A and C are the opposite vertices of a square ABCD, and have
coordinates (a,b) and (c,d), respectively. What are the coordinates
of the vertices B and D? What is the area of the square?
A 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
Find the vertices of a pentagon given the midpoints of its sides.
Published December 2011,February 2011.
The person who is credited with our coordinates system is a man
called Descartes. If you look
here then you can find out how he was inspired by a fly!
Coordinates are one way of describing the position of something.
If you are trying to find your way to a place you've never been to
before, chances are, you'll use a map. Has it ever occurred to you
how that map was made? Or perhaps who first thought of the idea of
designing maps? We're here to answer these questions for you.
The person designing a map (the cartographer ) may use colour, size,
shape, numerical value, texture, pattern and orientation to help
get across information.
Here we will concentrate on maps which help us to locate a
position on the earth's surface.
However, another map was found during
excavations at the site of a town called Catal Hy"uk in Turkey. It
is thought that this map shows Catal Hy"uk itself and maybe dates
back to even earlier -- around 6000 BC:
Map of Catal
Used with permission http://www.henry-davis.com/MAPS/
In 500BC, Greek scholars were the first to suggest that the
world was spherical. They based this on the altitude of stars and
how ships appear on the horizon. Aristotle saw that the earth casts
a round shadow on the moon. This had great impact on map drawing
and led Eratosthenes to calculate a surprisingly accurate figure
for the circumference of the earth. He went on to produce a world
Have you ever looked closely at a globe or a world atlas? In a
similar way to the x and y coordinates that we are familiar with, a
special coordinate system is used. Eratosthenes was the first
person to try to draw a map of the world based on a system of
lines of latitude and
Lines of latitude are
imaginary horizontal lines that circle the globe. We can think of
this as circles of different sizes:
The longest is at the equator.
The shortest are at the north and south poles where they shrink
to a point.
Lines of latitude are also called parallels as they run parallel to the
Lines of longitude are
imaginary lines reaching from pole to pole. We can think of them as
looking like segments of an orange.
Lines of longitude are also known as meridians .
Hipparchus built on the work of Eratosthenes and was the first
to give the positions of places on the earth as coordinates using
latitude and longitude. In fact, these coordinates are angles:
The latitude of the equator is 0 degrees.
The latitude of the north pole is +90 degrees.
The latitude of the south pole is -90 degrees.
All lines of longitude cross the equator and as this is
circular, it can be divided into 360 degrees. The longitude of a
point can be taken as where its meridian line crosses the equator.
However, this depends on which line of longitude is taken as 0
There were many twists in the tale to follow. Ptolemy, another
Greek, wrote a book in eight volumes called "Geography" which
contained maps of the known world at that time. All these were
based on a curved globe. Ptolemy showed how to represent a sphere
on a flat piece of paper:
Used with permission www.henry-davis.com/MAPS
Map-making did not really develop further until after the 12th
century AD. During the Age of Exploration (15th century) sailors
were discovering new lands and it was a complicated task to draw up
the maps. However, this meant that the position of new places on
earth had to be worked out.
In 1514, Werner, a German, translated Ptolemy's Geography and
suggested a way of measuring longitude using the moon.
Unfortunately, he was not able to be accurate enough at that time.
Many European explorers tried to tackle the problem of calculating
longitude. Gemma Frisius, from the Netherlands, used a clock to
keep time as he travelled in an east-west direction. He then
compared this time with the local time at his destination to work
out the total distance. This in fact proved to be a very good
method but unfortunately the clocks at that time were not accurate
Producing world maps became a great challenge and countries
started offering rewards to encourage people to solve the problem.
It was believed that maths and astronomy were the key to finding
longitude. Observatories were set up in both Paris and London.
Each country had different approaches. The French used the idea
of the movement of Jupiter's moons which had initially been
proposed by Galileo in 1616. The English however concentrated on
the position of our moon. Both groups continued to work on
developing an accurate clock.
Between the two, methods were eventually found and at last
longitude could be measured accurately.
In 1884, an International Meridian Conference was held. The
meridian which passed through the Royal Observatory in Greenwich,
London was given 0 degrees longitude. Greenwich was chosen as it
was already used as zero on over 70% of navigation maps at that
time, so this decision would benefit the most people. The line at 0
degrees longitude is known now as the Prime Meridian or Greenwich Meridian .
Places to the east of this line are measured from +0 degrees to
Places to the west of this line are measured from -0 degrees to
-180 degrees This means that the lines at +180 degrees and -180
degrees are in fact the same.
With improved technology, map making has become much easier.
Computers, space science, aerial photography and electronic
distance-measuring instruments have all helped to make cartography
more accessible. There is much more to a map than meets the eye,
wouldn't you agree? It just goes to show how useful maths is! These
websites will give you more information: