Chapter 12 — धरातल पर किसी स्थान की स्थिति का निर्धारण

On a sheet of paper a point can be fixed by saying how far it is from the left edge and from the bottom edge. On a round earth there are no edges: a ship in the middle of the Indian Ocean sees only water and sky to every horizon, and an explorer in the Sahara sees only sand. Yet the sailor must know where he is and the map-maker must be able to put every place in its right position on the globe. What is needed is a reference system — a set of fixed lines drawn on the earth from which the position of any point can be measured.

The earth itself supplies the starting points. Because it rotates, it has two fixed points where the axis meets the surface, the North Pole and the South Pole. These are the only two points on the earth that do not move when the earth turns, and the sky provides a marker for one of them: the Pole Star stands almost exactly above the North Pole. Halfway between the two poles, at equal distance from each, a great circle can be drawn round the earth — the equator, from the Latin for equaliser, because it divides the earth into two equal halves, the northern and southern hemispheres. The equator is the natural zero line for measuring distance north or south.

For distance east or west the earth gives no natural zero: every circle through the poles is like every other. So one of them has to be chosen by agreement, and the world agreed in 1884 on the circle through the observatory at Greenwich in London, the Prime Meridian.

Angles, not distances. Because the earth is a sphere, positions are measured not in kilometres from the reference lines but in angles at the centre of the earth, in degrees, minutes and seconds. A full circle is 360°, a degree has 60 minutes (') and a minute 60 seconds ("). One degree of arc on the earth's surface is about 111 km, one minute about 1.85 km (the nautical mile) and one second about 31 m. Angles are used because they are the same whether the globe is a school model or the real earth, and because they can be measured directly from the sky, from the height of the sun or a star.

The two coordinates. The angular distance of a place north or south of the equator is its latitude; its angular distance east or west of the Prime Meridian is its longitude. Together they form the geographic coordinates of the place and fix it as surely as a street number fixes a house. Kolkata is 22° 34' N, 88° 22' E: there is exactly one point on the earth with that address. Everything in this chapter unfolds from these two numbers — how the lines are drawn, what the important lines mean, and how the numbers are found by observation.

Before the system existed, sailors kept close to coasts and explorers were often lost; with it, Columbus could estimate his position, Cook could chart the Pacific, and today a mobile phone can tell its own latitude and longitude to a few metres.

Latitude of a place is its angular distance north or south of the equator, measured in degrees along a meridian from the centre of the earth. If a line is drawn from the centre of the earth to a point on the equator and another from the centre to the place, the angle between the two lines is the latitude of the place. The equator has latitude 0°; the North Pole is 90° N and the South Pole 90° S; every other place lies between, and its latitude is written with N or S to say which hemisphere it is in. Kolkata is 22° 34' N; Cape Town is 33° 56' S.

Parallels of latitude. If all the points with the same latitude are joined, they form a circle round the earth parallel to the equator; such a circle is a parallel of latitude or simply a parallel. Parallels are drawn on globes and maps at intervals of 10°, 15° or 5°, but in principle there is a parallel for every latitude, so their number is infinite; conventionally 90 are counted north of the equator and 90 south, 181 including the equator.

Characteristics of the parallels.

• They are all complete circles running east–west, parallel to the equator and to one another, and they never meet.
• They are of unequal length: the equator, the longest, is a great circle of 40,075 km; the others are small circles that shrink towards the poles, the 60th parallel being half the length of the equator and the 90th a point. The length of a parallel is the equator's length multiplied by the cosine of the latitude.
• The distance between any two consecutive parallels one degree apart is nearly constant, about 111 km, because it is one 360th of the polar circumference of 40,008 km; it is slightly larger near the poles (111.7 km) than near the equator (110.6 km) because of the earth's flattening.
• All parallels lie in planes at right angles to the axis, and every parallel cuts every meridian at right angles.
• Places on the same parallel have the same latitude, the same length of day and the same noon altitude of the sun, and so, other things being equal, similar climates; but they have different times.

Latitude and climate. Because the angle of the sun's rays depends on latitude, latitude is the first control of temperature: low latitudes near the equator are hot, middle latitudes temperate, high latitudes cold. The heat zones are bounded by parallels, and so are many crops — tea, rice and sugarcane in the low latitudes, wheat and apples in the middle, only tundra beyond 66½°.

Writing latitude. A latitude is written in degrees, minutes and seconds with the hemisphere letter: 22° 34' 12" N. In decimal form the same latitude is 22.57° N. Southern latitudes are sometimes written as negative numbers in computer systems: −33.9° for Cape Town. Latitude is also the angle used to name the important parallels of the next topic.

Five parallels have a special meaning fixed by the 23½° tilt of the earth's axis; two more are used constantly in India.

1. The Equator (0°). The great circle midway between the poles, 40,075 km long, dividing the earth into the northern and southern hemispheres. The sun is overhead here at the two equinoxes, day and night are always 12 hours, and the climate is hot and wet all year. It crosses South America (Ecuador is named for it), central Africa, Sumatra, Borneo and the Indian Ocean south of Sri Lanka; India lies wholly north of it.

2. The Tropic of Cancer (23½° N). The northernmost parallel on which the sun is ever overhead, on 21 June. It is named after the constellation Cancer in which the sun stood at the solstice in ancient times. It passes through Mexico, the Sahara, Egypt, Arabia, India, southern China and Taiwan. In India it crosses eight states — Gujarat, Rajasthan, Madhya Pradesh, Chhattisgarh, Jharkhand, West Bengal, Tripura and Mizoram — and divides the country into a tropical south and a sub-tropical north. In West Bengal it passes through Purulia, Bankura, Bardhaman and Nadia, close to Krishnanagar and Nabadwip.

3. The Tropic of Capricorn (23½° S). The southernmost parallel with an overhead sun, on 22 December, named after Capricornus. It passes through Chile, Argentina, Brazil, Namibia, Botswana, South Africa, Madagascar and central Australia (Alice Springs). The belt between the two tropics is the Torrid Zone.

4. The Arctic Circle (66½° N). The parallel north of which the sun does not set on 21 June and does not rise on 22 December. It passes through Alaska, northern Canada, Greenland, Iceland, Norway, Sweden, Finland and Siberia. Its latitude is 90° − 23½°.

5. The Antarctic Circle (66½° S). The corresponding parallel of the south, encircling Antarctica, with the midnight sun on 22 December and a sunless day on 21 June.

The two poles, 90° N and 90° S, complete the set; they are points, not circles.

Parallels important for India. India stretches from 8° 4' N (Kanyakumari, or 6° 45' N at Indira Point in the Nicobars) to 37° 6' N (in Ladakh), about 29° of latitude, and its climate ranges from equatorial in the south to alpine in the north because of this spread. West Bengal lies between 21° 38' N and 27° 10' N. The 8° N parallel is the southern tip; the 37° N parallel the northern.

Using the parallels. Two rules follow from the tilt: the tropics are at latitude equal to the tilt (23½°), and the polar circles at 90° minus the tilt (66½°). If the tilt were 30°, the tropics would move to 30° and the polar circles to 60°. The parallels also bound the heat zones — torrid between the tropics, temperate between tropic and polar circle, frigid within the polar circles — and mark where the sun can be overhead (between the tropics only) and where the sun can stay up or down for 24 hours (within the polar circles only).

Latitude fixes how far north or south a place is, but every point on a parallel has the same latitude, so a second coordinate is needed to fix how far east or west it lies. This is longitude: the angular distance of a place east or west of the Prime Meridian, measured in degrees along the equator (or any parallel) from the centre of the earth.

Meridians. A meridian of longitude is a half-circle drawn from the North Pole to the South Pole through all the places that have the same longitude. The name comes from the Latin meridies, midday, because every place on one meridian has noon at the same instant. Together, a meridian and the one directly opposite it (180° away) form a full great circle round the earth through both poles.

The Prime Meridian. Since the earth gives no natural starting line for longitude, one had to be chosen. Old maps used Ferro in the Canary Islands, Paris, Cadiz or Philadelphia; India's ancient astronomers reckoned from Ujjain. In 1884 an international conference in Washington chose the meridian through the Royal Observatory at Greenwich, near London, as the Prime Meridian, 0°, because most of the world's shipping already used British charts. Longitudes are measured up to 180° east of it and 180° west of it; the 180° meridian is common to both, and the Prime Meridian with the 180° meridian divides the earth into the eastern and western hemispheres. The Prime Meridian runs from the North Pole through Greenwich, France, Spain, Algeria, Mali, Ghana and the Atlantic to Antarctica. India, being wholly east of Greenwich, has east longitudes: Kolkata is 88° 22' E, Mumbai 72° 50' E.

Characteristics of the meridians.

• They are all half-circles of equal length, about 20,004 km from pole to pole, and all are halves of great circles.
• They run north–south and all pass through both poles, so they converge towards the poles and are farthest apart at the equator.
• The distance between two meridians 1° apart is about 111 km at the equator but decreases with latitude in proportion to the cosine of the latitude: about 96 km at 30°, 78.5 km at 45°, 55.5 km at 60°, and zero at the poles, where all meridians meet. At Kolkata (22½° N) it is about 103 km.
• Every meridian cuts every parallel at right angles.
• The number of meridians, like parallels, is infinite in principle; 360 are counted at 1° intervals (180 E + 180 W), and 24 standard meridians 15° apart mark the time zones.
• All places on the same meridian have the same local time — noon at the same instant — but different climates, since they lie at different latitudes.

Longitude and time. The chief use of longitude, apart from fixing position, is time. The earth turns 360° in 24 hours, so meridians 15° apart differ in time by one hour and meridians 1° apart by four minutes; places east are ahead. This is developed in the next chapter but is needed here for finding longitude at sea.

Writing longitude. 88° 22' E, or 88.37° E; western longitudes are sometimes negative in computer form (New York −74.0°). A full position is written latitude first, then longitude: 22° 34' N, 88° 22' E.

When the parallels of latitude and the meridians of longitude are drawn together on a globe or map, they form a network of lines crossing at right angles, the geographic grid or graticule. Every place on earth stands at the crossing of one parallel and one meridian, and its position is stated by naming the two. This is the address system of the whole planet.

Stating a position. The rule is: latitude first, with N or S, then longitude, with E or W. Kolkata is 22° 34' N, 88° 22' E; Chennai 13° 05' N, 80° 17' E; Cape Town 33° 56' S, 18° 25' E; Rio de Janeiro 22° 54' S, 43° 12' W. The four combinations of hemisphere letters — N/E, N/W, S/E, S/W — say at once in which quarter of the earth the place lies. A place on the equator has latitude 0° with no letter; a place on the Prime Meridian has longitude 0°.

Finding a place from its coordinates. Given 28° 37' N, 77° 13' E, run a finger along the 28° N parallel on the map of India until it crosses the 77° E meridian; the crossing is Delhi. On an atlas map with the grid drawn every 5° or 10°, the position between the drawn lines is estimated by proportion: 28° 37' is a little more than a third of the way from 25° to 35°.

Finding the coordinates of a place. Read the parallel nearest the place, estimate the fraction of the interval to the next, and do the same for the meridian. Practice with the atlas is the only way to become quick at this, and the board's map-pointing questions test exactly this skill.

Comparing places. The grid answers many questions at once. Two places with the same latitude (Kolkata and Mecca, both near 22° N) have the same length of day and noon sun and broadly similar temperatures; two places with the same longitude (Kolkata and Lhasa, both near 88° E) have the same local time. Kolkata is south of Delhi and east of it; Sydney is south-east of Kolkata.

Distances along the grid. Along a meridian, distance = difference of latitude × 111 km. Delhi (28° 37' N) to Kanyakumari (8° 4' N) along the same meridian is about 20.55 × 111 = 2,281 km. Along a parallel, distance = difference of longitude × 111 × cos(latitude). Kolkata (88° 22' E) to Mumbai (72° 50' E), both near 20° N for the purpose, is about 15.5 × 111 × 0.94 = 1,617 km along the parallel. For places not on the same line the shortest distance is along the great circle between them, which is found by spherical trigonometry or measured with a thread on a globe.

Extent of a country. The grid also expresses the size of a country: India lies between 8° 4' N and 37° 6' N and between 68° 7' E and 97° 25' E, so it spans about 29° of latitude (about 3,200 km north–south) and about 29° of longitude (about 2,900 km east–west at its widest, less than the latitude span because the meridians converge). West Bengal lies between 21° 38' N and 27° 10' N and 85° 50' E and 89° 50' E, so it is about 600 km long and, at most, about 400 km wide.

Grid references on topographical sheets. Survey of India sheets carry, besides the latitude and longitude in the margins, a numbered grid of kilometre squares; a feature is referred to by its easting and northing (2347, or the six-figure 234478), a quick local form of the same idea.

The great advantage of measuring position in angles is that the angles can be read directly from the sky. The oldest method of finding latitude, used by sailors for two thousand years, is to measure the height of the Pole Star above the horizon.

The principle. The Pole Star (Polaris, Dhruva tara) lies almost exactly on the extension of the earth's axis, less than 1° from the true celestial north pole. It is so far away — over 400 light years — that its rays reach every part of the earth parallel to the axis. For an observer at the North Pole the axis points straight up, so the Pole Star is overhead, at an altitude (angle above the horizon) of 90°. For an observer at the equator the axis lies along the horizon, so the Pole Star is on the horizon, altitude 0°. For an observer in between, the altitude of the Pole Star above the horizon is exactly equal to the latitude: at Kolkata (22½° N) it stands 22½° above the northern horizon, at Delhi 28½°, at London 51½°. The geometry is simple: the observer's horizon is tilted from the axis by (90° − latitude), so the star, which is in the direction of the axis, is (90° − (90° − latitude)) = latitude above the horizon.

The method. At night, face north and find the Pole Star: it is the last star in the handle of the Little Bear, or, more easily, follow the line of the two pointer stars at the end of the bowl of the Great Bear (Saptarshi) about five times their separation. Measure the angle between the horizon and the star with a sextant (at sea), a theodolite (on land), a clinometer, or, roughly, an astrolabe or even an outstretched hand (a fist at arm's length is about 10°). The angle read is the latitude of the place. A small correction of up to about 0.7° is applied for the star's slight distance from the true pole, using tables that depend on the time of night.

Worked example. A sailor in the Bay of Bengal measures the altitude of the Pole Star as 19° 30' above the horizon. The ship is at latitude 19° 30' N, roughly off Visakhapatnam.

Limitations. The method works only in the northern hemisphere, since the Pole Star is not visible south of the equator; it cannot be used on cloudy nights or when the horizon is invisible; and near the equator the star is so low that haze hides it. In the southern hemisphere there is no bright star at the pole; navigators use the Southern Cross, whose long axis points to the south celestial pole, and take the altitude of that empty point.

Historical note. Arab and Indian navigators of the Indian Ocean used a simple board on a knotted string, the kamal, to measure the Pole Star's height and sail along a chosen latitude; Vasco da Gama's pilot from Malindi to Calicut in 1498 used it. Columbus, Magellan and Cook all fixed their latitudes by the Pole Star at night and by the sun at noon. The Pole Star method is the origin of the word latitude for the north–south coordinate: it was the one that sailors could always find.

By day the Pole Star is invisible, and in the southern hemisphere it is never visible, so navigators and surveyors use the sun at noon. The method needs the altitude of the sun at local noon and one piece of information from an almanac: the latitude at which the sun is overhead that day, called the sun's declination.

The principle. At local noon the sun is on the observer's meridian and at its highest for the day. Its altitude above the horizon depends only on how far the observer is from the parallel where the sun is overhead. If the observer were on that parallel the altitude would be 90°; for every degree of latitude away from it the sun stands one degree lower. So the difference between 90° and the noon altitude, called the zenith distance, equals the difference in latitude between the observer and the overhead sun.

The formula. Latitude = (90° − noon altitude of the sun) ± declination of the sun. The declination is added when the sun is overhead in the same hemisphere as the observer and subtracted when in the opposite hemisphere; the zenith distance is taken north if the sun is to the south of the observer at noon, and south if it is to the north. The declination changes daily between 23½° N on 21 June and 23½° S on 22 December and is 0° at the equinoxes.

Worked example 1 — equinox. On 21 March the noon sun at a place is 67½° above the southern horizon. The sun is overhead at the equator (declination 0°). Zenith distance = 90° − 67½° = 22½°. The sun is to the south, so the place is 22½° north of the equator: latitude 22½° N (Kolkata).

Worked example 2 — summer solstice. On 21 June the noon sun at a place is 61° above the southern horizon. The sun is overhead at 23½° N. Zenith distance = 90° − 61° = 29°; the place is 29° north of the Tropic of Cancer: latitude = 23½° + 29° = 52½° N (near London).

Worked example 3 — winter solstice. On 22 December the noon sun at a place is 44° above the southern horizon. The sun is overhead at 23½° S. Zenith distance = 46°; the place is 46° north of the Tropic of Capricorn: latitude = 46° − 23½° = 22½° N.

Worked example 4 — southern hemisphere. On 22 December the noon sun is 80° above the northern horizon. Zenith distance = 10°, and the place is 10° south of the Tropic of Capricorn (the sun being to the north): latitude = 23½° + 10° = 33½° S (Sydney).

Worked example 5 — reverse. What is the noon altitude of the sun at Kolkata (22½° N) on 21 June? The sun is overhead at 23½° N, 1° north of Kolkata; so the sun is 1° from the zenith, altitude 89°, seen slightly to the north. On 22 December it is overhead at 23½° S, 46° away: altitude 44°, to the south.

The instruments. At sea the sextant measures the angle between the sun's lower edge and the sea horizon; on land a theodolite or a vertical pole and its shadow serve. The shadow method is the oldest: on the equinox a vertical stick 1 m tall at Kolkata casts a noon shadow of tan(22½°) × 1 m = 0.41 m, and from the shadow length the sun's altitude, and hence the latitude, can be worked out. Eratosthenes used exactly this in 240 BCE.

Longitude was the great unsolved problem of navigation until the eighteenth century. Latitude could be read from the sky, but nothing in the sky tells directly how far east or west one is, because the sky looks the same from every point on a parallel — only the time at which things happen differs. The solution was to carry the time of a known meridian and compare it with local time.

The principle. The earth rotates through 360° of longitude in 24 hours, so a difference of 1 hour in local time means a difference of 15° of longitude, and 4 minutes means 1°. If one knows the exact time at Greenwich at the moment of local noon, the difference gives the longitude: if local noon comes before Greenwich noon, the place is east; if after, it is west.

The method. (1) Carry a very accurate clock, the chronometer, set to Greenwich Mean Time. John Harrison's marine chronometer of 1761, accurate to a few seconds a month at sea, won the British Longitude Prize and made the method practical; today a radio time signal or a satellite does the same. (2) Observe local noon, the moment the sun reaches its highest point, by measuring its altitude with a sextant until it stops rising. (3) Read the chronometer at that moment. (4) Convert the difference between 12:00 and the chronometer reading into degrees at 15° per hour.

Worked example 1. Local noon is observed when the chronometer reads 6:08 a.m. GMT. Local time is ahead of Greenwich by 5 h 52 min, so the place is east. 5 h 52 min = 352 min; 352 ÷ 4 = 88°. Longitude 88° E (Kolkata).

Worked example 2. Local noon is observed when the chronometer reads 4:56 p.m. GMT. Local time is behind Greenwich by 4 h 56 min = 296 min, so the place is west; 296 ÷ 4 = 74°. Longitude 74° W (New York).

Worked example 3. A ship's local time is 3:30 p.m. when the chronometer shows 10:00 a.m. GMT. Difference 5 h 30 min = 330 min = 82½°, local time ahead, so 82½° E — the ship is on India's standard meridian.

Worked example 4 — the reverse. What is the local time at 30° W when it is 2 p.m. at Greenwich? 30° × 4 min = 120 min = 2 hours; west is behind: 12 noon.

Worked example 5 — between two places. When it is 8 a.m. at 20° E, what is the local time at 65° E? Difference 45° = 180 min = 3 hours; 65° E is farther east and ahead: 11 a.m.

Precision. An error of one minute of time gives an error of a quarter of a degree, about 28 km at the equator; this is why the chronometer had to be so accurate and why ships carried several. On land, longitude was found by the same principle using the telegraph to compare clocks in two cities, and later by radio.

Today. The Global Positioning System solves the same problem in the same way at far higher precision: each satellite broadcasts the exact time, the receiver measures how long the signals took to arrive, and from the delays to four satellites it computes latitude, longitude and height to a few metres. Longitude is still, in the end, a matter of time.

Counting time by longitude leads to a puzzle. Starting from Greenwich at noon on Monday and going east, the clock is set forward one hour for every 15°; after 180° it is midnight, the start of Tuesday; after a full 360° it is noon again — but noon on Tuesday, a day ahead of the Greenwich Monday one left behind. Going west, the clock is set back, and the traveller arrives home at noon on Sunday, a day behind. Two travellers who leave together and go round the world in opposite directions meet at home two days apart in their reckoning. Magellan's crew discovered this in 1522, returning with their logbook one day short of the Spanish calendar. The solution is to fix a line where the date changes.

Position. The International Date Line follows the 180° meridian, the opposite of Greenwich, because that meridian runs almost entirely through the empty Pacific Ocean and so inconveniences the fewest people. It is not perfectly straight: it bends east round the Chukchi Peninsula of Siberia so that all of Russia keeps one date, west round the Aleutian Islands so that all of Alaska keeps the American date, and east round Fiji, Tonga, Kiribati and Samoa so that each island group keeps a single date. Kiribati moved its whole date line eastward in 1995 so that its eastern islands could be the first to see the new millennium.

The rule. Crossing the line from west to east (from Asia and Australia towards America), the traveller goes back one day: Monday becomes Sunday and the same date is lived twice. Crossing from east to west (from America towards Asia), the traveller goes forward one day: Monday becomes Tuesday and a date is skipped. The clock time is not changed at the line, only the date. A useful way to remember: the day begins at the date line and sweeps westward with the midnight meridian; when it is Tuesday just west of the line it is still Monday just east of it.

Examples. A flight leaves Tokyo at 5 p.m. on Monday and reaches Los Angeles after ten hours; it lands at 10 a.m. on Monday local time, seven hours before it left by the clock, because it crossed the line eastward and gained a day. A ship sailing from San Francisco to Sydney crosses the line westward and skips a day: Tuesday follows Sunday in the ship's log. New Zealand, just west of the line, is among the first countries to greet each new day, and Samoa, just east, among the last; in 2011 Samoa moved itself to the western side to trade more easily with Australia and New Zealand, and 30 December 2011 never happened there.

Why not a line through Greenwich? The date must change somewhere, and the change is least troublesome where few people live. A date line through London would split Europe and Africa into two dates; the 180° meridian splits only the ocean. It is also elegant that the Prime Meridian and the date line together form one great circle, dividing the earth into the eastern and western hemispheres, with time reckoned from one and date from the other.

Relation to time zones. The zone immediately west of the date line is GMT + 12 and the zone immediately east is GMT − 12; they have the same clock time but dates one day apart. A few islands use GMT + 13 and + 14 so as to share the date of their trading partners, which is why the same instant can be, somewhere on earth, three different calendar dates.

Any circle drawn on the surface of a sphere is one of two kinds, and the difference matters for navigation and for understanding the grid.

Great circle. A great circle is a circle on the sphere whose plane passes through the centre of the sphere. It is the largest circle that can be drawn on the sphere, it divides the sphere into two equal halves (hemispheres), and its radius and circumference equal those of the sphere itself — on the earth about 40,000 km. An infinite number of great circles can be drawn, and any two of them cut each other into two equal halves at two opposite points. On the earth the equator is a great circle, every meridian together with its opposite meridian is a great circle, and so is the circle of illumination; but the only parallel that is a great circle is the equator.

Small circle. A small circle is a circle whose plane does not pass through the centre of the sphere; it is smaller than a great circle and divides the sphere into two unequal parts. All parallels of latitude except the equator are small circles, shrinking towards the poles. The Tropic of Cancer, the Arctic Circle and the 22½° N parallel through Kolkata are all small circles.

The great circle route. The most important property of a great circle is that the arc of a great circle between two points is the shortest distance between them on the surface of the sphere — just as a straight line is the shortest on a plane. A thread stretched tight between two points on a globe always lies along a great circle. Ships and aircraft therefore follow great circle routes wherever they can. Two places on the equator or on the same meridian are already joined by a great circle, so the shortest route runs along the equator or the meridian. But two places on the same parallel (other than the equator) are not most closely joined by that parallel: the great circle between them bows towards the nearer pole. This is why a flight from Kolkata to San Francisco flies over the Arctic and not along the 37° parallel, why the Delhi–New York route passes over Greenland, and why on a Mercator map the aircraft's track looks like a long curve. The saving can be large: along the 50° N parallel from London to Vancouver is about 7,800 km, while the great circle route, reaching 68° N over Greenland, is about 7,600 km; between places farther apart on the same high-latitude parallel the saving can be thousands of kilometres.

Finding a great circle route. On a globe, stretch a thread between the two places; the thread marks the route and its length, read against the globe's scale, is the distance. On a gnomonic projection, a map made by projecting the globe from its centre on to a flat sheet, every great circle appears as a straight line, so a ruler gives the route; navigators then transfer it to a Mercator chart, where it becomes a curve, and sail it as a series of short straight legs (rhumb lines).

Great circles on the grid. The grid is built of great and small circles: meridians are halves of great circles, so distances along them are the shortest possible and one degree is always 111 km; parallels other than the equator are small circles, so distances along them are not the shortest and one degree of longitude shrinks with latitude. This is the reason behind the two different distance formulas of the earlier topic, and behind the shape of the world's air and shipping routes.

Since the 1990s the sextant and chronometer have been replaced for most purposes by satellite navigation, and every smartphone now carries a receiver that finds its own latitude and longitude in seconds. The system is worth understanding because it uses exactly the ideas of this chapter — angles on a sphere and the measurement of time.

What GPS is. The Global Positioning System is a constellation of about 31 satellites, run by the United States, orbiting the earth at a height of about 20,200 km in six orbital planes, so that at least four are above the horizon from any point on earth at any time. Each satellite carries an atomic clock accurate to a billionth of a second and continuously broadcasts its own position and the exact time. Other countries run similar systems: Russia's GLONASS, Europe's Galileo, China's BeiDou, and India's own NavIC (Navigation with Indian Constellation), seven satellites covering India and 1,500 km around it, built after India was refused GPS data during the Kargil conflict of 1999.

How it works. Radio signals travel at the speed of light, about 3 lakh km per second. A receiver notes the time each satellite's signal was sent (contained in the signal) and the time it arrived (from its own clock); the difference, multiplied by the speed of light, is the distance to that satellite. Knowing its distance from one satellite places the receiver somewhere on a sphere round that satellite; from two satellites, on the circle where two spheres meet; from three, at one of two points; the fourth satellite fixes which point and corrects the receiver's cheap clock. The method is called trilateration, fixing position by distances, and the computer in the receiver converts the answer into latitude, longitude and height above sea level. Ordinary receivers are accurate to 5–10 m; survey receivers using ground corrections reach a few centimetres.

Uses. Navigation of ships, aircraft, cars and mobile phones; mapping and surveying, where the Survey of India now fixes control points by GPS instead of by triangulation; tracking of vehicles, fishing boats and wildlife; precision farming; measuring the movement of the earth's plates and the rise of the sea; timing of power grids and banking; and disaster management, guiding rescue to a stranded fisherman in the Bay of Bengal whose phone gives his coordinates. The board's chapter connects this to the old methods: the satellite replaces the star, and the atomic clock replaces the chronometer, but position is still latitude and longitude on the same grid.

Limitations. The signal is weak and is blocked by buildings, forest canopy, tunnels and mountains; it can be jammed or switched off by its owner; it gives position but not a map, so it must be combined with digital maps; and the reference ellipsoid used (WGS 84) differs slightly from the older Everest spheroid of Indian maps, so old and new coordinates of the same point can differ by a few hundred metres — a caution for anyone comparing an old topographical sheet with a phone.

Other modern aids. Radar and radio beacons, inertial navigation in aircraft and submarines, and remote-sensing satellites that photograph the earth and let a map be drawn without visiting the ground complete the modern toolkit. Yet every one of them delivers its result in the language invented for the sailor with his Pole Star: a latitude and a longitude.

The board's questions on this chapter are mostly numerical. This topic gathers the standard forms with their solutions so that the method becomes automatic.

Type 1 — latitude from the Pole Star. The Pole Star is seen 35° above the horizon. Latitude = altitude = 35° N.

Type 2 — latitude from the noon sun on an equinox. On 23 September the noon sun is 55° above the southern horizon. Zenith distance = 90° − 55° = 35°; sun overhead at the equator; the place is 35° north: 35° N. If the sun had been to the north, the answer would be 35° S.

Type 3 — latitude from the noon sun on a solstice. On 21 June the noon sun is 50° above the southern horizon. Zenith distance = 40°; sun overhead at 23½° N; place = 23½° + 40° = 63½° N. On 22 December the noon sun is 70° above the southern horizon: zenith distance 20°, sun overhead at 23½° S, place = 20° − 23½° = −3½°, that is 3½° S — the negative sign means the place is on the other side of the equator from the expected direction; check: at 3½° S on 22 December the sun is 20° to the south, altitude 70°, correct.

Type 4 — noon altitude from latitude. Find the noon altitude of the sun at Kolkata (22½° N) on 22 December. Sun overhead at 23½° S; difference = 22½ + 23½ = 46°; altitude = 90° − 46° = 44°, to the south. On 21 March: difference 22½°, altitude 67½°. On 21 June: difference 1°, altitude 89°, to the north.

Type 5 — longitude from time. Local time is 5 p.m. when GMT is 11 a.m. Difference 6 h, local ahead → east; 6 × 15 = 90° E. Local noon is observed when the chronometer reads 2:20 p.m. GMT: local behind by 2 h 20 min = 140 min → 35° W.

Type 6 — time from longitude. It is 9 a.m. at Greenwich; find the local time at 82½° E. 82½ × 4 = 330 min = 5 h 30 min ahead → 2:30 p.m. (This is IST.) Find the local time at 120° W: 120 × 4 = 480 min = 8 h behind → 1 a.m.

Type 7 — time between two places. When it is 6 a.m. at 45° W, what is the time at 60° E? Difference = 45 + 60 = 105° (the places are on opposite sides of Greenwich, so the longitudes are added) = 420 min = 7 h; 60° E is east, ahead → 1 p.m. When it is 10 p.m. at 150° E, what is the time at 30° E? Difference 120° = 8 h; 30° E is west of 150° E, behind → 2 p.m.

Type 8 — crossing the date line. A ship at 170° E, where it is 6 a.m. on Friday, sails east across the date line to 170° W. Longitude difference 20° = 80 min; 170° W is east and ahead by 80 min in clock time, but crossing the line eastward sets the date back a day: local time 7:20 a.m. on Thursday.

Type 9 — distance along the grid. Distance from Kolkata (22½° N) to a place at 47½° N on the same meridian: 25° × 111 = 2,775 km. Distance along the 60° N parallel between 20° E and 50° E: 30° × 111 × cos 60° = 30 × 111 × 0.5 = 1,665 km.

Type 10 — identifying a place. A place has the sun overhead on 21 June and 12-hour days on 21 March; the Pole Star is 23½° above its horizon. It lies on the Tropic of Cancer. A place has six months of daylight beginning 21 March: the North Pole. A place where the Pole Star is on the horizon: the equator.

Checks that catch errors. Latitude never exceeds 90° or longitude 180°; if a calculation gives more, a sign or hemisphere has been mixed. Local time east of Greenwich is always ahead. The noon sun is to the south for any observer north of the parallel where it is overhead. And 1° is 4 minutes, not 15 — the two numbers are confused more often than any others in the examination.