Beryl over , and its dimensions were long and in diameter. The largest cut aquamarine gem is the Dom Pedro aquamarine, now housed in the Smithsonian Institution's National Museum of Natural History. The ancient Romans believed that aquamarine would protect against any dangers while travelling at sea, and that it provided energy and cured laziness. Emerald is green beryl, colored by trace amounts of chromium and sometimes vanadium. Most emeralds are highly included, so their brittleness (resistance to breakage) is classified as generally poor. The modern English word "emerald" comes via Middle English "Emeraude", imported from modern French via Old French
History of manifolds and varieties spaces with a differentiable or topological structure, while "variety" refers to spaces with an algebraic structure, as in algebraic varieties. In Romance languages, manifold is translated as "variety" – such spaces with a differentiable structure are literally translated as "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties". Thus for example, the french word "" means topological manifold. In the same vein, the Japanese word "" (tayōtai) also encompasses both manifold and variety. ("" (tayō) means various.) Ancestral to the modern concept of a manifold were several important results of 18th and 19th century mathematics. The oldest