ObscuripediaThe Free Obscure EncyclopediaA sight does not give a position, and the article does not pretend otherwise until the third section. Each measured altitude yields a circle of position, and a circle is not a fix. Two circles give two intersections, of which one is usually absurd; three give a small triangle, which is not a point either and is called a cocked hat. Editors are asked not to write that a navigator "takes a sight and gets a position", which is the single commonest way of getting this subject wrong.
| Question answered | Where am I |
|---|---|
| Answered from | The angle from a body to the horizon |
| One sight gives | A circle, not a point |
| Two sights give | Two crossings, one absurd |
| Three give | A small triangle |
| What it needs | |
| Instrument | A sextant |
| Tables | An almanac[1] |
| Clock | Set to Greenwich, and right |
| Horizon | Visible, or faked with a bubble |
| Standing of the art | |
| A one-minute error costs | One nautical mile |
| Dropped by the USNA | 1998 |
| Restored | 2015[8] |
| Can be jammed | No |
Celestial navigation, or astronavigation, is the determination of position on the Earth from measured angles between celestial bodies and the visible horizon.[1] It is the oldest method of fixing a position out of sight of land, and the only one that cannot be switched off by anybody else.
Its central fact is geometric and is worth having before anything else. At any instant, a body stands directly overhead at exactly one point on the Earth, called its geographical position. An observer who measures that body's altitude above the horizon has measured, in effect, their angular distance from that point. Every place at the same distance from it sees the same altitude, and those places form a circle. The sight gives the circle. It does not give the observer.
The circle of equal altitude is centred on the geographical position and has a radius equal to the body's zenith distance: ninety degrees minus the altitude.[2] A body forty degrees up puts the observer fifty degrees from its geographical position, which is three thousand nautical miles, and the circle is six thousand miles across.
Nobody plots that circle, because at the scale of a chart a three-thousand-mile arc is indistinguishable from a straight line. The navigator plots a few miles of it, near where they suppose themselves to be, and calls the result a line of position. Two of them cross. The crossing is the fix.
This is the whole architecture, and everything that follows is a technique for getting a short segment of an enormous circle onto a chart without drawing the circle.
A sextant reading is not an altitude. It is a number that becomes one only after a fixed sequence of admissions, and the sequence has a shape: the reading is Hs, the corrections are applied in two groups, and what comes out is Ho.[3]
The order is not a convention. The first group – the instrument's error and the observer's height – gives the apparent altitude, Ha, and the second group has to be entered with Ha, because how much the air bends a ray depends on how high the ray is. Correct in the wrong order and you look up the refraction for an altitude the body never had.
Index error is the instrument's own, and is the first thing applied. If the drum reads high the error is "on the arc" and the correction is negative. A sextant reading zero when it should read zero is a sextant that has been checked recently, and no other kind exists.
Dip is the observer's height. From any height at all the sea horizon lies below true horizontal, so every altitude measured from it is too large: see the dip of the horizon. It is subtracted, and it depends on nothing but the height of the eye.
Refraction is the atmosphere's. The air bends light downward, so bodies appear higher than they are, and near the horizon the effect is largest and least predictable. It is subtracted.
Semidiameter is the body's own size. A navigator measuring the Sun or Moon does not measure the centre, because the centre is not marked; the lower or upper edge is brought to the horizon and the radius of the disc is then applied.
Parallax is the observer's displacement from the centre of the Earth. For the stars it is nil; for the Sun the mean equatorial horizontal parallax is 8.8 seconds of arc; for the Moon, which is close, it is 57′ 02.7″ at mean distance and varies by minutes either side of that as the Moon comes and goes.[3]
What comes out is the observed altitude, Ho, and it is worth saying exactly what that is, because the definition is the joke the discipline never makes. Ho is the altitude of the centre of the body above the celestial horizon, as it would be seen by an observer at the centre of the Earth.[3] Every correction in the list is a step away from the person who took the sight and towards a vantage point nobody has ever occupied, and it is only from there that the geometry is simple enough to use.
The arithmetic is small and unforgiving. Sumner's own worked example of 1843 gives a sextant reading of 12° 02′ for the Sun's lower limb, from an eye seventeen feet up, and applies dip 4′ 03″ and refraction 4′ 23″ against semidiameter 16′ 08″ and parallax 8″, for a net correction of eight minutes of arc and a true altitude of 12° 10′.[4] Eight minutes of arc is eight nautical miles of position.
The line of position was discovered by accident, in bad weather, by a man who did not know where he was.
Thomas Hubbard Sumner sailed from Charleston on 25 November 1837 for Greenock. After passing 21° west he got no observation at all until near the land. Approaching midnight on 17 December he was, by dead reckoning, within forty miles of Tuskar light when the wind hauled south-east and made the Irish coast a lee shore; he stood off on tacks until daylight, saw nothing, and ran east-north-east under short sail in heavy gales.[5]
At about ten in the morning he got an altitude of the Sun. His latitude was dead reckoning after six hundred miles without a sight, and he did not trust it. So he worked the longitude three times: once with his own latitude, once with a latitude ten minutes further north, once with twenty.
The three positions came out evenly spaced – the second twenty-seven nautical miles east-north-east of the first, the third twenty-seven further again – and lying in the direction of Smalls Light. The regularity is what gave it away. In his own words:
"It then at once appeared, that the observed altitude must have happened at all the three points, and at Small's light, and at the ship, at the same instant of time; and it followed, that Small's light must bear E. N. E., if the Chronometer was right."
– Thomas H. Sumner, 1843
He held that course, and made the light in under an hour, bearing east-north-east and close aboard. His latitude had been eight miles out.[5]
The importance is not that he found the lighthouse. It is that he never did learn his latitude, and did not need to. Not knowing it was what produced the line, and the line was enough to steer on. A quantity he could not measure had been converted into a direction he could.
Sumner's method requires solving the sight two or three times over, which is laborious. The modern replacement inverts the problem, and is due to a French naval officer, Adolphe-Laurent-Anatole Marcq de Blond de Saint-Hilaire (1832–1889), who set it out in Calcul du point observé in the Revue maritime et coloniale for 1875.[6]
The navigator chooses an assumed position – not the true one, and known not to be – and calculates from the almanac what the altitude and bearing of the body would be from there. That computed altitude, Hc, is compared with the one actually observed, Ho. The difference is called the altitude intercept, and because one minute of arc is one nautical mile it is already a distance:[3] it says how much nearer to or further from the body's geographical position the observer really is than the guess supposed.
Which way is settled by a rule short enough to be remembered in the dark. If the observed altitude is the greater, the body is higher than the guess predicted, so the observer is nearer to the point beneath it, and the offset is laid off towards the body. If the computed altitude is greater, away from it. Generations of watchkeepers have carried this as Ho More, Toward, which is not elegant and has never once been misremembered.
The line of position is then drawn through that offset, at right angles to the bearing of the body, because it is a piece of the circle and the circle is perpendicular to its own radius.
It is worth stating what has happened here. The navigator invents a position known to be wrong, computes what that fiction would have seen, compares it with what was seen, and plots the discrepancy. The fix is made out of the error in a deliberate guess, and the better the guess the smaller the error, and the answer does not depend on the guess at all.
Latitude alone, to about 1770. Latitude is easy: the Sun at noon, or the Pole Star at any hour, gives it directly. Longitude requires knowing the time somewhere else, and there was no way to carry it. Ships therefore sailed north or south to the latitude of the destination and then ran down it east or west, which is safe, slow, and puts every ship on the same track for the pirates.
Lunar distances, 1767 onwards. The Moon moves against the stars fast enough to be a clock, and Maskelyne's Nautical Almanac tabulated its distances from selected stars years ahead. The observation is hard and the arithmetic long, but it needs no machinery, and the tables were printed until the volume for 1906.
The chronometer, from the 1760s. A clock that keeps Greenwich time makes the Moon unnecessary and the calculation short: see the longitude problem. The two methods were carried together for a century, the clock for daily use and the Moon to check the clock.
Sumner's line, 1837, and the intercept, 1875. These change what a sight is. Before them a sight yielded a latitude or a longitude; after them it yields a line, and any two lines from any two bodies give a fix, which is why the practice moved from noon-and-Pole-Star to a rapid round of three or four stars at twilight.
Tables, then machines. The trigonometry was reduced to addition and subtraction by printed sight-reduction tables, so that a competent officer could work a sight in a few minutes without understanding the spherical geometry underneath it.
Aircraft, to the 1960s. An aeroplane has no sea horizon, so the sextant grew a bubble level to supply an artificial one. Early Boeing 747s were built with sextant ports in the roof; they were phased out as inertial and Doppler systems arrived.[7]
Satellites, and the return. The United States Naval Academy discontinued its course in the spring of 1998. It reinstated instruction in the academic year 2015–16, citing the vulnerability of satellite navigation to hostile interference.[8] Nothing about the method had improved in the interval. What had changed was the confidence of the thing that replaced it.
Everything above rests on knowing the time at Greenwich, and an error there is an error in position that no care with the sextant can recover.
The Earth turns fifteen degrees of longitude an hour, which is one minute of arc every four seconds. Four seconds of clock error is one nautical mile of longitude at the equator, and a chronometer a minute out puts a ship fifteen miles east or west of where it thinks it is, with every sight agreeing perfectly.[9] Sumner's own conclusion has the condition in it: Smalls Light must bear east-north-east, if the chronometer was right.
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