From a sight to a fix

Out of sight of land, nobody hands you your position. You have what you have done (a course steered, a speed logged, hours elapsed) and what you can measure (the angle of a star above the horizon, at a time you can read off a clock). Each of those is a piece of evidence, and each one, on its own, only narrows down where you could be. A fix is the place where several pieces of evidence agree. A good fix also tells you how far to trust it.

This chapter follows that reasoning from start to finish: from the ship’s own guess, through the circle a single sight draws across the world, to the straight pencil line a navigator actually plots, and on to the small, honest ellipse that several imperfect sights can give you.

Where you think you are§

Every ocean passage starts from a known point (a harbor, a headland, a last good fix) and carries that point forward by arithmetic. Bowditch’s American Practical Navigator, the standard American reference since 1802, defines dead reckoning as “determining the estimated position of a vessel by advancing from a known fix of position along the vessel’s ordered course and speed”^[1]. For centuries the inputs were a compass for the course and a log for the speed: originally a chip of wood watched as it passed down the side, later a weighted board on a reel of line with knots tied along it, so that the number of knots running out in a measured interval gave the speed and the unit its name^[1].

The chart-room conventions are old and still exact. Plot a DR at least every hour, after every change of course or speed, after every fix, and after plotting a single line of position. Mark a DR with a semicircle and write its time diagonally; mark a fix with a small circle and write its time horizontally^[1]. The figures in this chapter follow those conventions.

The trouble with dead reckoning is not the arithmetic. It is that the ocean moves under the ship and the ship does not go quite where it is steered. Bowditch is blunt: a DR “does not allow for the many errors that can cause a vessel to veer off course such as helmsmen error, compass error, or current and wind”^[1]. Those errors do not average out on a single passage. An unknown current of half a knot is still there in the twentieth hour, so the error it causes grows in step with time.

Press Play in the figure below. The cyan line is the DR. Each faint line is a voyage the ship might really have made under the same orders, with a current nobody measured and a helm that wandered by a few degrees. Watch the cloud of possible positions spread, and notice how the coral ellipse that describes it grows: after a day without a sight, the honest answer to “where are we?” is a patch of ocean some 80 miles across.

So the navigator’s first piece of evidence is a guess with a growing ellipse around it. The second comes from the sky.

One sight, one circle§

At any instant, every star, planet, and the Sun stand directly overhead somewhere on Earth. That point is the body’s geographic position, or GP. Its latitude equals the body’s declination; its longitude follows from the body’s Greenwich hour angle, which the Nautical Almanac tabulates hour by hour through the year^[1]. Stand at the GP and the body is at your zenith, 90° up. Walk away from it and the body sinks toward the horizon, one degree of altitude for every degree of arc you walk.

That gives the central fact of celestial navigation. If you measure a body’s altitude Ho, your angular distance from its GP is the zenith distance:

The factor of 60 is the reason navigators use nautical miles at all: one arcminute (′) of great-circle arc on Earth’s surface is, by convention, one nautical mile. That is an arcminute of angle, not a minute of time; the two meet again in the clock error below. Every point at distance z from the GP sees the body at the same altitude, so one sight puts you somewhere on a circle of equal altitude, a small circle centered on the GP^[1].

Try it in the figure. The Sun stands over the South Atlantic in the early afternoon of 3 November 2026. Slide the time and the Sun’s GP runs west at 15° an hour, dragging its circle with it. Slide the measured altitude and the circle shrinks or grows. Press “Take the sight from the ship” to set the altitude the ship would actually measure: the circle snaps onto the ship.

You have met one special case of this already. In Latitude from the sky, the noon Sun and Polaris gave latitude directly. Polaris sits almost on the pole, so its circle of equal altitude is very nearly a parallel of latitude. At local noon the Sun’s GP lies on your own meridian, so the part of its circle near you runs east and west. A noon latitude is a line of position that happens to lie along a parallel. What the noon sight cannot do is fix you east or west along it.

The bit of the circle near you§

A circle 3,400 miles in radius sounds useless for finding a ship. It is useful because you are never completely lost. Your DR is uncertain by tens of miles, not thousands, so you only need the part of the circle within a few tens of miles of where you think you are. Seen at that scale, the circle is straight.

The first navigator known to have used this idea at sea did so in desperate circumstances. Captain Thomas H. Sumner, a 30-year-old Harvard graduate, sailed from Charleston, South Carolina, on 25 November 1837, bound for Greenock^[1] documented. After passing 21° W he had no observations. Around midnight on 17 December, by dead reckoning within 40 miles of the Tusker light off southeastern Ireland, the wind hauled into the southeast and made the Irish coast a lee shore. At about 10 a.m. he caught one altitude of the Sun and noted the chronometer time. In his own account:

having run so far without any observation, it was plain the Latitude by dead reckoning was liable to error, and could not be entirely relied on.

So he worked the sight for longitude three times, with his DR latitude and with latitudes 10′ and 20′ farther north. The three positions came out 27 miles apart, each farther east-northeast than the last, and they lay on one straight line that, extended, passed through the Smalls lighthouse off Wales. Sumner reasoned that the ship had to be somewhere on that line, so he held his course east-northeast along it, and in less than an hour made the Smalls light “bearing ENE ½ E, and close aboard”^[2][1]. He published the method in Boston in 1843 as A New and Accurate Method of Finding a Ship’s Position at Sea, by Projection on Mercator’s Chart^[2][3]. Matthew Fontaine Maury called the Sumner line “the commencement of a new era in practical navigation”^[1].

Sumner had found a line of position: a short, straight stand-in for the circle of equal altitude. Zoom the figure below from the whole globe down to a plotting sheet a dozen miles across. Watch the cyan circle and the sage straight line merge.

How straight is “straight”? On the sphere, a small circle of angular radius z curves like a planar circle of radius ρ = R tan z, where R is Earth’s radius, about 3,440 NM. A distance d along the tangent, the circle has fallen away by about d²/2ρ. For a body 52° high, 30 NM from the point of tangency, that is about 0.17 NM. Bowditch gives the same order of magnitude for the worst case: with the assumed position within 30′ of the truth, the straight line departs from the circle by at most 3/8 mile for a 70° altitude and 3/4 mile at 80°^[1]. That is why very high sights are treated with care, and why everything below is plotted on a flat sheet.

The intercept method§

Sumner’s construction needed the same sight worked out twice, with two trial latitudes, to get two points on the line. In 1875 the French naval officer Adolphe-Laurent-Anatole Marcq de Blond de Saint-Hilaire published a more economical route in the Revue maritime et coloniale^[4][5] documented. Instead of asking “what longitude fits this altitude?”, ask “what altitude would I have seen from a position I choose, and how different is the altitude I actually saw?” It became known as the intercept method, or the Marcq Saint-Hilaire method, and it is still how sights are reduced today^[1].

The steps are short. Choose an assumed position (AP) near your DR. Solve the navigational triangle (pole, zenith, body) at the AP for the computed altitude Hc and the true azimuth Zn of the body:

Here φ is the AP’s latitude, δ the body’s declination, and LHA, the local hour angle, is the Greenwich hour angle plus the AP’s east longitude. The circle through the AP and the circle through you share a center, the GP, so the difference between their radii is the difference between the two altitudes:

Plot the AP, draw the azimuth line through it, measure a miles toward the body if Ho is greater (“Ho Mo To”) or away if Hc is greater (“Computed Greater Away”), and draw the line of position at right angles to the azimuth^[1]. That is the whole method.

The figure below steps through Bowditch’s own worked example, a twilight sight of the star Deneb on 9 March 2016 from a DR of 39° 00′ N, 45° 26.0′ W^[1]. The Nautical Almanac gives Deneb’s Greenwich hour angle at the moment of the sight as 351° 36.3′ and its declination as N 45° 20.2′; the corrected sextant altitude is 50° 25.8′. Press Next to go through the six steps. Then switch the assumed position from the table AP to the DR and look at the last step again.

Why the odd-looking AP of 45° 36.3′ W? Before electronic calculators, the triangle was solved with printed tables, and tables are entered with round numbers. Pub. 229, Sight Reduction Tables for Marine Navigation, is entered with whole degrees of latitude, declination, and local hour angle, so the navigator picks the AP that makes the latitude and LHA whole degrees and interpolates only for declination^[1]. The first volume of the American tables of this kind, H.O. Pub. 214, appeared in 1936; Pub. 229 later replaced it, and Pub. 249 was produced for air navigators^[1]. Bowditch gives the practical precision of these methods: about 0.1′, or 180 meters, for the Nautical Almanac with Pub. 229, and 1′ for the almanac’s concise tables or Pub. 249^[1].

The figure computes the triangle directly rather than from tables, so its numbers differ slightly from Bowditch’s: Hc 50° 14.3′ against the tables’ 50° 14.1′, and an intercept of 11.5 NM toward rather than 11.7. At the whole-degree declination of 45° the engine reproduces the tabulated Hc of 50° 10.5′ to within 0.1′ and the azimuth of 063.3° to within 0.1°. The remaining difference of about 0.2′ comes from rounding and interpolation in the tables, and it is the scale of error that Bowditch’s distinction between 0.1′ and 1′ methods is about.

Two circles, two crossings§

One line of position tells you that you are somewhere along it. A second body, at a different azimuth, gives a second line, and you are where they cross. Star sights are usually taken in nautical twilight, when the Sun is between 6° and 12° below the horizon, the sky is dark enough to show the brighter stars, and the horizon is still sharp enough to measure against^[1]. A navigator can shoot several stars in a few minutes and treat them as simultaneous.

On the globe, two circles of equal altitude cross twice. The two crossings are mirror images across the great circle through the two GPs, usually thousands of miles apart. Nothing in the sights says which crossing is yours; your DR does. This is the old evidence and the new evidence working together. The DR alone is too vague and the sights alone are ambiguous, but together they give one answer.

The figure shows Bowditch’s three stars of that March morning (Deneb, Antares, and Alkaid) as seen at 08:58 UT. On the left are the full circles; on the right, the plotting sheet around the DR, where each circle has become a line. Drag the true ship on the sheet and watch both views respond. Then tick Alkaid.

The readout labeled “angle of cut” matters more than it looks. Lines that cross at a right angle pin you down equally well in every direction. Lines that cross at a shallow angle pin you down well across them and badly along them. Bowditch’s advice on sight planning is to choose bodies whose azimuths differ by at least 30°, with 90° ideal for two bodies^[1].

Noise, bias, and geometry§

No observation is perfect. Bowditch puts shipboard sights “accurate to about an arc minute” under typical conditions, a few tenths of an arcminute for an experienced observer in ideal conditions, and a few arcminutes for someone inexperienced^[1]. The sextant itself is better than any of these: a good marine sextant has an instrument error no greater than 0.1′, enough for lines of position good to about 200 yards. The rest of the error comes from the horizon, the motion of the deck, the timing, and the observer^[1].

Errors come in two kinds, and they behave differently on the chart^[6]. Random errors scatter each line independently, so three lines from three stars no longer meet in a point. They form a small triangle, the cocked hat. Systematic errors push every line the same way. A wrong height of eye (and so a wrong dip correction), an index error, or abnormal refraction shifts every line toward or away from its star by the same number of miles. An error in the clock shifts every GP east or west, because Earth turns 15′ of arc in every minute of time, so the whole pattern of lines slides in longitude together.

The error lab below lets you produce all of these. Press “Take new sights” a few times and watch the cocked hat change. Then tick “Repeat 300 times” and read the note under the equation. Finally, try the presets.

Three things in the lab are worth stating plainly.

The cocked hat is not a confidence region you can trust case by case. If each line is equally likely to fall on either side of the truth, the true position lies inside the triangle only a quarter of the time on average, a result taught for decades and examined carefully by R. G. Stuart in 2019^[7]. The lab’s 300 repeats land close to 25 %. Bowditch warns in the same spirit that a small triangle “may not necessarily be an accurate reflection of the ship’s position” and that a large one can come from a constant error rather than careless sights^[1][6].

Geometry can cancel a constant error, or expose it. When three bodies are spread 120° apart in azimuth, an error common to all altitudes moves each line outward by the same distance, and the center of the figure does not move. With the bodies all on one side, the same error moves the fix, and the true position ends up outside the figure^[6][1]. With three or more well-spread sights you can also solve for the common error as an extra unknown. The “Solve for a common error” box does that; notice that the ellipse grows, because one sight’s worth of information has gone into estimating the bias.

A clock error cannot be seen on the chart. It slides every line together in longitude, so the hat keeps its size and the fix is wrong by about 1′ of longitude for every 4 seconds of time. This is the longitude problem of Longitude is a clock problem reappearing in modern form, and it is why a navigator’s time standard matters as much as the sextant.

The best estimate from several lines is a least-squares fix. Write each line of position as the set of points X on the plotting sheet whose distance along the unit azimuth vector n_i = (sin Zn_i, cos Zn_i) equals the line’s offset p_i:

The rows of A are the azimuth vectors and W weights each line by 1/σ². The covariance is the error ellipse: its axes are the square roots of the eigenvalues, scaled by 2.45 for a 95 % region. For two lines of equal error σ crossing at an angle θ, the semi-axes are σ/(√2 sin θ/2) and σ/(√2 cos θ/2). At θ = 90° both are σ. At 15° the long axis is almost eight times the short one, which is the long, thin ellipse of the “two stars 16° apart” preset. Because each line is only a tangent, the engine repeats the reduction from the new fix until the fix stops moving, which takes two or three passes. This is the kind of computation that navigation calculators and software carry out.

A ship that won’t sit still§

Twilight gives you several stars at once. In daylight you usually have only the Sun, and a single body gives a single line at a time. You can still get a fix, provided the Sun’s azimuth changes enough between two sights and you account for the ship’s movement in between. This is the running fix. Take a morning Sun line, sail on, take a second line hours later, and advance the first line along the DR track by the distance run. Its direction stays the same; only its position moves. Where the advanced line crosses the second line is the running fix^[1].

Bowditch asks for the Sun’s azimuth to change by at least 30° between the sights, with 90° ideal, and warns that a running fix is less accurate than a fix from simultaneous sights “due to accumulated errors from sources such as steering errors and currents”^[1]. The figure shows why. Slide the time between sights and watch the change in azimuth readout. Then raise the unknown current.

The common daytime pattern combines both ideas from Latitude from the sky: a morning sight when the Sun is well to the east, advanced to local noon and crossed with the noon latitude line. The noon sight is no longer a separate method. It is a line of position that happens to run east and west.

The practice today§

The mathematics has not changed since Marcq Saint-Hilaire. The arithmetic has. Bowditch notes that calculators and computer programs now do the reduction, given only the DR, the time, and the measured altitude, and that they are more accurate than tables because they avoid the tables’ rounding^[1]. The U.S. Navy’s STELLA program, developed by the Naval Observatory, uses algorithms good to about one arcsecond on Earth’s surface, roughly 30 meters, which is far better than any sextant can measure^[1]. The limiting factor is the observation, which is why sight planning matters: choosing bodies well spread in azimuth, at moderate altitudes, with a sharp horizon below them.

The last figure is an exercise. A true position is hidden within a dozen miles of your DR. Six stars are available. Choose your sights, think about where each new line will run, place your fix on the chart, and then reveal how far off you were. The least-squares fix and its 95 % ellipse appear alongside yours for comparison.

Every observation is a constraint§

Step back from the arithmetic and the pattern of this chapter is short to state. A dead reckoning is a prior: a best guess with an ellipse that grows with time. A sight is a measurement that confines you to a curve, which near you becomes a line. A fix is where several such constraints agree. An honest fix is also a statement about uncertainty, with an ellipse whose size depends on the quality of each measurement and, as much, on the geometry of how the constraints cross.

Nothing in that pattern depends on stars. In The invisible lighthouse, a radio bearing gives a straight line from a transmitter, a timed pulse gives a circle of range, and a time difference between two transmitters gives a hyperbola. Each is a new kind of line of position, and each is crossed with others, weighed for geometry, and checked against the DR in the same way. The intercept, the cocked hat, and the error ellipse carry over unchanged.