is only half glass with the right half of the frame filled with a mirror. The horizon is viewed through the glass, the reflected image of the celestial object viewed in the mirror. If this horizon glass is not perpendicu­lar to the frame, the error is referred to as side error. If a star is viewed both in the glass and in the mirror with the sextant set near zero, by adjusting the altitude, the star should pass over itself, become superimposed. If instead the reflected image of the star passes to the right of the direct image, side error exists and can be minimized by adjusting the two screws at the base of the horizon glass. Other celestial bodies may be used for this test as well as reasonably distant terrestrial objects.

The third test is for parallelism of the index mirror and horizon glass when the index arm is set exactly at zero. If at this setting the horizon or a celestial body appear higher or lower in the mirror than in the glass, the mirrors are not parallel and should be adjusted until they are. This error is called index error. This is an error in the sextant itself and can be found by setting the sextant to read exactly zero and observing the sea horizon, a distant mountain top (a reason­ably flat one), or a celestial object. At zero reading, the objects observed should appear the same height in the horizon glass and mirror. If this is not the case, in other words, if the horizon or object in one side is above or below that in the other side, adjust the micrometer drum or the tangent screw until the objects are level with each other. Note the sextant reading. This is the amount of index correction. If the arrow is to the left of the zero or "on the arc", the I.C. is negative. If the arrow is to the right of the zero or "off the arc", the I.C. is positive. An easy way to remember this, though at first confusing, is to memorize. If it's on, it's off. If it's off, it's on. With a plastic sextant, the index correction should be ascertained for each set of sights since plastic will expand and contract with varying temperatures and will have different instrument errors. With a brass or aluminum framed instrument, the index correction should always be the same barring tampering with the mirrors or dropping the instrument.

Errors and Adjustments of the Sextant
The sextant is subject to a number of errors and adjustments. To find the true altitude of a celestial body from the observed these must be allowed and adjusted for. These are:
1. Index Error
2. Dip
3. Refraction
4. Parallax
5. Semi-diameter

Index error is an instrumental error. When looking through a sextant at the horizon the exact level will seldom be seen to be at 0°. Before you use the sextant the index error should be determined. If the error is less than 0° it should be added to whatever reading is obtained, if more subtracted.
1. If its off, its on, ADD.
2. If its on, its off SUBTRACT.

Refraction is extracted from the Nautical Almanac. It allows for the bending of light rays as they travel through layers of varying density air. Parallax corrections are needed if the observed body is a planet, the sun or the moon, from the almanac. Semi-diameter correction is needed if the observed body is the sun or the moon. In this case either the top or bottom of the celestial object (upper or lower limb) is made to touch the horizon. To obtain the center of the body this correction is applied, from the Almanac. Once all the corrections are applied we have the true altitude. And this subtracted from 90 gives you the zenith distance. Which means you know exactly how far you are from that point on the earth which is at right angles to your observed celestial body. Remember the more sights you take the better you will get, so get lots of practice.
The objective of this is to describe the procedure to use the Rude Starfinder and to be able to identify an unknown body, plot the planets on the Starfinder, and to select the best stars or planets for a fix. To solve the navigational triangle for a computed altitude and true azimuth, the navigator must know beforehand or be able to determine afterwards the name of the celestial body observed, so that you can obtain its GHA and declination from the Nautical Almanac. Several aids are available to the navigator to assist in identifying and locating celestial bodies, among which is the starfinder. Starfinders are intended to furnish the approximate altitude and true azimuth of celestial bodies either before or after navigational observations. One of the best and most common is the Rude Starfinder.

Star Identification
As a navigator, you may be required to obtain a fix from two or more stars. Actually, only a few of the of stars are used regularly for celestial navigation, and they are not too difficult to locate and identify. No matter where you may be navigating, you can manage very well if you are able to recognize 20 or so. The Nautical Almanac consists of 57 principal stars as well as tables for finding latitude by the North Star (Polaris). Relative brightness of stars is called magnitudes the lower the magnitude, the brighter the star. Sirius, brightest of them all, has a magnitude of - 1.6, Acamar, dimmest of the navigational stars, is listed at + 3.1 magnitude.

First magnitude stars range from magnitude - 1.6 to magnitude +1.50.
Second magnitude stars are those from +1.51 to +2.50.
Stars of third magnitude range from +2.51 to +3.50, and so on. Stars of the sixth magnitude are barely visible to the unaided eye. The magnitudes given here of principal stars are only a fraction of the navigational celestial bodies. Selected navigational planet magnitudes vary due to atmospheric conditions. Mars magnitude, for example, varies from + 1.6 to -2.8. The moon usually has a magnitude of 12.6, however, its "phase" must be considered prior to use. The king of celestial bodies, with a magnitude of -26.7, is the sun, limited only by nighttime and atmospheric conditions. The magnitude of the planets is listed at the top of daily pages and stars at the end of the white pages in the Nautical Almanac.

One or more of the stars in a constellation may be navigational stars. Obviously, if you can recognize a constellation and know which of its stars may be used, you can identify them whenever the group is visible in the sky. The stars and constellations that might be familiar to you are not always visible from where you may happen to be. For this reason, you must have some means of identifying navigational bodies when nothing you know by sight can be seen overhead. One method by which you can identify those celestial bodies is to use the Star Finder. The Rude Starfinder consists of the star base, an opaque white plastic circular base plate fitted with a peg in the center, and ten circular transparent templates. On one side of the star base the north celestial pole appears at the center, and on the other side the south celestial pole.

Description of the Rude Starfinder
The Rude Starfinder, 2102-D, is designed to permit the determination of the approximate apparent altitude and azimuth of any of the 57 selected navigational stars tabulated in the Nautical Almanac. All of the 57 navigational stars are shown on each side at their positions relative to the appropriate pole in a type of projection called an azimuthal equidistant projection. In this projection, the positions of the stars relative to one another are distorted, but their true declinations and azimuths relative to the pole are correct, the pattern of the stars on the star base does not correspond to their apparent positions as seen in the sky. Each star on the base is labeled, and its magnitude is indicated by its symbol, a large heavy ring indicates first magnitude, an intermediate sized ring second magnitude, and a small thin ring third magnitude. The celestial equator appears as a solid circle about four inches in diameter on each side of the star base, and the boundary of each side is graduated to a half-degree of LHA of Aries. There are 10 templates included for use with the star base. Nine of these are printed with blue ink and are designed for apparent altitude and azimuth determinations, while the tenth, printed in red ink, is intended for the plotting of bodies other than the 57 selected stars on the base plate. There is one blue template for every 10° of latitude between 5° and 85°, one side of each template is for use in north latitudes, the other for south latitudes. Each of these "latitude" templates is printed with a set of oval blue altitude curves at 5° intervals, with the outermost curve representing the observer's celestial horizon, and a second set of radial azimuth curves, also at 5° intervals. The red template is printed with a set of concentric declination circles, one for each 10° of declination, and a set of radial meridian angle lines. The appropriate template is snapped in place on the star base.

Using the Star Finder
The star finder may be used either to:
1. Identify an unknown body whose altitude and azimuth have been observed.
2. Make a list of the stars and planets available for observation at morning or evening twilight for a fix. To use the star finder, first determine GHA of Aries for your time of observation from the Nautical Almanac. Next, determine LHA of Aries by subtracting your longitude from GHA of Aries if in west longitude or by adding your longitude of GHA Aries if in east longitude. Select the template nearest your DR latitude and place it on the northern or southern base, depending on whether you are north or south of the equator. Ensure that the proper side of the template is up, north to north, south to south. Rotate the blue template until the 000° to 180° arrow on the template is over the LHA Aries on the base plate. The stars or planets available to you at that time, will be under the grid system of your blue template. DIRECTLY OVERHEAD (Zenith) then is represented by the cross at the center of the open space on the template. The sky overhead or dome is now shown in the part of the base covered by the curves on the template. The approximate azimuth and altitude of any navigational star within these curves can be found by following the lines on the template.

Finding an Unknown Star or Planet
After a long period of heavy weather, you may see the navigator out on the bridge wing scanning the heavens, his sextant in hand. He is hoping that the overcast will break long enough for him to have a shot at even a single star. If the navigator should manage to pull a star down, the star's identity may not be known. This is where one uses the star finder. An azimuth (bearing) of the star should be taken at the instant of observation. When the correct template is oriented properly on the star base, the name of the star can be read at the intersection of the azimuth and altitude lines on the grid. This Star Finder is designed to help locate and identify, by altitude and azimuth, the 57 stars listed in the Nautical Almanac or any other celestial bodies that may be plotted on the star base. Because the unit uses an Azimuthal Equidistant Projection, it can not be compared directly with the heavens due to distortion. The complete unit consists of one star base, ten templates, and instructions.

To Find or Identify Celestial Bodies
1. Convert GHA Aries to LHA Aries by subtracting DR longitude if west, or adding DR longitude if east. When this answer is negative add 360°, or if the answer is over 360° subtract 360°.
2. Select blue-line template for latitude nearest your DR position. Center selected template over star base so that template and star base both conform to hemisphere (N or S) of observer. Rotate the template until arrow is over LHA Aries. The approximate altitudes and azimuths of celestial bodies above the horizon are then indicated by the curves. To plot the Sun, Moon, Planets, or additional Stars From the Nautical Almanac, determine the body's declination and right ascension (RA). The body's RA is obtained by: When GHA body is zero, GHA Aries equals RA. Center red-line template over star base, use correct hemisphere on both, then rotate until arrow (0°) points to RA body. If the body's declination is the same as the hemisphere in center of base, then position will be plotted towards center from celestial equator. If declination is opposite, then position will be plotted away from celestial equator towards edge of base. With a pencil through the cut-out slot, mark the body's declination.

Identifying Unknown Bodies
Using the appropriate blue-line template and base side, align index arrow to LHA Aries for the time of sighting. Locate intersection of altitude and azimuth of shot. If no star is near intersection, the body may be a planet or unmarked star. Keeping blue-line template in place, put red-line template on top and rotate until the cut-out slot is over the altitude / azimuth intersection of sight. Determine declination and SHA of body, then refer to Almanac for identification.

Star selection problems are one of the more difficult and time consuming of the problems found on USCG license exams. In star selection problems, you have four choices of combinations of three stars, planets, and or the moon in each problem. You must determine which of the four combinations would result in the most reliable fix for the date, time, and DR position. In making a choice the following three things must be considered, listed from highest priority to least priority:

First Priority - The azimuth differences between the bodies should be sufficient to give a reliable fix. The ideal azimuth spread would be for the bearings of the bodies to differ by 120 degrees.
Second Priority - The bodies should be at altitudes between 15 degrees and 70 degrees. Unusual refraction can introduce large errors in low altitude sights, and accurate sights at very high altitudes are difficult to obtain.

Third Priority - The magnitude of the star. Obviously, first magnitude stars are easier to see and to shoot while the horizon is still clearly defined. In analyzing the four choices the following should be avoided.
1. Two of three bodies are very close together in bearing.
2. All three bodies azimuths fall within the same 180 degrees of bearing.
3. Two of the bodies are reciprocal in bearing or nearly so.

To make a list of stars and planets available for observation at morning or evening twilight for a fix, you would setup the starfinder with the LHA (Aries) and have the planets plotted. Then use the following guide lines for selection of bodies:
1. Altitude 10° to 65°
2. 1st magnitude
3. Bearing 120° apart
4. Always try to get Polaris, it gives you a latitude line of position if you are in the northern hemisphere, and any body who has a bearing of 000° - 180° will do the same thing.
5. To check your course, select bodies with bearings perpendicular to your course.
6. To check your speed, select bodies that are parallel to your course.
In working these Coast Guard exam problems, some groups can be eliminated if one of the bodies in the group is below the horizon for that observer's DR position and time of star time.

Here is a Coast Guard exam problem.
On 23 July 1981, your 1700 zone time DR position is Lat. 27° 29.0' N, Long. 129° 26.0' W. You are on course 079° T at a speed of 20 knots. Considering their magnitude, azimuth, and altitude, which group includes the three bodies best suited for a fix at star time?

A. Arcturus, Jupiter, Denebola
B. Spica, Sabik, Vega (Answer)
C. Antares, Polaris, Altair
D. Jupiter, Saturn, Polaris

Step 1: Compute the ZT of evening twilight or star time (approx. 1857), use this DR position for your form.

Step 2: Compute the LHA (Aries) and the RA and Declination of the planets.

Step 3: Plot the planets on your starfinder, and then place your blue template over the star base on your LHA (Aries). This will be the stars and planets available at star time.

Step 4: Find the best answer. I use a diagram to help me locate the answers.

Latitude by Polaris
A second magnitude star called Polaris (north star) provides a reference for measurement in the Northem Hemisphere. Polaris has no counter part in the Southern Hemisphere. Polaris may be located in the northern sky between the constellation Ursa Major (big dipper) and Cassiopeia. The two stars in the bowl of the dipper at the greatest distance from the handle, point toward the north star. Polaris travels in a diurnal circle of a small radius around the celestial north pole. This causes a special circumstance in celestial navigation. If you think of the four arguments in celestial navigation, Ho, Lat, Dec, and LHA. Polaris has some constants that make it a special case in celestial navigation. The declination of Polaris is 90° North, and as long as your are in the northern hemisphere, your LHA will be 0°. Which means, there are only two arguments left, Ho and Latitude. For Polaris, the Nautical Almanac has a special (Polaris) table at the end of the white pages.

You can determine your latitude in the Northern Hemisphere by observing the Hs of Polaris, at a known time. From the time, and the DR or estimated longitude, compute the LHA of Aries. Correct Hs to Ho, and using the LHA Aries, approximate latitude, and date, determine corrections from Polaris tables ao, a1, and a2. Add total correction to Ho, and subtract 1 degree to obtain latitude. An good example is in the in the Nautical Almanac down at the bottom of the page.

Amplitudes of the Sun
An amplitude is a bearing of the sun or other celestial body taken when the body is either rising or setting. The amplitude angle is the angle between the east point (090°) of the horizon and the body at rising, between the west point (270°) of the horizon and the body at setting, The amplitude angle is converted into true azimuth and then lets you find gyro compass error or deviation. When a celestial body is actually on the celestial horizon, it appears to be above the visible horizon. In the case of the sun, the lower limb is two-thirds of the sun's diameter above the visible horizon when the sun's center is on the celestial horizon. The reason for this is the refraction of the earth's atmosphere, the atmosphere bends the line of sight making bodies appear higher than they are. When the sun is seen half-way above the horizon it is actually below the horizon.

The celestial horizon differs from the one you see (the visible horizon) because it runs through the center of the earth. There are a lot of computations that must be done to determine the celestial horizon of a body, but for now we will just say that it is the horizon that a navigator uses for all celestial computations. When the center of the sun is on the celestial horizon, its lower limb (lower edge) is about two-thirds of the diameter of the sun above the visible horizon. When planets and stars are on the celestial horizon, they are a little more than one sun diameter above the visible horizon. If the amplitude were taken on the visible horizon, the compass bearing would require a correction from Table 28 in Bowditch (Green Book).

The amplitude of a body can be taken directly from table 27 of Bowditch (Green Book), volume II, if the body is observed when its center is on the celestial horizon. First of all, to observe the sun when it is on the celestial horizon, its lower limb should be about two-thirds of the diameter above the visible horizon, note the time and your compass bearing as observed by a bearing or azimuth circle to the sun. Next, with the Greenwich Mean Time (GMT) of your observation, you can use the right-hand daily pages of the Nautical Almanac to determine the sun's declination. From this known information, you can use table 27 of Bowditch to determine the amplitude.

Amplitude on the Visible Horizon
When the center of the sun is on the visible horizon, its center is below the celestial horizon. Since table 27 of Bowditch is used when the sun's center is on the celestial horizon, you must make additional corrections to an observation of the sun on the visible horizon. The corrections come from table 28 of Bowditch. The values needed to enter the table are latitude and declination. The correction obtained from this table is applied to the amplitude that is taken from table 27. The correction should be applied to the observed amplitude (observed bearing, the bearing off the compass). Table 27 allows you to find the true bearing. So what do you do with this correction? Is it added or subtracted? At the bottom of the table is the rule. For the sun, a planet, or star apply the correction to the observed amplitude in the direction away from the elevated pole. For the moon apply half the correction toward the elevated pole. First, you must understand the term "elevated pole" if your latitude is North, then your elevated pole is North, likewise, if your latitude is South, then your elevated pole is South. Secondly, to apply the correction away from the elevated pole, you have the following set of rules:

1. Rising sun, north latitude - add table 28 correction.
2. Setting sun, north latitude - subtract table 28 correction.
3. Rising sun, south latitude - subtract table 28 correction.
4. Setting sun, south latitude - add table 28 correction.

Azimuth of the Sun
Computation of compass error at sea depends upon the observation of the azimuth of celestial bodies. The Sun is the most commonly used for this purpose. The observed azimuth is recorded, the time (to the nearest second) and the DR position are also noted. With DR position and time, the navigator computes Zn by using the Nautical Almanac and PUB 229 Sight Reduction Tables. The difference between pgc bearing and Zn (true bearing) is the gyro error (G.E.), and the difference between psc bearing and the magnetic bearing is the deviation. It should be appropriately labeled. Keep in mind that accuracy depends on the navigator's knowledge of position and the correct time. In taking a azimuth of a celestial body, the azimuth circle is used. A azimuth circle is a nonmagnetic metal ring sized to fit on a 7-inch compass bowl or on a gyro repeater. The inner lip is graduated in degrees from 0° to 360° in a counterclockwise direction for the purpose of taking relative bearings. Two sighting vanes (the forward or far van containing a vertical wire, and the after or near vane containing a peep sight) facilitate the observation of bearings and azimuths. Two finger lugs are used to position the instrument exactly while aligning the vanes. A hinged reflector vane mounted althe base and beyond the forward vane is used for reflecting stars and planets when observing azimuths.

Beneath the forward vane a reflecting mirror and the extended vertical wire are mounted, enabling the navigator to read the bearing or azimuth from the reflected portion of the compass card. For observing azimuths of the Sun, an additional reflecting mirror and housing are mounted on the ring, each midway between the forward and after vanes. The Sun's rays are reflected by the mirror to the housing where a vertical slit admits a line of light. This admitted light passes through a 45° reflecting prism and is projected on the compass card from which the azimuth is directly read. In observing both bearings and azimuths, two spirit levels, which are attached must be used to level the instrument. A azimuth is similar to a amplitude but it is taken at anytime, not at sunrise or sunset. When taking a azimuth it requires the use of Pub. 229 Sight Reduction Tables for Marine Navigation to obtain the Zn (true bearing).

Pub 229 Sight Reduction Tables for Marine Navigation
Pub. 229 Sight Reduction Tables for Marine Navigation is a set of six volumes of precalculated solutions for the computed altitude (Hc) and the azimuth angle (Z) of the navigational triangle. Each volume covers a 15 degree band of latitude with a 1° overlap occurring between volumes. When taking the Coast Guard exam you will be using Volume 2 - Latitudes 15° - 30°.

Entering arguments for the tables are local hour angle (LHA), latitude, and declination expressed in whole degrees. Values of Hc and Z are tabulated for each whole degree of each of the entering arguments. Tables inside the front and back covers of each volume allow for interpolation. Each volume contains two sets of tabulation for whole degrees of LHA between 0° and 360° The front half is for the first eight degrees of latitude (15° - 22°) covered by that volume, and the second half is for the remaining eight degrees of latitude (22° - 30°). The values of LHA are at the top and bottom of each page. The eight degrees of latitude form the horizontal argument and the declination is the vertical argument. Instructions at the top and bottom of each page indicate whether the tabulations on that page are for the latitude which is the same or contrary to the declination:

If both latitude and declination are north or both south, same name page. If they are of opposite names, north and south or vice versa south and north contrary page. The normal practice of navigation at sea is that the ship's compasses be checked frequently, it has been a custom to check for compass error at least once a day. There are two main celestial navigation methods of determining compass error, which are azimuths and amplitudes.

Azimuth observations are simply bearings taken of celestial bodies using one of the ship's compasses. Normally, it is best to take an azimuth when the body's altitude is less than 20 degrees. Azimuths may be taken of any celestial body but the sun is preferred because it is the easiest to observe. The sight reduction calculations for solving azimuths are very similar to determining computed altitude (Hc) and azimuth (Zn) when solving a line of position sight.

Sunrise and Sunset
The navigator uses morning and evening twilight for star observations because during twilight the darkness makes the stars visible, yet permits sufficient light to define the horizon. Both conditions are necessary if an accurate sextant altitude (Hs) is to be obtained. There are four stages of twilight, based upon the position of the sun with respect to the horizon. They are:
Astronomical Twilight - The sun is 18° below the horizon, too dark for observations.
Nautical Twilight - The sun is 12° below the horizon, favorable for observations, recorded in Nautical Almanac.
Observational Twilight - The sun is 10° below the horizon, best for observations.
Civil Twilight - The sun is 6° below the horizon, too light for observations, also recorded in the Nautical Almanac.

It is necessary to know the times of sunrise and sunset because it allows the navigator to take a amplitude (bearing) of the sun to help check his deviation or gyro error of the compass. Also, when to turn on and off your running lights. Twilight is of importance because during these periods of time the navigator will be taking sights. You will need to know how to determine the most advantageous time for taking sights. Basically, all three times depend on the observer's position with respect to latitude, longitude, and the location (east or west) of the nearest standard time meridian. Remember that the times shown for rising and setting of the sun and the moon, and twilight in ALL publications are standard times. If that zone is keeping Daylight Savings Time (DST) it must be accounted for after the final time is figured.
Now the question arises, where to look for this information? you may find it in at least three places. The Nautical Almanac, the Air Almanac, or you may use the SunriseSunset table (Table 4) in the back of the Tide Tables. The information is listed in tabular form in columns and unless the position being determined for is on one of the parallels listed and on one of the standard time meridians, interpolation must be carried out before the exact time desired may be obtained.

Sunrise and sunset are computed in these tables as follows: The American Nautical Almanac computes SR and SS once for three days for each 10 of latitude from the equator to 30 North and South. Then each 5 to 50 North and South. Then after, each 2 of latitude to 60 South and to 72 North. Information for computing sunrise, sunset, morning and evening twilight is located on the daily pages of the Nautical Almanac.

Interpolation Tables
The interpolation tables are located on the inside and back side of every Pub 229. The front part of the table covers DECLINATION INCREMENTS from 0.0' to 31.9'. The back covers DECLINATION INCREMENTS from 28.0' to 59.9'. Instructions on how to use the table is listed on page xi and xii. The entering arguments for the interpolation tables are the declination increment (Dec. Inc.) the remaining minutes and tenths of the exact declination, and the altitude difference (d) between the two tabulated Hc's bracketing the exact declination. The value of "d" between successive tabulated Hc's is pre-calculated and appears in the center of each column of tabulations. If Hc decreases in value with decreasing declination the sign of the altitude difference is negative. If Hc increases with increasing declination the sign is positive. The correction is extracted from the interpolation tables in two increments, one for the tens of minutes and the other for the remaining units and tenths. Adding the two parts together gives the total correction. The total correction is added algebraically to the tabulated Hc to obtain the final computed altitude.

Double Second Difference Correction
In some instances a third increment is necessary. It is called the "double second difference" correction (DS corr.). When the DS correction is necessary, it will be indicated in the tables by the "d" value being printed in italic type followed by a dot. An example is given on page xiv of Pub 229. To find a double second difference correction, the difference between the two "d" values directly above and directly below the d value corresponding to the whole portion of the actual declination is mentally computed. Using the double second difference as an entering argument, the right-hand side of the interpolation table is used to find the correction. There are several complete DS interpolation sections on each page, opposite the original declination increment. If a DS correction is necessary, it is always added to the total of the tens and units increments to form the total interpolation correction. The total correction is applied to the Hc to obtain the final computed Hc. Using the example on page xiv.

How to Determine your Latitude by Local Apparent Noon (LAN)
The objective of this is to define how to compute a latitude line by using the sun at Local Apparent Noon (LAN). Since the latitude of a position may be determined by finding the distance between the equinoctial and the zenith, you only need to know the declination and zenith distance (coaltitude) of a body to determine latitude. This has been used by mariners for centuries because of it's simplicity. Before the discovery of the Sumner line, and prior to the Harrison chronometer, longitude was difficult to compute. Early mariners used the technique of "latitude or parallel sailing," by which they traveled north or south to the known latitude of their destination, then east or west as often using the meridian sight as their only celestial computation. The meridian sight is applicable to all celestial bodies, in practice it is mostly used with the sun.

Latitude by Meridian Sight
When the altitude of a celestial body is measured as it transits the meridian, we think of the observation and the solution for a line of position as a "meridian sight." This sight includes observations of bodies on the lower branch of the meridian (lower transit) as well as on the upper branch (upper transit). Circumpolar stars may be observed on either branch of the celestial meridian. In practice bodies are seldom observed on the lower branch, and the sun is normally the only body observed. In polar latitudes, when the declination of the sun corresponds in name to the latitude of the observer, the sun may be observed when in lower transit, but generally meridian sights of the sun are made when it is in upper transit (LAN). The meridian sight is important for these reasons:

1. It provides a celestial LOP without resorting to trigonometry.

2. The intersection of the LOP obtained at LAN, and advanced to morning sun lines establishes a celestial running fix.

3. It is practically independent of time.

4. The knowledge of the approximate position is unnecessary.

5. The LOP is a latitude line, and is useful in latitude or parallel sailing.

6. A latitude observation is obtained when the celestial body is either bearing due north or south of the observer. When reduced, this observation yields a LOP extending in an east / west direction. This is called a latitude line.

Maximum Altitude
The oldest and most common method of determining meridian altitude of the sun is known as maximum altitude. It is recommended because of its adaptability to various conditions, and because its use develops an insight into how the altitude varies near the time of apparent noon. Approximately 10 minutes before watch time of LAN, the observer sights the sun's lower limb with the horizon in the sextant. The observer then swings the sextant from side to side, and adjusts it until the sun is seen moving in an arc, just touches the horizon at the lowest part of the arc. This is called "swinging the arc." As the sun continues rising, a widening space appears between its lower limb and the horizon. Adjusting the tangent screw, the observer keeps this space closed and maintains the sun in contact with the horizon. The change in altitude becomes slower and slower, until the sun "hangs." While it is hanging, the observer swings the sextant to make certain of accurate contact with the horizon. This observation is continued until the sun dips, which is a signal that the sun is now beginning to lose altitude. The sextant then shows the maximum altitude.

Taking Numerous Sights
The method of taking numerous sights is a modification of the maximum altitude method. It is useful under conditions where heavy seas, clouds, and the like may make steady observations impossible. Well before watch time of LAN, the observer begins taking a series of altitudes. Their number depends on the difficulties of the situation and the possible error in computed time of transit. The observer reads off the altitudes to a recording assistant, turning the tangent screw slightly after each observation to ensure that the next altitude is an independent sight. Observations are discontinued when the altitude shows signs of decreasing. Under the best conditions, even a series of skillfully taken observations may show an occasional, erratic deviation from the normal gradual rise and fall. After sights showing a radical difference from the preceding or succeeding series are discarded, the hang should be evident, and it should be possible to judge the maximum altitude. The reading will probably be less than the altitude shown in one observation and more than the one below it. The result should give latitude with an error no more than 1'. This reading is more accurate than one obtained by a single sight.

Astronomical Triangle
When you calculate and plot your line of position, you are, in effect solving the astonomical triangle. Thirty-odd methods of arriving at solutions have been devised since the first edition of Bowditch appeared in 1802. Although the methods may vary, two basic idea's remain the same:

1. A single observation gives only a single line of position (LOP) which is at right angles to the azimuth (Zn) of the observed body.

2. To establish an LOP, the observer must observe time, and correct the altitude (Ho). From the Nautical Almanac, the observer must determine the declination only. LHA of the observed body is 0° or 360°. You only need the Ho, Lat, and Dec as arguments to work out the sight.

Local Apparent Noon (LAN)
The noon sight is called LAN (LOCAL APPARENT NOON), is a simple element in navigation dating back several centuries. It results in a line of position which gives you the ship's latitude. Before methods of calculating the longitude were devised, many shipmasters made port by running down the latitude of the desired landfall verified daily by their noon sight. Accuracy of the latitude obtained by LAN depends only upon the accuracy of the sun's maximum true altitude, and the accuracy with which its declination can be determined. When the sun is on the observer's meridian, the ship's latitude is the sum or difference of the declination and the distance the ship is north or south of the sun, depending upon certain rules. There are three possible situations which may be encountered in solving LAN:

1. When the latitude is greater than the declination, but of the same name.
2. When the declination is greater than the latitude, but of the same name.
3. When latitude and declination are of opposite names. Each of these cases has a different rule, and you can't do much but try to memorize each of them.

How to Use a Sextant
The marine sextant is designed to measure angles, either horizontally or vertically. The most common use of the sextant is for celestial observations using vertical angles between celestial objects and the horizon. It can also be used to measure the horizontal angle between two terrestial objects to get a line of position.

Sun Sights
For a Sun sight, hold the sextant vertically and direct the sight line at the horizon directly below the Sun. Move the shade glasses into the line of sight, move the index arm outward along the arc until the reflected image appears in the horizon glass near the direct view of the horizon. Rock the sextant slightly to the right and left to make sure it is perpendicular. As you rock the sextant, the image of the Sun appears to move in an arc. The sextant is vertical when the Sun appears at the bottom of the arc. This is the correct position for making the observation. The Sun’s reflected image appears at the center of the horizon glass, one half appears on the silvered part, and the other half appears on the clear part. Move the index arm with the drum or vernier slowly until the Sun's exactly on the horizon. This takes some practice and some people will get a error by bringing the sun too far down. Some navigators will let the body contact the horizon by its own motion, bringing it slightly below the horizon if rising, and above if setting. At the instant the body is touching the horizon, note the time. This is the uncorrected reading of the sextant.

Moon Sights
When observing the Moon, use the same procedure as for the Sun. Sights of the Moon are best made during daylight hours or twilight when it is easier to see it and the horizon. At night you can get false horizons below the Moon because the Moon illuminates the water below it.

Star and Planet Sights
Most people find the Sun and Moon are easy to find in the sextant, but the stars and planets are harder to locate because your field of view is so small. There are several ways to help you locate a star or planet.

1. Set the index arm and micrometer drum on 0° and direct the line of sight at the body to be observed. Then keep the reflected image of the body in the mirrored half of the horizon glass, swing the index arm out and rotate the sextant down. Keep the sighted image in the mirror until the horizon appears in the clear part of the horizon glass. This way is hard to do, if the body is lost while it is being brought down, you might not find it, which means you will have to start over again.

2. Direct the line of sight at the body while holding the sextant upside down. Slowly move the index arm out until the horizon appears in the horizon glass. Then invert the sextant and take the sight in the usual manner. This is my preferred method, you will have less of a chance of losing the body.

3. Determine in advance the approximate altitude and azimuth of the body by the starfinder 2102-D. Set the sextant at the indicated altitude and face in the direction of the azimuth. The image of the body should appear in the horizon glass with a little searching. When measuring the altitude of a star or planet, bring its center down to the horizon. Stars and planets have no upper or lower limb that you can see, you must observe the center of the point of light. Because stars and planets don't have a limb and because their visibility may be limited, the method of letting a star or planet intersect with the horizon by its own motion is not recommended. As with the sun and moon, rock the sextant to get the perpendicularity.

Taking a Sight
Get the altitudes and azimuths for several stars or planets when preparing to take celestial sights. Choose the stars and planets that will give you the best bearing spread. Try to select bodies with a altitude between 20° and 65°. Take sights of the brightest stars first in the evening, take sights of the brightest stars last in the morning. Sometimes weather or other ships can obscure the horizon directly below a body that you want to observe. You could try a back sight using the opposite point of the horizon as the reference. For this you must face away from the body and observe the supplement of the altitude. If the Sun or Moon is observed, what appears in the horizon glass to be the lower limb is the upper limb, and vice versa. In the case of the Sun, it is usually best to observe what appears to be the upper limb. The arc that appears when rocking the sextant for a back sight is inverted, that is the highest point indicates the position of perpendicularity. If more than one telescope is furnished with the sextant, the erecting telescope is used to observe the Sun.

The collar that the sextant telescope fits may be adjusted in or out in relation to the frame. When moved in, more of the mirrored half of the horizon glass is visible to the navigator, and a star or planet is best observed when the sky is bright. Near the darker limit of twilight, the telescope can be moved out, giving a broader view of the clear half of the glass, and making the less distinct horizon more easily seen. When measuring an altitude, have an assistant note and record the time, with a "stand-by" warning when the measurement is almost ready, and a "mark" at the moment a sight is made. If a flashlight is needed to see the comparing watch, your assistant should be careful not to interfere with the navigator’s night vision. If an assistant is not available to time the observations, the observer holds the watch in the palm of his left hand, leaving his fingers free to turn the tangent screw of the sextant. After making the observation, note the time as quick as possible. The delay between completing the altitude observation and noting the time should not be more than one or two seconds.

In celestial navigation, lines of position are rarely obtained simultaneously. This is especially true during the day when the sun may be the only available celestial body. A celestial line of position may be advanced for 3 or 4 hours, if necessary, to obtain a celestial running fix. It may also be advanced by advancing the AP in the direction and distance an amount consistent with the ship's travel during the interval between two successive observations. In the latter procedure, the azimuth line is drawn through the advanced AP without any change in direction. The advanced LOP is drawn perpendicular to the azimuth, a distance from the AP equal to the intercept, and toward or away from the GP, as appropriate.

How to Advance or Retard a Line of Position
At morning and evening twilight, the navigator may succeed in observing the altitudes of a number of celestial bodies in a few minutes and establish a celestial fix. If 2 or more minutes elapse between observations, the navigator must consider:

1. Elapsed time
2. Speed of ship
3. Scale of the chart or plotting sheet

To determine whether or not a more accurate fix can be obtained by advancing AP's to a common time. It is possible during the day to obtain a celestial fix rather than a celestial running fix if two or more of the three following bodies are visible:

1. Sun
2. Moon
3. Venus

LOP from Celestial Observations
A ship has many possible locations on a line of position. In other words, the ship's position must be somewhere along that line. A fix, by position is the intersection of two or more lines of position, but this is not the ship's exact position, because one can always assume some errors in observation, plotting. The celestial navigator must establish the lines of position by applying the results of the observations of heavenly bodies. A line of position obtained at one time may be used at a later time. All you need to do is move the line parallel to itself, a distance equal to the run of the ship in the interim, and in the same direction as the run. Such a line of position cannot be as accurate as a new line, because the amount and direction of its movement can be determined only by the usual DR methods. If two new lines cannot be obtained, however, a old line, advanced and intersected with a new one, may be the only possible way of establishing a fix. Naturally, the distance an old line may be advanced without a substantial loss of accuracy depends on how closely the run can be reckoned. In celestial navigation, as in piloting, you essentially are trying to establish the intersection of two or more lines of position. A single observation is insufficient to obtain a fix, however it can be used with a loran line, etc. to provide a fix.

Two Circles
Observation of two bodies at the same time gives the navigator two circles of equal altitude. The circles intersect each other at two points, and because the ship is somewhere on each one of them, she must be at one or the other points of intersection. A circle on the surface of the earth, on every point of which the altitude of a given celestial body is the same at a given instant, the pole of this circle is the geographical position of the body, and the great-circle distance from this pole to the circle is the zenith distance of the body.

Line of Position
In practice, you may not be able nor will you need to plot the whole of a circle of equal altitude. The position is usually known within 10 miles and possibly even less than that. Inside these limits, the curve of the arc of a circle of equal altitude is hardly perceptible, and the arc is plotted and regarded as a straight line. Such a line, comprising enough of the arc of a circle of equal altitude to cover the probable limits of a position, is called a Sumner line of position or just a line of position.

Two Lines of Position
The prefered method of establishing two lines of position is by observing two different bodies, although two lines may be obtained from the same body by observations taken at different times. As in piloting, the nearer the two lines approach at right angles to each other, the more accurate the fix. When two lines are determined by observing the same body, the first line established is brought forward the distance run on the course steered. For example, if a ship steams 27 miles on course 315° between the first and second observations, obviously the position is on a line parallel with the first one established, but drawn 27 miles away (to scale) on the course line 315°. Intersection of the line established by the second observation with the advanced line of the first observation is a fix. The fix progressively decreases in accuracy, depending on how far the first line is advanced. You should not advance such a time for more than 5 hours of a run.

Complete Solution for Celestial Navigation Sights
1: After reading this you will have a beeter idea on how to compute a complete celestial observation of the Sun using the Nautical Almanac and the Sight Reduction Tables for Marine Navigation, Pub 229 and plot a line of position. The December 2008 blogs have dealt with all the aspects needed for determining a line of position from an observation of a celestial body. This will give you the complete solution using the Nautical Almanac and the Sight Reduction Tables for Marine Navigation, Pub 229. The steps involved will be covered in the order in which they are to be taken.

PUB. 229 Method
Pub. 229 Sight Reduction Tables for Marine Navigation is a set of six volumes of pre-calculated solutions for the computed altitude (Hc) and the azimuth angle (Z) of the navigational triangle. Entering arguments for the tables are local hour angle (LHA) expressed in whole degrees. This is done by using an assumed longitude, vice a DR longitude, assumed latitude in whole degree, and declination. Values of Hc and Z are tabulated for each whole degree of each of the entering arguments. Tables inside the front and back covers of each volume allow for interpolation of Hc and Z for the exact declination. No interpolation is necessary for LHA or assumed latitude.

Working Sights With Pub. 229
To work a sight with Pub. 229, you enter the tables by selecting the proper volume and turning to the page with the appropriate LHA. Using the assumed latitude and declination extract the tabulated values for Hc and Z. You then determine the exact value of Hc and Z corresponding to the time of observation by interpolation by using the interpolation tables or using the formula that the tables are based upon. For our purposes I will use the formula method. To find the intercept distance (a), this final Hc is compared to the observed altitude (Ho). If the computed altitude Hc is greater than observed altitude Ho the intercept is AWAY from the direction of the GP of the body. If the Ho is greater than the Hc it is TOWARDS the direction of the GP of the body. When working out a celestial sight a form should be followed so that there will be less chance for leaving out any pertinent information.

Sight Reduction
You will need a Sextant, a Watch, Nautical Almanac the Tables Ho 249 or Ho 229 marine or air sight reduction tables, I prefer Ho 229. For sighting stars, a Star-Finder No. 2101-D, Parallell Rulers or Triangles, Dividers, 0.5 mm Pencil with a good eraser, Scratch Paper, and Universal Plotting Sheets. With the "intercept" method, you will be comparing the position you think you might be in from dead reckoning on a boat, with what you actually observe. Your observed altitude is compared to a calculated altitude, calculated to be what altitude you would get if you were actually at the position you chose as your assumed position. You have to observe an altitude with the sextant and put your figures on your worksheet and along with the tables what the altitude would be if seen from the assumed position. The "v" is an extra correction for additional longitude movement of the body, and "d" is an extra correction for additional declination movement. The sun has no "v" correction and the stars have no "v" or "d" correction. The sun needs the "d", and the planets and moon need both "v" and "d".

These steps are for a SUN LINE only
Step 1: Setup a plotting sheet, DR ahead and enter it in your format.
Step 2: Apply your IE, if it is on the arc, subtract it, if it is off the arc add.
Step 3: Using the DIP Table on the inside cover of the Nautical Almanac, enter with your height of eye. Dip correction is always a minus correction.
Step 4: On page A2 "Altitude Correction Tables 10° - 90° Sun, Stars, Planets" Enter with your Ha and find the Altitude Correction. Remember that Lower limb are always + corrections and upper limbs are - corrections for the sun.
Step 5: Compute your corrected chronometer time.
Step 6: Using GMT, and Greenwich date of observation, enter the Nautical Almanac and record tabulated hourly value of GHA and Tab. Dec. in your format.
Step 7: At the foot of each declination sub-column, get the"d corr". This number is called the "d" correction. This is the average over the three day period that the declination changes per hour of GMT. The "d" is recorded on the "d corr." line off to the side. This is a correction, as with any correction, it is either a + or - . If the declination is increasing (getting larger), then it is a plus ( + ) correction, if the declination is decreasing, then it is a minus ( - ) correction.
Step 8: Turn to the yellow pages of the Nautical Almanac, and find the minute page, enter with your seconds. Then under the Sun and Planets column find the increase in the sun's GHA since the last tabulated (hourly) value, this is your M & S correction and enter this in your format. Always add the GHA and "M & S ''tabulated value together to get the total GHA.
Step 9: While on the minute page under the "v" or "d" correction column, find the "d" on the left hand side. This will be the declination (dec) of the sun at the time of sight.
Step 10: In full sight reduction, LHA has to end in the whole degree and YOUR ASSUMED LONGITUDE has to be within 30' of your DR Longitude.

Celestial Observations of the Moon
The moon is easy to identify and is often visible during the day. Yet, the navigator taking the sight must follow the moon across the horizon. It appears to move very fast, so the navigator must be quick in obtaining the sextant reading. The moon's proximity to the earth requires only some additional corrections (Parallax) to (Ha) to obtain (Ho). The rest of the sight reduction is the same process as an observation of the sun.

Parallax (P)
Parallax is the difference in the direction of an object at a finite distance when viewed simultaneously from two different positions. It enters into the sextant altitude corrections because (Hs) is measured from the earth's surface, but Ho is calculated from the earth's center. Since the moon is the celestial body nearest the earth, parallax has its greatest effect on lunar observations. If the moon is directly overhead, such as with an altitude of 90°, there is no parallax, as its direction is the same at the center of the earth as for the observer. As the moon decreases in altitude its direction from the observer begins to differ with its direction from the earth's center, and the difference in direction increases continuously until the moon sets. The same effect occurs in reverse when a body is rising. Parallax ranges from zero for a body with an altitude of 90° to a maximum when the body is on the horizon, with 0° altitude. At altitude 0°, it is called horizontal parallax (HP). In addition to increasing as altitude decreases, parallax increases as distance to a celestial body decreases. Venus and Mars, when close to the earth, are also affected by parallax. The sun is slightly affected, the parallax correction for the sun being +0.1' from zero altitude to 65°. All other celestial bodies are too far from the earth to require correction for parallax when observed with the sextant.

The correction for parallax is always positive. and is applied only to observations of the moon, sun, Venus, and Mars. When it is applied to (Hs), the sextant altitude is corrected to the value it would have if the observer were at the center of the earth. The value of horizontal parallax (HP) is found on the daily pages Nautical Almanac under Moon for every hour of GMT. As with other observations the sextant corrections for dip and altitude are in the front of the Nautical Almanac. But for the Moon, the altitude and dip tables are in the back of the Nautical Almanac. Which means you cannot correct your (Hs) to (Ho) until you get the HP from the daily pages.This procedure goes against the format that I use, but you have to be flexible as you will see in the next blog titled "How to Compute the Intercept and Azimuth of the Moon".

Universal Plotting Sheets
In order to plot celestial lines of position you need a Universal Plotting Sheet, the first thing that probably sticks out on the sheet is the compass rose, just like conventional charts. All the compass rose bearings are true bearings. You will notice the vertical line running due north and south through the center of the compass rose. Next, are the horizontal lines which will represent your desired latitudes. And last on the lower right, a longitude scale, which will allow you along with your dividers a way to lay off minutes of longitude. Your first observation that you make is the scale between any two horizontal latitude lines. It's a scale breaking down one degree of latitude into 60 minutes. If you designate the latitude line 28 degrees north and your indicated position is 28 degrees put your dividers on 28 you now have your latitude.

Now that you have an understanding of how to plot a latitude position without having to do anything to the sheet other than fill in the latitude and scale up the from the vertical compass rose, you only need to set up the sheet for longitude. Your assumed position is 28 degrees north latitude, sixty two degrees thirty six minutes west longitude. The first step is your latitude, whatever whole degree of latitude that you are currently located in or centered on as part of a plot will be indicated as the central latitude line, In this case 28 degrees. Since we are in North latitude the one below will be 27 degrees the one above 29 degrees. If you are in the Southern Hemisphere, it will be reversed. So now all you need to do is to account for your longitude. Setting up your longitude lines. On the edge of the compass rose there are two tick marks just below 30, two tick marks the top thirty, the other tick mark below thirty.This is important the two tick marks below 30 and above 30 on the compass rose are actually on the 28 degree marks, the same 28 as our central latitude. Next, draw a vertical line connecting the two points. Looking at a plotting sheet you will see that this gives you your second longitude line and your longitudinal spacing. To get additional longitude lines, use your dividers one point on the compass rose central line the other on the constructed line and lay them off as needed.

Since the plot on the sheet required a longitude of 62 degrees West, pick one convenient to your position and labeled it.The next one that would have obvious chart significance is 63 degrees West. Your position is 62 degrees 36 minutes West. How do you lay off the minutes? Look to the longitude scale lower right corner. Draw a line between 25 degrees and 30 degrees latitude, approx. 28 with the construction of this line, you put one point of my dividers on 30, extending the other point back to the zero mark would indicate only 30 minutes. The lines to the right are further broken down into increments of two degrees. Use your dividers and walk out 6 more minutes this should give you 36. This sheet is prepared and published by the US Defense Mapping Agency, normally comes in a pad of 100, and you can use both sides. I like to save mine for a reference if I want to go back and check.

Circles Of Equal Altitude
Celestial bodies are so far away that rays of light from them reach earth as parallel lines. In solving celestial navigation problems you can assume that all bodies are at the same infinite distance from earth on the celestial sphere. Corrections for bodies close to earth are taken into account when determining the actual altitude of a body in a celestial sight. The geographical position of a celestial body is the point on the earth directly below the body at any given instant. The geographical position is one of the three corners of the "navigational" triangle, and the side of the triangle between the GP and an observer's position is equal to 90° minus altitude, or co-altitude. Because the light rays from celestial bodies arrive on earth as parallel lines, you measure the same angle between the body's light rays and the horizon are located the same distance from the GP of the body. All observers who measure a celestial body at the same altitude at an instant of time are located on a common circle. This circle is called a circle of equal altitude, and the distance from the center to the circle (radius) is equal to co-altitude.

This circle of equal altitude is a celestial "line of position" like a radar range line of position from piloting. In this case the co-altitude, converted from degrees and minutes of arc to miles, is similar to the radar range. The geographical position (GP) of the body is the center from which you swing an arc with a pencil compass. A celestial line of position can be established by:

1. Observing the altitude of a celestial body.
2. Recording the instant of time (GMT) of the observation.
3. Determining the GHA and declination of the body from the Nautical Almanac.
4. Locating the GP of the body on a chart using GHA and declination.
5. Subtracting altitude from 90° and converting the resulting co-altitude to minutes of arc or miles.
6. Swinging an arc with a radius equal to co-altitude in miles using the GP as the center. A celestial fix may be obtained by crossing two or more circles of equal altitude based on observations of celestial bodies.

Plotting Fixes Based On Circles Of Equal Altitude
In actual practice fixes based on circles of equal altitude are impractical and are seldom used in marine navigation. The main reason for not using them is that the radii of these circles is too large to plot the circular line of position on the chart unless very high altitude sights are used, and they are difficult to obtain. The altitude of the body must be very close to 90° (87° is the practical minimum) in order for the co-altitude to be small enough to be plotted on a normal celestial plotting sheet or navigation chart. In taking a sight on a moving ship, the navigator would have difficulty determining the actual point on the horizon, below the body vertically, from which to measure altitude when the body is nearly at the zenith. The only real advantage of high altitude sights is that the same body may be used for a fix because the location of the GP will move a considerable distance in only a few minutes, giving a good angle of intersection between the two arcs when plotted. The USCG requires all unlimited license candidates to be proficient in solving high altitude "running" fixes using the sun.

Solving Navigational Triangles
This will give you an idea on how to solve the navigational triangle to find computed altitude (Hc) and azimuth (Zn) for a celestial sight. By comparing (Hc) with observed altitude (Ho), you can find the altitude intercept (a). With the assumed position coordinates, azimuth, and inter­cept, a line of position can be found. The two basic methods for solving a navigational triangle or other spherical tri­angle are by sight reduction table or by mathematical solution. A sight reduction table gives solutions of the navigational triangle for com­putation of the altitude and azimuth of a celestial body. The variable components are co-latitude, polar distance, and meridian angle "t". Their values depend on time of the observation and the observer's position. The parts of the triangle to find are computed co-altitude (in order to find Hc) and azimuth angle (in order to determine Zn). Mathematical solutions use the same formulas used to pre-compute the values found in the sight reduction tables. Final results, found through the correct use of either method, seldom differ by more than one-tenth minute of arc for intercept values, or one tenth degree for azimuth values.

Sight Reduction Table Pub. 229
The Sight Reductton Table for Marine Navigation (Pub 229) is a form of sight reduction table which is produced by the Defense Mapping Agency. They are designed to help in the solution of navigational triangles to find computed altitude (Hc) and azimuth (Zn). There are six volumes of Publication 229, each volume covering 15 degrees of latitude. All of the problems used in USCG license exams, which may be solved with Publication 229, are in latitudes 15 degrees to 30 degrees covered by Volume II of the tables. The entering arguments for 229 are assumed latitude, the whole degree LHA which results when assumed longitude is applied to GHA, and declination. The main part of the tables is divided in half. The first half of Volume II covers latitudes 15 degrees through 22 degrees, and the second half covers latitudes 23 through 30 degrees. The latitude which is used to enter the table may be either north or south. Each half covers LHA's from 0 degrees through 360 degrees. There are two pages for each LHA value from 0 through 90 degrees, and 270 through 360 degrees. Left-hand pages are for situations where declination and latitude are both north or both south, they are said to have the "same name." Each "same name" page covers declinations from 0 through 90 degrees.

The adjacent right-hand pages cover "contrary name" situations (meaning the declination is north and latitude south or vice versa) for the LHA values at the top of the page, and "same name" situations when the LHA values are between 90 degrees and 270 degrees located at the bottom of the page. The dividing line between "contrary" and "same name" situations is the step-like line across the right-hand page. The step-like line across the right-hand page represents the horizon. Entry values of LHA, latitude, and declination (same or contrary) should never result in crossing this line, as this would mean the body is below the horizon. The inside front and back covers of each 229 are an interpolation table to be used for correcting the the altitude and azimuth angle obtained in the main part of the table for the declination increment. Declination increment is the minutes and tenths of minutes part of the declination of the body. For example, if the body's declination is N 43° ­23.7', the declination increment is 23.7'. In other words, declination is tabulated on each page in whole degrees, and you must interpolate between those whole degrees for your actual declination value. Declination increments 00.0 through 31.9 minutes are located inside the front cover, and declination increments 28.0 through 59.9 minutes are inside the back cover.

Longitude and time are related, without time longitude by normal methods cannot be established correctly. The earth is a circle 360° with a rotation of approx. 24 hours of time, one zone or hour of time is 15° of longitude, one hour of rotation of the earth is 15° angle of rotation of the earth. Reducing this even further, one second of time corresponds to one quarter of a minute of longitude, equal to one quarter of a mile at the equa­tor, less of a distance as latitude increases. An error of time is a error of longitude. A system of time zones has been established worldwide, most zones being one hour or 15° wide. Everyone within the zone keeps the same time called zone time or ZT. The centers of each zone have longitudes divisible by fifteen such as 0°, 15°, 30°, 45°, etc. The edges of the zones extend 7 1/2 to each side of the center. The zone whose center is 45° would include longitudes from 37 1/2 degree's to 52 1/2 degree's.

Longitude and Time
In celestial navigation, zone time must be converted to Greenwich Mean Time, time kept at the zone whose center is 0° or the Prime Meridian. This is done by the Zone Description or ZD, a number obtained by dividing your longitude by 15 and rounding the answer off to the nearest whole number. If the longitude is west, the ZD is positive. If the longitude is east, the ZD is negative. If your longitude was 40° W, dividing by 15 would give 2.66. Rounding to the nearest whole number would give you 3 as a ZD. It would be a +3 since your longitude is west. This means that GMT is three hours later than your zone time. If your longitude was 153° E, dividing by 15 would give you 10.2. Rounding to the nearest whole number would give you 10, a negative ZD since your longitude is east. If addition or subtrac­tion of the ZD to your ZT puts your GMT over 24h or less than 00h, a date change must be made. If daylight time is used instead of standard time, a negative one hour (- l hour) is used to the ZD to get your GMT. For example, if ZD were 6h for standard time, it would be +5h for daylight time. If ZD were - 4h for standard time, it would be - 5h for day­light time.

Watch error (WE) is the error that your time varies from true zone time. If your chronometer is 12 seconds slow, the error should be added to the time to give true zone time. A fast error should be subtracted from the chronometer reading to give true zone time. A chronometer doesn't have to tell the exact time. If its rate of loss or gain is one second per day, and is consistent then the known error can be applied to get true zone time. If you don't have a chronometer a stopwatch can be used that is set to a radio time signal. In this case, the time the stopwatch is started is added to the stopwatch reading at the time of sextant sight to obtain true zone time. The watch time (WT) is the time read on your time piece to the nearest second at the time your sextant sight is taken.

Example: Watch Time (WT) is the reading of your watch at the instant you make the sextant reading. The watch is set to the standard time of the time zone your vessel is in.
Zone Time (ZT) is the time of the zone you are in.
Watch Error (WE) is the amount of time the watch is slow (S) or fast (F). If the watch is slow you add the error. If it's fast subtract.

Choosing a Sextant
When looking to buy a sextant your biggest decision is quality and purchase price. The two basic sextants to look at are plastic and metal, with the price ranges varying. Their is quite a bit of difference in the price range between the two, choose whats best suited for you and how much it will be used. The plastic sextant's advantage is price, disadvantage is plastic will expand and contract with varying temperatures, the index correction (instrument error) is constantly changing. This can be partially compensated for by obtaining an index correction each time you take a set of sights. The navigator will find that even between the first and last sight during a twilight series, the change can be considerable. Remember, a minute of error in sextant altitude corre­sponds to one nautical mile on the plot.

Secondly, plastic sextants weigh less than a pound and some have considerable wind resistance, making it more difficult to hold the sextant vertical when sighting in windy conditions. Third, the quality of the components is less, the filters, the mirrors, the zero to three power viewing scopes. And last, the life of a plastic sextant is shorter, depending on the amount of use. Filters break off, the plastic gearing wears out, the micrometer drum develops slop. Any of the plastic sextants make excellent teaching aids where principle, not accuracy, is important. Also, as a back-up sextant, the micrometer plastic sextants can be very valuable. As a primary sextant when a celestial fix is important, I would recommend investing in the better sextant. The advantages of a metal sextant are obvious after reading the disad­vantages involved in using a plastic sextant. Index correction is always the same unless the sextant is dropped or mirror adjustments are made. The weight (2 to 4 pounds) and open work frame reduce windage problems, the better optics and filters give better accuracy and the life of the instrument is long if care is used in usage and storage. The metal frame may be made of either brass or an aluminum alloy, lightening the weight from roughly four to three pounds. The size of the frame varies too, also changing the weight.

The telescope power varies from three to eight power, the advantage of the higher power being mainly its ability to pick up the light of a star earlier in the twilight when the naked eye still cannot see it. The disadvantage of greater power is reduced field of view, and this is critical when the navigator is trying to keep the celestial body in the field while bouncing around on a small vessel. A four power scope is a good compromise. Lighting is another option, of course this isn't needed during the day but near the end of twilight, it is convenient to press a button or turn a switch to illuminate the arc and micrometer drum. The battery case, wires and bulb socket are all subject to corrosion at sea and batteries tend to wear down when when needed. Cases usually come with the sextant and are included in the price. Second hand metal sextants are a rare and not much of a price bargain. Some sextants are sold as antiques and application of this prices them beyond their value. The true antiques, the vernier sextants or octants, are nice as display items but the difficulty of reading a vernier versus a micrometer drum is a disadvantage. Sometimes Navy surplus sextants can be found at reasonable prices. In any of these situations, instrument cleaning or mirror resilvering may be necessary but this cost will be minimal compared to the price of the instrument. The final choice of instrument to buy comes down to how much you can afford, how essential celestial navigation is to you, how comfortable the instrument is to use, and how experienced the navigator is in handling the sex­tant.

Celestial Navigation and How it Works
Celestial navigation is a position fixing technique that was devised to help sailors cross the oceans without having to rely on dead reckoning to enable them to find land. Celestial navigation uses angular measurements (sights) between the horizon and a celestial object. You can use the Sun, Moon, Planets or one of the 57 navigational stars whose coordinates are tabulated in the nautical almanac. Celestial navigation is used to measure angles between celestial objects in the sky and the horizon to locate your position. At any given time, any celestial object the Sun, Moon, Planets, and Stars will be located directly over a particular geographic position on the Earth. Your location (latitude and longitude) can be found by referring to tables in the nautical almanac. The measured angle between the celestial object and the horizon is used to define a circle on the surface of the Earth called a celestial line of position (LOP). The size and location of this circular line of position can be found using mathematical or graphical methods. The LOP is significant because the celestial object would be observed to be at the same angle above the horizon from any point along its circumference at that instant.

Angular Measurement
Using a marine sextant to measure the altitude of the sun above the horizon has developed over the years. One method is to hold your hand above the horizon with your arm stretched out. The width of a finger is an angle just over 1.5 degrees. The need for more accurate measurements led to the development of a number of increasingly accurate instruments, including the kamal, astrolabe, octant and the sextant. The sextant and octant are more accurate because they measure angles from the horizon, eliminating errors caused by the placement of an instrument's pointers, and because their dual mirror system cancels relative motions of the instrument, showing a steady view of the object and horizon. Navigators measure distance on the globe in degrees, arc minutes and arc seconds. A nautical mile is defined as 1852 meters, it is also one minute of angle along a meridian on the earth. Sextants can be read accurately to within 0.2 arc minutes. So the observer's position can be determined within (theoretically) 0.2 miles, about 400 yards (370 m). Most ocean navigators, shooting from a moving platform, can achieve a practical accuracy of 1.5 miles (2.8 km), enough to navigate safely when out of sight of land.

Practical Navigation
Practical celestial navigation usually requires a marine chronometer to measure time, a sextant to measure the angles, an almanac giving the coordinates of celestial objects, a set of sight reduction tables to compute the height and azimuth, and a chart of your area. With sight reduction tables, the only math required is addition and subtraction. Most people can master simpler celestial navigation procedures after a day or two of instruction and practice, even using manual calculation methods. Modern practical navigators usually use celestial navigation in combination with satellite navigation to correct a dead reckoning track, that is, a course estimated from a vessel's position, angle and speed. Using multiple methods helps the navigator detect errors, and simplifies procedures. When used this way, a navigator will from time to time measure the sun's altitude with a sextant, then compare that with a precalculated altitude based on the exact time and estimated position of the observation. On the chart, you can use the straight edge of a plotter to mark each position line. If the position line shows you to be more than a few miles from the estimated position, you can take more observations to restart the dead-reckoning track. In the event of equipment or electrical failure, one can get to a port by simply taking sun lines a few times a day and advancing them by dead reckoning to get a crude running fix.

Latitude was measured in the past either at noon (the "noon sight") or from Polaris, the north star. Polaris always stays within about 1 degree of celestial north pole. If a navigator measures the angle to Polaris and finds it to be 10 degrees from the horizon, then you are on a circle at about North 10 degrees of the geographic latitude. Angles are measured from the horizon because locating the point directly overhead, the zenith hard to do. When haze obscures the horizon, navigators use artificial horizons, which are bubble levels reflected into a sextant. Latitude can also be determined by the direction in which the stars travel over time. If the stars rise out of the east and travel straight up you are at the equator, but if they drift south you are to the north of the equator. The same is true of the day-to-day drift of the stars due to the movement of the Earth in orbit around the Sun; each day a star will drift approximately one degree. In either case if the drift can be measured accurately, simple trigonometry will reveal the latitude.

Longitude can be measured in the same way. If you can accurately measure the angle to Polaris, a similar measurement to a star near the eastern or western horizons will provide the longitude. The problem is that the Earth turns about 15 degrees per hour, making the measurements dependent on time. A measure only a few minutes before or after the same measure the day before creates navigation errors. Before good chronometers were available, longitude measurements were based on the transit of the moon, or the positions of the moons of Jupiter. For the most part, these are too difficult to be used by anyone except professional astronomers.

Use of Time
The most popular method was, and still is to use an accurate timepiece to directly measure the time of a sextant sight. The need for accurate navigation led to the development of progressively more accurate chronometers in the 18th century. Today, time is measured with a chronometer, a quartz watch, a shortwave radio time signal broadcast from an atomic clock, or the time displayed on a GPS. A quartz wristwatch normally keeps time within a half-second per day. If it is worn constantly, keeping it near body heat, its rate of drift can be measured with the radio, and by compensating for this drift, a navigator can keep time to better than a second per month. Traditionally, three chronometers were kept in gimbals in a dry room near the center of the ship. They were used to set a watch for the actual sight, so that no chronometers were never risked to the wind and salt water on deck. Winding the chronometers was the duty of the navigator, logged as "chron. wound." for checking by line officers. Navigators also set the ship's clocks and calendar.

Modern Celestial Navigation
The celestial line of position concept was discovered in 1837 by Thomas Hubbard Sumner when, after one observation he computed and plotted his longitude at more than one trial latitude in his vicinity and noticed that the positions lay along a line. Using this method with two bodies, navigators were finally able cross two position lines and obtain their position in effect determining both latitude and longitude. Later in the 19th century came the development of the modern (Marcq St. Hilaire) intercept method, with this method the body height and azimuth are calculated for a convenient position, and compared with the observed height. The difference in arc minutes is the nautical mile "intercept" distance that the position line needs to be shifted toward or away. Two other methods of reducing sights are the longitude by chronometer and the ex-meridian method. Celestial navigation is becoming increasingly redundant with the advent of inexpensive and highly accurate satellite navigation receivers (GPS), it was used extensively in aviation until 1960s, and marine navigation until recently. But since a prudent mariner never relies on any sole means of fixing his / her position, many national maritime authorities still require deck officers to show knowledge of celestial navigation in examinations, primarily as a back up for electronic navigation. One of the most common current usages of celestial navigation aboard large merchant vessels is for compass calibration and error checking at sea when no terrestrial references are available.

Great Circle Tracking Charts
You are planning a voyage from departure Seattle (Lat. 48°30.0' N, Long. 125°00.0' W) to a position at Lat. 44°00.0' N, Long. 161°00.'0 E. Which of the following statements is true? (Use gnomonic chart WOXZC 5270)

A. You must plot a composite sailing to remain south of the Aleutian Islands.
B. The northern hemisphere vertex lies to the west of your arrival position.
C. Military exercises north of 53° N between longitudes 150° W to 165° W will not effect your voyage.
D. At your highest latitude, the sun will be visible at upper and lower / transit if the voyage occurs near 21 June.

Step 1: Use a straightedge to draw a line bewteen the points of departure and arrival which were given.
Step 2: Eliminate choice A. The great circle track passes south of the southern most of the Aleutian Islands.
Step 3: Eliminate choice B. The northern hemishere vertex is at about Lat. 52°45.0' N, Long. 155°00.0' W between the points of departure and arrival.
Step 4: Choice C is the correct choice. The statement that the voyage will not be effected by military exercises is true. The great circle track passes just south of the exercice area above 53° N between 150° W tp 165° W.
Step 5: Eliminate choice D. In order to see the sun at upper and lower transit about June 21, the track would have to go above latitude 66°30.0' N (the Arctic Circle).

Example: You are planning a voyage by great circle from Lat. 59°00.0' N, Long. 07°00.0' W via Lat. 38°00.0' N, Long. 61°30.0' W. Which of the following statements is true? (Use gnomonic tracking chart WOXZC 5274)

A. You are east of the northern hemisphere vertex.
B. When plotted on a Mercator chart the track line will be concave to Cape Farwell.
C. All courses are in the southwest quadrant of the compass.
D. Distance is measured by using the length of a degree of latitude at the midpoint of the trackline.

Step 1: Use a straight edge to draw a line between the coordinates of the points given.
Step 2: Eliminate choice A. The northern hemisphere vertex is east of the point of departure. It is located approximately 59°15.0' N, 01°00.0' E.
Step 3: Eliminate choice B. Great circles between points which lie in the northern hemisphere are concave to the equator. Therefore, this trackline is convex, not concave, to Cape Farwell (south tip of Greenland) which lies to the north of the track.
Step 4: Choice C is the true statement. Inspection of the angle at which the track­line intersects each meridian between departure and arrival indicates all the courses to be steered will fall in the southwest quadrant (between 180° T and 270° T).
Step 5: Eliminate choice D. Distance may not be measured directly on gnomonic charts. The method given for measurements is true for Mercator chart pro­jections not gnomonic projections.

The Sailings
The sailings get their name from the days of sail when charts were reference sources for the navigator and were almost never used for plotting. In those days, determination of future DR position, voyage plan­ning, prediction of ETA, and most other evolutions which are now done on charts or plotting sheets, were done mathematically. These mathemati­cal methods are known as the "sailings". In actuality there are seven different types of sailings. The choice of which would be used was primarily dependent on the distance and direction to be traveled and some other factors. While the sailings were used extensively in times past, they are seldom used today. The availability and use of relatively inexpensive and highly accurate charts in coastal navigation, along with great circle sailing charts (Gnomonic projections) and modern ship routing services have relegated the sailings to a matter of academic interest. While seldom used in actual practice today, they do remain in one place the deck officer license exam.

Terminology of Sailing Problems
The various values which are given or found in sailing problems have been given certain abbreviation symbols. Some of them might be familiar to you or self explanatory, others will be new. You should be familiar with the symbol and brief descriptions given below.
L1 - Latitude of the point of departure (position the vessel is leaving from)
Lo1 - Longitude of the point of departure
L2 - Latitude of the point of arrival (position the vessel is going to)
Lo2 - Longitude of the point of arrival
Lm - The latitude mid-way between L1 and L2
I - Difference of latitude between L1 and L2. It is assigned a sign, "N" or "S," depending on the direction of L2 from L1 or course of the vessel (N for courses in the NW and NE quadrants, S for courses in the southerly quadrants).
Note: When L1 and L2 are on different sides of the equator, the difference of latitude (l) is the sum of the two latitudes.
Dlo - Difference of longitude between Lo1 and Lo2. It is assigned a sign "E" or "W," depending on the course of the vessel or the direction of L2 from L1.
Note: Dlo is always less than 180°. When one point is in east longitude and the other in west, Dlo is measured either across the prime (0°) meridian or the 180th meridian by the shortest distance. For example, when Lo1 is 165°00.0' E and Lo2 is 170°00.0' W is Dlo is 25-00.0' (E).
p - Departure - The number of miles measured east or west between Lo1 and Lo2 (it has the same sign, E or W, as Dlo).
Cn - The vessel's true course.
C - The vessel's course angle.
The course angle (C) is necessary for solution of all but parallel sailings problems. Conversion of true course (Cn) to course angle depends on the directional quadrant of the ship's heading. quadrant.
D - The distance the vessel travels. It may be determined from the vessel's speed and the time travelled.
M1 - The number of meridional parts of L1.
M2 - The number of meridional parts of L2.
m - The meridional parts difference between M1 and M2. Meridional parts are assigned to each minute of latitude from the equator to the pole in table 5 of Bowditch, Volume II. Meridional parts aid in constructing of Mercator charts and solving problems in Mercator sailing.
Note: When L1 and L2 are on different sides of the equator, meridional difference (m) is equal to the sum of Ml and M2.

Methods Of Solving Sailing Problems
There are two basic methods which may be used to solve the sailings problems which might be in the license exam. One is by mathe­matical formulas and the other is by the tables in Bowditch, Volume II. The sailing formulas are all listed in Bowditch, Volume II, a publication which is available for you to use in the exam room. The sailings chapter contains explanations on how to solve all types of sailings. You should remember where these formulas are in Bowditch in case your memory starts to fail while taking the exam. A hand-held calculator is the preferred method of solution. Your calculator must have the capability of multiplying and dividing with Sine, Tangent, and Cosine trigonometric functions. All the formulas are very basic and a modestly priced calculator is all that you need for these problems.

Parallel Sailing
Parallel sailing is used for vessels that are sailing along a parallel of lati­tude. Its course must be either east (090° T or west (270° T. There are two categories of parallel sailing problems:
1. Problems where the latitude and longitude of the point of departure are given. The vessel then proceeds on course 090° T or 270° T for a given dis­tance. You must find the longitude of the point of arrival.
2. Problems where the coordinates of two positions along the same paral­lel of latitude are given, and you must find the course and distance between them.

Middle Latitude Sailing
Middle Latitude, or Mid-Latitude sailing problems are used when the vessel is proceeding along a course other than north (000°), south (180°), east (090°) or west (270°). There are two different types of Mid-Latitude problems which the Coast Guard can give you in the exam.
1. The coordinates of the points of departure and arrival are given. The course and distance between them must be found.
2. The coordinates of the point of departure and the vessel's course and the distance travelled are given. The coordinates of the point of arrival must be found. In up coming blogs I will show you how you can solve Mercator, Parallel, and Mid-Latitude sailings problems found on the Coast Guard exams.

Latitude and Longitude
The earth can be regarded as a spherical object, and you will most likely be found on the surface of this sphere while using another system of coordinates, that covers our planet with imaginary lines called meridians and parallels. All these lines together provide the grid which would enable you to describe any position in Latitudes and Longitudes. It takes the earth 24 hours for a full rotation of 360°, every hour we rotate 15° of longitude.
When it is 12:00 Greenwich Mean Time (GMT) anywhere in the world, it is 12:00 Local Time in Greenwich and 24:00 Local Time at the other side of the planet: 180° E or 180° W of the date line. Crossing this meridian changes not only the hour but also the date. The North Pole has a latitude of 90° N and the South Pole 90° S. The meridians cover twice this angle up to 180° W or E.

Meridians converge at the poles, where as parallels run parallel to each other and never meet. All meridians and the equator the biggest parallel form great circles, and the remaining parallels form so-called small circles. A great circle divides the earth in two exact halves. On small scaled charts you want to be accurate within one minute or one nautical mile. On larger scaled charts the accuracy is more likely to be within a tenth of a mile (a cable).

If the earth were a perfect sphere with a circumference of roughly 40000 kilometres all great circles, meridians plus the equator, would have the same length and could be used as a distance unit when divided into 360 degrees, or 360° x 60' = 21600' minutes. In 1929, the international community agreed on the definition of 1 international nautical mile as 1852 metres, which is roughly the average length of one minute of latitude one minute of arc along a line of longitude (a meridian). You should now able to describe any position in latitudes and longitudes. Now you can state the distance between two of those positions using nautical miles or minutes. All you need now is a way to define speed. For that, you use the term knots, the number of nautical miles an hour.

Parallels: Circles parallel to the equator, ranging from 0° to 90° N or S. Only the equator is a great circle.

Meridians: Half-circles converging at the poles, ranging from 0° to 180° E or W. Each pair of opposing meridians forms a great circle.

Prime meridian: 0° or the Greenwich meridian which, together with the date line meridian, divides the Western and Eastern hemispheres.

Great circle: The intersection of a sphere and a plane that passes through the sphere's center.
Small circle: The intersection of a sphere and a plane that doesn't pass though the sphere's center.

Time zones: By convention 24 zones, each 15° longitude wide. Noon at Greenwich gives you midnight at 180° E.

GMT: Greenwich Mean Time, UTC or Zulu, which is the local time at Greenwich. For example, local time in Seattle, Wa. = GMT + 8.

Date line: The 180° meridian which extends from or is opposite to the prime meridian. Here, not only the hour changes when crossing the meridian, but also the date.

Latitude: Position defined by the number of degrees north or south of the equator, varies from 0° to 90°.

Longitude: Position defined by the number of degrees east or west of the prime meridian, varies from 0° to 180°.

Position: Latitude first and longitude second. For example: 42° 21'.5 N , 71° 03'.6 E.

Nautical mile: One nm is one minute (') on the vertical scale on the chart. 1' equals 1852 metres. Nautical miles are divided into 10 cables. A cable is equal to one tenth of a nautical mile.
Knots: Nautical miles per hour.