tangent line

In geometry, a line that intersects a circle exactly once; in calculus, a line that touches a curve at one point and whose slope is equal to that of the curve at that point. Particularly useful as approximations of curves in the immediate vicinity of the point of tangency, tangent lines are the basis of many estimation techniques, including linear approximation. The numerical value of the slope of the tangent line to the graph of a function at any point equals that of the function's derivative at that point. This is one of the keystones of differential calculus. Seealso differential geometry.

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Type of sailing warship, the principal vessel of the West's great navies from the mid-17th to the mid-19th century. It evolved from a tactic in naval warfare known as the line of battle, in which two opposing columns of ships maneuvered to fire their guns broadside against each other. Since the largest ships carrying the biggest guns usually won these battles, this led to the construction of more big line-of-battle ships, or ships of the line. These three-masted ships were often 200 ft (60 m) long, displaced 1,200–2,000 tons (1,100–1,800 metric tons), and had crews of 600–800 men; they usually had 60–110 cannons and other guns arranged along three decks. They eventually gave way to the steam-powered battleship.

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Industrial arrangement of machines, equipment, and workers for continuous flow of workpieces in mass-production operations. An assembly line is designed by determining the sequences of operations for manufacture of each component as well as the final product. Each movement of material is made as simple and short as possible, with no cross flow or backtracking. Work assignments, numbers of machines, and production rates are programmed so that all operations performed along the line are compatible. Automated assembly lines (see automation) consist entirely of machines run by other machines and are used in such continuous-process industries as petroleum refining and chemical manufacture and in many modern automobile-engine plants. Seealso Henry Ford, interchangeable parts, Taylorism.

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In mathematics, the integral of a function of several variables defined on a line or curve that has been expressed in terms of arc length (see length of a curve). An ordinary definite integral is defined over a line segment, whereas a line integral may use a more general path, such as a parabola or a circle. Line integrals are used extensively in the theory of functions of a complex variable.

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Originally, the boundary between Maryland and Pennsylvania. The 233-mi (375-km) line was surveyed by Charles Mason and Jeremiah Dixon in 1765–68 to define the disputed boundaries between the land grants of the Penns, proprietors of Pennsylvania, and the Baltimores, proprietors of Maryland. The term was first used in congressional debates leading to the Missouri Compromise (1820) to describe the dividing line between the slave states to its south and the free-soil states to its north. It is still used as the figurative dividing line between the North and South.

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Elaborate defensive barrier in northeastern France built in the 1930s. Named after its principal creator, Andre Maginot, it was an ultramodern defensive fortification along the French-German frontier. Made of thick concrete and supplied with heavy guns, it had living quarters, supply storehouses, and underground rail lines. However, it ended at the French-Belgian frontier, which German forces crossed in May 1940. They invaded Belgium (May 10), crossed the Somme River, struck at the northern end of the line (May 12), and continued around to its rear, making it useless.

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Chain of islands, central Pacific Ocean, south of the Hawaiian Islands. The Line Islands extend 1,600 mi (2,600 km) and have a land area of 193 sq mi (500 sq km). Of the northern group, Teraina (Washington) Island and the Tabuaeran (Fanning) and Kiritimati (Christmas) atolls belong to the Republic of Kiribati, while Kingman Reef, Palmyra Atoll, and Jarvis Island are U.S. territories. Kiribati also holds the central group (Malden and Starbuck islands) and the southern group (Vostok and Flint islands and Caroline Atoll).

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Life-Line is Heinlein's first published science fiction story (1939), about a man who builds a machine that will predict how long a person will live. It does this by sending a signal along the world line of a person and detecting the echo from the far end.

Professor Pinero's invention has a powerful impact on the life insurance industry, as well as on his own life. Pinero is mentioned in passing in the novels Time Enough for Love and Methuselah's Children when the practically immortal Lazarus Long mentions having been examined and being sent away because the machine is "broken".

Heinlein was motivated to write the story by a contest in Thrilling Wonder Stories magazine promising $50 US to the winner, but ended up submitting it to a rival magazine, Astounding, and was paid $70. It made a later appearance in The Worlds of Robert A. Heinlein, a collection of short stories published in 1966, and his Expanded Universe in 1980.

In Grumbles from the Grave, on receiving the check for the story Heinlein is reported to have said, "How long has this racket been going on?" The amount was the equivalent of about $500 in 1984, or approximately one month's rent on a nice apartment. (Update: $70 in 1939 equates to $1085 in 2008 U.S. dollars. Source: U.S. Federal Reserve Bank)


An often quoted passage from this story is relevant to modern discussions of intellectual monopoly:

There has grown in the minds of certain groups in this country the idea that just because a man or corporation has made a profit out of the public for a number of years, the government and the courts are charged with guaranteeing such a profit in the future, even in the face of changing circumstances and contrary to public interest. This strange doctrine is supported by neither statute or common law. Neither corporations or individuals have the right to come into court and ask that the clock of history be stopped, or turned back.

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