Celestial Navigation 101 with Bowditch

Lesson 1

Bowditch: the boy who taught himself Latin to read Newton

Nathaniel Bowditch was born in Salem, Massachusetts, in 1773, into a seafaring family fallen on hard times. School ended early for him. At about twelve years old he was indentured, bound by contract to work for years in a ship chandlery, a shop selling rope, charts, and supplies to sailors. It could have been the end of his education. Instead it became the beginning, because the shop put the tools of navigation in his hands and the sea in his imagination, and the boy taught himself everything the schoolroom would have denied him: algebra, geometry, astronomy, and, hardest of all, Latin, learned on purpose so he could read Isaac Newton's Principia in the original. No teacher assigned it. Hunger did.

At twenty one he finally went to sea, as a clerk and then supercargo on merchant voyages, carrying his books and instruments with him and working navigation problems at night below deck. There he discovered something shocking. The standard English navigation tables that sailors trusted with their lives, Moore's Practical Navigator, were full of errors. Bowditch checked the calculations himself, one by one, thousands of them, the way Euler checked an equation, by doing the work rather than trusting the print. A printed error in a lunar table was not a typo. In fog, off a rocky coast, it could be a wreck.

So he wrote his own book. The New American Practical Navigator, first published in 1802, gathered the rules, tables, and worked examples a sailor needed into one volume, and in his preface Bowditch could state plainly that in checking the standard works he had found and corrected above eight thousand errors in Moore alone, and above two thousand more in Maskelyne's Requisite Tables. Harvard gave the indentured shop boy an honorary degree. This course teaches navigation from his book, the 1837 edition, and honors his method as much as his results: observe carefully, calculate honestly, and check everything.

Lesson 2

Latitude from the North Star

Latitude is your distance north or south of the equator, and it is the easier half of your position, because the sky itself is labeled for it. The Earth's axis points, by a lucky accident of our age, very near a bright star: Polaris, the North Star. Stand anywhere in the northern hemisphere and Polaris sits above the northern horizon at an altitude, an angle, that matches your latitude. The reason is pure geometry. At the North Pole, latitude 90 degrees north, Polaris is straight overhead, altitude 90 degrees. At the equator, latitude 0, it sits on the horizon, altitude 0. Everywhere between, the two numbers track each other degree for degree.

So the measurement is beautifully simple. On a clear night, find Polaris at the end of the Little Dipper handle, or by following the two pointer stars of the Big Dipper upward. Measure its altitude above the horizon with your sextant, the subject of the next lesson. Suppose you read 42 degrees. Then you are at about 42 degrees north latitude, the latitude of Boston, give or take the small corrections a careful navigator applies. Polaris is not exactly on the celestial pole; it circles it slowly at a distance of roughly a degree in Bowditch's day, so his book carries tables to correct the reading by the star's position that night. An uncorrected sight is good to about a degree. A corrected one is far better.

Be honest about the limits, as Bowditch always is. This works only in the northern hemisphere, only at night, and only under a clear northern sky. South of the equator Polaris never rises, and southern sailors steered by other stars and by the Sun. Clouds can hide the star for a week of dangerous sailing. And one line of latitude is only half a position: it tells you which east to west line you sit on, but not where along that line. For the other half, the north to south line of longitude, you need the Sun, a clock, and the hardest problem in the history of navigation, coming in Lesson 4.

Lesson 3

The sextant

Every measurement in this course is an angle, and the instrument that measures it at sea is the sextant. Bowditch describes it carefully in his book: a sextant is built on the same principles as the older quadrant, and its arc, true to its name, is one sixth of a circle, 60 degrees. But by the cleverness of double reflection, the arc is divided into 120 degrees, so the instrument reads angles twice the size of its frame. Two mirrors do the work. One, the index glass, turns with a movable arm. The other, the horizon glass, is half clear and half mirrored, so you can see the horizon straight through it and a reflected star at the same time, in the same view.

To take a sight, you hold the sextant by its wooden handle, look through the small telescope at the horizon, and slide the index arm until the reflected image of the Sun or star comes down, in the old sailors phrase, to touch the horizon line. The body seems to rest right on the edge of the sea. Then you read the angle off the arc, with a fine vernier scale and a small screw for the last delicate adjustment, exactly as Bowditch instructs, loosening the clamp for big movements and using the tangent screw for the final touch. That reading, after corrections, is the body's altitude: its height in degrees above the horizon.

Corrections matter, and a Bowditch navigator applies them in order, every time. Dip: you stand above the water, so your visible horizon drops slightly below the true one, a few minutes of arc depending on your height of eye. Refraction: the air bends light, making bodies look a little higher than they are, badly near the horizon, hardly at all overhead. For the Sun, also semidiameter, since you sight its edge, its limb, not its center, and parallax, a tiny shift because you observe from Earth's surface, not its center. Bowditch prints tables for all of them. Measure, correct, calculate: the rhythm never changes.

Lesson 4

Time and longitude

Longitude is your distance east or west of a chosen starting line, settled in Bowditch's world as the meridian of Greenwich, and it resisted sailors for centuries for one reason: latitude can be read off the sky's geometry, but longitude is really a question about time. The Earth turns 360 degrees in 24 hours, which is 15 degrees every hour, 1 degree every 4 minutes, and 1 minute of arc, one nautical mile at the equator, for every 4 seconds of time. Read that rate again, because it rules this lesson. If your clock is wrong by just 4 seconds, your plotted position is wrong by a full mile at the equator. A minute of time error is 15 miles.

Here is the method. The Sun crosses your local meridian at your local noon, the moment it stands highest. If you also know what time it is at Greenwich at that same instant, the difference between the two times is your longitude in disguise. Suppose your shipboard clock, set to Greenwich time, reads 2 hours 20 minutes past Greenwich noon when the Sun culminates for you. Your noon came later, so you are west of Greenwich, by 2 x 15 degrees for the hours, which is 30 degrees, plus 20 minutes of time at 15 degrees per hour, which is 5 degrees more: 35 degrees west longitude. In 1800 a good marine chronometer was rare and expensive, so most sailors in Bowditch's book found Greenwich time the hard way, by the Moon: measuring its distance from known stars, the famous lunar distances, and untangling the result with Bowditch's tables.

That is why his book's full title promises longitude by lunar observations, and why his corrections to the lunar tables mattered so much. A lunar sight was fiddly, the arithmetic was long, and every bad table value flowed straight into the answer. Franklin's era had joked that longitude would be found by a clockmaker or an astronomer; in practice Bowditch's generation needed both, plus flawless arithmetic at a rolling table. Today a chronometer, and then GPS, retired the lunar method. But the principle you just used, comparing local time from the Sun with home time from a clock, is the true idea of longitude, and it never changed.

Lesson 5

Sun sights at noon

The single most useful sight at sea is the Sun at noon, the meridian altitude, because it gives latitude with the simplest arithmetic in the book, and Bowditch builds a whole chapter on finding latitude by observation around it. You begin a little before noon. You set your sextant near the Sun's expected height, find the Sun's lower edge in the glass, and follow it as it climbs, adjusting the arm as the image rises, until it stops rising, hangs for a moment at the top of its arc, and begins to fall. That highest reading is the meridian altitude. You do not need to know the exact minute of noon. The Sun tells you itself, by pausing.

Now the arithmetic, in full. Correct the reading for dip, refraction, semidiameter, and parallax, as you learned with the sextant. Say your corrected meridian altitude is 50 degrees. Subtract from 90 degrees to get the zenith distance, how far the Sun passed from directly overhead: 90 - 50 = 40 degrees. Then look up the Sun's declination for the day in the almanac, its distance north or south of the equator. Suppose the declination is 10 degrees north. When the noon Sun bears south of you, as it does for most northern sailors, latitude equals zenith distance plus declination: 40 + 10 = 50 degrees north. Check the sense of it: at latitude 50 north, with the Sun 10 north, the Sun culminates 40 degrees from your zenith, at altitude 50. It fits.

Bowditch is careful about the cases: if the Sun passes north of you, or you sail in southern latitudes, the signs in that little addition change, and his rules walk through each one. But the heart of it never changes: altitude to zenith distance, zenith distance combined with declination, latitude out. A noon sight takes five minutes and one page of figures, needs no clock at all, and was repeated every clear day of every voyage, the daily heartbeat of navigation. Generations of sailors who could not have named a theorem trusted their lives to this sum. After Lesson 2's star and this lesson's Sun, you hold both of latitude's classic methods.

Lesson 6

Dead reckoning

Between sights, the ship sails blind, and dead reckoning is how you keep your place anyway. The name is probably short for deduced reckoning, reckoning by deduction, and the ingredients are three: the course steered, read off the compass; the speed, measured by the log line, a knotted rope heaved over the stern and timed with a sandglass, whose knots gave us the word knots for nautical miles per hour; and the time run on that course. Multiply speed by time and you have distance. Sail due east at 6 knots for 8 hours and you have made good, in the ideal case, 48 nautical miles of easting from your last known position.

The ideal case is a lie the sea tells politely, and Bowditch's honesty about it is a lesson in itself. The compass course is not the course made good: the helmsman wanders, the ship makes leeway, sliding sideways under wind pressure, and unknown currents carry the whole ocean, ship and all, miles off the reckoning while the log happily reports speed through the water, not over the ground. A current of just 1 knot, unseen for a full day, puts you 24 miles from where your figures swear you are. That is the size of a harbor entrance, a reef, a coastline in fog. Dead reckoning does not drift because sailors were careless. It drifts because its inputs can only ever be estimates.

So the navigator treats a dead reckoning position the way a good scientist treats a prediction: as the best current estimate, to be replaced the moment a real observation arrives. Every clear noon, the Sun sight resets the latitude. Every star sight and lunar or chronometer check resets more. Then the reckoning starts fresh from the new fix and begins, slowly, to drift again. Your chart at any moment is a braid of the two: observed positions, solid as ink, joined by reckoned tracks that sag a little more with every hour of cloud. Hold both ideas at once, the arithmetic is exact and the position is approximate, and you think like Bowditch.

Lesson 7

Plotting a course

Now put it together on the chart, because a position you cannot plot cannot steer a ship. Plotting begins with the two lines you have earned. A noon sight gives a line of latitude: draw it straight across the chart, east to west, at 50 degrees north in our running example. A morning or evening Sun sight, worked with your Greenwich time, gives longitude, a north to south line: draw it at, say, 35 degrees west. Where the lines cross is your fix, the ship's observed position at that time, the most trustworthy point on the whole chart. Between fixes, you plot the dead reckoning track: from the last fix, lay off the course with parallel rulers, measure the distance run with dividers against the latitude scale at the chart's edge, and mark the reckoned position with its hour.

Course plotting is also where navigation turns from finding yourself to choosing your future. Given a destination, the navigator lays the intended track on the chart, reads off the course to steer, and estimates the distance: at 6 knots, a 300 mile passage is 50 hours of sailing, a little over two days, weather permitting. Each day the noon fix is compared with where the plan said the ship should be, and the course is corrected for the difference, a quiet daily argument between intention and reality. Bowditch's book even exemplifies the whole system in a worked journal of a voyage from Boston to Madeira, every rule introduced in the order a real navigator meets it, which is exactly the spirit this course has followed.

One tool deserves its sentence: the chart's compass rose, printed with its star of points, is not decoration. You walk your course angle from the rose to your track line with parallel rulers, and the rose reminds you that every course is an angle against the Earth's own directions, magnetic north wobbling a few degrees from true north by an amount, the variation, that Bowditch tabulates port by port. Plot carefully, label every line with its time, never erase a reckoning until a fix replaces it. The chart is the ship's memory, and at the inquiry after a wreck, it is also the evidence.

Lesson 8

Find your position like a sailor of 1800

You have every piece now, so sail a day as a navigator of 1800. Dawn: the reckoning, carried forward from yesterday's noon fix through a night of compass courses and log readings, says where you probably are. Morning: if the sky is kind, a Sun sight worked with the chronometer, or a lunar if you must, checks the longitude. Noon: the whole ship's routine bends around your meridian altitude; you call the turn of the Sun, work the little sum, and ink the new latitude. Afternoon: courses, speeds, leeway estimates, another advance of the reckoning. Night: Polaris confirms the latitude a second way, and the lookouts earn their pay, because all this mathematics still ends with eyes watching for breakers.

Let us run one full position, start to finish, with the numbers from this course. Your noon Sun, corrected, stands 50 degrees high, declination 10 degrees north, so latitude is 90 - 50 + 10, which is 50 degrees north. Your Greenwich clock read 2 hours 20 minutes at that culmination, so longitude is 2 and a third hours at 15 degrees per hour, 35 degrees west. Tonight Polaris should stand about 50 degrees above your northern horizon, and if it does, two independent methods agree and you may sleep. Your fix: 50 north, 35 west, in the broad Atlantic, on a Bowditch chart, found with a brass sextant, a printed almanac, and arithmetic you did yourself.

Honesty to finish, as Bowditch would insist. Today GPS fixes a position to a few meters, instantly, in all weather, and it is rightly the primary system; celestial navigation survives as the backup that needs no satellite, no battery, and no signal, and navies and ocean sailors still learn it for exactly that reason, and for the seamanship it teaches. What you have learned is the backup, and something more: a way of knowing where you are by looking up and thinking clearly. An indentured shop boy from Salem put it in one book so any sailor could learn it. Now you have learned it from him.

Sources

The Books Behind This Course

This course is taught from Bowditch great Navigator, free for anyone to read in full, together with the modern United States edition that still bears his name.

• The New American Practical Navigator, by Nathaniel Bowditch (1837 edition), Smithsonian Libraries
• The American Practical Navigator, Pub. 9, National Imagery and Mapping Agency (2002, U.S. government work), the modern edition still bearing Bowditch name, everyspec

Graduation

You finished Bowditch navigation

You crossed an ocean the slow way: a star for latitude, the noon Sun for a second line, a clock for longitude, a knotted rope for speed, and a chart where every line is labeled and every fix earns its ink. Bowditch learned all of it without a teacher and wrote it down so any sailor could. Congratulations, navigator. Keep your sextant dry and your arithmetic honest.

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