Лењир

A variety of rulers

A 2 m (6 ft 6 ³⁄₄ in) carpenter's rule

Retractable flexible rule or tape measure

A closeup of a steel ruler

A ruler in combination with a letter scale

Школски лењир са скалом дужине 30 cm

Лењир (од нем. Lineal) је инструмент, који се користи у геометрији, техничком цртању и инжењерству за повлачење правих линија одређених или неодређених дужина као и за мерење димензија у простору или на површима. Лењир је најчешће квадарни комад пластике мале дебљине са урезаном скалом која је подељена на центиметре и милиметре или на инче и делове инча. Поред пластике, за израду се користе дрво и метал.

Варијанте

Gilded bronze ruler. 1 chi = 231 cm (91 in). Western Han (206 BCE – 8 CE). Hanzhong City, China

Bronze ruler. Han dynasty, 206 BCE – 220 CE. Excavated in Zichang County, China

Rulers have long been made from different materials and in multiple sizes. Some are wooden. Plastics have also been used since they were invented; they can be molded with length markings instead of being scribed. Metal is used for more durable rulers for use in the workshop; sometimes a metal edge is embedded into a wooden desk ruler to preserve the edge when used for straight-line cutting. 12 in or 30 cm in length is useful for a ruler to be kept on a desk to help in drawing. Shorter rulers are convenient for keeping in a pocket. Longer rulers, e.g., 18 in (46 cm), are necessary in some cases. Rigid wooden or plastic yardsticks, 1 yard long, and meter sticks, 1 meter long, are also used. Classically, long measuring rods were used for larger projects, now superseded by tape measure, surveyor's wheel or laser rangefinders.

Desk rulers are used for three main purposes: to measure, to aid in drawing straight lines, and as a straight guide for cutting and scoring with a blade. Practical rulers have distance markings along their edges.

A line gauge is a type of ruler used in the printing industry. These may be made from a variety of materials, typically metal or clear plastic. Units of measurement on a basic line gauge usually include inches, agate, picas, and points. More detailed line gauges may contain sample widths of lines, samples of common type in several point sizes, etc.

Measuring instruments similar in function to rulers are made portable by folding (carpenter's folding rule) or retracting into a coil (metal tape measure) when not in use. When extended for use, they are straight, like a ruler. The illustrations on this page show a 2 m (6 ft 7 in) carpenter's rule, which folds down to a length of 25 cm (10 in) to easily fit in a pocket, and a 5 m (16 ft) tape, which retracts into a small housing.

A flexible length-measuring instrument which is not necessarily straight in use is the tailor's fabric tape measure, a length of tape calibrated in inches and centimeters. It is used to measure around a solid body, e.g., for a person's waist measurement, as well as for linear measurement, e.g., for the inside leg length. It is rolled up when not in use, taking up little space.

A ruler software program can be used to measure pixels on a computer screen or mobile phone. These programs are also known as screen rulers.

Примене лењира у геометрији

Главни чланак: Конструкције лењиром и шестаром

In geometry, a ruler without any marks on it (a straightedge) may be used only for drawing straight lines between points. A straightedge is also used to help draw accurate graphs and tables.

A ruler and compass construction refers to constructions using an unmarked ruler and a compass. It is possible to bisect an angle into two equal parts with a ruler and compass. It can be proved, though, that it is impossible to divide an angle into three equal parts using only a compass and straightedge — the problem of angle trisection. However, should two marks be allowed on the ruler, the problem becomes solvable.

Историја

The Nippur cubit-rod, c. 2650 BCE, in the Archeological Museum of Istanbul, Turkey

A wooden carpenter's rule and other tools found on board the 16th-century carrack Mary Rose

In the history of measurement many distance units have been used which were based on human body parts such as the cubit, hand and foot and these units varied in length by era and location. In the late 18th century the metric system came into use and has been adopted to varying degrees in almost all countries in the world.

The oldest preserved measuring rod is a copper-alloy bar that dates from око 2650 BCE and was found by the German Assyriologist Eckhard Unger while excavating at the Sumerian city of Nippur (present day Iraq).

Rulers made of ivory were in use by the Indus Valley civilization period prior to 1500 BCE. Excavations at Lothal (2400 BCE) have yielded one such ruler calibrated to about ¹⁄₁₆ in (1,6 mm). Ian Whitelaw holds that the Mohenjo-Daro ruler is divided into units corresponding to 132 in (3.352,8 mm) and these are marked out in decimal subdivisions with amazing accuracy, to within 0.005 in (130 mm). Ancient bricks found throughout the region have dimensions that correspond to these units.

Anton Ullrich invented the folding ruler in 1851. Frank Hunt later made the flexible ruler in 1902.

Закривљени и флексибилни лењири

The equivalent of a ruler for drawing or reproducing a smooth curve, where it takes the form of a rigid template, is known as a French curve. A flexible device that can be bent to the desired shape is known as a flat spline, or (in its more modern incarnation) a flexible curve. Historically, a flexible lead rule used by masons that could be bent to the curves of a molding was known as a lesbian rule.

Филозофија

Ludwig Wittgenstein famously used rulers as an example in his discussion of language games in the Philosophical Investigations. He pointed out that the standard meter bar in Paris was the criterion against which all other rulers were determined to be one meter long, but that there was no analytical way to demonstrate that the standard meter bar itself was one meter long. It could only be asserted as one meter as part of a language game.

Галерија

• 
  Три месингана лењира у кутији
• 
  Месингани угаоник са либелом
• 
  Месингани угаоник са либелом у кутији

Литература

• Cherry, Dan. "Collector's guide to rules", Furniture & Cabinetmaking, no. 259, July 2017, ISSN [https://www.worldcat.org/search?fq=x0:jrnl&q=n2:1365-4292 1365-4292], pp. 52–6
• Rees, Jane and Mark The Rule Book: Measuring for the Trades. Lakeville, MN. . Astragal Press. 2010. ISBN 978-1-931626-26-2. OCLC 907853704. 
• Russell, David R.; with photography by James Austin and foreword by David Linley Antique Woodworking Tools: Their Craftsmanship from the Earliest Times to the Twentieth Century, Cambridge: John Adamson. 2010. ISBN 978-1-898565-05-5. стр. 64-74.
• Whitelaw, Ian (2007). A Measure of All Things: The Story of Man and Measurement. Macmillan. ISBN 978-0-312-37026-8. OCLC 938084552. 
• Wantzel, Pierre-Laurent (1837). „Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas.” (PDF). Journal de Mathématiques Pures et Appliquées. 1. 2: 366—372. Приступљено 3. 3. 2014. CS1 одржавање: Формат датума (веза)
• Weisstein, Eric W. „Trigonometry Angles--Pi/17”. MathWorld. 
• Neumann, Peter M. (1998), „Reflections on Reflection in a Spherical Mirror”, American Mathematical Monthly, 105 (6): 523—528, JSTOR 2589403, MR 1626185, doi:10.1080/00029890.1998.12004920 
• Highfield, Roger (1. 4. 1997), „Don solves the last puzzle left by ancient Greeks”, Electronic Telegraph, 676, Архивирано из оригинала 23. 11. 2004. г., Приступљено 2008-09-24 CS1 одржавање: Формат датума (веза)
• Kazarinoff, Nicholas D. (2003) [1970]. Ruler and the Round. Mineola, N.Y.: Dover. стр. 29—30. ISBN 978-0-486-42515-3. 
• Row, T. Sundara (1966). Geometric Exercises in Paper Folding. New York: Dover. 
• Simon Plouffe (1998). „The Computation of Certain Numbers Using a Ruler and Compass”. Journal of Integer Sequences. 1: 13. Bibcode:1998JIntS...1...13P. ISSN 1530-7638. 
• Hiram A. Farrand Inc. (децембар 1930). Popular Science. Popular Science Monthly. Приступљено 3. 11. 2011. CS1 одржавање: Формат датума (веза)
• Walter W. Jacob (септембар 2004). „Stanley Advertising and Imprinted Tape Rules”. The Chronicle of the Early American Industries Association. Приступљено 3. 11. 2011. CS1 одржавање: Формат датума (веза)
• „Talmeter — Hultafors”. Архивирано из оригинала 9. 2. 2013. г. CS1 одржавање: Формат датума (веза)
• "Range Finder (instrument)." Encyclopædia Britannica Online. Encyclopædia Britannica,
• Army Test and Evaluation Command, Aberdeen Proving Ground Maryland (1969) Laser Rangefinders Ft. Belvoir Defense Technical Information Center, U.S. Army, Ft. Belvoir, Virginia, OCLC 227620848 (early history of the use of lasers in rangefinders)
• Photographic and Imaging Manufacturers Association (1999) American national standard for photography (optics) : rangefinders and other focusing aids – performance specifications (revision and redesignation of "ANSI PH3.619-1988" as "ANSI/PIMA IT3.619-1998") American National Standards Institute, New York, OCLC 41501265
• Hicks, Roger and Schultz, Frances (2003) Rangefinder: Equipment, History, Techniques Guild of Master Craftsman, Lewes, UK, ISBN 1-86108-330-0
• Notes on rangefinders, compasses and on contouring with the Scale of Horizontal Equivalents (PDF), Infantry and Cavalry School Lectures 1902-1910, Staff College Press, U.S. Army, Fort Leavenworth, Kansas, 1905, OCLC 278057724, Архивирано из оригинала (PDF) 2016-03-03. г. — преко archive.org 
• Whitehouse, J. C. (2005) "Further considerations of defocus rangefinders" Transactions of the Institute of Measurement and Control 27(4): pp. 297–316
• Range-Finding in the Army. How to use range-finders to get results: the erect and inverted types, Popular Science monthly, February 1919, page 118–120, Scanned by Google Books
• Electro Optic Application Test Equipment, Berkeley Nucleonics Corporation, Архивирано из оригинала 2011-07-11. г. 

Спољашње везе

• Онлајн лењир
• Regular polygon constructions by Dr. Math at The Math Forum @ Drexel
• Construction with the Compass Only at cut-the-knot
• Angle Trisection by Hippocrates at cut-the-knot
• Weisstein, Eric W. „Angle Trisection”. MathWorld.