Alexander, Amir R. Geometrical Landscapes: The Voyages of Discovery and the Transformation of Mathematical Practice. Stanford, CA: Stanford University Press, 2002. 293 pp. ISBN 0804732604. Although the book is written by a philosophical mathematician and primarily aimed at academic mathematicians, there is much within its scope of interest to the student of early exploration and the cartography resulting from this exploration. The author reviews and ties the advances in fifteenth and sixteenthcentury mathematical theorems to the advances in knowledge of world geography resulting from explorations of the Columbian era. The book, true to its subtitle, covers selected early voyages of discovery and in a highly readable manner shows how these voyages brought about the transformation and advance of mathematical practice. Its value to the historian lies in the fact that the converse of this theme is also covered, to show how the advances in mathematics in the sixteenth century had a significant effect on exploration and the development of accurate cartography resulting from these explorations. The book treats this relationship between the advances in mathematics and the ocean navigation of early explorers chronologically from Columbus’s voyage in 1492 to the voyages of the late seventeenth century. Chapter 1 investigates the origin of the exploration narrative. Alexander discusses how the role of early cartography was similar to the exploration narrative in that it not only presented an accurate geographical description of the new land and its peoples, but was also intended to encourage followon exploration and settlement. In pursuing the latter idea, Alexander cites as examples the oftenromanticized illuminations on cartography resulting from the Mediterranean voyages of the Christian Crusades to the voyages of Columbus, Cortés, Verrazano, and Raleigh. Chapter 2, titled “Breaking Alien Coastlines,” will be of particular interest to students of cartography. An interesting and provocative thesis pictures early cartography as carrying forward the role of the exploration narrative (Chapter 1) in combining the depiction of realistic and proven coastlines with romantic and fictionalized coastlines and inland geography to encourage further interest in the area. Illustrations and detailed discussion of early cartography to support this point are centered on the English voyages of Frobisher to the Canadian arctic between 1576 and 1580, and Raleigh’s attempts to colonize his “Virginia” on the eastern seaboard of the USA between 1584 and 1587. The cartography related to Raleigh’s search for the lost kingdom of “El Dorado” in South America in the 1590s is discussed in detail to show how valuable geographical knowledge was obtained by searching for a fictional romantic destination that never existed.. And in like manner, the voyages of Foxe and James to Hudson’s Bay in the 1630s, searching for the nonexistent Northwest Passage, led to the later strong British presence in North America. There are several highquality reproductions of early maps that are used in the discussion of injection of fictionalized coastlines in early cartography. The best of these is John Ferrar’s map of Virginia (1651) which shows an accurate coastline and inland rivers from the Cape Hatteras area north to Delaware to include the Chesapeake Bay. The accuracy of this portion of the map would indicate that it was made from an actual and competent exploration and survey. Yet just beyond the inlet and bay correctly called “Delaware” the map abruptly changes to show a completely fictitious or speculative broad river containing small islands that runs west all the way to the sea of China on which sails the annotated frigate of Sir Francis Drake. Following the discussion of Ferrar’s map of Virginia, Alexander explains how the map was related in its construction to James Beare’s earlier map of Frobisher’s voyage, John White’s map of the Carolina coast, and Thomas Hariot’s map of Raleigh’s Guyana (El Dorado) in the OrinocoAmazon drainage area. Chapters 3 and 4 bring out the significant role the leading Elizabethanera mathematicians played in early English exploration. Prominent among these early mathematicians were Sir Francis Bacon, Edward Wright, Henry Briggs, John Dee, Thomas Blundeville, and Thomas Hariot. The mathematicians were not only involved in planning, support, and preparation of the voyage, but in several cases went along on the voyages as active participants. These mathematicians were invariably among the welleducated intellectual elite of the time and so had more influence in the court than the unlettered pilots and captains who conducted the exploration voyages. One underlying theme stressed by the English mathematicians was the emphasis on direct observation and experience, such as that of the explorers, over theoretical reasoning. The renowned theoretical mathematician, Sir Francis Bacon, is quoted on this point when he wrote: “if the new sciences are to follow the lead of the explorers, they should be based on actual experience and not on abstract disembodied reasoning or speculation.” And Thomas Hariot, who developed the tables used in modern Mercator projections, insisted on “the primacy of firsthand observation as the proper way to knowledge.” The remainder of the book, Chapters 5 and 6, is largely concerned with the advances in mathematics that produced the Mercator projection charts and maps. This part may be difficult to follow for a reader not versed in higher mathematics. However, the first four chapters are well worth purchase of the book as a valuable and innovative reference for the serious student of history to show the vital interaction between explorers, cartographers, and mathematicians. Douglas T. Peck Independent historian, Bradenton, Florida 
