length


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length

1. Prosody Phonetics the metrical quantity or temporal duration of a vowel or syllable
2. the distance from one end of a rectangular swimming bath to the other
3. Prosody the quality of a vowel, whether stressed or unstressed, that distinguishes it from another vowel of similar articulatory characteristics. Thus in English beat is of greater length than in English bit
4. Cricket the distance from the batsman at which the ball pitches
5. Bridge a holding of four or more cards in a suit

Length

A one-dimensional extension in space. Length is one of the three fundamental physical quantities, the others being mass and time. It can be measured by comparison with an arbitrary standard; the specific one in most common usage is the international meter. In 1983, at the meeting of the Conférence Général des Poids et Mésures, the meter was redefined in terms of time and the speed of light: “The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.” This definition defines the speed of light to be exactly 299 792 458 m/s, and defines the meter in terms of the most accurately known quantity, the second. See Light, Mass, Time

Length

 

a numerical characteristic of the extent of curves. The concept of length is defined differently for different cases. (1) The length of a line segment is the distance between its end points, measured by some segment taken as a unit of length. (2) The length of a polygonal line is the sum of the lengths of its components. (3) The length of a simple arc is the limit of the lengths of polygonal lines inscribed in the arc, when the number of components increases indefinitely and the maximum length of the components tends to zero. (4) The length of a continuous curve consisting of a finite number of simple arcs is equal to the sum of the lengths of these arcs. For example, the circumference of a circle can be obtained as the limit of the perimeters of inscribed regular polygons when the number of their sides is doubled indefinitely; it is equal to 2πR where R is the radius of the circle. Any continuous curve has finite or infinite length. If its length is finite, then the curve is said to be rectifiable. The graph (see Figure 1) of the function

for 0 <x ≤ 1 and f(x) = 0 when x = 0 is an example of a nonrectifiable curve; here the lengths of the inscribed polygonal lines increase beyond all bound as the length of the components tend to zero. If the equation of a plane curve has the form y =f(x) (a ^ x ^ b)m rectangular coordinates and the function f(x) has a continuous derivative f’(x), then the length of the curve is given by the integral

The length of a curve given in parametric form and the length of a space curve can be expressed in a similar manner.

Figure 1

To calculate the length of a curve the mathematicians of antiquity essentially used lengths of polygonal lines and passage to the limit. For them, however, such passage to the limit was only a method for calculating the length of a curve and not for defining the concept of the length of a curve, since they apparently perceived the latter as one of the elementary mathematical concepts. The necessity of defining the length of a curve became clear only in the first half of the 19th century. A full elucidation of the problem was achieved by C. Jordan. In differential geometry the length of a curve is also defined on a surface or in an arbitrary Riemannian space.

REFERENCES

Lebesgue, H. Ob izmerenii velichin, 2nd ed. Moscow, 1960. (Translated from French.)
Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’nogo ischisleniya, 7th ed. vol. 2. Moscow, 1969.

length

[leŋkth]
(mechanics)
Extension in space.
References in classic literature ?
When we in Flatland see a Line, we see length and BRIGHTNESS.
At length, one of the men, more exhausted than the rest, threw himself upon the grass, and declared he could go no further.
Sometimes slipping them through the slits in his ears, he would seize his spear--which in length and slightness resembled a fishing-pole--and go stalking beneath the shadows of the neighbouring groves, as if about to give a hostile meeting to some cannibal knight.
My outstretched hands at length encountered some solid obstruction.
He steered directly for it, and at length, with the aid of the surf, ran his boat high upon a sandy beach.
My agony was at length insufferable, and I awoke just in time to perceive that a rat had ran off with the lighted candle from the stand, but not in season to prevent his making his escape with it through the hole.
With some little difficulty I at length came within sight of him, approached, and followed him closely, yet cautiously, so as not to attract his attention.
It was about six feet in length by two and a half in breadth; I observed it attentively, and like to be precise.
From it there were outspread two pairs of wings- each wing nearly one hundred yards in length -- one pair being placed above the other, and all thickly covered with metal scales; each scale apparently some ten or twelve feet in diameter.
At length, in my old age, [these are the words of Sinbad himself, as retailed by Scheherazade] -- 'at length, in my old age, and after enjoying many years of tranquillity at home, I became once more possessed of a desire of visiting foreign countries; and one day, without acquainting any of my family with my design, I packed up some bundles of such merchandise as was most precious and least bulky, and, engaged a porter to carry them, went with him down to the sea-shore, to await the arrival of any chance vessel that might convey me out of the kingdom into some region which I had not as yet explored.
But she knew not what answer to make to this, and stood for a time in doubt; at length she thought: 'I will go home and ask if it be I, or if it be not I, they will be sure to know.
From the western gate aforesaid the highway, as every Wintoncestrian knows, ascends a long and regular incline of the exact length of a measured mile, leaving the houses gradually behind.