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 ?
In pompous language, however, which jumbled one sentence into another, and at length grew disconnected, he gave me to understand that I was to lead the children altogether away from the Casino, and out into the park.
From the lower part of the face he appeared to be a man of strong character, with a thick, hanging lip, and a long, straight chin suggestive of resolution pushed to the length of obstinacy.
Indeed, my only wonder was that he had not already been mixed upon this extraordinary case, which was the one topic of conversation through the length and breadth of England.
The Admiral pooh-poohed it at first as a piece of necessary but annoying garden work; but at length the ring of real energy came back into his laughter, and he cried with a mixture of impatience and good humour:
I can well believe that you are not of Barsoom," he said at length.
At length, after having passed a final wicket, so loaded with locks that a quarter of an hour was required to open it, they entered a vast and lofty vaulted hall, in the centre of which they could distinguish by the light of the torches, a huge cubic mass of masonry, iron, and wood.
He could not understand such a state of things, and was obliged to conclude that it was pride, the pride of an injured and imaginative woman, which had gone to such lengths that it preferred to sit and nurse its contempt and hatred in solitude rather than mount to heights of hitherto unattainable splendour.
They kept a distance of about one and a half lengths, so that the horns of the crescent were nearly thirty miles apart.
That she had a remarkable fondness for that young fellow is most certain; but whether she indulged this to any very criminal lengths is not so extremely clear, unless we will suppose that women never grant every favour to a man but one, without granting him that one also.
It will have been seen that the Heidelburgh Tun of the Sperm Whale embraces the entire length of the entire top of the head; and since --as has been elsewhere set forth --the head embraces one third of the whole length of the creature, then setting that length down at eighty feet for a good sized whale, you have more than twenty-six feet for the depth of the tun, when it is lengthwise hoisted up and down against a ship's side.
At length she could control herself no longer, so she sent a trusty servant to her old and faithful friend the Fairy of the Mountain, to beg her to devise some means by which she might get rid of her stepson.
On this coast we landed, with an intention of travelling on foot to Jubo, a journey of much greater length and difficulty than we imagined.