SOME MATHEMATICAL ASPECTS OF THE PLANET EARTH José Francisco Rodrigues (University of Lisbon) Article of the Special Invited Lecture, 6th European Congress of Mathematics 3 July 2012, KraKow. The Planet Earth System is composed of several sub-systems: the atmosphere, the liquid oceans and the icecaps and the biosphere. In all of them Mathematics, enhanced by the supercomputers, has currently a key role through the “universal method" for their study, which consists of mathematical modeling, analysis, simulation and control, as it was re-stated by Jacques-Louis Lions in [L]. Much before the advent of computers, the representation of the Earth, the navigation and the cartography have contributed in a decisive form to the mathematical sciences. Nowadays the International Geosphere-Biosphere Program, sponsored by the International Council of Scientific Unions, may contribute to stimulate several mathematical research topics. In this article, we present a brief historical introduction to some of the essential mathematics for understanding the Planet Earth, stressing the importance of Mathematical Geography and its role in the Scientific Revolution(s), the modeling efforts of Winds, Heating, Earthquakes, Climate and their influence on basic aspects of the theory of Partial Differential Equations. As a special topic to illustrate the wide scope of these (Geo)physical problems we describe briefly some examples from History and from current research and advances in Free Boundary Problems arising in the Planet Earth. Finally we conclude by referring the potential impact of the international initiative Mathematics of Planet Earth (www.mpe2013.org) in Raising Public Awareness of Mathematics, in Research and in the Communication of the Mathematical Sciences to the new generations. Ancient Mathematics and the Earth There is no doubt that the planet Earth is one a main ancient root of mathematics. Distancing, constructing, spacing, surveying or angulating led to Geometry, that means literally measurement of the earth (respectively, metron and geo, from ancient Greek). The Babylonian tablets and the Egyptian papyri, which are dated back about 4000 years, are the first known records of elementary geometry. Even if it may be controversial to attribute to Pythagoras the idea that the shape of the Earth is a sphere, this was clear already to Aristotle (384 – 322 BCE) in his “On the Heavens”: “Its shape must be spherical… If the earth were not spherical, eclipses of the moon would not exhibit segments of the shape they do… Observation of the stars also shows not only that the earth is spherical but that it is not no great size, since a small change of position on our part southward or northward visibly alters the circle of the horizon, so that the stars above our heads change their position considerably, and we do not see the same stars as we move to the North or South.” But if the Hellenistic scientists had observed the sphericity of the planet, they had also obtained a relatively accurate estimate of its radius. Indeed, we owe one of the first estimates of the circumference of the earth to Erastosthenes (276-194 BCE), a member of the Alexandrine school, who established it in 250,000 stadia. He measured in Alexandria the angle elevation of the sun at midday, i.e. the angular distance from the zenith at the summer solstice, and he found 1/50th of a circle (about 7°12’) making then the proportion, by knowing that Syene (Aswan) was on the Tropic of Cancer at a distance of about 5000 stadia. If the stadion meant 185 m, he obtained 46,620 km, an error of 16.3% too great, but if the stadion meant 157.5 m, them the result of 39,690 km has an error less than 2%! Erastosthenes of Cyrene, as Heath wrote [H], “was, indeed, recognised by his contemporaries as a man of great distinction in all branches of Knowledge”. He is remembered for his prime number sieve, still a useful tool in number theory, and was the first to use the word geography and to attempt to make a map of the world for which he invented a line system of latitude and longitude. Another old trigonometric technic, the basic principle of triangulation to determine distances of inaccessible points on earth, was used by Aristarchus of Samos (about 310-230 BCE) to estimate the relative sizes and distances of the Sun and the Moon. Even if these estimates were of an order of magnitude too small, this was a remarkable intellectual achievement of the Hellenic mathematician. He was also a precursor of Copernicus, as one of the philosophers of the Antiquity to suggest the heliocentric theory in Astronomy. http://en.wikipedia.org/wiki/File:PtolemyWorldMap.jpg (15th century redrawn of Ptolemy’s world map) Ptolemy (about 100-178), the most influential Hellenic astronomer and geographer of his time, credited Eratosthenes to have measured the tilt of the Earth's axis with great accuracy obtaining the value of 11/83 of 180° (23° 51' 15"). In his Guide to Geography he gave information on the construction of maps of the known world in Europe, Africa and Asia. However, as we may see from a world map redrawn in the 15th century, from the present point of view his representation of the earth is not accurate at all, in particular showing the Atlantic and the Indian Oceans as closed seas. Ptolemy used Strabo’s value for the circumference of the Earth, which was too small with an error of 27.7%. This crude estimate has been used to explain the Columbus’ error of looking Cipango (Japan) going West more than thirteen centuries later [RW], but historians have recently discovered other reasons for this fact. In its great astronomical treatise of the second century, the Almagest, which geocentric theory was not superseded until a century after Copernicus’ book De Revolutionibus Orbium Coelestium of 1543, Ptolemy describes, in particular, a kind of ‘astrolabe’, which is a combination of graduated circles that later became a more sophisticated chief astronomical instrument reintroduced into Europe from the Islamic world. The nautical adaptation of the planispheric astrolabe was one of the tools used by the Portuguese navigator Bartolomeu Dias in his ocean expedition rounding Africa and crossing the Cape of “Boa Esperança” in 1488. This has shown the connection between the Atlantic and the Indian Oceans, a discovery that would change dramatically the geographical vision of the world, and has happened four years before Columbus first travel to the Antilles. http://en.wikipedia.org/wiki/File:Martellus_world_map.jpg (Martellus world map of 1489 or 1490) This fact was immediately reflected in the world map made in 1489 or 1490 by Henricus Martellus and in the Nuremberg Globe of Martin Behaim, of 1492 [Da]. Atlas and globes are treasures of the Renaissance cartography that illustrate how useful mathematical techniques were necessary for map making in the late 15th century, for practical navigations or for helping the European minds to change their concept of the world, as did the famous Globus Jagellonicus of 1510 that is considered as being the oldest existing globe showing the Americas. The strategic importance of the new terrestrial representations and of the ocean navigations, as new key technologies, goes beyond their scientific meaning and consequences. They represented technological breakthroughs and were decisive tools for the European expansion in the period 1400-1700, as “the conquest of the high seas gave Europe a world supremacy that lasted for centuries” [B]. If shape, measure and representation of the Earth were key elements in ancient mathematics, the novel problems and concepts of Renaissance mathematics, in particular, those associated with a new geometric approach to the theory and practice of navigation, as well as to mapping techniques, were instrumental in the rise of modern science. As the Dutch historian of science R. Hooykaas has stressed, “the great change (not only in astronomy or physics, but in all scientific disciplines) occurred when, not incidentally but in principle and in practice, the scientists definitively recognized the priority of Experience. The change of attitude caused by the voyages of discovery is a landmarK affecting not only geography and cartography, but the whole of 'natural history'.” [Ho] Mathematical Geography and the Scientific Revolution(s) Recently historians of Mathematics have been recognizing the importance of Renaissance methods [K], often invoking the significant and countlessly repeated phrase of Galileo in “Il Saggiatore” (1623): “Philosophy is written in this grand book, the universe, (…) written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wonders about in a dark labyrinth.” The Elements of Euclides were first printed, in Latin, in Venice in 1482, and had several vernacular translations in the following century in Italian, German, French, English and Spanish. The English’s edition, printed in London in 1570, contains a “very fruitfull Preface made by M.I.Dee, specifying the chiefe Mathematicall Sciences” [AG]. In this influential text of the English scientist John Dee (1527-1608), after stating that “Of Mathematicall thinges, are two principall Kindes: namely, Number, and Magnitude”, he describes among the branches of his remarkable “Mathematicall Tree”, the “Arte of Nauigation, demonstrateth how, by the shortest good way, by the aptest Directiõ, & in the shortest time, a sufficient Ship, betwene any two places (in passage Nauigable,) assigned: may be cõducted: and in all stormes, & naturall disturbances chauncyng, how, to vse the best possible meanes, whereby to recouer the place first assigned” [D]. Geography and navigation were in fact extremely important in the 16th century [LA] and it became now clear that Dee’s mathematical program has roots in the works of the Portuguese mathematician and cosmographer Pedro Nunes (1502-1578).