NATARAJAAn imaginary sphere of effectively infinite radius centred on the Earth (or the observer), onto which every celestial body is projected. Distance is discarded; only direction is kept — which is exactly what a sextant measures. It is a geometric tool, not a physical shell; it appears to rotate east→west once per sidereal day because the Earth rotates west→east.
Glossary — the language of the sky
Every term is defined once, linked from the labs that use it — and every term can be simulated: press Simulate to watch the definition move, driven by the same validated engine as the labs. (98 terms)
The point of the celestial sphere vertically overhead — where your plumb line, extended upward, pierces the sphere. It is observer-specific: move, and it moves with you. Its declination equals your latitude.
The point of the celestial sphere vertically beneath the observer, diametrically opposite the zenith.
The great circle of the celestial sphere everywhere 90° from the zenith — the plane through the Earth's centre perpendicular to the observer's vertical. The reference circle for altitude. Bodies theoretically rise and set when crossing it.
The line where sea and sky appear to meet. It lies below the celestial horizon because the observer's eye is above sea level — that offset is dip, the reason for the dip correction to sextant altitudes.
The plane parallel to the celestial horizon but passing through the observer's eye rather than the Earth's centre.
The two points where the Earth's axis, extended, pierces the celestial sphere. The whole diurnal rotation of the sky pivots on them. The elevated pole (the one above your horizon) stands at an altitude equal to your latitude — the foundation of latitude by Polaris.
The great circle where the plane of the Earth's equator cuts the celestial sphere; every diurnal circle is parallel to it. Declination is measured from it, as latitude is from the equator.
The apparent annual path of the Sun on the celestial sphere — the projection of Earth's orbital plane — inclined at about 23° 26′ (the obliquity) to the celestial equator.
The point where the Sun crosses the celestial equator from south to north at the March equinox — the zero of right ascension and SHA, and the reference the sidereal day is measured against.
Angular distance of a body north (+) or south (−) of the celestial equator: the celestial twin of latitude — but belonging to the BODY, not to a place. They meet through one fact: your zenith's declination equals your latitude.
A great circle through both celestial poles and a body — the celestial twin of a meridian. A body's hour circle sweeps westward with the diurnal rotation.
Angle at the pole from the Greenwich celestial meridian westward (0°–360°) to a body's hour circle. Tabulated in the almanac; ties sky positions to Earth longitudes.
Angle from YOUR celestial meridian westward to the body's hour circle: LHA = GHA + east longitude (− west). LHA 0° means the body is on your meridian — transit.
Angle from the First Point of Aries westward to a star's hour circle. The almanac lists stars by SHA so that GHA★ = GHA♈ + SHA★.
Angle from the First Point of Aries EASTWARD to a body's hour circle, usually in hours (24h = 360°). RA and SHA describe the same thing in opposite directions: SHA = 360° − 15·RA.
The great circle through both celestial poles and your zenith. Bodies culminate — reach greatest altitude — when they cross it (upper transit).
Any great circle through the zenith and nadir, cutting the horizon at right angles. Azimuth says WHICH vertical circle a body is on; altitude says how far up it.
The particular vertical circle passing through the east and west points of the horizon.
Angular height of a body above the celestial horizon, 0° at the horizon to 90° at the zenith, measured along the body's vertical circle.
Direction of a body's vertical circle, measured from true north clockwise 0°–360° along the horizon.
90° minus the (corrected) altitude: the arc from your zenith to the body. Its magic: zenith distance equals your great-circle distance from the body's geographical position — the whole basis of the position line, and the ZX side of the PZX triangle.
The daily path a body traces across the sky: a circle parallel to the celestial equator at the body's declination. Its tilt relative to your horizon equals your co-latitude — stars rise vertically at the equator, obliquely in mid-latitudes.
A body whose whole diurnal circle stays above the horizon (never sets): |lat + dec| > 90° with latitude and declination the same name. The mirror case (opposite names) never rises.
Angular distance of a place north or south of the equator (0°–90°), measured at the Earth's centre. One minute of latitude = one nautical mile, everywhere.
Angular distance of a place's meridian east or west of Greenwich (0°–180°). An ANGLE, not a distance: a degree of longitude spans 60 nm at the equator and shrinks with the cosine of latitude — ~30 nm at 60°.
Differences of latitude and longitude between two places, named N/S and E/W by the direction of travel. DLong is always taken the SHORTEST way round — crossing the date line is ordinary arithmetic, not a special case.
A circle on a sphere whose plane passes through the centre — the largest possible circle and the shortest path between two points. All meridians are great circles; among parallels only the equator is.
Any circle on the sphere whose plane misses the centre — every parallel of latitude except the equator.
The distance subtending one minute of arc at the Earth's centre — in practice, one minute of latitude. 1 nm = 1.852 km by definition; speed in knots is nautical miles per hour.
Local Apparent Time follows the real Sun (sundial time: apparent noon = Sun on your meridian). Local Mean Time follows the fictitious mean sun that moves uniformly. LAT − LMT = the equation of time.
Apparent minus mean time: how far the sundial is ahead (+) or behind (−) the clock. Caused by the ellipse (Kepler speed-up, ±7.7 min, one cycle/yr) and the obliquity (ecliptic→equator projection, ±9.9 min, two cycles/yr) — not by any unevenness of Earth's rotation.
Time by the stars: the hour angle of the First Point of Aries. Runs at a different RATE from solar time (its day is 23h 56m 04s), gaining ~3m 56s daily — not a time zone. GAST as an angle IS GHA♈.
The point of the Moon's (or any Earth satellite's) orbit nearest the Earth — about 356,500 km for the Moon. The disc looks widest (SD ≈ 16.7′), parallax is greatest, and coinciding new/full moons amplify spring tides ('perigean springs').
The farthest point of the Moon's orbit from Earth — about 406,700 km. Smallest disc (SD ≈ 14.7′), least parallax. The perigee→perigee cycle is the anomalistic month, 27.55 days.
Nearest and farthest points of an orbit around the SUN (Earth: early January / early July). Same Greek construction as perigee/apogee — peri- near, apo- away — with the suffix naming the body orbited (-gee Earth, -helion Sun).
Either of the two points where the Moon's orbit, tilted 5.14°, pierces the ecliptic plane (ascending ☊ and descending ☋). Eclipses can only happen when a new or full moon falls near a node; the node line points sunward twice a year — the eclipse seasons.
The angle at the Earth's centre between the Sun's direction and the Moon's direction — the Moon's angular distance from the Sun in the sky, 0–180°, named EAST while the Moon is east of the Sun (waxing) and WEST after full (waning). Phase IS elongation: illuminated fraction = (1 − cos E)/2, so E = 0° is new (syzygy), 90° quadrature (quarters), 180° full. Growing ~12.2°/day (Moon 13.2 − Sun 1.0), it is the reason moonrise retards ~50 min/day and the synodic month outruns the sidereal.
Sun, Earth and Moon in a line — new moon or full moon (elongation 0° or 180°). Spring tides occur at syzygy; eclipses occur at syzygy near a node.
Moon 90° from the Sun as seen from Earth — first and last quarter. Solar and lunar tidal bulges partly cancel: neap tides.
Phase to phase (new moon to new moon): 29d 12h 44m — longer than the sidereal month because Earth moved ~27° round the Sun meanwhile and the Moon must catch up. The tide and phase calendar.
The Moon's return to the same star: 27d 07h 43m — its true orbital period. The synodic/sidereal difference is the same catch-up logic as solar vs sidereal DAY, one storey up.
The angle the Earth's radius subtends at a body — the maximum shift between the geocentric position (tabulated) and what an observer measures. For the Moon HP runs 54′–61′ (about a degree!), which is why lunar sights need a parallax correction and stellar sights don't.
Half the angular width of a body's disc: asin(radius ÷ distance). Sun ≈ 15.7′–16.3′ through the year; Moon 14.7′–16.7′ through the month. Applied to limb observations to reach the body's centre.
The daily lag of the Moon's meridian passage — the lunar day averages 24h 50m because the Moon moves ~13° eastward daily. It varies tens of minutes with orbital speed, and high water follows it: tides run ~50 minutes later each day.
Inferior planets (Mercury, Venus) orbit INSIDE Earth's orbit; superior planets (Mars outward) orbit outside it. The split decides everything you see: inferior planets show phases, stay chained to the Sun within a capped elongation, and can cross its face; superior planets can stand at opposition on your midnight meridian.
The widest angle an inferior planet can reach from the Sun — Mercury 18–28°, Venus 45–47°, set by the size of its orbit seen from ours. At greatest elongation the planet shows a half phase and lingers longest in twilight: the sighting window. The engine finds each event; the golden suite pins them within minutes of the independent ephemeris.
One planet, two ancient names. EAST of the Sun a planet sets after it — an evening star in the western twilight. WEST of the Sun it rises first — a morning star in the east. The sign of elongation is the entire rule; Venus was Hesperus and Phosphorus before the Greeks realised both were her.
A line-up with the Sun as seen from Earth. An inferior planet has two kinds: INFERIOR conjunction, passing between Earth and Sun (huge thin crescent — occasionally a transit across the disc); SUPERIOR conjunction, passing behind it (small, full, lost in glare). A superior planet has only the far-side kind and is unobservable for weeks around it.
Sun–Earth–planet in that order: elongation 180°, possible only for superior planets. The planet rises at sunset, crosses your meridian at LOCAL MIDNIGHT, and is nearest and brightest — with its retrograde loop centred exactly here.
The apparent backward (westward) drift of a superior planet against the stars around opposition. Nothing reverses: the faster Earth overtakes on the inside track and the line of sight swings back. Ptolemy needed epicycles to fake it; a moving Earth explains it in one sentence.
The angle at the BODY between the directions of Sun and Earth. It fixes the illuminated fraction we see: (1 + cos i)/2. Near 0° the disc is full (superior conjunction, or opposition); near 180° it is new (inferior conjunction). Only the inferior planets — and the Moon — can run the whole range.
One of the 57 stars the Nautical Almanac selects for sight work — bright (all magnitude ~2.8 or better), spread over both hemispheres, each with a number, an SHA and a declination. Polaris is carried separately for latitude. The engine computes their apparent places rather than copying them.
A star's angular distance WEST of the First Point of Aries: SHA = 360° − RA. Stars barely move against ♈ (~1′ a month, from precession), so the almanac prints one fast column — GHA♈ — and a slow SHA per star: GHA★ = GHA♈ + SHA. One angle serves all fifty-seven.
The brightness scale, running BACKWARD: each step of 1.0 is ×2.512 fainter, five steps exactly ×100. Sirius −1.5, Vega 0.0, Polaris 2.0, faintest naked-eye ~6. For identification, magnitude is your first filter — bearing and altitude come second.
Bright-star signposts that survive patchy cloud: the Plough's leading edge runs to Polaris; Orion's belt runs one way to Sirius, the other to Aldebaran; the long axis of the Southern Cross, extended 4½ times, marks the blank south celestial pole.
The altitude of the celestial pole equals your latitude — and Polaris rides a small circle (radius ≈ 40′, shrinking until ~2100) around that pole. Correct the sextant altitude, allow for where Polaris sits on its circle (the almanac's a₀+a₁+a₂; the engine solves the triangle exactly), and the north star hands you your latitude.
Sun 0–6° below the horizon: civil — horizon sharp, stars scarce. 6–12°: NAUTICAL — the sight window: horizon still visible, bright stars out. 12–18°: astronomical — stars everywhere, horizon gone. Star sights live in that middle band, morning and evening.
The point on Earth directly beneath a body — where it is at the zenith this instant. Its latitude is the body's declination; its longitude is the body's GHA (west of Greenwich). Every sight is really a distance measurement from the GP.
All observers measuring the same altitude of a body stand on one circle centred at its GP, of radius equal to the zenith distance (90° − alt, in minutes = nautical miles). A sight puts you ON that circle; near the DR its arc is so vast it plots as a straight line — the position line.
The workable piece of the circle of equal altitude: a straight line drawn PERPENDICULAR to the body's azimuth, through the intercept point. One sight gives one line — you are somewhere on it. Crossing two or more lines gives the fix.
Compute what the altitude SHOULD be at the DR (Hc, with its azimuth Zn), compare with what you OBSERVED (Ho): a = Ho − Hc in minutes of arc = nautical miles, plotted toward the azimuth when positive. 'Ho More, Toward.' The whole method in one subtraction.
Your position carried forward from the last known fix by course and speed alone — the best guess the sights will correct. The intercept method needs one: every Hc/Zn is computed AT the DR, and the fix is expressed as a displacement from it.
The small triangle where three position lines fail to meet at a point. Random errors put the ship near (often inside) it — but a SYSTEMATIC error moves all three lines alike, keeping the hat tight while displacing it whole. A small hat shows consistency, never accuracy.
The angle between TRUE north and the north your compass believes in: error = true bearing − compass bearing, named EAST when positive. The sky supplies the true bearing (azimuth or amplitude of a body); one observation names the error. 'Compass least, error East; compass best, error West.'
The magnetic field's own disagreement with true north at your position — magnetic north lies east or west of true by this angle. It belongs to the PLACE, is printed on the chart's compass rose with its annual change, and is the V of the T-V-M-D-C ladder.
The ship's personal compass error — her iron and electrics pulling the needle off MAGNETIC north. It belongs to the SHIP and changes with her heading, which is why it is tabled against the ship's head. Found at sea as: deviation = total error − variation.
A body's bearing at rising or setting, measured from due EAST or WEST toward its declination: sin A = sin dec / cos lat, taken with the centre on the TRUE horizon (lower limb ~1½ semidiameters above the visible one). At the equator amplitude equals declination; at high latitudes cos lat inflates it dramatically. The quickest compass check the sky offers.
A body's greatest altitude, taken as it crosses your meridian bearing due N or S. There the zenith distance lies entirely north–south, so lat = dec ± ZX with the naming rules — and because the altitude 'hangs' at transit, the sight forgives your clock. Latitude's daily bread since antiquity.
The classic sum: a morning or afternoon altitude, a trusted latitude, and GMT from the chronometer. The PZX cosine rule turns Ho into the meridian angle t; you decide east/west of the meridian (morning = east); LHA − GHA = longitude. Time becomes place at 15° per hour — Harrison's prize, worked on one sheet of paper.
The clock that carries Greenwich to sea. Its ERROR on GMT is logged daily (radio time signal) and applied before any almanac lookup; its daily RATE predicts the error between checks. Four seconds of neglected error is a mile of longitude at the equator — latitude barely cares, longitude lives and dies by it.
The spherical triangle joining the elevated Pole, your Zenith and the body X. Its sides are co-latitude (PZ), polar distance (PX) and zenith distance (ZX); its angles hold the hour angle (at P), azimuth angle (at Z) and parallactic angle (at X). Every sight-reduction method is a way of solving it.
90° minus your latitude — the side PZ of the navigation triangle, the arc from the elevated pole to your zenith. At the equator it is 90°; near the pole it shrinks and the triangle degenerates.
The arc from the elevated pole to the body: PX = 90° − declination when latitude and declination share a name, but 90° + declination when they are CONTRARY — the sign trap of the examination hall.
The angle of the navigation triangle at the body X, between the body's hour circle and its vertical circle. It governs how position lines rotate as a body crosses the sky and appears in advanced sight analysis.
The sextant's zero offset: with both horizons aligned the reading should be zero; what it reads instead is IE. The correction has the OPPOSITE sign: on the arc, take it off; off the arc, add it on. Measured before or after every round of sights.
The atmosphere bends light downward, lifting every body's image — about 34′ at the horizon, ~1′ at 45°, near zero overhead — so refraction is always subtracted. It is also why low sights are avoided: near the horizon its value is large and weather-uncertain.
The three altitudes of a sight: Hs (sextant altitude, as read), Ha (apparent altitude, after index correction and dip), Ho (observed/true altitude, after refraction, semidiameter and parallax). The ordered stack between them is the whole craft of altitude correction.
At altitude you stand up to one Earth-radius closer to the Moon than the Earth's centre does, so its semidiameter grows slightly — up to ~0.3′ at the zenith. One more way Moon corrections differ from the Sun's.
The intersection of the sphere with any plane through its CENTRE — the largest circle that can be drawn on it, and the shortest track between two points on the surface. Meridians are great circles; parallels (except the equator) are not. On a great-circle passage the true course keeps changing; only a rhumb line holds one course.
A triangle whose three sides are arcs of great circles. Its sides are ANGLES (measured at the centre of the sphere), its angle sum exceeds 180°, and one triangle carries all of celestial navigation: as Pole–A–B it solves the sailings, as PZX it reduces every sight.
The point where a great circle reaches its highest latitude — one in each hemisphere, 180° of longitude apart. At the vertex the track runs exactly east–west (course 090°/270°). Exam favourite: find the vertex and check whether it lies between departure and destination.
Along any great circle, cos(latitude) × sin(course) is CONSTANT. It is the great circle's fingerprint: one number that ties course to latitude everywhere on the track, gives the vertex latitude as acos|K|, and lets an examiner (or this engine) check a whole passage with one line.
A great-circle passage capped by a LIMITING LATITUDE: great circle to the parallel, run along the parallel, great circle down to the destination. At the tangent points the track touches the parallel at course 090°/270°, making the triangles right-angled — Napier's rules territory. The saving of the great circle, without the high-latitude weather.
The point where the vertical through the shifted centre of buoyancy cuts the centreline as the ship heels a SMALL angle. For a box form BM = B²/12d — beam squared: half a metre of beam buys more stiffness than a metre of anything else. Above G it rights you; below G it wrecks you.
GM = KM − KG, the distance from the centre of gravity up to the metacentre — the single metre that sets initial stability. Positive: stiff, she snaps upright (short, harsh roll if too large). Small: tender, slow and comfortable but watch the free surfaces. Negative: angle of loll, and a genuinely dangerous morning.
The horizontal arm between weight acting down through G and buoyancy acting up through the heeled B — righting moment = W × GZ. Small angles: GZ ≈ GM·sinθ; wall-sided: (GM + ½BM·tan²θ)·sinθ. The GZ CURVE against heel is the ship's whole transverse character on one sheet.
A slack tank's liquid runs downhill as the ship heels, moving its own G the wrong way — equivalent to a VIRTUAL RISE of the ship's G by i·ρ/W, where i = l·b³/12 of the liquid surface. The cube on the breadth is the whole story: one longitudinal bulkhead quarters it. Depends on the surface, not the amount of liquid.
The tonnes needed to sink the ship one centimetre: waterplane area × water density / 100. The workhorse of loading arithmetic and the simple draft survey — sinkage in centimetres × TPC = cargo aboard.
How much deeper the load line may dip in fresh water, since the ship rises again reaching salt: FWA = W/(4·TPC) millimetres. For dock water in between, scale it: DWA = FWA·(1025 − ρ_dock)/25. Engine-checked to be EXACT for a constant waterplane, not an approximation.
The volume one tonne of a cargo occupies as stowed, m³/t — iron ore ~0.4, bulk grain ~1.3, packaged timber ~2.4. Below the ship's own cubic-per-deadweight ratio you run out of allowed WEIGHT first; above it you run out of SPACE. The first number of every stowage plan.
The space in a compartment that cargo cannot use — between packages, around frames and brackets, over the top of an uneven stow — expressed as a percentage of the space occupied. Grows the volume a parcel needs: V = w·SF/(1 − BS%). Bagged and cased cargo suffer it; bulk barely does.
The longitudinal stiffness of the ship: W·GM_L/(100·L) tonne-metres per centimetre of trim. Divide any longitudinal moment by it and you have the change of trim; distribute that about the centre of flotation and you have the new drafts. The workhorse of every loading problem.
Weighing GM itself: shift a known weight a known distance across the light ship, read the heel on a long pendulum, and GM = w·d/(W·tanθ). From GM and the known KM falls the light-ship KG — the founding number every later stability sum stands on. Done at build and after major conversion.
What an UNSTABLE ship does instead of sitting upright: with GM negative she flops to the angle where the wall-sided term rebuilds a righting arm, and lolls there — flopping to either side. The cure is weight DOWN (or free surfaces out), never ballasting the high side first: that can roll her through the loll to the other side, hard.
The temperature at which a parcel of air becomes saturated and its moisture starts condensing. THE number of cargo ventilation: compare dew points, never temperatures — warm air can be drier than cold air. Ventilate only while the outside air's dew point is below the hold's.
Two condensations, one rule. Warm→cold passage: the hold's moist air finds the ship's cooling steel — SHIP sweat, dripping from frames onto the stow (ventilate when dew points allow). Cold→warm passage: admitted outside air finds the still-cold cargo — CARGO sweat on the goods themselves (keep the hold shut until the cargo warms).
The IMDG Code's graded distances between quarrelsome cargoes, in rising severity: 'away from' (≥3 m or a bulkhead's protection), 'separated from' (different compartments; ≥6 m on deck), up to a complete compartment or hold between. The terms are physical requirements in steel and metres, not paperwork.