Our present-day knowledge of the principal research trends in geometry during the nineteenth century may aptly be characterized as fragmentary and, in many resepects, thin. Reasons for this are not far to seek. Some mathematicians might be inclined to the ungenerous view that those few classic results which survived the conceptual upheaval that gave birth to modern mathematics during the early decades of this century constitute the vital core of the previous century's accomplishments. Historians of mathematics would presumably take a more liberal stance, but they, too, have done rather little to illuminate the vast body of geometrical knowledge that accrued during the period spanned by two nearly forgotten giants, Gaspard Monge and Sophus Lie. If modern mathematicians often take comfort in knowing that today's standards in geometry are infinitely more rigorous than the ones that guided the best work of the mid-nineteenth century, historians should be especially sensitive to the kinds of conceptual and technical problems that preoccupied the geometers of the distant period.
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