Line
The word line apparently derives from the Latin linum, meaning flax plant from which linen is produced; at one time, a stretched linen thread was the most reliable way to determine a straight line. Also see liner[?] and lining[?].
In telecommunications, a telephone line is a single-user circuit on a telephone system. More generally, a line is a circuit or loop in any communications system.
Mathematics
A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object. Given two points, one can always find exactly one line that passes through the two points; the line provides the shortest connection between the points. Two different lines can intersect in at most one point; two different planes can intersect in at most one line. This intuitive concept of a line can be formalized in various ways.
If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's Foundations of Geometry[?]), then lines are not defined at all, but characterized axiomatically by their properties. "Everything that satisfies the axioms for a line is a line." While Euclid did define a line as "length without breadth", he did not use this rather obscure definition in his later development.
In Euclidean space Rn (and analogously in all other vector spaces), we define a line L as a subset of the form
- <math>L = \{\mathbf{a}+t\mathbf{b}\mid t\in\mathbb{R}\}</math>
where a and b are given vectors in Rn with b non-zero. The vector b describes the direction of the line, and a is a point on the line. Different choices of a and b can yield the same line.
One can show that in R2, every line L is described by a linear equation of the form
- <math>L=\{(x,y)\mid ax+by=c\}</math>
with fixed real coefficients a, b and c such that a and b are not both zero. An important property of these lines is their slope.
More abstractly, one usually thinks of the real line as the prototype of a line, and assumes that the points on a line stand in a one-to-one correspondence with the real numbers. However, one could also use the hyperreal numbers for this purpose, or even the long line of topology.
The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesics on differentiable manifolds.
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Ears, who heard the truffles growing.
"In accordance with the principles of general economy," said my
two-tenths of his income; now our apartment and our attendance cost
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twenty-five louis; I take for myself only eight hundred francs;
there does not remain, as you see, more than six hundred francs for
draw a thousand crowns from our capital, and if once we take that
Paris which you love so much, and at once take up our residence in the
increase fast enough! Come, try to be reasonable!"
"I suppose I must," she said, "but you will be the only husband in
duty. My master made a gesture of relief. When he saw the door close
to the Rue de Provence, little knowing that I had received the first
the conquest of Constantinople by General Diebitsch. I arrived at.
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