500 years ago on 5^{th} March 1512 Gerard de Kremer was born in Rupelmonde, in those days a town in the Spanish Netherlands today in Belgium. He is of course much better known through his Latinised pseudonym Mercator.

In German a Kremer or Krämer is a shopkeeper, a grocer and Latinised it becomes Mercator, the root of course of the English words merchant, mercantile, merchandise etc. In modern German a Krämerladen is a corner-shop. Mercator turns up for the first time in a scientific context as an assistant globe-maker to Gemma Frisius in Leuven between 1534 and 1537. Gemma Frisius was professor of medicine at the University of Leuven and was a leading European mathematicus.

Under his tuition Mercator learnt mathematics, astronomy, astrology, surveying, instrument and globe making and cartography. As a pupil and later colleague of Frisius Mercator became part of an important circle of mathematical practitioners that included the Englishman John Dee and the Dutchman Abraham Ortelius.

In his own right Mercator became one of the leading European instrument and globe makers, surveyors, and cartographers but in this post I just want to take a brief look at the thing for which he is most famous the Mercator projection.

As you probably know it is impossible to cut open a sphere and spread it out flat. This is a major problem for cartographer because of course the world is a sphere, at least theoretically. In reality it resembles a rather lumpy distorted potato but cartographers treat it as an idealised oblate spheroid. Over the centuries cartographers developed mathematical methods of transferring the surface of the sphere onto a flat sheet of paper, parchment or what ever. These methods are known as projections because that is exactly what they are. On selects a viewpoint inside or outside the sphere and projects the points on its surface along the lines of sight onto a flat surface. One such projection is the cylindrical projection in which the sphere is conceived of as being inside a cylinder and all points on the spheres surface are projected from the middle of the sphere onto the inside of the cylinder which is then rolled out flat.

The problem is that all projections distort if some way the surface of the sphere. A cartographer has to choose the projection, which best conserves that aspect of the map that he wishes to emphasise.

In the Renaissance the age of exploration, which had been kick-started by the Portuguese King Henry the Navigator (who coincidentally was born on 4^{th} March 1394) in the 15^{th} century,

the mathematicians were called upon to develop new methods of astral and mathematical navigation and new forms of cartography. Mercator’s teacher Gemma Frisius for example invented triangulation for surveying and cartography and first suggested the chronometer method of determining longitude. One of the central problems that needed to solved was what path does a compass bearing follow on a globe and how would it be possible to represent that compass bearing as a straight line on a sea chart?

The first part of the problem was solved by the Portuguese mathematicus Pedro Nunes one of the leading mathematical practitioners of the age. Nunes was professor of mathematics at the University of Coimbra and Royal Cosmographer to the Portuguese Crown.

Nunes demonstrated that a constant compass bearing on a globe follows a segment of a spiral and not a great circle as had been previously assumed. Such lines are technically known as loxodromes or rhumb lines.He also knew that in order to make a constant compass bearing on a sea-chart a straight line then the lines of longitude and latitude must be straight parallel lines but he was not able to work out how the lines of latitude needed to be spaced. This was the problem that was first solved by Mercator and led to the Mercator projection.

The Mercator projection is basically a cylindrical projection in which the distances between the lines of latitude are adjusted according to a special mathematical formula.

Mercator printed and published a world map constructed according to this method of projection in 1569 but he did not explain the mathematical rules on which it was based. He was a professional cartographer and globe maker and he probably hoped that if he kept his method secret then the people who wished to take advantage of this new development would have to order their maps and charts from him.

We know from their unpublished papers that both of the English mathematicians John Dee

and Thomas Harriot

independently solved the mathematical problem of the projection but like Mercator neither of them made the knowledge public. We can however assume that both of them made use of this knowledge when teaching navigation and cartography, Dee to the pilots of the Muscovy Company and Harriot to Walter Raleigh’s sea captains.

The first person to publish the mathematical method of constructing such a chart was another English mathematicus Edward Wright in his book *Certaine Errors in Navigation*, first published in 1599.

It is because of this that modern historians of cartography say that the correct name for this type of map projection is the Mercator-Wright Projection.

If you go to John D. Cook’s Endeavour Blog and follow the links you can find out all about the maths of the Mercator-Wright projection. There’s a link to a Mercator projection xkcd cartoon too!

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Have I ever mentioned that your blog is amazing?

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One of the most interesting historical tidbit vis-a-vis the Mercator projection, to my mind, concerns the formula

integral sec(φ) dφ = log |sec(φ) + tan(φ)|

which, as explained in John D. Cook’s blog, governs the spacing of the lines.

Eli Maor’s book

Trigonometric Delightsdevotes a whole chapter to the Mercator projection. Let us take up the story right when Wright published his book, in 1599.Wright essentially computed the integral of sec(φ) using numerical integration, extremely tedious without a calculator. In 1614 Napier published his invention of logarithms, and in 1620 that Edmund Gunter (mathematician and clergyman) published a table of logarithms of tangents. Around 1645, Henry Bond (math teacher and navigation expert) noticed, by comparing Wright’s tables with Gunter’s, that

integral sec(φ) dφ = log |tan(π/4 + φ)|

which is equivalent to the formula given above.

Bond’s conjecture became one of the outstanding unsolved math problems of the 1650s. Finally in 1668 James Gregory came up with a proof. It was so complicated that it was considered a major advance when Isaac Barrow, Newton’s predecessor, devised a simpler proof in 1670. Barrow’s proof may have been the first use of partial fractions in an integration problem.

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It’s hard to believe that it was widely believed (among people who understood what a great circle is) that a line of constant bearing is a great circle. It is patently obvious without any resorting to equations that any great circle not through the poles must have bearing of 0 degrees at its highest and lowest points, so has constant bearing only it it is the equator.

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