Every technique here reduces to the same trick: measure an altitude, compare it with what a known or assumed position predicts, and let the difference do the work. This chapter walks the arithmetic all the way through, correction by correction.
A celestial fix does not point straight at your position the way a GPS fix does — it works by comparison. You choose an assumed position (AP) close to your dead reckoning, work out what the altitude of the body would have been had you actually been standing at that AP at the instant of the sight, and set that calculated altitude against the altitude you actually measured. The gap between the two is the intercept, and it tells you how far, and in which direction, your true position lies from the AP along the line to the body.
Direction alone is not enough — you also need which way "toward" points. That comes from the azimuth (Zn), the true bearing of the body from the AP, worked from the same sight reduction. Plot the intercept distance from the AP along Zn (or its reciprocal, if the intercept is away), and draw the position line at right angles through that point. One sight gives a line; two or three bodies, sighted close together in time and reduced to the same fix, give lines that cross and bound a small "cocked hat" around your best estimate.
Every sight actually defines a circle, not a line — a circle of position centred on the body's geographical position, with a radius equal to the zenith distance. Near your AP, over the short arc a ship covers between sights, that circle is close enough to straight to draw as a position line, which is the only reason the intercept method works with a ruler and pencil instead of a globe.
Almost every mark lost in celestial work is lost here, before the sight reduction even begins. The sextant gives you an angle read off a mechanical instrument, against a horizon that is not quite horizontal, of a body that is not quite where the eye perceives it — three separate errors, each corrected in turn, and each correcting a different stage of the measurement.
Work the chain in that order — instrument, then horizon, then physics — and each stage only ever corrects the error proper to it. Try to shortcut it, or apply dip after the main correction, and errors that should cancel start compounding instead.
At the instant a body crosses the observer's meridian — bearing due north or due south, exactly — it is at its highest altitude of the day and its bearing is not changing at all near that moment. That second fact is what makes the meridian altitude so forgiving: because altitude barely changes for a minute or two either side of the peak, a small timing error costs you almost nothing, unlike a normal sight where four seconds of clock error already costs a mile.
No sight reduction tables, no assumed position, no azimuth — the body's declination for that moment (from the almanac) and the true altitude you measured are all the working requires. The zenith distance (90° minus true altitude) is added to the declination when latitude and declination share the same name and the zenith distance is the smaller of the two; when they are contrary in name the declination is subtracted from the zenith distance instead. Getting that sign right is entirely a matter of sketching which side of the equator, and which side of the observer's zenith, the body actually sits.
A meridian altitude is not really a sight-reduction shortcut — it is a different, simpler kind of observation, and that is why it has stayed useful long after tables and calculators made ordinary sight reduction fast. It is quick, robust to timing error, and needs nothing beyond the declination for the moment of local apparent noon.
A gyro or magnetic compass drifts, and the only way to know by how much is to compare its reading against a bearing you know to be true. The sky supplies that true bearing without needing a shore mark or a second vessel, which is the whole reason this branch of celestial work has survived the retreat of the rest of it — checking the gyro is still, in practice, done once a watch.
Amplitude applies specifically at the theoretical moment of rising or setting, when true altitude is zero, and is reckoned as an angle from the East or West point of the horizon rather than from north as a three-figure bearing. Reading it off needs no tables beyond the day's declination and your latitude, which is what makes it a fast check at sunrise or sunset and nothing else. Azimuth, worked from tables or by calculation for any time of day and any altitude, does the same job whenever a sight of the Sun, a star or a planet happens to be convenient — more work to obtain, but not tied to the horizon.
Star and planet sights need a horizon you can see and stars you can see at the same time, which only happens for a limited window either side of sunrise and sunset. Twilight is graded by how far the Sun's centre sits below the horizon: civil twilight out to 6° of depression, nautical twilight out to 12°, and astronomical twilight out to 18°, after which the sky is properly dark. Nautical twilight is the practical window for star sights — the horizon is usually still distinct enough to bring a star down to it, and enough stars are already visible to choose a good spread of bearings for the fix.
Predicting the times of sunrise, sunset and the start or end of each twilight band is a table lookup against the observer's latitude and the date, adjusted for longitude to convert from the tabulated Greenwich event to local time. The habit worth building is planning backwards from it: work out nautical twilight for the evening, and you know the short window in which the evening star sights have to be taken, which stars will be well placed, and how much time there is to get sights of three or four of them before the horizon is lost.
Every almanac entry — declination, GHA, the times themselves — is tabulated in UTC. Convert to the ship's zone time only at the very end, for briefing the watch on deck; feed zone time into a sight reduction by mistake and the whole fix comes out wrong by whole degrees of longitude, not by a forgivable mile or two.
A well-worked sight is only as good as the observation feeding it, and the observation is vulnerable in ways the arithmetic afterwards cannot fix. Personal error — a habitual tendency to bring a body down slightly too far or not quite far enough — is worth finding out about yourself and allowing for. A residual index error, not checked recently, quietly biases every sight taken that day in the same direction. Dip is calculated for a level ship; heavy rolling or a confused sea state makes the horizon itself unsteady under the body, and no formula corrects for that.
Timing matters more than it looks. The earth turns roughly 15 minutes of arc a minute, so four seconds of clock error already costs about a mile of position error along the line of the sight — trivial for a meridian altitude, where altitude is barely changing, but not trivial for an ordinary sight taken well away from the meridian. Abnormal refraction near the horizon, more common in cold, calm conditions than the tables assume, is the reason sights are best avoided below about 10° of altitude if a higher body is available instead.
None of this makes celestial navigation obsolete as a backup. It works with nothing electronic at all, and the discipline of taking a sight, clearing it correctly and judging the resulting cocked hat is exactly the skill that catches a GNSS position that has quietly gone wrong.
The three examples below carry a sight all the way from a raw sextant reading to a usable result — a latitude, a plotted position line, and a compass error — with every correction shown rather than assumed.
At local apparent noon the Second Officer takes a meridian altitude of the Sun's lower limb, bearing due south of the ship. The sextant reads 64 00.0′ with the index error and height of eye noted below. The Sun's declination at the moment of meridian passage is 15 00.0′ N. Find the ship's latitude.
Sextant altitude (Hs), Sun's LL: 64° 00.0′ Index error: 2.0′ off the arc Height of eye: 16 m Main correction (Sun, LL, from the altitude correction tables): +15.0′ Declination at meridian passage: 15° 00.0′ N (Sun bears south of the observer)
Find the ship's latitude
Clear the index error first.
Off the arc means the error reads low on the scale, so it is added back to the sextant reading.
Take out dip.
The visible sea horizon sits below the true horizontal by an amount that grows with height of eye, so dip is always subtracted, never added.
Apply the main correction.
For the Sun's lower limb this one figure folds together refraction, semi-diameter and parallax; it is positive here, so it is added.
Convert to zenith distance.
This is simply the complement of the true altitude — the radius, in effect, of the position circle centred on the Sun's geographical position at that instant.
Combine with declination for latitude.
The Sun bears south of the observer and its declination is north and smaller than the latitude being sought, so latitude and zenith distance fall on the same side and the declination is added to it.
AnswerLatitude 40° 50.0′ N.
The trap: applying dip the wrong way round, or leaving it out altogether, is the single commonest way to lose marks in this correction chain — it is subtracted every time, whatever the index error is doing.
During evening twilight the Third Officer sights a star and reduces the sight against an assumed position. The sextant altitude, the instrument errors and the sight-reduction results already worked for that assumed position are given below. Find the intercept and state how the position line is to be laid off.
Sextant altitude (Hs): 35° 20.0′ Index error: 1.0′ on the arc Height of eye: 4 m Main correction (star, from the altitude correction tables): −1.5′ Calculated altitude (Hc), from the assumed position: 35° 26.0′ Calculated true azimuth (Zn): 128.0°
Find the intercept and state how the position line is to be laid off
Clear the index error.
On the arc, the correction reads high, so it is subtracted from the sextant reading.
Take out dip for the height of eye actually in use at the pelorus or gyro repeater where the sight was taken.
Apply the main correction.
A star sight carries refraction only — no semi-diameter, negligible parallax — so the tabulated figure is smaller than a Sun sight's, and here, well clear of the horizon, it is negative.
Compare true and calculated altitude.
The calculated altitude is the larger of the two, which means the true position lies on the far side of the assumed position from the star, not the near side.
Lay off the position line.
'Away' is plotted along the reciprocal of the calculated azimuth, and the position line itself is drawn at right angles through that plotted point.
AnswerIntercept 12.0′ (= 12.0 n.mile) away; plotted along 308.0°T from the assumed position, with the position line perpendicular to that.
The trap: mixing up toward and away drops the position line onto the wrong side of the assumed position altogether — check the sign of the intercept before the pencil goes anywhere near the chart, not after.
At sunrise the OOW takes an amplitude of the Sun as its centre crosses the visible horizon and reads the gyro-compass bearing at the same instant. Using the observer's latitude and the Sun's declination for that morning, find the compass error and state whether the gyro is reading high or low.
Observer's latitude: 45° 00.0′ N Sun's declination: 14° 00.0′ N Gyro-compass bearing of the Sun at sunrise: 076.0°
Find the amplitude.
At the theoretical moment of rising the body's true altitude is, by definition, zero, so amplitude links declination and latitude directly with no sight reduction at all.
Name the amplitude.
The Sun is rising, so the bearing is reckoned from the East point; the declination is north, so the bearing lies north of east — amplitude N20.0°E in quadrant notation.
Convert to a three-figure true bearing.
East itself is 090.0°T, and swinging 20.0° towards north reduces that figure.
Compare with the gyro reading to find the error.
The gyro read a larger bearing than the true one, so it is running fast of the truth — reading high.
AnswerGyro error 6.0° high (west); apply −6.0° to gyro bearings until the check is repeated.
The trap: taking the sight after the Sun's centre has visibly lifted clear of the horizon — amplitude only holds good at the instant of rising or setting, and a late sight quietly bakes extra error into the result.
True alt = Sext alt ± IE − dip + main corrOrder mattersDip = 1.76√h (minutes)h = height of eye in metresLatitude = 90° − true alt ± declinationMeridian altitude, sign by name and bearingIntercept = true alt − calculated altPositive: toward the bodyZenith distance = 90° − true altitudeThe radius of the position circleAmplitude: sin(amp) = sin(dec)/cos(lat)At theoretical rising or setting onlyCompass error = true bearing − compass bearingPositive: compass reading lowLHA = GHA ± longitudeAdd east, subtract westTwilight limits: civil 6°, nautical 12°, astronomical 18°Sun's depression below the horizon4 seconds of time error ≈ 1′ of position errorFrom the ~15′-per-minute rotation rate