Looking at the Origin of Gerardus Mercator's Projection
Looking at the Origin of Gerardus Mercator's Projection
In my previous post, Leibniz States that Nicolaus Mercator Introduced the Word Infinitesimal, I wrote about the coining by Nicolaus Mercator [Nicolaus Kauffman] (1620-1687) of the word ‘infinitesimal’ and of the first use of infinite series, at least according to Gottfried Wilhelm Leibniz (1646-1716).
There is another Mercator, more famous, who lived in the previous century. His name was Gerardus Mercator [Geert de Kremer] (1512-1594), with whom we associate the Mercator projection, first used in his 1569 world map, entitled Nova et aucta orbis terrae descriptio ad usum navigantium emendate accomodata [New and augmented description of the Earth corrected for use in navigation].
The Mercator projection is well known to most of us, but its utility is perhaps not.
The horizontal lines on the map correspond to parallels on a globe, and the vertical lines correspond to meridians. Now, on a globe, meridians all meet at the North and South poles, which obviously is not possible on a rectangular map. So how does it work? Starting from the equator, and moving north (or equivalently south), the distance between successive meridians is less on each parallel. So, as we move north, the distance between meridians is augmented on the map to allow the meridians to remain vertical. And, since the horizontal distance is augmented, so must the vertical distance be augmented correspondingly. As a result, the distance between successive parallels increases as we move north. And the same holds true heading south from the equator.
What is the result of such a map? A straight line on the map will cross successive meridians at exactly the same angle, and so the map can be used by navigators heading on a fixed course, guided by the sun and the stars, anywhere on the globe. Of course, the navigator must bear in mind that the projection does not preserve distances.
It is still unclear how exactly Mercator, who had previously created many maps and globes, developed this particular map. This topic has been addressed by many researchers, and two of the most recent, Joaquim Alves Gaspar and Henrique Leitão, claim to have found the source123. Gaspar is himself a former navigator and Leitão was the principal editor of the recent new edition of the works of the Portuguese mathematician Pedro Nunes (1502-1578)4.
They define the problem as follows:
The problem consists of finding a general expression for the ordinate y = y(φ) of a parallel of latitude φ, measured from the equator, so that the proportion between the lengths of a parallel and a meridian arc on the spherical surface of the Earth is conserved in the projection. It can be shown that such an expression is the solution of the equation dy/dφ = μ sec(φ), where μ is an arbitrary linear scale. That is,
y(φ) = μ ψ(φ)
ψ(φ) = ln(tan(π/4 + φ/2)) [Reference 1, p.45]
Today the problem is completely straightforward, and we just type in the equations into a computer, which can spit out the results to the needed accuracy. But in Mercator’s day, not only were there no computers, nor was there calculus, nor logarithms.
Here is a description of the extant maps:
Mercator’s map of the world is composed of eighteen printed sheets that, when assembled, measure about 125 × 203 centimetres. Three exemplars are extant: one in the Bibliothèque nationale de France, which is mounted on a frame…; one in the Universitätsbibliothek Basel, which is conserved as three longitudinal bands of six sheets each…; and one in the Rotterdam Maritiem Museum…. This last copy has been hand coloured, and the original sheets have been cut and re-assembled to make an atlas, in a way that the main land masses on Earth are shown without interruption. An additional copy once in the municipal library of Wrocław (Breslau), Poland…, lost during the Second World War, survives in facsimile only. All exemplars were printed from the same copper plates (now lost) engraved by Gerard Mercator himself in 1569. [Reference 2, p.2]
After a meticulous examination of these maps, Gaspar and Leitão conclude that Mercator used a table of rhumbs to create his map. Rhumb lines, also known as loxodromes, are lines placed on a globe that intersect successive meridians at the same angle. When these are placed on a Mercator projection, they become straight lines. The standard rhumb lines correspond to the seven angles at 11¼º from each other between 0º and 90º, in other words, 11¼º, 22½º, 33¾º, 45º, 56¼º, 67½º and 78¾º. Gaspar and Leitão, in Reference 2, were able to determine that the rhumb line for 45º was most likely used to derive the projection, creating a succession of small right-angled triangles approximating the line on the projection.
So now that we have an idea of the method by which the projection was created, we need to understand that this invention was not simply of importance for navigation, but also an important step in the development of the calculus. Even Mercator understood this:
Mercator, according to his biographer Walter Ghim, considered his invention to ‘correspond to the squaring of the circle in a way that nothing seemed to be lacking save a proof’. [Reference 1, p.48]
And history has clearly recorded this. For example, Dirk Jan Struik, in his A Source Book in Mathematics, 1200-1800, begins the section on the Fundamental Theorem of Calculus as follows5:
The so-called inverse-tangent problem consisted in finding the curve, given a law concerning the behavior of the tangent. An early example was the search for loxodromes on the sphere, which are curves intersecting the meridians at a given angle; this problem was originated by Pedro Nuñez and Simon Stevin in the sixteenth century. [p.253]
I have previously written about Simon Stevin (1548-1620) on two occasions:
In Reference 3, Gaspar and Leitão demonstrated that the idea of using rhumb lines was known in Europe, with Pedro Nunes having played a key part therein. Nevertheless, they explain in Reference 2 that the data that Mercator used could not have come from Nunes, since the latter’s data does not fit.
One possibility would be that Mercator did all of the calculations himself, but were that the case, why would there be no trace therefore in his notes? Another possibility is that there was a non-European source. This idea is not so far fetched. C.K. Raju writes6:
Mercator was actually imprisoned by the Inquisition [in 1543, some 26 years before he published his famous map]. Revealing his pagan sources would have definitely been fatal to him. Naturally enough his sources have not been found. But the similarity of his maps to projections used in Chinese star maps of the 10th c. is well known. So summary and brutal were the ways of the Inquisition, and such was the atmosphere of terror created by it, that people were intensely afraid of being associated with anything that might even faintly be theologically incorrect, for any rival could have denounced them, leading to painful and fatal consequences. Thus, in the days of the Inquisition there was little likelihood that even an otherwise honest European would have acknowledged knowledge from any non-Christian sources. [pp.295-296, my emphasis]
Raju continues:
There is also strong circumstantial evidence that transmission of Indian mathematics and astronomy texts did take place to Europe in the 16th and 17th c. CE. In the first place there is Mercator’s mysterious source of trigonometric values. Had Mercator obtained his values from some source like Regiomontanus there would have been no need for him to hide his sources, nor any possibility of doing so. On the other hand, since he had been arrested by the Inquisition, he had strong reason to keep any “pagan” sources a secret. Therefore, the very fact that he kept his sources a secret, combined with the fact that his map was similar to a Chinese map, is strong circumstantial evidence that his sources were non-Christian sources. As pointed out above, calculation of loxodromes is equivalent to the fundamental theorem of calculus, and Indian texts were the best possible source for this information. [p.347, my emphasis]
But to which Chinese map is Raju referring to? For this, we can refer to Joseph Needham (1900-1995), the great historian of Chinese science and technology7:
Rather crude star charts have been found in tomb cupolas of the Thang period (seventh to ninth century) both in China and Japan. A manuscript star-map from Tunhuang, dating from A.D. 940…, is almost certainly the oldest extant star chart from any civilisation, if we exclude of course the highly stylised carvings and fresco paintings of antiquity…. Again, the maps incorporated in the Hsin I Hsiang Fa Yao (New Description of an Armillary Clock) by Su Sung must be the oldest printed star charts which we possess. The maps of the book, begun in A.D. 1088 and finished in 1092, are remarkable in several ways. Two use a ‘Mercator’ method of charting…, already employed a century earlier, although not so accurately; while one of the charts of the polar regions makes use of the latest observations by, it seems, Shen Kua. Other wood block star-map prints of 1005 and 1116 have also been discovered recently. [pp.121-125, my emphasis]
The dating given by Needham has been revised to an earlier period. For example, Zi Xexong8 states that the Dunhuang star map is “Dated to Tang Dynasty, circa 700.” [p.464]
It is clear that the introduction of the Mercator projection played an important part in the development of European cartography and navigation, ultimately allowing the European powers to take over as the dominant powers of the globe. But at the same time, this projection was understood as one of the first steps towards the development of the calculus in Europe in the following century. What kind of direct access Mercator had to Chinese or Indian sources is still, at least to this author, an open question.
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Joaquim Alves Gaspar and Henrique Leitão. How Mercator Did It in 1569: From Tables of Rhumbs to a Cartographic Projection. EMS Newsletter 99:44-49, 2016.
Joaquim Alves Gaspar and Henrique Leitão. Squaring the Circle: How Mercator Constructed His Projection in 1569. Imago Mundi: The International Journal for the History of Cartography 66(1):1–24, 2013.
Henrique Leitão and Joaquim Alves Gaspar. Globes, Rhumb Tables and the Pre-History of the Mercator Projection. Imago Mundi: The International Journal for the History of Cartography, 66(2):180–195, 2014.
Pedro Nunes. Obras. Edited and commented by Henrique Leitão. Lisboa: Academia das Ciências de Lisboa & Fundação Calouste Gulbenkian. 8 volumes, 2002-2015.
Dirk Jan Struik. A Source Book in Mathematics, 1200-1800. Princeton Legacy Library, Princeton University Press, 1986.
C.K. Raju. Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE. New Delhi: Centre for Studies in Civilizations, 2007.
Colin A. Ronan. The Shorter Science and Civilisation in China. An Abridgement of Joseph Needham's Original Text. Volume 2. Cambridge University Press, 1981.
Xi Zezong. Chinese Studies in the History of Astronomy, 1949-1979. Isis 72(3):456–470, 1981.