Lost in the mists of history until comparatively recently was the Elizabethan- and Jacobean-era mathematician Thomas Harriot who, in many respects, anticipated the discoveries of Galileo, Kepler, Descartes and even Newton. That he was little known to students of the history of science had much to do with his failure to publish his work, a reticence owning to political and religious tensions of the day as well as to circumstance and his own indifference to the limelight.Robyn Arianrhod, a research fellow at the School of Mathematical Sciences at Monash University in Melbourne, adds the latest cornerstone to the edifice of Harriot's resurrected reputation. Hers is an authoritative, often engrossing marriage of history and science.At age 24, Oxford graduate Harriot (1560-1621) already had captured the attention of Sir Walter Raleigh (or Ralegh, in the author's spelling), who served as a navigational theorist on the latter's first reconnaissance to the New World. Harriot was indispensable to its success, both in his tutoring of pilots and through his open-minded, friendly encounters with native peoples.It was an amity that, sadly, would not last. Exploration was a commercial enterprise, and despite Raleigh's admonitions to his men to act humanely toward the indigenous populations, some were more than prepared to resort to violence and plunder. Later expeditions, not least the one ending in the mysterious Lost Colony of North Carolina, were typical of the disasters that befell Raleigh’s New World ambitions.Yet Raleigh the man, a complex figure, is rendered here quite sympathetically. The book is almost as much a biography of him (and his remarkable wife Bess) as it is of Harriot.But Harriot is the linchpin. Before age 30, the insatiably curious mathematician already was adding to his studies of astronomy, optics and physics the then-unnamed disciplines of ethnology and linguistics. Arianrhod demonstrates how he made original contributions to each field, and not just theoretical ones. Harriot also had a practical side to his inquisitive character.“Harriot's 'inclination' first and foremost was to experiment, rigorously and repetitively, then to describe his results mathematically,” Arianrhod writes. “For many of his forerunners and contemporaries, the process was reversed.”Harriot could be naive and credulous, and especially vulnerable to political intrigues or dangerous accusations of atheism, but these were exceptions to the rule for a man who balanced the theoretical and technical. His discoveries were so diverse and often so seminal that the head spins. With so many ideas to investigate and research projects to complete, frequent interruptions and crises involving Raleigh and his other benefactors — notably, Henry Percy, the ninth Earl of Northumberland — Harriot was often prevented from focusing on one subject at a time, much less ordering his voluminous work for publication.Arianrhod plugs the gaps in the historical record with exhaustively researched background, showing how science developed as an independent disciple during a transitional period in human affairs. The author intends that her book be clear and accessible to the general reader, but those chapters that depart from pure history to present an array of tables and formulae amid the prose tend to compete with the narrative flow and the richness of biographical and historical detail.Like Raleigh, Harriot retained a life-long interest in the New World. But he “yearned to go deeper into the foundations of pure mathematics and physics — to discover 'new worlds' in numbers and the laws of nature rather than through geographical exploration.”By studying his life and career, Harriot helps us understand how modern mathematics and science began to emerge. Arianrhod's is a significant achievement.