5000 Years Of Geometry – Christoph J Scriba

5.2.6 Heart-shaped world map according to the principle of Stab(ius) and Joh. Werner [Published by Peter Apianus: Tabula orbis cogniti universalior, Ingol- stadt, 1530] 5.2 Geometry in astronomy, geodesy and cartography Stab also designed a map of the terrestrial hemisphere by means of vertical parallel projection onto a tangential plane. Besides, Albrecht D¨urer helped with the design of this map printed in 1515 by order of Emperor Maximilian (Illus. 5.2.7). Further cartographic illustration first proposed or used in this time are, amongst others: • Azimuthal equidistant projection (Cusanus, Snellius), in which the merid- ians are represented by rays originating in N and, the circles of latitude by concentric circles in the map, whilst making sure that its radius equals the pole distance measured in arc measure (the name of the method means that all points are represented in their true distance from the North Pole = centre of map).

• The draft first used by Gerard Mercator and later named after Sanson and Flamsteed, in which the circles of latitude are represented by parallel line segments true to distance and meridians stay true to area. The climax of Renaissance cartography is, without a doubt, symbolised by the work of the Flemish cartographer Gerard Mercator, who later worked in Duisburg. The excellent mathematician Pedro Nunes (Nonius), who lived in Portugal, first addressed the curves of constant course (later called rhumb lines or loxodromes) on the globe, which are so important for seafaring; in other words, curves that are defined by the fact that they intersect all meridi- ans in a constant angle.

(In 1624, Snellius introduced the still customary name ‘loxodrome’ for these curves in his theory on navigation Tiphys Batavus and likewise the name ‘orthodrome’ for great circle arcs, i.e., shortest curve on the globe.) Nunes showed by means of approximate construction (only us- ing eight meridians and approximation until the next meridian by means of circular arcs, Illus.

5.2.8) that these loxodromes, called “curvas dos rombos” (rhumb curves) by him, approximate both poles in a spiral manner without ever being able to reach them. Some of these curves are displayed on a globe made by Mercator in 1541. In 1568, he finalised the first world map, which shows these curves as straight lines – and the Mercator projection was born.

However, this projection was only circulated in 1595 in the printed world atlas, after Mercator’s death. Ever since then, the literature has speculated as to how he could have achieved his map [K¨oberer 1982]. The reasons for this speculation are twofold. On one hand, the exact law, which says that the intervals of the images of the circles of latitude grow into the infinite with increasing latitude from the image of the equator, can only be found by solving an infinitesimal equation.

in History and Culture Christoph J. Scriba • Peter Schreiber 5000 Years of Geometry Mathematics in History and Culture Christoph J. Scriba Peter Schreiber University Hamburg University Greifswald Hamburg, Germany Greifswald, Germany Translated by Jana Schreiber ISBN 978-3-0348-0897-2 ISBN 978-3-0348-0898-9 (eBook) DOI 10.1007/978-3-0348-0898-9 Library of Congress Control Number: 2015935235 Springer Basel Heidelberg New York Dordrecht London © Springer Basel 2015 This work is subject to copyright.

All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication.

Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com) Graphic on the cover: © Helmut Schwigon Editor: Project Group “History of Mathematics” of Hildesheim University H.W. Alten, K.-H. Schlote, H. Wesemüller-Kock Originally published in German in the series “Vom Zählstein zum Computer” under the title: “5000 Jahre Geometrie.

Geschichte-Kulturen-Menschen” (ISBN 978-3-642-02361-3) Springer-Verlag Berlin Heidelberg 2010 Preface of the editor of the German edition Geometry (from the Greek word for ‘measuring the Earth’, the modern sci- entific discipline of which is now called geodesy), branch of science which deals with regular patterns, shapes and solids, was one of the first human attempts, after counting, to concern themselves with the emerging science mathematics.

This is evident from the spirals on megalithic graves, incisions in stone and patterns on clay fragments. In this book, you will learn how geometry has developed over the millennia from these earliest origins in distant times and much more.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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