**Mercator **

**His life…**

Gerard Mercator
was born as Gerard de Cremer on March 5th in 1512 in Rupelmonde, which is located close to
Antwerp and Sint-Niklaas in the northern part of Belgium. However
he always signed his most important work as
Gerardus Mercator Rupelmondanus |

In 1537, he published his first map, which was immediately successful : a map of Palestine meant to illustrate texts from the Bible. The following year, he published a small map of the world in double hart-shaped projection. Mercator became famous in 1540 with his Map of Flanders that was dedicated to the emperor Charles the 5th. This map was quite accurate because of the use of the triangulization method. He made terrestrial and celestial globes respectively in 1541 and 1551.

In the meantime, Mercator went through hard times : in 1544, he was
imprisoned for seven months on accusation of heretic sympathies in the
Rupelmonde citadel or Counts tower located at what is now called the Mercator island.

At the age of 42,
Gerard moved to Duisburg on the Rhine, where he made his most important cartographic
works.

In 1554, he produced one of his first masterpieces - a milestone in the history of
cartography - namely a map of Europe.

On an astronomical-mathematical basis, he improved on the work of Ptolemy and in 1569 appeared his world map "Ad usum navigation'(for the use of navigation) which was Mercator's last map in large format and the first(and only) map on which he used the angle preserving cylindrical projection bearing his name. With this it was possible for sailors to chart their course on the map as a straight line . The importance of the Mercatormaps(later improved by the maps of Edward Wright) for the navigation cannot be underrated. With this kind of map, seafarers knew at any time their position and navigation became more safe.

Fascinated by his life's dream : the study and the publication of a Cosmography, a synthesis of the history of heaven and earth, he looked for the best representation of our planet making himself earth globes (in 1541 fist covered with the loxodromes) and later also a celestial globe. The name of Mercator will always remain linked to the publication of the Atlos, a cartographic overview of the modern world. The first part appeared in 1585, the third and final one not until 1595, after his death. Together with the monumental wall maps, the earth and celestial globes, his research on the magnetic pole of the earth, these and other publications give a picture of Mercator's versatility.

When Mercator died at the age of 82, five of his children were already deceased. His grandchildren inherited his work but they couldn’t do anything with it themselves. So, they sold the carts to a Dutch cartographer Jodocus Hondius who made al lot of money with Mercators legacy.

This technique was published in the book "Libellus de locorum describendorum ratione" in 1533 by the dutch mathematician Gemma Frisius. Calculation by triangulization starts from measuring one side of a triangle and the angles in order to calculate with trigonometry the other sides. Using the spires of churches or constructed reference points, surveyors can elaborate a chain of triangles starting from one basic segment and measured angles.

The explanation of Gemma Frisius, enclosed with this drawing, is the following :

He starts drawing a circle with a meridian(the diameter) on two different pieces of paper. Afterwards he climes on a the tower in Antwerp and puts one circle in the right position (the meridian north-south) using the compass needle. He draws the directions to the adjacent cities Middelburg, Gent, Brussel, Leuven , Mechelen and Lier taking that tower in Antwerp as the center of the circle. With the circle on that other piece of paper he repeats the procedure looking out from a suitable tower spire in Brussels towards the other cities. Coming home he puts the two papers at an arbitrary distance from each other but keeping the two meridians parallel. Then prolongating the lines of the different directions (which he drew at the top of the towers) he gets the exact location of the cities at the points of intersection. In this example he writes that starting from the distance between Antwerp and Mechelen (four units), dividing this segment in four and comparing with the other segments, one can find all the distances between the different cities on the map. |

In a later example he explaines how, starting from a given distance at an accessible place and an angle-measuring instrument, we can calculate the distance to non-accessible or distant points by using the sine rule.

Before Mercator, seafarers had a problem: there
weren’t any dependable carts. The
indications of the compass didn’t agree with the indications of the carts. As a consequence the seafarers ran ashore hundreds
of kilometres from their destination. But Mercator had a solution. He wanted to give the seafarers an reliable cart. But how to represent a globe at a flat level ? In 1569 Mercator found the solution: he projected
the world on a cylinder having the parallels and meridians intersect each other
perpendicularly and stretching the distances on the parallels with the same factor as the
distances on the meridians.

When Mercator presented his new world map in 1569, he immediately solved one of the most
urgent problems of navigation: to draft a map on which a rhumb can be represented as a
straight line. A rhumb or loxodrome is a
curve which cuts every meridian under the same angle.

It took a
while before the maps of Mercator were introduced in navigation because Mercator kept the
mathematical background as a secret. Therefore
other mathematicians tried on other way to solve the problem. Michiel Coignet, a mathematician of Antwerp,
tried to give it a solution by calculating the length of the rhumb piece by piece(one
piece = 1 degree of latitude) for some given courses.
These numbers didn’t mean anything for seafarers as long they couldn’t
mark them out on their maps. Simon Stevin had
similar ideas and he suggested to use moulds, shaped like a loxodrome, and
appropriate to each map. The real
mathematical approach came from Edward Wright which he published in his book ‘Certain
errors in Navigation’ in 1599 using the basic idea for a Mercator map : the degrees
of latitude have to be stretched out as much as the degrees of longitude. This idea gives the mathematical basis for the
Mercatormap, namely: proj(1'mer.)=proj(1').sec j, where j is the latitude. Starting
from this you can calculate at which distance of the equator you have to draw the
parallelcircle.

Mercatormap |
Petersmap |

Mercator made his
carts only for navigation and they were used for a long time almost everywhere. However sometimes the cart gave a wrong image of
the world. The surfaces didn’t always
agree with the reality. For example Greenland
(=2.175.600 km²) seems a lot bigger than South-Africa ( 17.795.000 km²), thus the latter
being nine times smaller than the first. The
Mercatorprojection deforms surfaces and at the most for the surfaces located far from the
equator.

But still, a world
globe stays the most correct reproduction of our planet. The scientists of the 16th
century studied mostly in their room. For the
cartographers it wasn’t any different. Usually the practical work, like the drawing
of carts and the building of globes, was done by craftsman.
But Mercator wanted to do it all by himself.
That’s the reason why he served apprenticeship with a goldsmith, an engraver
and a globe maker. In 1541 Mercator made a
globe of chips of wood, covered with linen and plaster.
On top of it he stuck 12 pieces of paper which he coloured himself. |

But seafarers
gravitated towards the starry sky, so Mercator
made also an celestial globe.

Mercator worked to order, he even got assignments from the emperor Charles the 5th . But in the meantime he stayed working on the dream
of his life: a study of heaven and earth. His
‘cosmography’ would exist of five big parts, a great work he wouldn’t be
able to complete by himself but fortunately was finished by his son Rumoldus, after his
father's dead.

The basic principle of the Mercator projection (the degrees of latitude towards the poles become bigger in the same relation as the parallel circles in their relation towards the equator) is decisive for the construction. This means that we get an anglepreserving cylinderportrayal when we enlarge a piece of the meridian arcs on a latitude j with regard to the equator with a factor 1/cos j. At that time Mercator couldn’t lean on the professional knowledge for a mathematical argumentation and we don’t know how he has calculated his factors for the meridian division. Since Mercator didn’t left us tables of the meridian division, no other sea maps could arise based on this projection which is still the foundation of the modern sea map.

Just 40 years
after the publication of the world map of Mercator the English mathematical astronomer
Edward Wright gave the initiative for the general application of Mercator ’s
principle. He established the relation with the secansfunction and in that way he found
the basis of the right construction. The Mercator projection reproduces meridians like
parallel straight lines perpendicular on the equator. That’s why every latitude arc
is enlarged and this arc has then the same length as the corresponding equator arc.

The factor
according to which the meridian arcs are enlarged can be calculated based on the following
consideration: ABC and MNP are similar 'triangles' because the angle in C respectively in
P have the same size and always applies that

AC = BC, respectively NP = MP. Therefore all the
according sides (or arcs) have to be in the same prportion thus we get:

MN/AB = NP/BC = NP/NC because BC = NC being the radius of the sphere. The angle in
N, as a reverse, has the same magnitude as the angle in C. NPC is a right-angled triangle,
thus : NP/NC = cos j. Then the real length of the arc MN
is ABcos j. In the projection MN has to have the same
length as AB, so MN = M'N'cos j or M’N’ = MN/cos j = MNsec j.

The factor according to which the latitude arcs is enlarged must be secj. This means that this factor depends on the degree of the latitude. By the following steps, we demonstrate that we can speak of an anglepreserving image when the longitude circles are enlarged according to the same factor. This is just the property we need. For that, we take an arbitrary point on the sphere and we determine his latitude and longitude and also the angle a and the connecting loxodrome through that point. Locally, in a small neighbourhood of the point, we observe that the image reflects the same situation as the projection. Dz*cosa is enlarged according to the factor secj. When the angle has to be preserved, we have to enlarge Dz*sina with the same factor.

Now, we are still missing the calculation of the length of the meridian arcs. We have already said that the enlargement factor depends of the parallel of latitude. Let's imagine an arc along a meridian between the equator and the latitudecircle with the angle f. At the starting point the length stays the same, but just a little bit part further, for example one degree further, the factor of enlargement factor sec 1 degree, again a degree further, the factor is sec 2 degrees,… . Approximately we can calculate the length, when we accept that along such small parts the enlargement can be regarded as constant and that we can add these little pieces. The more the partition is refined, the more precise becomes the approximation of the correct length of the arc. It was Wright who made the first table containing the meridiandivision factors from degree to degree. Of course with our present knowledge of differential and integral calculus (which Wright didn't use at the end of the 16th century) we find that the latitude circles must be located at

distance from the equator |

*Drawing yourself a map following the
Mercator projection. *

*
You can get a fairly good Mercatormap using the following method

__Necessary materials :
__a pair of compasses, a quadrant scale

__Different
steps :__

1. Stick the 2 A4 sheets
of paper together along the longest side.

2. Hold the tape at the bottom. We consider the dividing line between both sheets
as the equator.

3. Draw a circle with the pair of
compasses with a diameter of 15 cm (source z situated on the equator)

4. Mark a point T, starting from
the center z, measured at a length of 2/5 of the radius to the left (3cm)

5. Draw a perpendicular line on the
equator, touching to the circle.

6. Divide the half circle into
equal parts of 10 degrees each, and draw straight lines which connect point T via the 10

degree points to the straight line.

7. Now, draw horizontal lines
(parallel to the equator) through the intersecting points on the perpendicular line.

8. Draw now 18 other parallel lines
(about 2 cm distance) perpendicular to the equator. Every
line represents a

meridian with a length difference from 10
degrees.

9. The map only represents half a
planet now (180 degrees). To span the whole
earth, you better make a copy, and stick

both hemispheres together.

10. Now bring degrees of length and latitude on the map

11. Indicate the scale at the bottom of
the paper. To calculate the scale, we divide
the earth circumference 40 075 km,

in 360. We
represent 10 degrees of latitude on our map with a distance of 2 cm; so that 1 cm agrees
with

(5 x 40 075 : 360) km = 556.6 km.

* Of course substituting j in | for different latitude values f.e. 0°, 10°, 20°....80° |

will give you the
right 'Mercator' distance to the equator of the resp. parallel for each choosen value(not
equal to 90°) of j .

The historically
remarkable town of Rupelmonde is situated on the banks of the Scheldt. Most places of interest are situated on the
Mercator Isle. |

He studied at the
university of Cambridge. There, he became
famous for his mathematical and cosmographical knowledge.
In 1589 he participated at an expedition to the Azores. During this trip he experimented with new
navigationtechniques. Thanks to this it
became possible to write his masterwork in his book "Certain Errors in
Navigation." In his foreword he
complained: “I wish I had been as wise as he (=Mercator) in keeping it more charily
to myself." It was the Mercatormap of
1569 who gave him the idea to construct a map with a larger latitude, but " the how
this should be done I learned neither from Mercator nor any man else.' Wright died in
1615.

*Sources*

Gerard Mercator en de geografie in de zuidelijke Nederlanden, Stad Antwerpen, Museum Plantin-Moretus en Stedelijk Prentenkabinet, Antwerpen 1994

Gerardus Mercator Rupelmundanus, Marcel Watelet, Mercatorfonds Paribas

Land Van Waas, Mercator, Toerisme Oost-Vlaanderen

Matematikk 1MA (5-timersfaget), Erstad-Bj**Ø**rnsgård-Heir-Bielorentzen,
H. Aschehoug & Co

Uitwiskeling, mei 2000

Computer programme CABRI

http://www.scholen.vlaanderen.be/schoolprojecten/nom0113/NLCHRT.htm

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