MERCATOR

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 :

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. For this kind of calculation you can use now the integral. But, in the time of Wright there wasn't any integralcalculation which means that he needed to do long calculations.

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. Actually the Mercatorprojection is at is best for the equatorterritory. It was Arno Peters who tried to make a more 'honest' cart. In 1973 he published his Petersprojection. But he exaggerated in the other direction. The surfaces were right this time, but the shape was totally wrong. If you see now a world map in a book or on television it is a compromise between the Mercator- and the Petersprojection. The shapes, surfaces and the distances between countries are almost right.

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.
twaalfdelen.gif (29831 bytes)

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

* You can get a fairly good Mercatormap using the following method
Necessary materials :
a pair of compasses, a quadrant scale, a ruler, 2 sheets of paper(A4), tape and a pencil

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)

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.

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.

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 Mercator Isle

The historically remarkable town of Rupelmonde is situated on the banks of the Scheldt. Most places of interest are situated on the Mercator Isle.
The Counts of Flanders resided in the Gravenkasteel, the Counts Castle, a watercastle on the banks of the Scheldt. The Graventoren, what remains of this 13th century water castle, was arranged as Museum of the Scheldt, but there is also a hall dedicated to Mercator. In its shade you will find the 18th century manor. The 16th century tidal mill is unparalleled in Europe. And don’t forget the grand view over the Scheldt.

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.

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

Matematikk 1MA (5-timersfaget), Erstad-BjØrnsgård-Heir-Bielorentzen, H. Aschehoug & Co

For more information

You can pay a visit to :
the Mercator Museum in Sint-Niklaas(Belgium)

or contact us :
Annelies Verlee, Nele Waterschoot,
Anke De Wilde en Esther Vermeulen