Classes in celestial navigation and related topics

Lunars - the Other Longitude The geometry of lunars A traditional lunars sight sequence The Moon as a Great Clock Margetts Tables for Clearing Lunars Joshua Slocum - the last of the lunarians Shooting lunars by Lake Michigan

Easy Lunars

"Lunar Distances" or lunars for short were used to determine longitude at sea in the period from about 1767 until 1850 (and rarely after that date). This was a time when chronometers were not yet widely available at a reasonable price and were considered unreliable on long voyages. The principle behind lunars is simple. The Moon in the sky is the hour hand of a great clock. The stars along its path are the numbers on the face of the clock. If we can measure the Moon's position relative to the stars, we can read the time from the clock. Comparing that absolute time with local time, usually determined by a time sight, we have our longitude. A one hour difference in time corresponds to a 15 degree difference in longitude. I wrote the essay below in 2004. It's getting a bit dusty, and I'll re-write it soon, but the principles and details are all in good shape.


First, you have to get out your sextant and shoot a lunar. I shot a set on April 26, 2004 on Lake Michigan. Since my Plath sextant had been banged around a bit recently, I started with an index correction sight. This has to be done very carefully for a lunar sight since you're trying for fraction of a minute of arc accuracy. My IC was -1.0 minutes. Next, I shot an altitude of the Moon's Upper Limb above the horizon of our inland sea and noted the time:
Moon UL: 45°46.8' at 20:09:51 GMT.

Then I shot a very rough altitude of the Sun (no horizon but easy to estimate from converging lines (Chicago is flat):
Sun LL: 47°49' at 20:11:14 GMT.

And finally one lunar sight (it's best to take four or five and average them or draw a line through them but I didn't have time to spare). The Moon was just shy of half full so I estimated that its elongation from the Sun would be about 75 degrees. I preset the sextant to that angle and aimed at the Moon. The line perpendicular to a line drawn through the Moon's "horns" always points to the Sun, so I rotated the sextant until the Moon appeared "horizontal" in the horizon glass. After a minute or so of sweeping around, I found the Sun. Then I brought the Sun into rough contact with the Moon so that they were touching limb-to-limb. When you're shooting Sun-Moon lunars, you're always bringing the "near" limbs together. From here, it's a slow process of adjustment. I usually lower the sextant from my eye, give the micrometer a slight turn, and then look through it again holding the sides of the frame with both hands. This makes for a stable platform. It also helps to sit while doing all of this. Note that you don't have to adjust the micrometer "live" as you're used to doing with altitude sights since the lunar changes very slowly. Finally, when the limbs were just touching, I noted the distance:

Moon-Sun Near: 80°09.3' at 20:16:37 GMT.
After taking the lunar, I again measured the altitude of the Moon:
Moon UL: 47°13.8' at 20:17:57 GMT.

I didn't bother getting a second altitude for the Sun, but it was probably about a degree and a half lower by this time. Finally, note that my height of eye was close to ten feet, and temperature was 59°F and pressure was 29.93 inches Hg.

Calculating the Geocentric Lunar Distances
With the observations in hand, it's time to get out the Nautical Almanac. First, there is one long calculation to do so that we have proper geocentric lunar distances to compare against the observations. This is exactly like the case of star-star distances except the Moon and the Sun move around much more rapidly, so we have to calculate two distances bracketing the time of the observations. In the 19th century, these distances were pre-calculated in the Nautical Almanac for every three hours of GMT. Today, it may make more sense to do the calculation for every hour (you can follow the steps here or you can also get pre-computed distances here on my web site (

We go to the almanac for April 26, 2004 at 20:00 and 21:00 GMT and dig out the GHA and Dec for the Sun and the Moon. Then the LD follows from the usual cosine formula:
LD = arccos[sin(Dec1) · sin(Dec2) + cos(Dec1) · cos(Dec2) · cos(GHA2-GHA1)].

I find:
geocentric_LD = 79°59.1' at 20:00:00 GMT,
geocentric_LD = 80°26.7' at 21:00:00 GMT.

We'll also need the horizontal parallax for the Moon for those hours and the semidiameter for the Sun on that date. They are:
HP Moon = 54.7, SD Sun = 15.9.

Pre-Clearing the Lunar
Unike a star-star sight, there is a "pre-clearing" step in every lunar. With stars, you can measure the distance from center-to-center since the star's image is basically a point. But with lunars, you're always measuring to a limb. So the first step is to add in the semi-diameter of the Sun and the Moon (you add for a "Near Limb" lunar and subtract for a "Far Limb"). We've got the Sun's SD from the almanac. You can take the Moon's SD from there, too, by interpolating. The listed SDs are correct for noon GMT. It's also easy to calculate it since it's proportional to the HP:
Moon SD = 0.2724 · HP = 14.9'.

For the Moon, we also need the "augmentation" of the semidiameter. This is the slight increase in the Moon's size when it's overhead (you're closer to the Moon on average when it's high overhead). You can calculate this from 0.3 · sin(H_moon), or you can use this short table:

<10°   0.0'
<30   0.1
<56   0.2
<90   0.3

For the sights above, the augmentation is 0.2' so the Moon's augmented SD is 15.1 minutes of arc. The sum of the two semidiameters is 31.0 minutes. That means our center-to-center lunar measurement is:
observed_LD = 80°39.3'.

Clearing the Distance
And now to clear the distance... We need to remove the local, observer-centric effects of refraction and parallax so that we can compare our measurement with the predicted geocentric distances above. As in the case of star-star sights, we clear the altitude by determining very carefully how much each object's altitude is modified by refraction (and parallax now, too) and then figure out what percentage of that change in altitude acts along the arc from one object to the other. For star-star sights, the equation was:
corrected_dist = observed_dist + dh1 · A + dh2 · B

where dh1 and dh2 are the altitude corrections and A and B are the percentages or "corner cosines" for the arc in question. With lunars, the change in the Moon's altitude can be as much as a degree. This is large enough compared to the other angles involved that we need to go one more step in this calculation. There is a "quadratic" correction, call it Q, that has to be added into the puzzle. That is, the equation to correct a lunar distance is:
corrected_LD = observed_LD + dh_Moon · A + dh_Sun · B + Q
  where Q is given by
Q = 0.5 · (dh_Moon)² · (1-A²) / tan(dist) / 3438.

and dh_Moon is the Moon's change in altitude. A is the same "corner cosine" factor from above [note: if dh_Moon is measured in minutes of arc and you want the result in minutes of arc, then Q, as indicated, has to be divided by 3438 (the number of minutes in a unit angle)]. This "Q" correction is usually less than a minute of arc, and it is often small enough to be ignored. In the observation I've described above, the measured lunar is about 80 degrees, so the factor cot(dist) is a small number which guarantess that Q is less than 0.1 minutes of arc. Q is also very small when the Moon's altitude is above 60 degrees and when the Moon and the other object are roughly aligned vertically.

The Linear Terms
Let's calculate A and B first. These "corner cosines" tell us how much of an object's altitude correction acts along the arc that we are measuring. So, for example, when you're shooting a lunar, if the Moon is exactly above the Sun or other object (so that their azimuths are identical), then the A and B factors would both be 100% (though opposite in sign). These factors are exactly the same as in the case of star-star distances:
A = [sin(H_sun) - cos(dist) · sin(H_moon)] / [cos(H_moon) · sin(dist)],
B = [sin(H_moon) - cos(dist) · sin(H_sun)] / [cos(H_sun) · sin(dist)].

The Altitudes
We've already got the distance, 80°39.3'. We also need the corrected altitudes. These altitudes do not have to be especially accurate. For the Moon's altitude, I took one sight before the lunar and one after the lunar. We can interpolate between those two values to get an altitude that is effectively simultaneous with the lunar distance observation. You can correct the altitudes for dip and semidiameter separately but it was common in the 19th century to use a "cheap" correction that combined these: add 12' for a Lower Limb sight, subtract 20' for an upper limb sight. Applying the various corrections, I get:
Sun LL corrected = 47°12'
Moon UL corrected = 46°39.5'

Crunching the numbers on a handheld calculator, the results are:
A = 0.909, B = 0.90.
It's only a coincidence that the values are almost the same in this case. There's no need to bother writing down more than 3 digits of A and 2 digits of B. These number are telling us that 90% of the Sun's altitude correction acts along the arc of the lunar distance and almost 90.9% of the Moon's altitude correction acts along the arc.

Altitude Corrections from the Almanac
To complete the clearing process, we need accurate values for dh_Moon and dh_Sun. These are the combined effects of parallax and refraction. And here we can use the standard tables in the Nautical Almanac. Opening the front cover of the Nautical Almanac, the altitude correction for the Sun at 47 degrees altitude is -0.8 minutes. This is the average of the corrections for UL and LL. Opening the back cover of the Nautical Almanac, the altitude correction for the Moon is 36.5 minutes (again found by taking the average of the corrections for UL and LL --this is the same procedure that's used for bubble sextant sights). I also checked the temperature-pressure correction table in the front of the almanac. The conditions were close to standard, so there was no additional correction for either object. And there's one more important step: change the signs on both dh_Moon and dh_Sun. Finally putting all the pieces together, we get the corrected distance:

corrected_LD = observed_LD + dh_Moon · A + dh_Sun · B + Q
= 80°39.3' - 36.5 · 0.909 + 0.8 · 0.90 + 0 (Q is negligible)
= 80°39.3' - 33.2' + 0.7'
= 80°06.8'.

Interpolating to find the Longitude
At last, we can compare this cleared lunar distance with the predicted geocentric lunars for 20:00 and 21:00 GMT. They were 79°59.1' and 80°26.7' respectively. Just eyeballing the numbers, our cleared measurement is about 25% of the way in between the two predicted distances, so this is telling us that the time we've measured must be close to 20:15. That's good. Interpolation yields a time for the lunar of 20:16:44. This is the GMT implied by the measurement. Since the sight was actually taken at 20:16:37, this means we've determined GMT to within just 7 seconds. Not bad! In fact, that's probably just a very lucky sight. If you can determine GMT to within 30 seconds of time, you're doing well. To convert the difference in time to a difference in longitude, divide the difference in seconds of time by 4. That is, an 8 second error in GMT is equivalent to a 2 minute error in longitude.

This Approach in Historical Context
The method for clearing a lunar that I've described here is very similar to the old methods of Witchell, Mendoza Rios, Bowditch, Thompson and others which were popular in the 19th century. Those methods depended on logarithms for their calculation of A and B and they used various trigonometric identities to put those calculations in a form suitable for logarithms, but that's just a re-arrangement of terms. There is no fundamental difference. The quadratic term which I have called "Q" was usually calculated in a special table that was included in Bowditch, Norie, etc. (table 35 in Norie, for example). The method I've described above is also very similar to the approach that Letcher used in his 1978 "Self-Contained Celestial Navigation with H.O. 208". He chose to calculate refraction instead of using the tables in the Nautical Almanac. This has some advantages, but also a few disadvantages. Specifically, that approach is unable to handle the parallax of the Sun, Venus, and Mars (all of which are included in the tables in the front of the almanac) and it also cannot handle non-standard temperature and pressure which can easily amount to a few tenths of a minute of arc difference in the cleared lunar. I prefer including the complete altitude correction from the almanac since they can accomodate these various small factors.

Little details to worry about...
The method I've described here does not include a correction for the oblateness of the Earth. This can be added easily but it's a fairly minor issue. This method is also prone to error when the lunar distance itself is below 15 degrees or the altitudes of the objects are below 15 degrees, and it should not be used under those circumstances. This is an intrinsic weakness of any "series expansion" solution of the lunar distance problem. This method also does not account for the flattening of the Moon and Sun by refraction when they are close to the horizon, but this effect is only significant at altitudes below 20 degrees which are already excluded on other grounds.

© Copyright 2004-2022, Frank Reed,, Conanicut Island USA.


Lucinda Fleeson wrote: 2/8/2021
"Celestial Navigation in the Age of Sail" was exactly what I was looking for to familiarize myself with historical marine navigation. Frank is an engaging instructor who weaved the story of a 19th century whaling voyage into hands-on practical calculations. He started by unveiling an incomprehensible historical document filled with numbers and promised that by the end of the course we could decode the figures. He's developed a step-by-step teaching method that is a marvel to watch and experience.

I must admit I dreaded two, back-to-back, 5-hour days, but the time sailed by. (Sorry can't resist.) Thank you.

He also included some 17th century data that I needed. I'm neither a mathematician nor a sailer, but a journalist and author.
Jeffrey Rock wrote: 11/9/2020
Lunar Distance and Age of Sail Classes:
Fascinating classes taught by a fascinating and capable instructor.
Lunars class is challenging and fun. Frank presents several different recipes for accomplishing lunars, one of which seems almost easy (kind of). Highly recommended for anyone interested in celestial navigation or nautical navigation history.

Jeffrey Rock
FAA Designated Pilot Examiner
Dr. Stanley J. Zawada & Ilona M. Kovacs wrote: 4/21/2022
If we could leave 5 stars, we'd leave 10 ! We have attended several courses and enjoyed each one! Mr. Frank Reed is as entertaining as is his complete mastery of the subjects he teaches. The level of preparedness shines like the stars he so loves. We thank you for the most excellent courses we have had the ultimate pleasure to experience.
Greg Rudzinski wrote: 11/2/2020
The online class "Lunars: Finding Longitude by Lunar Distance" was a very interesting introduction to the esoteric history of lunars as practiced at sea in the pre chronometer 18th century tall ship era. Practical instruction was also done demonstrating the physical process of observing a lunar with a sextant followed by a how to lunar sight reduction example using a pocket calculator, formulae, and tables. A very rewarding experience.

Greg Rudzinski
Retired Merchant Mariner
SUNY Maritime class of 80
Doug MacPherson wrote: 5/16/2020
I recently took online versions of Frank Reed's "Celestial Navigation in the Age of Sail", and "Lunars - Finding Longitude by Lunar Distances". I couldn't have been more happy with them. Having originally learned post WWII celestial methods as an officer in the United States Navy, and taken it up as a hobby, I was quite familiar with that era's procedures. However, I was intrigued by how they managed prior to then. Frank's two classes filled that void. His vast knowledge of the subject, both the technical aspects of the work as well as the historical significance were perfectly balanced. These are classes that can be thoroughly enjoyed by both the novice as well as the well versed practitioner. Recipe's for doing the work, the science behind those recipes, and actual voyages by the sailors that practiced the art were all presented with wonderful clarity. If "time sights", "cleared lunar distances" or "apparent time" have ever roused an interest, you owe it to yourself to take one of Frank's classes.

Doug MacPherson
Lieutenant, USN sep.
Samuel S Lyness wrote: 4/29/2019
Frank, a wonderful course in Lunars. I learned a lot. I admire your teaching skills and your astounding fund of knowledge. I would wish to emulate your style of instruction. Best regards, hope to sign up for your course in Cel. Nav. in Age of Sailing.
Sam Lyness
Captain Richard D. Buchanan wrote: 11/12/2019
I have taken Frank's Modern Celestial Navigation class twice. I am always inspired and I always come away with a few practical techniques and more than a few insights into the history and beauty of celestial navigation. You owe it to yourself to enjoy this class, whether or not you are a mariner.
John Workman wrote: 11/20/2018
I just took one of Franks classes and it was awesome!!

Frank taught an incredible class on celestial navigation that brought me from novice to some solid understanding of sextants, their history and most importantly their use as a aid to seeing the sea...and knowing where you are on this planet!

Hands on, wealth of knowledge, great resources at Mystic Seaport, he really covered a lot of ground! There was a lot of math but unlike in my youth, I was on the edge of my seat to soak up knowledge!! Frank made it relatable and real. The sextant which is such an iconic tool of the sea, was demystified. By the end of class i felt comfortable with it. I had mastered how it worked, how to read it and how to adjust it to insure its accuracy.

I came away with all of the cheat sheets and understandings of equations and concepts that breathe the life into what you capture through your sextant sightings.

I would highly recommend Frank and believe the Mystic Seaport with its planetarium, an ideal setting for my class with him discovering this timeless tool of the sea.

Frank did a great job keeping the class interesting with visual aids, both on screen and out on the seaport grounds. Frank had also noticed i was interested in the Draken. This is the Viking ship which had made its voyage across the Atlantic and up and down the east coast, resting for winter in Mystic as the troops regroup, gathering resources for another ocean voyage. He took extra time to talk about and show with polarized film the concept of the "Viking Sun Stone" which is a suspected navigational aide the Vikings may have used to traverse the globe as they had.

All in all i would highly recommend this class to any and all folks interested in learning about navigation and sextants. Informative and digestible, but most of all useful to the point where i am comfortable with the instrument and have the formulas needed to continuing to set my sights on the horizon!!

I look forward to more classes to learn more from Frank and strengthen my understandings of celestial navigation!

Thank you!!!
Mark Coady wrote: 6/6/2017
I have now done every course I think that has been offered so far at Mystic Seaport taught by Frank Reed in the last two years. I found the courses to all be extremely rewarding.

Several things stand out. The course material is presented in a balanced way, with a well thought mixture of detailed calculation, broken up by historical, factual, and hands-on aspects. This type of teaching is well suited to most, as it provides periods of more intense reasoning with relaxation and humor. Anyone can walk away with new-found knowledge. I also feel that the approach of understanding historical context and a simple practical approach is unique. It has gone a great way toward clearing up a lot of my preconceived ideas and confusions resulting from the many contradictory or esoteric approaches found in various volumes or on the internet.

Very simply, I learned a lot and it went a long way toward clearing up a mess. I was fascinated the whole time. The courses and NavList provide the tools to keep learning even after the course is over. I left able to measure what I see with a more calibrated eye for real world application, and a greater appreciation of human history. I can strongly recommend these classes for the curious, the fascinated, the historian, the hardcore navigator, or the armchair one. There is something in them for all.

I also found the NavList community to be helpful and encouraging as my journey continues. I hope I can undertake even more material in additional courses in the future.

"There is nothing - absolutely nothing - half so much worth doing as simply messing about in boats" (Kenneth Grahame, from the "Wind in the Willows")

Capt. Mark

Dr. Russell D. Sampson wrote: 6/22/2017
I took Frank's 19th Century Celestial Navigation class in April 2013 and really enjoyed it. Not only was the class interesting but my fellow classmates were too; a retired skipper of a ballistic missile sub, the son of the fellow who invented GPS, a teacher, a captain of a Panamax container ship and a fellow who crossed the Atlantic solo - twice!

The class was also a great resource for my teaching and my own research interests such as the visibility of celestial objects in the daytime (Jupiter and Venus) and the effects of astronomical refraction near the horizon. I hope to take more workshops with Frank.

Dr. Russell D. Sampson
Wickware Planetarium
Eastern Connecticut State University
Philip M. Sadler wrote: 6/22/2017
What a joyful and stimulating experience to enroll in Frank Reed's class, Celestial Navigation: 19th Century Methods. Frank is a skillful and engaging teacher, able to draw students into this fascinating subject, whether they be novice or experienced. His depth of knowledge is tremendous. Participants get a real taste of what it was like to be aboard a sailing ship of the day. I learned much to enliven my own teaching and decode 19th century ship's logs. It is a rare experience, indeed, to have so much thoughtfulness, enthusiasm, and fun packed into two days. This is the way to learn!

Philip M. Sadler, Ed.D.
F.W. Wright Senior Lecturer in Celestial Navigation
Harvard University Astronomy Department
Cambridge, MA

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