# Chapter 2. Latitude and Longitude

18th December 2017 by Dr Helen Klus

## 2.1 Latitude and longitude

People split the Earth into lines of latitude and longitude in order to help with navigation. Latitude lines are lines around the Earth moving from north to south, and a person’s latitude determines how far north or south they are in degrees. At the equator, the latitude is 0° and it is 90° at the poles. Longitude lines, or meridians, circle the Earth from east to west, and a person’s longitude determines how far east or west they are in degrees. The meridian that passes through Greenwich, in London, is set at 0° longitude for historical reasons.

 Figure 2.1Image credit Latitude and longitude lines.

## 2.2 Measuring latitude

You can work out your latitude by measuring the angle of the Sun, relative to a stick placed vertically in the ground at noon, when the Sun is highest in the sky.

At the equator, the Sun will be perpendicular to the ground, but as you move away, the Sun will seem tilted because you are on a surface that is tilted. At the pole, you are standing at a 90° angle to how you would be standing on the equator. This is not strictly true, however, because the Earth is titled at an angle of about 23.4°.

 Figure 2.2Image credit An illustration of the effects of a spherical Earth on shadows, where the yellow lines represent individual rays of the Sun, and the brown lines represent objects that block the light. The effects of a spherical Earth on the atmosphere are also shown, where light travels through more of the atmosphere at sunrise and sunset. This makes the light appear redder. This is an approximation because the Earth is tilted by about 23.4°.
 Figure 2.3Image credit The tilt of the Earth.

### 2.2.1 Seasons and the tilt of the Earth

There is an angle of about 23.4° between the ecliptic, the path the Sun and planets appear to travel through, and the celestial equator, which runs through the centre of the Earth between the North and South Poles. This angle is known as the obliquity of the ecliptic.

The tilt of the Earth means that the light of the Sun arrives from different angles over the course of a year. This causes the seasons to change, where summer occurs on the side of the Earth that’s tilted towards the Sun, and winter on the side that’s tilted away.

 Figure 2.4Image credit The seasons are caused by the tilt of the Earth.

It’s hotter in the summer for the same reason that it’s hotter at the equator: light loses less energy - meaning it has a shorter wavelength and is therefore bluer (as discussed in Book II) - when it travels through less atmosphere.

The summer solstice occurs when the part of the Earth that you are on is tilted towards the Sun at the highest possible angle. This occurs on June 21st in the northern hemisphere, and on December 21st in the southern hemisphere.

The winter solstice occurs when the part of the Earth that you are on is tilted towards the Sun at the lowest possible angle. This occurs on December 21st in the northern hemisphere, and on June 21st in the southern hemisphere.

 Figure 2.5Image credit Eratosthenes measured the Earth’s axial tilt by measuring the angle of the Sun on the summer and winter solstices.

### 2.2.2 Eratosthenes and the tilt of the Earth

In order to measure your true latitude, you need to know the angle the Earth is tilted by.

Ancient Greek astronomer Eratosthenes first measured the tilt of the Earth in about 240 BCE.[1] He did this by measuring the angle of the Sun on the summer and winter solstices, where the difference between these two values is twice the angle that the Earth is tilted by.

The Earth is tilted by about 23.4°, and so to find your latitude at the summer solstice, you need to add 23.4° to the angle you measure in order to get your correct latitude. During the winter solstice, you need to deduct 23.4°, and at any date in between, you need to work out the angle, knowing that it changes by 23.4° every six months, and therefore about 0.13° per day.

In the northern hemisphere, you can also calculate the tilt of the Earth, and therefore your latitude, by looking at the angle of the North Star. At the North Pole, it will be straight above you, 90° from the horizon, and at the equator, it will be straight ahead, 0° from the horizon.

In order to know your full position, however, you need East-West as well as North-South coordinates, and for this, you need to know your longitude.

## 2.3 Measuring longitude

Longitude is much harder to calculate than latitude. The Earth rotates 360° per day, or 15° per hour, and so there’s a direct relationship between longitude and the time that the Sun rises and sets. The Greenwich meridian is designated 0° longitude. The Sun sets an hour earlier every 15° east of this, and will set one hour later every 15° west.

 Figure 2.6Image credit Time zones are caused by the rotation of the Earth.

If you know the difference between the time of the sunset in your location and another known location, then you can work out how far east or west you are from there.

Ancient Greek astronomer Hipparchus was the first to suggest that longitude could be calculated by comparing the time of a lunar eclipse at different locations in about 127 BCE, but the inaccuracy of timekeeping devices meant that longitude measurements were often wrong.[2]

### 2.3.1 The first mechanical clocks

Early clocks included sundials, which measure the movement of the Sun across the sky by marking where a shadow falls, and sand clocks, candles, incense sticks, and water clocks, which were all used to measure constant time durations.

Water clocks measure how long it takes for water to flow from one place to another, and existed in Ancient Egypt, Babylon, India, and China.

Mechanical clocks were first built in 13th century Europe. These used an escapement mechanism, which was first designed by Chinese polymath Su Song in 1088. An escapement is a device that supplies energy to the timekeeping element and allows each cycle to be counted. In Song’s clock, it was driven by water.[3]

In medieval clocks, the escapement mechanism was composed of a verge and foliot, where verge refers to a vertical rod connecting a wheel, shaped like a crown, to the foliot. The foliot is a horizontal beam with weights attached to either side. The rod has two metal plates, the pallets, placed at the top and bottom of the wheel.

 Figure 2.7Image credit Verge and foliot escapement mechanism.

The foliot moves back and forth, which causes the verge and pallets to rotate. This turns the wheel, which can be attached to the hands on the clock via a series of wheels, so that the hands on the clock move every time the foliot swings.

Ottoman Turkish polymath Taqi al-Din invented the first known alarm clock, the first watch, and the first clocks that measure time in minutes and seconds. In 1551, he invented a rudimentary steam engine.[4]

### 2.3.2 The first pendulum clocks

Italian natural philosopher Galileo Galilei became the first person to suggest using a pendulum to measure time in 1602. Galileo discovered that the time it takes for a pendulum to swing back and forth, its period, does not depend on how far it swings or how massive the pendulum is. The time it takes is, however, proportional to the square root of the length of the pendulum.[5]

 P ∝ √ ℓ (2.1)

Here, P is the period of the pendulum and is the length of the pendulum.

The pendulum was the first harmonic oscillator to be studied scientifically. A harmonic oscillator is a system that, when moved from equilibrium, experiences a force that restores it back to its original position. If the restoration force is the only force involved, then it is called a simple harmonic oscillator. Dutch natural philosopher Christiaan Huygens built the first pendulum clock in 1656.[6]

 Figure 2.8Image credit Diagram of Christiaan Huygens’ pendulum clock by Huygens, 1673.

In 1660, English natural philosopher Robert Hooke showed that a spring acts in a similar way to a pendulum, where the force needed to compress a spring is proportional to the length you compress. Hooke used a pendulum to model the orbits of the planets and, in about 1666, he suggested that the pendulum could be used to measure acceleration due to gravity (g).

 P = 2π √ℓ/g (2.2)

This equation would later be explained by English natural philosopher Isaac Newton’s theory of gravitation, which was published in 1687 (discussed in Chapter 5).

In 1671, French astronomer Jean Richer noticed that his pendulum clock was over two minutes slower per day when he was in Cayenne, in South America, than when he was in Paris, in Europe. He deduced that the force of gravity must be lower in Cayenne, allowing the pendulum to swing further.[7] Newton later showed that this is because the centrifugal force, the outwards force caused by rotation, is stronger there. The centrifugal force pushes things away from the Earth, in the opposite direction to the gravitational force and this is why the Earth and all other rotating spheres are slightly elongated at their equator. Gravity maps of the Earth could then be made by taking pendulums around the world.

 Figure 2.9Image credit The Earth’s equatorial diameter is slightly larger than its polar diameter because the centrifugal force is strongest at the equator.

### 2.3.3 The Royal Observatory at Greenwich

In 1514, German mathematician Johannes Werner suggested that time could be measured by observing the position of the Moon.[8] He may have been inspired by Italian explorer Amerigo Vespucci who wrote, in 1502, that the Moon moves west at about 11.5° per hour, which is slightly less than the movement of the stars.[9] This is because the Moon is also orbiting the Earth, completing one full orbit a month. Once an observer knew the position of the Moon, they could determine the time at Greenwich using a prepared table of lunar distances and times, and then compare this to the local time.

French explorer Jean-Paul Le Gardeur informed King Charles II of England of Werner’s technique in 1674. The King consulted his royal commissioners, who included Hooke, and they turned to English astronomer John Flamsteed.

Flamsteed stated that they didn’t have the means to measure the positions of the stars and Moon accurately enough to determine longitude in this way, and so King Charles II established the Royal Observatory at Greenwich and appointed Flamsteed as the first Astronomer Royal.[10]

### 2.3.4 The Board of Longitude

Werner’s method was modified to account for Newton’s theory of gravity, but errors in navigation led to so many shipwrecks that in 1714, the British government established the Board of Longitude. The Board promised to reward the first person who showed how longitude could be accurately calculated, following similar schemes in France, Spain, and Holland.

German astronomer Tobias Mayer and Swiss mathematician Leonhard Euler created a set of tables that predicted the position of the Moon more accurately than ever before, and allowed people to calculate their longitude to within half a degree.

Euler and Mayer submitted these to the Board of Longitude in 1755, and British astronomer Nevil Maskelyne, the fifth Astronomer Royal, suggested that tables like these should be published annually in an official book known as a nautical almanac. The nautical almanac was ready for 1767, and became the standard set of tables used worldwide.[11] The Greenwich meridian was designated 0° longitude, and this was accepted internationally in 1884.[12]

### 2.3.5 The marine chronometer

People were able to measure longitude more accurately after English clockmaker John Harrison devised a clock that would work at sea. This was known as a marine chronometer. Harrison made a number of attempts before he solved the problem in 1759. Harrison’s fourth design, known as H4, was only 13 cm in diameter, and looked like a large pocket watch.[5]

 Figure 2.10Image credit Harrison’s H4 chronometer.

Harrison had to fight to win the prize offered by the Board of Longitude, eventually gaining the prize money in 1773, after appealing to King George III. Harrison’s clocks were used by explorers like Captain James Cook, but they were expensive, and so did not become standard for another few decades.[5]

The British Nautical Almanac published lunar distance tables until 1906, two years after the time in any particular location was first broadcast by radio.

## 2.4 References

1. Agarwal, R., Sen, S., Creators of Mathematical and Computational Sciences, Springer, 2014.

2. O’Connor, J. J., Robertson, E. F., Hipparchus of Rhodes, MacTutor History of Mathematics archive, University of St Andrews, 1999.

3. Yan, H. S., Reconstruction Designs of Lost Ancient Chinese Machinery, Springer Science & Business Media, 2007.

4. Al-Hassani, S., The Astronomical Clock of Taqi Al-Din: Virtual Reconstruction, Muslim Heritage.

5. Matthews, M. R., Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion can Contribute to Science Literacy, Springer Science & Business Media, 2012.

6. Macey, S. L., Encyclopedia of Time, Routledge, 2013.

7. Good, G., Sciences of the Earth, Psychology Press, 1998.

8. Mörzer Bruyns, W. F. J., Dunn, R., Sextants at Greenwich: A Catalogue of the Mariner’s Quadrants, Mariner’s Astrolabes Cross-staffs, Backstaffs, Octants, Sextants, Quintants, Reflecting Circles and Artificial Horizons in the National Maritime Museum, Greenwich. OUP Oxford, 2009.

9. Sanders, R., 21st Century Science & Technology 2001, 14, 58–60.

10. Couper, H., Henbest, N., The Story of Astronomy: How the universe revealed its secrets, Hachette UK, 2011.

11. Reidy, M. S., Kroll, G. R., Conway, E. M., Exploration and Science: Social Impact and Interaction, ABC-CLIO, 2007.

12. Various, International Conference Held at Washington for the Purpose of Fixing a Prime Meridian and a Universal Day. October, 1884. Protocols of the Proceedings. Gibson Bros., Printers and Bookbinders, 1884.

### How We Came to Know the Cosmos: Space & Time

I Pre 20th Century theories

1. Constellations

2. Latitude and Longitude

3. Models of the Universe

4. Force, Momentum, and Energy

5. Newton’s theory of Gravity

6. The Age of the Universe

II 20th Century discoveries

7. Einstein’s theory of Special Relativity

8. Einstein’s theory of General Relativity

9. The Origin of the Universe

10. Galaxies

11. Stars

12. Red Giants and White Dwarfs

13. Supergiants, Supernova, and Neutron Stars

14. Black Holes

III Missions to planets

15. The planet Mercury

16. The planet Venus

17. The planet Earth

18. The Earth’s Moon

19. The planet Mars

20. The Asteroid Belt

21. The planet Jupiter

22. The planet Saturn

23. The planet Uranus

24. The planet Neptune

IV Beyond the planets

25. Comets

26. The Kuiper Belt and the Oort Cloud

27. The Pioneer and Voyager Missions

28. Discovering Exoplanets

29. The Search for Alien Life in the Universe

30. Where are all the Aliens?

V List of symbols

31. List of symbols