Lindsay, (Nicholas) Vachel (b. Nov. 10, 1879, Springfield, Ill., U.S.—d. Dec. 5, 1931, Springfield) U.S. poet. In his youth, he began traveling the country reciting his poems in return for food and shelter, in an attempt to revive poetry as an oral art form of the common people. He first received widespread recognition for “General William Booth Enters into Heaven” (1913), about the founder of the Salvation Army. His works are full of powerful rhythms, vivid imagery, and bold rhymes and express an ardent patriotism, a passion for progressive democracy, and a roman¬ tic view of nature. His collections include Rhymes to Be Traded for Bread (1912), The Congo (1914), and The Chinese Nightingale (1917). He was responsible for discovering the work of Langston Hughes. Depressed and unstable in later years, he committed suicide by drinking poison.
line Basic element of Euclidean geometry. Euclid defined a line as an inter¬ val between two points and claimed it could be extended indefinitely in either direction. Such an extension in both directions is now thought of as a line, while Euclid’s original definition is considered a line segment. A ray is part of a line extending indefinitely from a point on the line in only one direction. In a coordinate system on a plane, a line can be rep¬ resented by the linear equation ax + by + c = 0. This is often written in the slope-intercept form as y = mx + b, in which m is the slope and b is the value where the line crosses the y-axis. Because geometrical objects whose edges are line segments are completely understood, mathemati¬ cians frequently try to reduce more complex structures into simpler ones made up of connected line segments.
line integral In mathematics, the integral of a function of several vari¬ ables 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.
Line Islands 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 (Christ¬ mas) 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).
Linear A and Linear B Linear forms of writing used by Aegean civi¬ lizations during the 2nd millennium bc. Examples of Linear A, a syllabary (a writing system in which one character represents a whole syllable) written from left to right, date from 1850 bc to 1400 bc. The language written in Linear A remains unknown. Linear B, adapted from Linear A, was borrowed from the Minoan civilization by the Mycenaean Greeks, probably c. 1600 bc, and used to write the Mycenaean Greek dialect. Examples of Linear B script have been found on clay tablets and vases
from c. 1400-1200 bc. These texts represent the oldest known form of Greek. Linear B was deciphered as Greek in 1952 by Michael Ventris and John Chadwick.
linear accelerator or linac Type of particle accelerator that imparts a series of relatively small increases in energy to subatomic particles as they pass through a sequence of alternating electric fields set up in a lin¬ ear structure. The small accelerations add together to give the particles a greater energy than could be achieved by the voltage used in one section alone. One of the world’s longest linacs is the 2-mi (3.2-km) machine at the Stanford Linear Accelerator Center, which can accelerate electrons to energies of 50 billion electron volts. Much smaller linacs, both proton and electron types, have important practical applications in medicine and industry.
linear algebra Branch of algebra concerned with methods of solving systems of linear equations; more generally, the mathematics of linear transformations and vector spaces. “Linear” refers to the form of the equa¬ tions involved—in two dimensions, ax + by = c. Geometrically, this rep¬ resents a line. If the variables are replaced by vectors, functions, or derivatives, the equation becomes a linear transformation. A system of equa¬ tions of this type is a system of linear transformations. Because it shows when such a system has a solution and how to find it, linear algebra is essential to the theory of mathematical analysis and differential equations. Its applications extend beyond the physical sciences into, for example, biology and economics.
linear approximation In mathematics, the process of finding a straight line that closely fits a curve (function) at some location. Expressed as the linear equation y = ax + b, the values of a and b are chosen so that the line meets the curve at the chosen location, or value of x, and the slope of the line equals the rate of change of the curve (derivative of the func¬ tion) at that location. For most curves, linear approximations are good only very close to the chosen x. Yet much of the theory of calculus, including the fundamental theorem of calculus and the mean-value theorem for derivatives, is based on such approximations.
linear programming Mathematical modeling technique useful for guiding quantitative decisions in business, industrial engineering, and to a lesser extent the social and physical sciences. Solving a linear program¬ ming problem can be reduced to finding the optimum value (see optimi¬ zation) of a linear equation (called an objective function), subject to a set of constraints expressed as inequalities. The number of inequalities and variables depends on the complexity of the problem, whose solution is found by solving the system of inequalities like a system of equations. The extensive use of linear programming during World War II to deal with transportation, scheduling and allocations of resources under constraints like cost and priority gave the subject an impetus that carried it into the postwar era. The number of equations and variables needed to model real-life situations accurately is large, and the solution process can be time-consuming even with computers. See also simplex method.
linear transformation In mathematics, a rule for changing one geo¬ metric figure (or matrix or vector) into another using a formula with a specified format. The format must be a linear combination, in which the original components (e.g., the x and y coordinates of each point of the original figure) are changed via the formula ax + by to produce the coor¬ dinates of the transformed figure. Examples include flipping the figure over the x or y axis, stretching or compressing it, and rotating it. Every such transformation has an inverse, which undoes its effect.
linen Fibre, yarn, and fabric made from the flax plant. Flax is one of the oldest textile fibres used by humans; evidence of its use has been found in Switzerland’s prehistoric lake dwellings. Fine linen fabrics have been discovered in ancient Egyptian tombs. The fibre is obtained by subject¬ ing plant stalks to a series of operations, including retting (a fermentation process), drying, crushing, and beating. Linen is stronger than cotton, dries more quickly, and is more slowly affected by exposure to sunlight. Low elasticity, imparting a hard, smooth texture, makes linen subject to wrinkling. Because linen absorbs and releases moisture quickly and is a good conductor of heat, linen garments feel cool to wearers. Fine grades of linen are made into woven fabrics and laces for apparel and household furnishings.
linga or I ingam In Hinduism, the symbol of the god Shiva and of gen¬ erative power. Fashioned from wood, gems, metal, or stone, lingas are the main objects of worship in temples to Shiva and family shrines through-