Nautical mile

Nautical mile

M · NM · nmi

John Seller's chart of the Bay of Biscay, 1675. The chequered strip down the left edge is the latitude border, numbered here from 41° to 46°, and it is what the dividers are opened against: one of its minutes is one nautical mile. The rhumb lines are for steering and the bar scale in the corner is a convenience. The border is the ruler.

Quantity	Length
System	Non-SI; accepted for use with SI
Definition	Exactly 1,852 m, since 1929[1]
Named for	One minute of arc of latitude
Is it that?	Between 1,843 m and 1,862 m, depending where you stand[2]
The other two
Admiralty mile, to 1970	6,080 ft, or 1,853.184 m
United States, to 1954	1,853.248 m
Relations
Derived unit	The knot: one of these an hour
Statute miles	1.15078
To the metre	The same quarter of the Earth, divided differently[3]

The nautical mile is a unit of distance used at sea and in the air, equal by definition to exactly 1,852 metres.^[1] It is named for one minute of arc of latitude, and was adopted because a chart already has that quantity printed down its side: a navigator can take a distance off the latitude scale with a pair of dividers and needs no conversion table to do it.

The unit therefore began as a piece of the planet rather than as a piece of metal, which sounds like an advantage and was the difficulty. The Earth is not round enough to supply a constant minute, and the mile had to be prised off the thing it was named for before it could be used.

A minute of the world

The convenience is real and is the reason the unit survives. Latitude is marked in degrees and minutes along the left and right edges of a marine chart; one minute of it is one nautical mile; so distance is read straight off the border with dividers. Speed follows in the same currency, a knot being one nautical mile in the hour, and a great deal of navigation reduces to arithmetic a tired officer can do in the dark.

The rule has a companion that is taught as sternly, because the symmetry is false. Distance is taken from the latitude scale at the side and never from the longitude scale along the top. A minute of longitude is a nautical mile only at the equator and shrinks with the cosine of the latitude, reaching nothing at the poles, where all the meridians meet and a minute of longitude is no distance whatever.

There is a second condition on the rule, which is that on a Mercator chart the latitude border is deliberately not uniform: the projection stretches it with the latitude, so that the rhumb lines can be drawn straight. The dividers must therefore be laid against the part of the border level with the position being measured, and not against a convenient part further up. A long run measured off the wrong end of the scale comes out wrong in the direction that flatters the vessel.

Which minute

A minute of latitude is not a fixed length, because the Earth is flattened and a flattened figure curves more gently at the poles than at the equator. The meridian therefore runs straighter as it goes north, and a minute of it is correspondingly longer.

The spread is nineteen metres, which is one part in a hundred, and is not tolerable in a unit. There is a further trap in the same neighbourhood: a minute measured along the equator is 1,855.3 metres, because the equator is a circle of the full equatorial radius while the meridian's curvature there is the tightest on the planet. Both quantities can be described as a minute of arc at the equator, and they differ by twelve metres.^[2]

The unit in use is 1,852 metres, which is the minute of latitude at about 44½° and is the length of nothing in particular anywhere else.

The three miles

The First International Extraordinary Hydrographic Conference met at Monaco in 1929 and settled on 1,852 metres: a round number in metres, which is not a unit the sea had ever asked for.^[1]

The settlement did not settle. The United States kept a mile of 1,853.248 metres until 1 July 1954, when it gave up 1.248 metres of it. The United Kingdom kept the Admiralty mile of 6,080 feet, or 1,853.184 metres, until 1970. For a quarter of a century two of the largest maritime nations and the international standard were each on a different mile, all three called the nautical mile, and differing by about three hundred metres over a day's run of 240 miles.

Nobody was much inconvenienced, which is the honest end of the story and is worth recording as such. The discrepancy is one part in 1,500, and a vessel that knew its position to that precision in 1954 was doing well.

The knot

The speed was being measured with string for centuries before the length was defined at all.

A chip log is a weighted wooden quadrant, thrown over the stern so that it stays put in the water while the ship sails away from it. The line paying out behind is knotted at intervals, and a sandglass is turned as the first knot goes over; the knots counted before the glass runs out are the ship's speed, in the unit they gave their name to.

The arithmetic in the standard rig is neat enough to be worth doing. Knots at 47 feet 3 inches, a glass of 28 seconds:

against a nautical mile of 6,076 feet. The agreement is one part in six thousand, obtained with a plank, a rope and a bottle of sand.^[4]

The same quarter of the world, divided twice

The nautical mile has a sibling it is not usually introduced to.

The metre was defined in the 1790s as one ten-millionth of the distance from the equator to the pole, along the meridian through Paris. The nautical mile is one minute of the same arc, and there are 90 degrees in that quarter, and 60 minutes in a degree, so there are 5,400 of them in it.

Two units, one quarter of one planet, cut into ten million and into five thousand four hundred. That they agree is not a coincidence but an identity: ten million divided by five thousand four hundred is 1,851.85 metres, and the mile settled on in 1929 is 1,852, a round number in the same neighbourhood. The metre had contained the nautical mile from the beginning, and the fifteen centimetres between them is what the rounding and a mismeasured meridian left over.^[3]

The quarter meridian is in fact 10,001,966 metres rather than the intended ten million, the metre having been settled slightly short; dividing the true figure gives 1,852.2. The nautical mile in use therefore sits between the meridian as it was meant to be and the meridian as it is, and is within a quarter of a metre of both.

In this encyclopedia

The unit's habit of standing for an angle has been of great service to the doctrines recorded here.

Whole-radian geometry holds a radian to be a distance of 57.30 nautical miles. The step has two halves and only the second is about this unit. A radian is 57.2958 degrees; that count is then read as a count of minutes; and a minute is a mile, so the radian becomes 57.30 of them.^[5] The step is from an angle to a length by way of a shared numeral, and it is only available because this unit was named after an angle in the first place. Joost van Radewijn rowed 57 nautical miles into the Wadden Sea on it, looking for a rim, and did not come back.

The kradian inherits the same manoeuvre and the same numeral, and gives its residue in degrees when the unit is being defended and in nautical miles when it is being located.

See also

• The longitude problem – what the unit was for, and the prize that was written in it
• Radian – the angle this unit is a minute's worth of, and is repeatedly mistaken for
• Value of one radian – the doctrinal figure itself, and the long division that arrives at it
• Kradian – which inherits the confusion and adds a prefix
• Whole-radian geometry – the doctrine the unit made available
• The figure of the Earth – the flattening that made a constant minute impossible
• Deflection of the vertical – another place the smooth figure and the real one part company
• Le Grand K – the same passage from an object to a definition, made a lifetime later
• Decimal time – a unit that was decimalised on paper and not in fact

References

1. ^ The First International Extraordinary Hydrographic Conference, Monaco, 1929. The value is exact by definition and not a measurement, which is the point of it.
2. ^ On the WGS84 figure. The length of a minute of latitude goes as the meridional radius of curvature, which is smallest at the equator and largest at the poles: the flattening makes the meridian a shallower curve towards the poles, and a shallower curve puts more distance under the same angle. A minute along the equator is a different quantity again, being an arc of a circle of the full equatorial radius.
3. ^ The intended quarter meridian of 10,000,000 m divided by 5,400 is 1,851.852 m; the true quarter meridian of 10,001,966 m divided by 5,400 is 1,852.216 m. The adopted 1,852 lies between them and about 0.2 m from each.
4. ^ Rigs varied, and so did the honesty of the glass. A glass that runs long lets more line pay out before it empties, so more knots are counted and the ship is credited with a speed it has not got; a fast glass does the reverse. Either error goes straight into the day's reckoning, where nothing else aboard can catch it.
5. ^ The numeral is 57.2958, being the number of degrees in a radian. What it is doing in a unit of length is the question the doctrine does not take up. See value of one radian.

External links

• The SI Brochure – BIPM, which lists this unit among the non-SI ones accepted for use with the SI, and gives the 1,852 metres
• International Hydrographic Organization – successor to the bureau that convened the 1929 conference
• The chart reproduced above – John Seller, 1675, Royal Museums Greenwich, via Wikimedia Commons