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Portraits of the Earth: A Mathematician Looks at Maps

American Mathematical Society
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"We shall see later on that there can be no ideal flat maps of even a portion of a sphere. That is, no flat map can show all distances in all directions in their correct proportions. We shall also see that no flat map of a sphere can both show all shortest routes between arbitrary points on the sphere as straight line segments and at the same time, preserve all angles between routes." It was at this point (page 22) that I became hooked on this book. Portraits of the Earth is Volume 18 in the Mathematical World series published by the AMS. This book is interesting, entertaining, mathematical, and, so it seems to me, a labor of love. It came from an interdisciplinary course the author taught with the late Dr. Elaine F. Bosowski (to whom the book is dedicated) at Villanova University (75° 20' 31.0'' W; 40° 2' 24.2'' N).

The book is aimed at undergraduates who have an understanding of sophomore level mathematics, although I believe a good first year student or advanced high school student could master the material. Mathematically one sees derivatives (for example, with scale factors along a meridian), integrals (equal area maps), matrices (changing coordinates), and trigonometry (everywhere). The author shows an excellent touch in the mathematics included. He includes necessary details without getting bogged down in small details or putting in so much that it would turn off non-math majors. The writing is informal and engaging. The book can be used in a formal course with a group of students, independent study with one student, or just by a professor who wishes to learn more. At the end of every chapter there are exercises.

Worth special note are the historical references throughout the book. Names dropped include the obvious: Euclid, Columbus, and Ptolemy. In addition Gauss, Lambert, Halley, Riemann and others make appearances. This human side of the subject helps to motivate the origins of the results. Many "pure" topics are interesting in their own rights, but by adding these personalities and the problems that they worked on, the subject comes alive. I truly enjoyed the dose of history along with the map making.

If I have one criticism, it's the diagrams. I would have liked more. For example, more pictures would have helped in the explanation of spherical triangles. In addition, I'd like those pictures there are to be larger (of course, maybe that's just my eyes getting older).

All in all, this is a highly enjoyable book. It covers an application and a history of mathematics that does not get a lot of attention. I recommend this for yourselves, for your bookshelves, and for your students.

Robert W. Vallin ( is Associate Professor of Mathematics at Slippery Rock University of Pennsylvania.

Date Received: 
Thursday, June 6, 2002
Include In BLL Rating: 
Timothy G. Feeman
Mathematical World 18
Publication Date: 
Robert W. Vallin
  • Geodesy--measuring the earth
  • Map projections
  • Scale factors
  • Distances and shortest paths on the sphere
  • Angles, triangles, and area on a sphere
  • Curvature of surfaces
  • Classical projections
  • Equal-area maps
  • Conformal maps
  • Analysis of map distortion
  • Oblique perspectives
  • Other worlds: Maps of surfaces of revolution
  • Appendix A: Aspects of thematic cartography: Symbolization, data classification, and thematic maps
  • Appendix B: Laboratory projects
  • Appendix C: Portraits of the earth: How the maps in this book were produced
  • Bibliography
  • Index
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Monday, January 21, 2008