Celestial Navigation Simplified


I got the idea to do this link...as an update..., it must be about 15 years ago, while on a Great Circle back from China to the States. In 1964, if memory still serves me…maybe ’65, the Beatles did a song called "8 Days a Week." Sailing eastward, you inevitably will cross the International Dateline. You might, like me, hit the sack on a Tuesday night, only to wake up Tuesday morning...."on the long end of an eight day week".… This is indeed a fact for any sailor in these waters. A week ago when I began the project on this link, I just started typing, accessing a few of my files and worksheets that I use professionally. I never figured on the end product to be as lengthy as 28 typewritten pages; notwithstanding, I feel upon completion of Part One,... it is intended that you should be sufficiently qualified to do a Sunline and a Star fix. Theory has been presented on a need to know simplified basis only.… Basic theory, coupled with a thorough understanding of the nomenclature associated with the worksheets, will hopefully lead to a simple and organized application to solve the respective Celestial solutions. I mean "how difficult can it be" if each solution takes a navigator no more than 5 to 10 minutes. Part Two builds upon the redundancy of Part One..explaining simple modifications found upon the worksheets to solve the additional celestial solutions, associated and contained in the Nautical Almanac. The worksheets that I allude too can be accessed, and I would strongly suggest printed out, on a need to know basis. As we progress in Part 1, I'll inform you when we need the first worksheet. For want of a better word, which I just thought up, the worksheets and their appropriate excerpts from the Almanac and Tables, can be accessed in the "Library", at the end of this introductory. The excerpts come from the three reference sources associated with the discipline. The Nautical Almanac, HO 229 affectionately known by me as "Old Maroon" for its distinct maroon hard covers, used by mariners around the world, and finally, without peer, its inevitability and omnipresence, from Bowditch, I copied the appropriate excerpt from Table 27 and Table 28 in it’s brief entirety.. Keep looking skyward with wonderment…after all that’s what perhaps brought us to sail upon the seas in the first place.….Tomas


Irrespective of the 1441 pages of Volume 1 of American Practical Navigator and the 960 pages of Volume 2, both volumes generally found upon any American vessel that sails offshore, and the same which I might proudly add sits handsomely in my bookshelves…much of the information that we will be dealing with in this link will simply build upon that which you already know. Just the sheer volume of information familiarly known as Bowditch is enough to intimidate the curiosity of someone who would like to learn and apply Celestial Navigation to his routines at sea….The key is application…The discipline is surprisingly easy to apply. With increased application solutions generally take no more than five or ten minutes…perhaps twenty minutes in a three star fix… if well organized…and that’s the key. Learning celestial in all its theories and refinements, presupposes reading, understanding, and applying the two aforementioned volumes which could take a lifetime.

If you can plot a course on a nautical chart, by the end of the two Celestial links, you will be able to A: Chart a line of position from the Sun. B: Take an Azimuth from the sun. C: Do an Amplitude on a rising or setting sun D: Shoot and plot a three star fix. E: Calculate Meridian Passage a.k.a. Local Apparent Noon. F: Calculate Latitude by Meridian Altitude. G: Calculate Sunrise & Sunset. H: Determine Latitude & Azimuth by Polaris.

This might appear ambitious within the confines of an Internet link but really not so… Once you understand how to do "A" i.e. a Sunline the rest is somewhat redundant; the learning curve on in becomes rapid.

I’ve simplified theory on a need to know basis only. This is coupled with work sheets, the most important element in learning and attaining an accurate result…after all that is what we are after results not theory. Once you can get the results…Your study of theory will obviously flourish, and broaden your understanding of your acquired application . As I’ve stated several times if many technical books were read backwards, they would be easier to understand and perhaps easier to hold the readers attention.

First I’m going to presuppose that you have a familiarity with a nautical chart. I’m going to review over a few things to assure we are all on the same page. Let’s begin. If our voyage plan has us sailing out of a harbor, our first objective in clearing the harbor is to take a Fix from the sea buoy. A Fix is a known position expressed in latitude and longitude…Standard navigation practice suggests that we draw a line along our intended passage from this known position. On top of the line is our course expressed as three digits, below speed in two digits…and at the point of our fix…time of departure four digits. The question that we’ll raise at this point is "IF I know my compass course and I know the time I left… can I accurately predict my speed ? That’s why it’s necessary to try to get another fix within the hour. Hypothetically we are sailing along our course line, we are still in view of land…we take a compass bearing on the light house. We line up the light house, an imaginary directional line to the compass Eg..045 degrees…the Lighthouse bears NE 045…We also look at our watch at the time we took the bearing. What we have up to this point is a line of position LOP. On a chart if I used my parallel rulers drawing a line 045 from the lighthouse extended out toward the course line etc.…I know that my position is somewhere along that line exactly where though I can only make a good navigational guess… probably on the line somewhere close to the intended course line… after all that is my compass track. Now I happen to line up a geographical feature visually that is also indicated on the chart this feature is bearing 240 degrees…I draw another LOP. Where these two lines intersect tells us exactly where we are. We’ve fixed our position.… Since our chart visually indicates the plotting of two fixes.. we extend our dividers from one fix to the other…we put our dividers on the charts latitude scale to determine the distance and make note of the time it took to travel that distance. We now can determine our speed made good. This is how it‘s done…take the distance traveled multiply by 60 convert the time to minutes….e.g. one hr and thirteen minutes is 73 minutes…take the 60 times D and in our case divide by 73 and you have your speed make good over the last hr.. This determination and concept is very important. When you run out of objects to take bearings off to plot Lops…and ultimately fix a position you need to settle for something less.. not a fixed position but a dead reckoned position (DR). This is a position based upon knowing two arguments to find a third. As we continue along our intended track void of any LOP’s, we calculate the time from our last fix. We simply multiply our last estimation of speed and put a circle along our chart line of the result… We have established a dead reckoned position based upon information available. Our arguments speed and time to give us distance. If we faithfully transited along our intended course line we have a good idea as to where we are but not an exact location. An exact location is considered a fix. Our DR was not able to account for wind, sea and current. Although a DR is a good guesstimate it’s inferior to a fix. Knowing what a DR position is and how to calculate a DR is an extremely important concept in the application of celestial navigation. To review a (DR) is a position plotted on a chart, directly on our intended course line, based strictly upon the arguments of speed, distance and time. Knowing any two arguments will render a third.

Learning first the basics of Celestial Navigation through a Sunline will render you a plotable line of position (LOP). " Some might feel initially that they did a lot of work just to get a (LOP); rendering a similar result if one was to take merely a simple 10 second (LOP) from an available offshore buoy." At sea however, the word available is quite a big word. It most definately becomes important to have a clear concept of a running fix so that you can advance your celestial (LOP) allowing you the navigator to plot a fixed position. This is a simple concept that gets confusing, generally because the term itself is somewhat confusing. Therefore, if you are not thoroughly familiar with its application, you will find this concept explained and illustrated in the LIBRARY , used in conjunction with some of the aforementioned concepts relating to our treatment of our basic concepts of coastal navigation. Essentially knowing how to properly advance a line of position, to save it so to speak, will allow you use it in conjunction with another (LOP) taken at a later time rendering a position fix. While creating the illustration, it also points out and illustrates the concept and application of current set and drift, illustrating the difference between the DR and fixed position.

You might be thinking that I spent an undue amount of printed words on concepts that you already knew but that was partly my point however. Surprisingly we have established a descriptive vocabulary that is very important in celestial navigation. First of all our fix is based upon a known position Latitude and Longitude that pinpoints our position on the globe. Second of all we know how to chart a DR… this is a particularly important technique in determining LAN and sunrise/sunset. …Above I might have used two three dollar words called Azimuth and Amplitude …all they are is the celestial terminology for the concept we used above… taking a Bearing…"just substitute the words."…better still just get used to the words instead of taking a compass bearing of the sun you take an azimuth…I told you celestial navigation can be simplified…When you use a sextant. to shoot a Sunline your result will render one half the equation in determining an LOP…the result used in conjunction with pre-calculated tables will enable the navigator to plot an LOP….Right now we are about 25% there in solving the aforementioned solutions (A through G.) Get used to how I employed the word arguments……this will eliminate undue explanation at pivotal points….OK then

What does a sextant " actually do" to be of any practical value? Too many it is a mysterious instrument…that has at least a deserved curiosity which is supposed to tell "those guys with the rain hats and beards" where they are….well that’s only partially correct. What it does exactly is this… If I stand on the deck of a vessel….or even the parking lot in front of my beach…my body of course is perpendicular to the ground, facing the sun over the water. The line extending from my toes, across the beach, out on the water to the horizon is one line. The other line extending up through the ground, through my body up through my head, and up into the sky is another line…where these two lines meet of course, would be at my ankles…where my feet attach. This of course forms a 90 degree angle… after all I’m standing perpendicular to the ground. This angle is very important.. In essence we have established two lines of a celestial triangle. Now if I point the sextant toward the sun, besides attracting every curious kid on the beach….what I try to do is this…I manipulate the mirrors in such a way as to bring the sun down to the horizon…It’s about as difficult as working the remote on your VCR. (Access article "how to work a sextant" in the link's library). …When I do that I take a reading the reading is called HS… that stands for sextant angle…. I take a few corrections which has the same level of difficulty as using the TV guide previews… apply some forth grade math… and put the sextant back in the box. My corrected HS is termed HO… Once you get your HO you have half the celestial problem solved as it relates to the aforementioned problem A and B. This is what you need to know theory wise about HO…I was standing on the beach with my sextant an invisible line from my toes to the horizon, coupled with a line up through my head to the sky formed a right angel of 90 degrees at my ankles. When I looked up and pointed my sextant to the sun I bisected the angle… By using the sextant and bringing the image down to the visible horizon out over the water…I established a known bisected angle between the sun and the hypothetical line extended to the horizon from my toes…let’s say that my sextant angle HS …corrected to HO read 40 degrees. This means that the angle distance from the sun, back to my position is 50 degrees. I begin to acknowledge the curious good looking girl on the beach whose bikini is an obvious welcomed distraction (it invariably happens, when I bring my sextant to the beach for practice ) I explain… "You see the sun up there"…If it came down to earth it would be three thousand miles from us…(Now I got her full attention…plus a zillion kids on the beach). The whole idea of Celestial Navigation is to bring the celestial bodies down to earth in terms that we can measure. The sextant allows us to do just that. Since the sun, measured by my sextant to the visible horizon in our case measured 40 degrees, the sun’s angle distance back to me must be 50 degrees (40 degrees + 50 degrees = 90 degrees). When the sun is hypothetically brought to earth…(similar in effect to a ball one of the kid’s is holding, falling straight to the ground from it’s elevated position) this becomes known as the sun’s groundpoint. We can easily measure this GP distance which in our case is 3000 miles from my position. How do I know it’s three thousand miles? Easy solution….for our purposes each degree of Longitude equals 60 miles multiplied by the angle distance. (my position…to the sun’s GP) is 50 degrees 60 X 50=3000 miles…Now If you could visualize a giant pair of dividers…one point on the sun’s groundpoint, rotate the dividers creating a circle. Now if the other point extended a diameter of three thousand miles and if my HO is accurate, the divider point would pass directly over the position I was standing on.. "Look out everyone, off the beach hear comes the dividers)…What I have established is an LOP…similar in effect to a range off a light house. Now that we know what a groundpoint is, as referenced to a celestial body, if we can bring it to the ground, we should be able to define its position, and that’s exactly what we do. If I told you there’s a ship 3000 miles from you ….you might say "that’s nice where"? You could pinpoint its precise location if I gave you the Latitude & Longitude… its groundpoint…It’s the same principle in Celestial Navigation the sun’s groundpoint is defined not exactly in terms of Lat. & Long.. but two other three dollar words. Celestial Latitude is called Declination… Celestial Longitude is called GHA…Let me put it this way If the location of a ships position 3000 miles from me is defined at Latitude 42 degrees North, Longitude 20 West,…if the sun fell out of the sky through the ships funnel to it’s groundpoint the sun’s position would be defined as Declination North 42 degrees GHA 20 degrees….Get the point? You might suggest that perhaps I made a mistake? ..that I referenced declination as 42 North…the same as the ship but that I left off the WEST when I defined the sun’s groundpoint…"It is at 20 degrees West isn’t it?…It certainly is 20 degrees West but you can leave the words East and West out when you are referring to GHA because all GHA is WEST let me explain… this requires a fuller explanation of longitude.

First of all we know that Latitude measures distance north and south of the equator…A latitude position North of the equator is expressed in our example at 42 degrees North…It is exactly the same description in declination terms at the Sun’s groundpoint N 42 degrees…The N or S as the case may be, (you may have noticed), precedes the numbers in declination…just a slight insignificant difference "don’t ask me why?" Longitude measures distance East and West of the Prime Meridian. For those of you who don’t know where the Prime Meridian is… for our purposes it passes through Greenwich England. In Longitude as in Latitude… all distances are measured in two descriptive hemispheres In longitude’s case, East and West of the Prime Meridian. At 90 degrees North or 90 degrees South.. we run out of earth…that’s as far as it goes.. there is no 91 degrees North or South Latitude. West Longitude runs from 000 degrees at Greenwich west across the Atlantic Ocean.. …New York …Chicago ..Denver… LA…. Out across the Pacific past Hawaii. About 1100 miles west of Hawaii you come to the end of West Longitude this is 180 degrees West Longitude. Now east of Greenwich ..France…. Germany ..across Russia….China Japan….across the Pacific, Guam , Midway… etc. all these areas are found in East Longitude. Like West Longitude it ends when they meet up at !80 degrees…For our purposes what separates East Longitude from West Longitude is the International Date Line. "Does it need to be said that 180 + 180 = 360 degrees the circumference of the earth." . There’s a method to my madness in what seems a lengthy explanation leading nowhere…but consider this…in GHA there is no East or West… it is designated and understood to be ALL west of Greenwich… In other words if the ship in our example was defined at a position 42 degrees North 20 EAST….The declination would again be N42 but the sun’s groundpoint would be defined not at a GHA of 20 East but a GHA of 340 degrees…."Think about it a bit" If There is No East GHA similar to East Longitude then there is only ALL West GHA… "002 degrees west longitude would be equivalent to 002 degrees GHA" but 002 degrees East Longitude would be equivalent to 358 GHA…360-2=358.."That’s pretty easy to comprehend"… You now know well over 50 % of what you need to know about solving the aforementioned problems….Hang in there….

At this point I’m going to ask you to print out sheet A PRESS Sunline-Sight Reduction Now highlight the terms we have gone over…Hs is the sextant angle I pointed toward the sun and shot the angle…Ho is the corrected Hs…The position I was standing at is the Lat. and Long…"don’ worry about DR just highlight all these terms"…Now where it says Nautical Almanac …Highlight all the terms of familiarity…GHA, DEC…again highlight DEC a second time just underneath the words 229 tables…Toward the bottom highlight Ho again… "Now at least we have some structure to work with."

If you are like me you probably learned to tell time at about five. That acquired knowledge for most people became sufficient to answer the mundane question posed to millions each day… "hey buddy ..do you have the time"…You bring your arm up rotate your wrist read the numbers from your Timex LCD and respond 3:45…If the party is standing maybe three feet away he knows exactly what you mean; so we hardly found it necessary to investigate the subject further. Generally as we get a little older we learn that there are time differences between places such as New York, Chicago, Denver and LA…"Well so what" In Celestial Navigation our knowledge of time needs to become a bit more sophisticated …03:45 is three o’clock in the morning…but 3:45 in the afternoon is correctly expressed as 15:45….Obviously the guy standing three feet away knows weather it is morning or afternoon …but not the guy 10,000 miles who has a need to know basis. He doesn’t off hand necessarily know that where you are "is it very early morning or afternoon.. even with certitude what day it might be; so, your answer of 3:45 needs refinement.

The remedy of telling time anywhere in the world is so simple that it can become complicated. The first thing to understand about time …is that regardless of where you are…New York, China, or France… The only thing that changes in our example of 3:45-19sec is the 3…just the hour. Anywhere in the world it will be (00 to 23hrs:45mts 19 sec) Nothing else. It may be a different day or hour but minutes and seconds are the same world wide…So the only refinement we need to know is what hour is it. Even if we guess we have one in 24 of being right….let’s take the guess work out of it it’s quite easy…Let’s say you're laying on the sand in Palm Beach Fl….You look at your watch it says 3:45-19 seconds on your Timex LCD display…That’s your local time… Write that in the loc time on your worksheet "A"…Palm Beach is approximately 80 degrees West Longitude. Now divide 80/15…(do it the way you learned in the third grade ..the Division box.. no calculator…there’s a reason for this here) 80 divided by 15 = 5 with a 5 degrees remainder…where it says ZD put in +5…and add the +5…that’s the exact time it is in Greenwich England for our purpose…it’s 8:45-19 seconds. " Is it eight in the morning or eight at night? It’s five hours later than Eastern Standard Time…A little thought will tell you that if it is 3:45-19 in the afternoon…we translate the time to 15:45-19 + 5 makes it 20:45-19 so we know it’s 45 after eight at night at Greenwich. You telephone your brother who lives in New Orleans Central Standard Time… his watch would read 2:45-19 his longitude is 90 degrees West divided by 15 = +6 You know by custom when it’s 3:45 on the East Coast, Central Time is an hour earlier. His watch to repeat would say 2:45-19 but his ZD is + 6; so, he adds his ZD of + 6 to his local time on his watch and arrives of course at the same conclusion as you on Palm Beach it’s 20:45-19 …where.. In Greenwich England…The numbers go from +1 to +12 in the Western Hemisphere that is Longitude 15 West to 180 West and -1 to -12 from 15 East to 180 East Longitude…Let’s say you are somewhere in the "old U.S.S.R" Your ZD is - 8….It’s a few minutes past four o’clock in the afternoon Your Timex says only 4:10-22. You convert that to 16:10-22. You also know your ZD is -8 you subtract 8 from 16 it’s 08:10-22 AM at Greenwich (note the four digits of time are hr. & mts..it’s not 8 AM but 08) You are thinking of calling your brother in New Orleans…what time is it there? Since you know his ZD is + 6… and at Greenwich the time is 08:10-22 (@ Greenwich time back to local time just reverse the sign)… We subtract 08:10-22 -6 makes it 02:10- 22 in New Orleans "that’s still early in New Orleans" Just to check yourself out add the ZD back, visualizing a watch in New Orleans 02:10-22 +6 is 08:10-22 GMT .. Now reverse the - 8 to +8 @GMT to local 08:10-22 + 8 makes it 16:10-22 at your location somewhere in the Far East. We’ve all learned by custom that when it’s 3 in the morning, in New York it’s 12 midnight in LA… or more precisely 00:10-22 in LA. California’s ZD description is +8… 125 degrees W longitude. By knowing local time that is looking at your watch and knowing zone description ZD.. it’s a two step process to figure out any time in the world. First calculate what it is in Greenwich then apply the zone description reversed to the desired local location. If I called an hour sooner from my far east location, 15:10-22…-8 = 07-10-22GMT What time is it in LA… LA is +8…so at Greenwich reverse sign ..I can’t subtract -8 from 07 hr.…."yes I can" I add 24 hrs to Greenwich. It’s 31:10-22 -8 = 23-10-22. Local time at your phone in the Far East and GMT are the next day. For those still partying in LA it’s only 11:10 at night. Fifty more minutes however you will all be in the same day. "We’ll get into this a little more…down the road"…i.e. What day is it?

All the solutions relative to Celestial Navigation are based upon the Local Time of sextant or compass observation and using the ZD to calculate GMT. There is only one other refinement to the process. You know how to calculate time but you must have the right day… E.g.…You are off shore just west of San Francisco Bay…You just had dinner and you are going to do a sunline…According to your Timex LCD the time is 6:19 22 seconds…date is Monday April 3rd….Since it’s 6 19 in the evening your local time is more precisely defined as 18: 19- 22 You apply your ZD+ 8 and in Greenwich England it is precisely 26 o’clock 19 minutes 22 seconds….since the day ends at 23hrs 59 minutes and 59 seconds…26 o’clock must be the next day; so, you subtract 24 and you arrive at 02:19 22 seconds April 4th GMT.

Let’s say you are in Indian Ocean, It’s 6:19-22. April 4th You decide to do a sunline. You need to know with certitude not only what time it is in Greenwich GMT; but in addition, the correct date….First of all lets convert time to it’s standard 4 digits…06:19-22 Since you cannot subtract your ZD - 8 since it’s larger than - 06 add 24 hours to your time the lower number. Your local time now referenced to Greenwich is 30 o’clock 19 minutes 22 seconds. Now from 30 o’clock subtract 8..it’s 22:19-22 in Greenwich April 3rd…It won’t be April 4th for another 2 hrs….These types of calculations are crucial…Now highlight time and it’s references on your worksheet

You are now up to 85% of what you got to know in order to solve the Celestial Navigation problems on the worksheets A through G

The next 5% is a bit of theory as it relates to Time and Longitude. Latitude has only one function, it measures distance north and south of the Equator.. Equally Declination in Celestial Navigation is defined similarly, measuring distance, i.e. the GP of a celestial body north or south of the Equator. On the other hand, we know, Longitude measures distance east or west of the Prime Meridian. We know that when we have both Lat. & Long. we can pinpoint any vessel on the globe. We also stated that in the Celestial Sphere Longitude was defined as GHA, …defined not as West or East GHA but only GHA. Again all GHA is measured west of Greenwich from 001 to 360 rather than 180 West and 180 East.. as in the case of measuring Longitude. I know we already discussed this point earlier but I wanted to go over it one more time. Why? because unlike Latitude, Longitude serves double duty …it is the indicator of time.

OK this is the 5% needed to get you to 90%… We learned that in order to get our ZD which we find necessary to tell time particularly GMT as it relates to our local time…we divided by 15…Why?…Well if you multiply 15 X 24 you get 360. Since the earth is round, its 360 degrees is equally divided into 24 segments. We also know there are 24 hours in a day…if we multiply 15 X 24 segments we get 360 degrees…This begs the question what is the significance of 15? Well we break the earth longitudinally at 15 degree intervals in order to get our 24 segments. 23 are called Standard Meridians and one is called the Prime Meridian at Greenwich.. The Sun travels 15 degrees in one hour…or another way of expressing the point, it passes over each and every one of the Meridians, on the clock hour, in precisely a 24 hr period. We know the earth revolves around the Sun. "Forget that, for Simplified Celestial Navigation purposes".. To our eyes, the Sun travels around the earth. If I’m standing precisely on a Standard Meridian…The Sun will pass directly over my head at noon, or if not over my head, over the head of the Standard Meridian that I'm perched upon. How close to my head depends on the time of year.… E.g. If I’m standing at the Meridian 30 degrees West…and the sun passes overhead my watch says 12:00 PM more or less… My friend standing precisely on the 45 degree Standard Meridian, his watch would read 11:AM more or less…It will take precisely one hr for the Sun to be directly over his head i.e. to travel the distance West from the 30 degree Meridian to the 45 degree Meridian. Actually we have always had a fundamental idea of this concept. We always knew when it was noon in New York, it was an hour earlier in Chicago. That’s because Chicago’s Longitude is closer to the Standard Meridian which observes the Sun an hour later overhead, as opposed to the observation at the Standard Meridian further East relative to New York. I hope two things become apparent concerning time and longitude…We should now know why we divide longitude by 15. #1.To figure out our ZD #2. When the Sun passes over a Standard Meridian, it is always noon somewhere on the planet crossing one of the 24 meridian’s. When your watch reads …1-2-3-4-5 etc. etc.….6:00 -7:00- 8:00 more or less. "It is always noon somewhere". That central point is the conceptual basis of your calculations on Sunrise and Sunset, and Meridian Passage from the Nautical Almanac… "When you begin to grasp that point the rest is quite easy." One last point…It should have become apparent that all our times are referenced to Greenwich…There’s a very simple explanation for this. The Information in the Nautical Almanac is all based upon GMT. It is "somewhat important" to be aware that each new day i.e. 12:01 AM starts out at the International Date Line 180E./180W. We learned, by custom, "the dawning of a new day" begins for us with the Sun rising in the East. Simply you can’t get any more East than 180E. This of course is -12 ZD. When the Sun works it’s way East to Greenwich (ZD 0).. and is overhead at the The Prime Meridian…A day [BUT WHAT DAY] is ending at ZD+12 180W AND ZD -12 180E It’s the International Date Line that separates the day (ZD -12 from ZD +12). The difference between the two aforementioned ZDs is 24 hrs. (-12 to Greenwich around to +12)

This is what happens at the International Date Line if it is not yet quite obvious. When it is 23 hrs 59 minutes Sunday at ZD + 12, Sunday will end in one minute….and it will be Monday. On the other side of the line, the time is the same 23 hrs 59 minutes…but it’s not Sunday night that will be ending but Monday night at ZD-12. In one minute Tuesday will begin. All you need to do to put things into perspective is to know where the Sun is at both these almost (a minute shy) 24:00 hr locations… Using your ZDs @ 2400 local time, i.e. 180 E and 180 W the Sun will be found on the Prime Meridian. From our data it’s easily determined that it is Noon Monday at Greenwich. Logically we know this; at, ZD-12 noon was an event that happened 12 hrs before it occurred at Greenwich… We also know that at ZD + 12 noon is an event that occurs in Greenwich 12 hrs before it occurs at 180 W. Without fanfare just across the line it’s the same time, but logically and mathematically we know a 24 hr cycle has taken place. It’s Tuesday morning at 180E….. Monday morning 180W same time though at each side of the line.…To bring us truly to 90% of what we need to know to do our celestial problems…is just a small but important point. Remember I said to divide your longitude by 15 the way you did it in the third grade… that is because if the remainder is more than 7 degrees 30 minutes You add "1" either + or - as the case may be to your above acquired ZD number E.g. What is the ZD of a position 127 degrees 42.3 minutes…East or West whatever the case may be….? 127/ by 15 is + or - 8 the remainder is 7 degrees 42.3 minutes correct ?…15 X8 =120….since the remainder is larger than 7 degrees 30 minutes the ZD in West Longitude becomes + 9 and in East Latitude it becomes - 9…Don’t let a longitude position such as 127 degrees 42.3 minutes confuse you in calculating your ZD…get fancy with the calculator when you have plenty of practice… "Make up your own longitude positions and divide by 15 to calculate ZDs… "when you have nothing else to do". This is such an important area…accuracy is essential… yet it’s so easy to get careless as it relates to calculating the proper time and date; nevertheless the work sheets make it much easier…then contending with an oftentimes mind twisting sequence which I’ve been guilty of more than once or twice in calculating solutions at sea…These sheets are my own… that I use professionally…Everything is there to come up with a quick accurate simplified answer based upon experience at sea. Your own routines will eventually customize them to your own routine.

I said we were about 90% finished the next 10% is about as easy as using the TV guide, and applying a bit of grammar school math. You have your highlighted Sunline work sheet, now print out another blank worksheet. What we are going to do is go through a scenario of doing a sunline… First you are going to use your highlighted sheet; so, I can bring out some rather obvious preliminary conclusions to simplify our work.

If you're a bit confused at this point...don't worry about it your doing fine...you got this far..."Go back into the Library at this point"...

LIBRARY Read over or print out the article on "How To Work a Sextant"....The whole subject will start making more sense. Now if you haven't already done so, print out the four (Part One) sheets from the Nautical Almanac and the two (Part One) Sheets from HO 229...plus you should have your hightlighted worksheet and a fresh worksheet. Now we are organized and ready to nail it down.

I’m up on the bridge at 08:00 just finishing off a cup of coffee. I just took over the watch. My intention is to take a sunline about 08:15. My position puts me about 1000 miles NE of Miami steaming toward Southampton England.

I grab the sextant box and take it to a corner of the chart table…I make sure I have my Sunline-Sight Reduction worksheet. Besides my pencil and my Timex LCD watch that I’m wearing, and my stop watch, I ready two books The Nautical Almanac, a thin salmon colored book, and a maroon hard covered book called HO 229..My position suggests I need Volume 2 for latitudes 15 to 30 degrees. I have everything I need. Now follow along with me with your highlighted sheet (along with the suggested printouts, you have all the same information that you will need.)… Under the heading Time I put in the day 16… Date Nov…Now I put in my DR position Lat: 28 degrees 10 minutes N; Long 62 degrees 18 minutes West…I divide my longitude by 15 and since I’m West my ZD is + 4. Now I grab my sextant and go out on the bridge wing…I sight the Sun adjust the mirrors and bring the lower limb on the horizon…when I have taken the shot I mark the time and start my stop watch that I carry…I can now leisurely go inside put the sextant away and note my HS 19 degrees 18 minutes (fill it in on the highlighted sheet.) The reason I carry a stop watch is this…I don’t have to remember the sextant angle of the sun and the time…I can check my watch or a clock anytime mark that time… and deduct the stopwatch time…which of course is the time I took the sighting…It increases accuracy… the button is pressed when I have my sight…. Now I can fill in the local time of my HS… 08-15-12 "fill it in on your sheet" your + 4 is already in there so we add it to local time. The GMT is 12-15-12….Common sense tells us if it is 8 AM where we are and our zone is + 4…we are only 4 hrs behind Greenwich so the 12-15-12 is the same day….fill it in 16th Nov. "I told you it’s easy"

Nautical Almanac: Now I open the Nautical Almanac… As I would be using the TV guide…I go through the daily pages until I reach Nov 16th..I go across the first page on to the second where there is the heading SUN…Where it says on the left GMT I make sure I have the 16th….You will notice under the Sun there are two columns our old friends GHA and Declination…The page only gives the values for the whole hour Under 1200 hrs it says GHA 3 degree 48 minutes…Declination S 18 47.1 minutes. (fill it in) The rest of it is so easy that I can now pour another cup of coffee…and finish off the problem in two minutes…You are almost home free…I’m going to divert from the scenario for learning purposes. What I want you to do is this. Where it says HO on top I want you to put in 19 degrees 24.6 minutes write it again on the bottom under HO…What you learned is this The difference between your sextant shot of 19 degrees 18 minutes and HO…look at it 19 degrees 24.6 minutes are about 6 minutes in correction…When you get your HO called "Observed Altitude you are completely finished with all aspects of the problem as it relates to sextants……Look at it on the bottom of the sheet it’s three lines above the answer it’s half the answer…The rest of the problem concerns itself with HC…This is the Computed Altitude…The difference between what we observed with the sextant and what we are now doing i.e. calculating it through the tables is the answer… simple enough. OK Under Declination the Bold DEC I want you to fill in 18 degrees 47.3 minutes…Typical of this celestial stuff…some of these corrections require a lot of work for a useless result this is one of them as you can see…so we won't even bother with it….under the heading 229 DEC put S 18 degrees 47.3…If you pick up the explanation when I get to it fine ..if not don’t bother and stick to what you have 18 degrees 47.1..won’t affect the outcome…

Let’s now finish up with the corrections from the Nautical Almanac. First we will tackle the necessary corrections to convert HS to HO…IN fact this is so routine you don’t even have to go further than the cover of the Almanac. Where it says ic put in +1.0..That 1.0 by the way is one minute not one degree… OK This correction comes off your sextant…when and if you get a sextant and use it you will understand it.. so don’t worry about it… Dip is the height that I’m taking my HS ..Under the heading Dip on the inside cover it says height of Eye and correction…Since I took it off the bridge wing I’m 66 feet above the water it tells me to correct my HS by subtracting 7.9 minutes…(fill it in - 7.9minutes…) also noteworthy "dip will always be a minus correction" After you have done your addition and subtraction your result is HA 19 degrees 11.1 minutes…"One more correction". Two columns to the left of dip you have a heading Sun one side it says Oct -March the other side April -Sept. it’s further broken down upper limb (of the sun).. lower limb. Now we are ready for the final correction to our HS sextant angle. I shot the lower limb of the sun Nov 16th…So I’m in the first column Oct-March now take a look at HA 19 degrees 11.1..Go down the first column where it says app alt till you reach 19 degrees 11.1…Lower limb… The correction says + 13.5…add that to your HA and you have your HO 19 degrees 24.6.

Before we finish with the Nautical Almanac we need to make one more correction… Our GHA says 3 degrees 48.0 that would be fine if I took it at exactly 8AM local time but I took it at 08-15-12 added my + 4 to get GMT 12-15-12..My tables in the Nautical Almanac based upon GMT gave me a GHA for the whole hr only.. What about the 15 minutes and 12 seconds? That’s what the +m & s is on the sheet… What you do is this. You keep on thumbing through the Almanac, toward the back. You will start to see pages call increments and corrections… Find the page where you see 15 m…this is the 15 minute page…thumb down to 12 seconds.. Go across to Sun and planets.. By coincidence only… you will see 3 degrees 48.0 minutes..(That by the way is a 1 in a hundred shot the minutes and corrections value the same as the Hourly GHA). Fill in 3 degrees 48.0 minutes under +m & s. The addition of the two 7 degrees 36.0 minutes is your corrected GHA for 12-15-12.(fill it in). I decided not even to bother with the minute d correction to declination. You saw how small it was .2. So let’s just close up the Nautical Almanac and put it away with the sextant.

You’re about 10 minutes away of learning…that which you need to know for taking Sunlines, Star Fixes, Azimuth’s, Amplitudes, Lan’s, Sunrise and Sunset… If you begin to look at the other worksheets, you will see how redundant most of the required information is. Along with the few notes I made, and a little practice, there’s no reason not to be accomplished in all the included areas of Celestial… Again, these work sheets are my own that I take to sea for practical application. The key to Celestial is to be organized… Generally it takes me no more than 10 minutes to solve any one of the celestial solutions…Maybe 20 minutes to shoot and plot a three star fix…What’s important is practice. You’ve spent a lot of time in this link learning this stuff. What you will have is a navigational backup system in place should you be offshore and your GPS systems go down.. Your electronics may get wet, and the AA batteries for your hand help may be dead and useless… If you have a pencil, watch ,an Almanac aboard… Ho tables a sextant and a few worksheets, and have been adhering to good navigation practice, your attitude toward learning the discipline would quickly adjust to its practicality. If you go on an extended voyage…get in the habit of a few procedures each day…so in the eventuality of primary system failure your celestial Lop’s, fixes and azimuth’s can be relied upon…consistency and accuracy go hand and hand.

OK Now that I’m off the soapbox, let’s finish up the solution by giving attention to the 229 Tables…Before we can enter into the tables we need to have three arguments.. You see them listed there LHA… Ass Lat… Dec. We already filled one argument that we got right off the top from our almanac …Dec It should read S18 degrees 47.1 or .3 make it .3…I made the irrelevant correction… as mentioned not worth the time to explain it.. for such a small increment… What is this Ass Lat. It means Assumed Latitude…Take a look at your position on the worksheet it says DR Lat 28 degrees 10 minutes… In a Sight Reduction application we want an assumed position so the nearest whole degree of Lat is 28 degrees… Now we have the second argument put in 28 degrees N.

We are at a pivotal point here…stick with it and don’t get confused. Our GHA is 7 degrees 36 .0 minutes. Now look at the next row down are we E or W…look at your Longitude position you are at 62 degree 18 minutes West. Before you do anything, circle -W, and put in a minus sign… now write in your longitude…make sure you do it in pencil because I’m going to have you use your erasure after you put it in to modify it ..We have under Ass long - 62-18..I want you to erase the 18 and in it’s place put in 36… substituting 36 minutes for 18 minutes….. First just take a look at what you have done. Since 36-36= 0, I’ve negated any remainder…..In Assumed Latitude it was a bit easier I just erased the 10 minutes.. If it happened to be 28 degrees 31 minutes I would have erased the 31 and made it 29 degrees North…always to the nearest degree..

The question arises is this. If my GHA is 7-36.0 and my Assumed Long is -62-36.0 W, how can I subtract a greater number from a smaller number…The convention is this. You add 360 degrees to the smaller number ..it’s sought of a loan so to speak; so, what we have is this: GHA 367-36.0 minus our assumed Long 62- 36.0. This renders an even number LHA of 305 degrees. Now you have your three arguments to use 229..Which requires looking up two values writing them on your sheet.. correcting them and finish off your Sunline. Then plotting the LOP as you would for a lighthouse.

What If my position was not West Longitude but East Longitude. The work sheet tells me to add not subtract… OK let’s give it a try…make sure your erasure is handy…. Under assumed Long circle E + put the + sign in. Copy your position 62 degrees 18 minutes…GHA is 7-36.0… Again we use our erasure on the 18 minutes as we did for the subtraction method of West Longitude to get an even LHA. Unlike the other solution, instead of making it 36-36 = 0 the 18 minutes that we erased becomes 24 minute.. I heard someone yell WHOA…stick with it…remember I need an LHA argument with an even number. Here I’m adding not subtracting. What I have is an assumed Long of 62 degrees 24.0 minutes when I add this is what I have 7-36 .0 + 62-24.0 = 70 degrees. If I was in East Longitude my LHA argument going into the tables would be 70 degrees. ( Don’t think about it too much just stick with the convention) LHA is actually the distance the body is West of your position.

Back to finishing off our problem ..the home stretch…

Before we get into the 229 tables…Notice that your assumed Lat is 28 N and Dec is S 18-47.3…Write the word contrary in the column to the right of Dec. "You will see why in a minute"

I opened up Vol - 2, of its 365 pages of Tables. On page 294 I found the heading LHA 305…On page 295 I also found LHA 305.. PG 294 says Lat. Same as Declination…PG 295 says Lat. Contrary to Declination. "Guess which one we want"…that’s why you wrote it in the column. This may seem trivial but the mistake is made often. Most mistakes are careless mistakes in Celestial, not conceptual misunderstanding, that’s again why work sheets are so important…even more so to a professional or someone who has done a solution a million times… it eliminates CARELESSNESS. Our arguments say Lat. N… Dec S… Therefore the contrary page.

So now that we have found our page …You will notice on the left a column headed Dec in one degree intervals… In our case go down to Dec 18…Now go across the Latitudes on top to the column 28 degrees… Where 18 and 28 come together you will see HC 19 degrees 40.1 minutes…(copy that down on your sheet). Right next to it you will see a small d the indicated value -38.5 (copy that down under the HC. In it’s respective column d The next column you will see is Z 124.2 at the @ sign copy down 18 degrees Now go down one line and copy the Z for 19 degrees125.0@ 19 degrees…When you have done this…all the information needed out of Tables 229 is completed…but don’t put the book on the self just yet ..we need to confirm, and perhaps modify an important point in our solution…at the very end..

Now we are on the very last step in our problem i.e. correcting HC and correcting Z….One thing that immediately becomes apparent the HC is only about 16 minutes different than HO. Let’s take a minute here and reflect upon just what we have done…We stood on the bridge wing of a vessel at sea. We observed the lower limb of the sun by bringing the image to the visible horizon, by manipulating the sextants mirrors. We took a reading and carefully marked the time of our observation. We called this a sextant angle i.e. HS. We made a correction of our height called dip and a mirror correction called IC…After applying these corrections, we termed it HA. We then made our final correction an altitude correction.. and wound up with an HO…called our Observed Altitude or as we know better our corrected sextant angle of the sun observed through our sextant. The sextant did it’s job admirably, and we put it away.

The KEY to our Celestial Solution termed "sight reduction" is the TIME of the observation. By knowing our TIME of observation and our LATITUDE & LONGITUDE. We began the process of finding HC which is the Calculated Altitude. "That makes sense"… after all we made all those corrections in the Almanac and 229 Tables. In effect we calculated the position of the Sun pretty close to what we observed from our sextant…Before I say anything else or before I explain what Z is… let’s finish up our corrections through a process called Interpolation

Interpolation is a very easy convention…If the Declination was "exactly" 18 degrees no corrections would be necessary but our worksheet says it is 18 degrees 47.3 minutes….but the tables are adjusted to only one degree intervals… In our case @18 and @19. Get use to this convention because you will use it in making your final corrections, on practically all of your worksheets…After all most of the worksheets particularly with Star fixes and Azimuths use the same interpolation.. Amplitudes are Interpolated from tables 27 and 28 in Bowditch. Amplitudes are a particularly easy and compact process for small offshore vessels without the necessity of sextant or 229. You could just slip the four pages of tables in the Nautical Almanac with some worksheets and take an Amplitude at Sunrise and Sunset.. As I said before Azimuth & Amplitude are just fancy words for Celestial Bearings In this case bearings on the Sun…OK back to Interpolation

Working with what we have… we took the value of HC at 18 degrees declination to be 19 degrees 40.1 minutes. We made note of the d to be -38.5 minutes…."This in actuality is what the HC difference is between 18 and 19 degrees"…in other words HC is descending. Since our Declination is not 18 degrees or 19 degrees but 18.47.3…the 47.3 is called an increment… Put the increment value in X 47.3 minutes …that’s fairly simple. "That’s the value we want to solve for in (our difference X the increment) Now where it says int. put in 60. We said before our intervals are one degree…18 degrees to 19 degrees 60 minutes equals one degree. Remember all our interpolations are minutes… Let’s get our hand held calculator out -38.5 multiplied by the increment 47.3 then divided by 60 = -30.4… - 30.4 subtracted from 19 degrees forty .1 minutes equals a corrected HC of 19 degrees 9.7 minutes….Now put that in the final HC column at the bottom…

Now let’s Interpolate for Z… Z @ 18 degrees we noted was 124.2…unlike HC "the tables gave us the correction"… here we need to figure it out… probably because it’s a small correction.. OK we noted Z @ 19 degrees to be 125.0 that’s a 0.8 difference…we note Z is rising so we put a + 0.8..d on our sheet. We want to solve for 18 degrees 47.3… so that again becomes our increment…our interval is the same 60 minutes so again our calculator renders +.08 multiplied by 47.3 divided by 60 =s a correction of +0.6 we add that to 124.2 and our corrected Z is 124.8… "Are you thinking what I’m thinking"…It seems like a lot of work for a 0.6 correction. That’s why I didn’t care whether you had 18 degrees 47.1 or 18.47.3 for declination increment. If you corrected the d above, from the Almanac that’s fine. If you eliminated that step that’s fine too…it’s not going to radically change anything.

Now that we have a corrected Z and before I explain what it is…look at the next line down it says ZN + Z or…. "What does that mean"? I told you not to put the HO 229 on the shelf… one other adjustment may be necessary…If you have a copy of 229 handy any volume any page will do. There are instructions… Either on the upper right hand comer of the "Same" page….or the lower left hand corner found on the "Contrary Page". Since we are in North Latitude, the instructions North Latitude are applicable. It states that if "LHA is greater than 180 degrees ZN equals + Z ..If LHA is less than 180 degrees ZN equals 360-Z"…just follow the instructions…since our LHA is 305 degrees our ZN = Z or 124.8. Now you can put the books away we are finished…All we need to do is plot our LOP of the Sunline

So what is ZN or Z or ZN = Z… Very simple it is the compass bearing of the sun from our assumed position.. Some whiz kid might say "that’s kool" you could check your compass deviation or gyro error with Z as an added bonus.. correct? ..WRONG… That’s exactly what we do with an Azimuth & Amplitude…to check compass error (a hint on Azimuths and Amplitude worksheets) BUT in sight reduction we are using the vessel’s assumed position in solving the Computed Altitude not an actual position as in determining HO

Let me clarify this we took the HO from the ships position…compare the ships position 28-10 N 62-18 W to assumed position 28N 62-36W Just off the top of my head my assumed position is "at least 10 miles" from the actual position I took the sextant angle…..enough of theory let’s finish the problem and plot the line of position…

I’m assuming you all know how to plot a position.. "Don't worry...follow along all the information you will need to know about plotting, including this plot, will be found in the Library under the heading Diagrams." Plot your assumed position LAT 28 N Long 62-36W Draw a little rectangular box with a dot in the middle of your plotted position….Use you compass rose and parallel rules. Set them at 125 degrees. Put a DASHED 125 degree line through the center dot…The line should be about eight inches…4" each side of the center dot of the assumed position’s rectangular box. That portion of the head of the line bearing 125, put an arrow head on it…Why? Because that’s the direction of the sun from your assumed position…… 125 degrees. Now take a look at HO and HC and subtract the difference… the answer comes out to 14.9… Now check with Admiral Homoto… "he will always give you the right answer" …HOMOTO…this means that if HO more than HC (To = "T") HOMO( HO more)TO(toward)..Again HO..MO..TO… "get it". Since HO is indeed more than HC in our solution by 14.9 minutes, our answer is 14.9 T…T means toward the arrow head….Now that portion of the line that has the arrow head mark off 14.9 miles from the center dot. Make a small mark there…now at the mark draw a solid line 90 degrees bisecting the dashed line…That’s your line of position…That’s what you worked so hard to learn…that’s what your sextant accomplished for you…You are somewhere along that line…I hope you don’t feel let down…If you observed another body…maybe a late moon set, and followed the same procedure we just did…where the two Lop’s cross, you would have a position fix…"What else can I say"? One other point, If HC is more than HO the answer would have been 14.9 A (A = Away) This means you would have measured your distance from the center dot to that portion of the dashed line away from the arrow head; again, mark the distance, then draw a solid line bisecting the dashed line at 90 degrees… That’s your LOP. At this point you may appreciate the value of the concept of a running fix illustrated in the library, I illustrated also the intercept plot. WE ARE FINALLY FINISHED…"With the Mother of all Celestial Solutions" The rest of the problems are small modifications to the body of knowledge you already have learned. Before moving on, take a look at how a Universal Plotting Sheet is constructed. Learning how to construct a plotting sheet will put your celestial plotting skills in perspective. Now all you need is a good erasure and you can plot and navigate around the world on one piece of paper. "You may not want to do it, but you will know how to do it."

From here on out, when you access another worksheet "B through G, Refer to the notes on each section in order to learn the solution. You will find that you are already familiar with 90% of the stuff.

If you understand what we just did …you should have no problem in doing a 3 Star Fix, Azimuth, Amplitude, Meridian Transit (LAN) Sunrise-Sunset, Lat & AZ by Polaris.

If you have a good working grasp of what has just been presented, you should have little problem with Part Two. The Star Fix worksheet will re-inforce what you have already learned. Calculating Local Apparant Noon, along with Sunrise and Sunset, should reaffirm our explaination on celestial theory as it bears upon Longitude and Time. Amplitude and Azimuth are simply different solutions to the same thing. By the time you reach Latitude and Azimuth by Polaris... no explaination will be hardly necessary, you... "Just Do It"