Main Page: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older editNewer edit →
No edit summary
No edit summary
Line 1: Line 1:
{{Other uses|Square the Circle (disambiguation){{!}}Square the Circle}}
The following are examples of [[orders of magnitude]] for different [[length]]s.
{{multiple image
  | width    = 250
  | image1    = Squaring the circle.svg
  | caption1  = Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized [[compass and straightedge]].
  | image2    = Hipocrat arcs.svg
  | caption2  = Some apparent partial solutions gave false hope for a long time. In this figure, the shaded figure is the [[Lune of Hippocrates]]. Its area is equal to the area of the triangle {{math|ABC}} (found by [[Hippocrates of Chios]]).
  }}
{{Pi box}}
'''Squaring the circle''' is a problem proposed by [[classical antiquity|ancient]] [[geometers]]. It is the challenge of constructing a [[square (geometry)|square]] with the same area as a given [[circle]] by using only a finite number of steps with [[compass and straightedge]]. More abstractly and more precisely, it may be taken to ask whether specified [[axiom]]s of [[Euclidean geometry]] concerning the existence of lines and circles entail the existence of such a square.


In 1882, the task was proven to be impossible, as a consequence of the [[Lindemann–Weierstrass theorem]] which proves that [[pi]] ({{pi}}) is a [[Transcendental number|transcendental]], rather than an algebraic irrational number; that is, it is not the [[root of a function|root]] of any [[polynomial]] with rational coefficients. It had been known for some decades before then that the construction would be impossible if pi were transcendental, but pi was not proven transcendental until 1882. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to {{pi}}.
{| align="right"
|
|}
{| class="wikitable" border=1 cellpadding=3px
! rowspan=2 | Section !! colspan=2 | Range ([[meter|m]]) !! rowspan=2 | Unit !! rowspan=2 | Example Items
|-
! ≥ !! <
|-
| Planck area || [[#lt 1E-15|？]] || 10<sup>−35</sup> || || [[Quantum foam]]
|-
| [[#Subatomic|Subatomic]] || [[#lt 1E-15|10<sup>−35</sup>]] || 10<sup>−15</sup> || [[attometre|am]] || [[electron]], [[quark]], [[String (physics)|string]]
|-
| rowspan=3 | [[#Atomic and cellular|Atomic and cellular]] || [[#1E-15|10<sup>−15</sup>]] || 10<sup>−12</sup> || [[femtometre|fm]] || [[atomic nucleus]], [[proton]], [[neutron]]
|-
| [[#1E-12|10<sup>−12</sup>]] || 10<sup>−9</sup> || [[picometre|pm]] || [[wavelength]] of [[gamma ray]]s and [[X-ray]]s, [[hydrogen]] atom
|-
| [[#1E-9|10<sup>−9</sup>]] || 10<sup>−6</sup> || [[nanometre|nm]] || [[DNA]] [[helix]], [[virus]], wavelength of [[optical spectrum]]
|-
| rowspan=4 | [[#Human scale|Human scale]] || [[#1E-6|10<sup>−6</sup>]] || 10<sup>−3</sup> || [[micrometre|µm]] || [[bacterium]], [[fog]] water droplet, human hair<ref name="Physics Factbook">
According to ''The [[Physics]] Factbook'', the [[diameter]] of human hair ranges from 17 to 181 µm. {{cite web|url=http://hypertextbook.com/facts/1999/BrianLey.shtml|title=Width of a Human Hair|last=Ley|first=Brian|work=The Physics Factbook|year=1999}}</ref>
|-
| [[#1E-3|10<sup>−3</sup>]] || 10<sup>0</sup> || [[millimetre|mm]] || [[mosquito]], [[golf ball]], [[football (association football)|football]]
|-
| [[#1E0|10<sup>0</sup>]] || 10<sup>3</sup> || [[metre|m]] || [[human being]], [[association football#Pitch|football field]], [[Eiffel Tower]]
|-
| [[#1E3|10<sup>3</sup>]] || 10<sup>6</sup> || [[kilometre|km]] || [[Mount Everest]], length of [[Panama Canal]], larger [[asteroid]]
|-
| rowspan=7 | [[#Astronomical|Astronomical]] || [[#1E6|10<sup>6</sup>]] || 10<sup>9</sup> || [[megametre|Mm]] || the [[Moon]], Earth, one [[light-second]]
|-
| [[#1E9|10<sup>9</sup>]] || 10<sup>12</sup> || [[gigametre|Gm]] || [[Sun]], one [[light-minute]], Earth's orbit
|-
| [[#1E12|10<sup>12</sup>]] || 10<sup>15</sup> || [[terametre|Tm]] || orbits of [[outer planets]], [[Solar System]]
|-
| [[#1E15|10<sup>15</sup>]] || 10<sup>18</sup> || [[petametre|Pm]] || one [[light-year]]; distance to [[Proxima Centauri]]
|-
| [[#1E18|10<sup>18</sup>]] || 10<sup>21</sup> || [[exametre|Em]] || [[spiral arm|galactic arm]]
|-
| [[#1E21|10<sup>21</sup>]] || 10<sup>24</sup> || [[zettametre|Zm]] || [[Milky Way]], distance to [[Andromeda Galaxy]]
|-
| [[#1E24|10<sup>24</sup>]] || <math>10^{10^{10^{122}}}</math><ref name="exponents" group=note/> || [[yottametre|Ym]] || [[Huge-LQG]], [[Hercules-Corona Borealis Great Wall]], [[visible universe]]
|}


The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.<ref>{{cite web|last=Ammer|first=Christine|title=Square the Circle. Dictionary.com. The American Heritage® Dictionary of Idioms|url=http://dictionary.reference.com/browse/square%20the%20circle|publisher=Houghton Mifflin Company|accessdate=16 April 2012}}</ref>
==Detailed list==
To help compare different orders of magnitude, the following list describes various lengths between 1.6{{E|−35}}&nbsp;meters and <math>10^{10^{10^{122}}}</math>meters.


The term ''[[numerical integration|quadrature]] of the circle'' is sometimes used synonymously or may refer to approximate or numerical methods for finding the area of a circle.
===Subatomic===
{| class="wikitable" border=1 cellpadding=3px
! Factor ([[metre|m]])
! Multiple
! Value
! Item
|-
<div id="lt 1E-15"/>
|[[1 yoctometer|10<sup>−35</sup>]]
| 1 [[Planck Length]]
|0.0000000000162&nbsp;ym&nbsp;(1.62{{E|−35}}&nbsp;m)
|[[Planck length]]; typical scale of hypothetical [[loop quantum gravity]] or size of a hypothetical [[String (physics)|string]] and of [[branes]]; according to [[string theory]] lengths smaller than this do not make any [[physics|physical]] sense.<ref name="Burgess_and_Quevedo">{{cite news | author=[[Cliff Burgess]] |author2=[[Fernando Quevedo]]  | title=The Great Cosmic Roller-Coaster Ride | url= | type=print | work=[[Scientific American]] | publisher=Scientific American, Inc. | page=55 | date=November 2007 |accessdate=2008-06-27 }}</ref> [[Quantum foam]] is thought to exist at this level.
|-
|[[1 yoctometre|10<sup>−24</sup>]]
|1 [[yoctometer]] (ym)
|20 ym (2 × 10<sup>&minus;23</sup> meters)
|effective [[Cross section (physics)|cross section]] radius of 1 [[MeV]] [[neutrino]]s<ref>{{cite web |author=Carl R. Nave |url=http://hyperphysics.phy-astr.gsu.edu/hbase/particles/cowan.html#c1 |title=Cowan and Reines Neutrino Experiment |accessdate=2008-12-04}} (6.3 × 10<sup>&minus;44</sup> cm<sup>2</sup>, which gives an effective radius of about 2 × 10<sup>&minus;23</sup> m)</ref>
|-
|rowspan=3|[[1 zeptometre|10<sup>−21</sup>]]
|rowspan=3|1 [[zeptometer]] (zm)
|
|[[Preons]], hypothetical particles proposed as subcomponents of quarks and leptons; the upper bound for the width of a [[cosmic string]] in string theory.
|-
|7 zm (7 × 10<sup>&minus;21</sup> meters)
| effective cross section radius of high energy [[neutrino]]s<ref name="NaveN3">{{cite web |author=Carl R. Nave |url=http://hyperphysics.phy-astr.gsu.edu/hbase/particles/neutrino3.html#c2 |title=Neutron Absorption Cross-sections |accessdate=2008-12-04}} (area for 20 GeV about 10 × 10<sup>&minus;42</sup> m<sup>2</sup> gives effective radius of about 2 × 10<sup>&minus;21</sup> m; for 250 GeV about 150 × 10<sup>&minus;42</sup> m<sup>2</sup> gives effective radius of about 7 × 10<sup>&minus;21</sup> m)</ref>
|-
|310 zm (3.10 × 10<sup>&minus;19</sup> meters)
|[[de Broglie wavelength]] of [[protons]] at the [[Large Hadron Collider]] (4 TeV as of 2012)
|-
|rowspan=3|[[1 attometre|10<sup>−18</sup>]]
|rowspan=3|1 [[attometer]] (am)
|rowspan=3|
|upper limit for the size of [[quark]]s and [[electron]]s
|-
|sensitivity of the [[LIGO]] detector for [[gravitational wave]]s
|-
|upper bound of the typical size range for "fundamental [[String (physics)|strings]]"<ref name="Burgess_and_Quevedo"/>
|-
|rowspan=1|10<sup>−17</sup>
|rowspan=1|10 am
|rowspan=1|
| range of the [[weak force]]
|-
|}


==History==
===Atomic and cellular===
Methods to approximate the area of a given circle with a square were known already to [[Babylonian mathematics|Babylonian mathematicians]]. The Egyptian [[Rhind papyrus]] of 1800BC gives the area of a circle as (64/81)&nbsp;{{math|''d''}}<sup>&nbsp;2</sup>, where {{math|''d''}} is the diameter of the circle, and pi approximated to 256/81, a number that appears in the older [[Moscow Mathematical Papyrus]] and used for volume approximations (i.e. [[hekat (volume unit)|hekat]]). [[Indian mathematics|Indian mathematicians]] also found an approximate method, though less accurate, documented in the ''[[Sulba Sutras]]''.<ref>{{Cite web|author=O'Connor, John J. and Robertson, Edmund F. |year=2000|url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html|title=The Indian Sulbasutras|work=MacTutor History of Mathematics archive|publisher=St Andrews University}}</ref> [[Archimedes]] showed that the value of pi lay between 3&nbsp;+&nbsp;1/7 (approximately 3.1429) and 3&nbsp;+&nbsp;10/71 (approximately 3.1408). See [[Numerical approximations of π]] for more on the history.
<!-- No more than three examples per row: choose examples from a range of sizes, and add extra examples to the sub-articles instead -->


The first known Greek to be associated with the problem was [[Anaxagoras]], who worked on it while in prison. [[Hippocrates of Chios]] squared certain [[Lune (mathematics)|lunes]], in the hope that it would lead to a solution — see [[Lune of Hippocrates]]. [[Antiphon the Sophist]] believed that inscribing regular polygons within a circle and doubling the number of sides will eventually fill up the area of the circle, and since a polygon can be squared, it means the circle can be squared. Even then there were skeptics—[[Eudemus]] argued that magnitudes cannot be divided up without limit, so the area of the circle will never be used up.<ref>{{Cite book| last = Heath | first = Thomas | year = 1981 | title = History of Greek Mathematics | publisher = Courier Dover Publications| isbn = 0-486-24074-6}}</ref> The problem was even mentioned in [[Aristophanes]]'s play ''[[The Birds (play)|The Birds]]''.
{| class="wikitable" border=1 cellpadding=3px
!Factor ([[metre|m]])
!Multiple
!Value
!Item
<div id="1E-15"/>
|-
|rowspan=3|[[1 femtometre|10<sup>−15</sup>]]
|rowspan=3|1 [[femtometer]] (fm)
|1.5 fm
|size of an 11 MeV [[proton]]<ref name="Nav">{{cite web |author=Carl R. Nave |url=http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/crosec.html |title= Scattering Cross Section |accessdate=2009-02-10}}
(diameter of the [[Cross section (physics)|Scattering Cross Section]] of an 11 MeV [[proton]] with a target proton)<!-- To verify, use online form with these figures: Z=1 KE=11 A=1 and obtain cross section of 1.758 fm^2; use Area=Pi*R^2 to derive diameter (2R) --></ref>
|-
|2.81794 fm
|[[classical electron radius]]<ref>[[National Institute of Standards and Technology|NIST]]. [http://physics.nist.gov/cgi-bin/cuu/Value?re CODATA Value:    classical electron radius]. Retrieved 2009-02-10</ref>
|-
|
|scale of the [[atomic nucleus]]<ref name="Burgess_and_Quevedo"/><ref>{{cite web |title=The Scale of the Universe |url=http://cass.ucsd.edu/public/tutorial/scale.html |author=H. E. Smith |publisher=[[University of California, San Diego|UCSD]] |quote=~10<sup>-13</sup>cm |accessdate=2009-02-10}}</ref>
|-
|rowspan=1|[[10 femtometers|10<sup>−14</sup>]]
|rowspan=1|10 fm
|
|
|-
|rowspan=1|10<sup>−13</sup>
|rowspan=1|100 fm
|
|
<div id="1E-12"/>
|-
|rowspan=3|[[1 picometre|10<sup>−12</sup>]]
|rowspan=3|1 [[picometer]] (pm)
|...
|longest [[wavelength]] of [[gamma ray]]s
|-
|2.4 pm
|[[Compton wavelength]] of [[electron]]
|-
|5 pm
|wavelength of shortest [[X-ray]]s
|-
|rowspan=3|[[10 picometres|10<sup>−11</sup>]]
|rowspan=3|10 pm
|25 pm
|radius of [[hydrogen atom]]
|-
|31 pm
|radius of [[helium]] atom
|-
|53 pm
|[[Bohr radius]]
|-
|rowspan=3|[[100 picometres|10<sup>−10</sup>]]
|rowspan=3|[[100 picometres|100 pm]]
|100 pm (0.1&nbsp;nm)
|1 [[Angstrom|Ångström]] (also [[covalent radius]] of [[sulfur]] atom<ref>{{cite web |author=Mark Winter |title=WebElements Periodic Table of the Elements / Sulfur / Radii |url=http://www.webelements.com/sulfur/atom_sizes.html |year=2008 |accessdate=2008-12-06}}</ref>)
|-
|154 pm (0.154&nbsp;nm)
|length of a typical [[covalent bond]] ([[carbon|C]]–C).
|-
|500 pm (0.50&nbsp;nm)
|width of [[protein]] [[alpha helix|α helix]]
<div id="1E-9"/>
|-
|rowspan=3|[[1 nanometre|10<sup>−9</sup>]]
|rowspan=3|1 [[nanometer]] (nm)
|1&nbsp;nm
|diameter of a [[carbon nanotube]]<ref>{{Cite journal |first=E. |last=Flahaut |author2=Bacsa R |author3=Peigney A |author4=Laurent C.  |year=2003 |title=Gram-Scale CCVD Synthesis of Double-Walled Carbon Nanotubes |journal=Chemical Communications |volume=12 | pages=1442–1443 |url=http://www.rsc.org/Publishing/Journals/CC/article.asp?doi=b301514a |doi=10.1039/b301514a |accessdate=2008-11-14 |pmid=12841282 |issue=12}}</ref>
|-
|2.5&nbsp;nm
|Smallest [[microprocessor]] [[transistor]] [[Gate (transistor)|gate]] oxide thickness (as of Jan 2007)
|-
|6–10&nbsp;nm
|thickness of [[cell membrane]]
|-
|rowspan=3|[[10 nanometres|10<sup>−8</sup>]]
|rowspan=3|10&nbsp;nm
|10&nbsp;nm
|thickness of [[cell wall]] in [[Gram staining|gram]]-negative [[bacteria]]{{Citation needed|date=February 2009}}
|-
|40&nbsp;nm
|extreme [[ultraviolet]] wavelength
|-
|90&nbsp;nm
|[[HIV|Human immunodeficiency virus (HIV)]] (generally, [[virus]]es range in size from 20&nbsp;nm to 450&nbsp;nm)
|-
|rowspan=3|[[100 nanometres|10<sup>−7</sup>]]
|rowspan=3|100&nbsp;nm
|121.6&nbsp;nm
|wavelength of the [[Lyman-alpha line]]<ref name="ber">Cohn, J. [[University of California, Berkeley]] [http://astro.berkeley.edu/~jcohn/lya.html Lyman alpha systems and cosmology]. Retrieved 2009-02-21</ref>
|-
|380–435&nbsp;nm
|wavelength of [[violet (color)|violet]] light—see [[color]] and [[optical spectrum]]<ref name="hyp">[http://hyperphysics.phy-astr.gsu.edu/hbase/vision/specol.html#c1 Color]</ref>
|-
|625–740&nbsp;nm
|wavelength of red light<ref name="hyp"/>
|}


It is believed that [[Oenopides]] was the first Greek who required a plane solution (that is, using only a compass and straightedge). [[James Gregory (astronomer and mathematician)|James Gregory]] attempted a proof of its impossibility in ''Vera Circuli et Hyperbolae Quadratura'' (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was incorrect, it was the first paper to attempt to solve the problem using algebraic properties of pi. It was not until 1882 that [[Ferdinand von Lindemann]] rigorously proved its impossibility.
===Human scale===
<!-- No more than three examples per row: choose examples from a range of sizes, and add extra examples to the sub-articles instead; prefer examples that are well-cited and well-known throughout the world -->


[[Image:Quadrature of Circle Cajori 1919.png|right|thumb|A partial history by [[Florian Cajori]] of attempts at the problem.<ref>{{cite book|author=Florian Cajori|title=A History of Mathematics|edition=2nd|page= 143|location= New York|publisher= The Macmillan Company|year=1919}}</ref>]] The famous Victorian-age mathematician, logician and author, [[Lewis Carroll|Charles Lutwidge Dodgson]] (better known under the pseudonym "Lewis Carroll") also expressed interest in debunking illogical circle-squaring theories. In one of his diary entries for 1855, Dodgson listed books he hoped to write including one called "Plain Facts for Circle-Squarers". In the introduction to "A New Theory of Parallels", Dodgson recounted an attempt to demonstrate logical errors to a couple of circle-squarers, stating:<ref>{{Cite book
{| class="wikitable" border=1 cellpadding=3px
|author=Martin Gardner
!Factor ([[metre|m]])
|title=The Universe in a Handkerchief
!Multiple
|isbn=0-387-94673-X
!Value
|publisher=Springer
!Item
|year=1996
<div id="1E-6"/>
}}</ref>
|-
|rowspan=3|[[1 micrometre|10<sup>−6</sup>]]
|rowspan=3|1 [[micrometer]] (µm)
|1&nbsp;µm
|also called one [[micron]]
|-
|1–3&nbsp;µm
|particle size that a [[surgical mask]] removes at 80–95% efficiency{{Citation needed|date=December 2008}}
|-
|6-8&nbsp;µm
|diameter of a [[red blood cell]]<ref>{{Cite web | url=http://www.wadsworth.org/chemheme/heme/microscope/rbc.htm | title=Through the Microscope: Blood Cells - Life's Blood | publisher=Wadsworth Center, New York State Department of Health | accessdate=2011-09-13 }}</ref>
|-
|rowspan=3|[[10 micrometres|10<sup>−5</sup>]]
|rowspan=3|10&nbsp;µm
|10&nbsp;µm
|typical size of a fog, mist or cloud water droplet. Chip [[10 µm process]] in 1971.
|-
|12&nbsp;µm
|width of [[acrylic fibre]]
|-
|25.4&nbsp;µm
|1/1000&nbsp;inch, commonly referred to as one [[Thou (unit of length)|thou]] or one [[Thou (unit of length)|mil]]
|-
|rowspan=3|[[1 myriometre|10<sup>−4</sup>]]
|rowspan=3|100&nbsp;µm
|100&nbsp;µm
|average width of a strand of human hair<ref name="Physics Factbook"/>
|-
|200&nbsp;µm
|typical length of ''[[Paramecium|Paramecium caudatum]]'', a ciliate protist
|-
|750&nbsp;µm
|maximum diameter of ''[[Thiomargarita namibiensis]]'', the largest bacterium ever discovered
|-
<div id="1E-3"/>
|-
|rowspan=3|[[1 millimetre|10<sup>−3</sup>]]
|rowspan=3|1 [[millimeter]] (mm)
|2.54&nbsp;mm
|1/10th inch; distance between pins in [[dual in-line package|DIP]] (dual-inline-package) electronic components
|-
|5&nbsp;mm
|length of average red [[ant]]
|-
|7.62&nbsp;mm
|common military ammunition size
|-
|rowspan=3|[[1 centimetre|10<sup>−2</sup>]]
|rowspan=3|1 [[centimeter]] (cm)
|1.5&nbsp;cm
|length of a large [[mosquito]]
|-
|2.54&nbsp;cm
|1 [[inch]]
|-
|4.267&nbsp;cm
|diameter of a [[golf ball]]
|-
|rowspan=3|[[1 decimetre|10<sup>−1</sup>]]
|rowspan=3|1 [[decimeter]] (dm)
|10&nbsp;cm
|wavelength of the highest [[Ultra high frequency|UHF]] radio frequency, 3&nbsp;GHz
|-
|30.48&nbsp;cm
|1 [[foot (length)|foot]]
|-
|91.44&nbsp;cm
|1 [[yard]]
<div id="1E0"/>
|-
|rowspan=3|[[1 metre|10<sup>0</sup>]]
|rowspan=3|1 [[meter]]
| 1 m
|wavelength of the lowest [[Ultra high frequency|UHF]] and highest [[Very high frequency|VHF]] radio frequency, 300&nbsp;MHz
|-
|1.7 m (5 feet 7&nbsp;inches)
|average height of a human
|-
|8.38 m
|The length of a London Bus ([[Routemaster]])
|-
|rowspan=3|[[1 decametre|10<sup>1</sup>]]
|rowspan=3|1 [[decameter]] (dam)
|10 m
|wavelength of the lowest [[Very high frequency|VHF]] and highest [[shortwave]] radio frequency, 30&nbsp;MHz
|-
|33 m
|length of longest [[blue whale]] measured, the largest animal<ref>{{Cite web | url=http://nationalzoo.si.edu/Animals/AnimalRecords/ | title=Animal Records | publisher=Smithsonian National Zoological Park | accessdate=2007-05-29 }}</ref>
|-
|93.47 m
|height of the [[Statue of Liberty]] (foundation of pedestal to torch)
|-
|rowspan=3|[[1 hectometre|10<sup>2</sup>]]
|rowspan=3|1 [[hectometer]] (hm)
|100 m
|wavelength of the lowest [[shortwave]] radio frequency and highest [[medium wave]] radio frequency, 3&nbsp;MHz
|-
|137 m (147 m)
|height (present and original) of the [[Great Pyramid of Giza]]
|-
|979 m
|height of the [[Salto Angel]], the world's highest free-falling waterfall ([[Venezuela]])
<div id="1E3"/>
|-
|rowspan=4|[[1 kilometre|10<sup>3</sup>]]
|rowspan=4|1 [[kilometer]] (km)
|1&nbsp;km
|wavelength of the lowest [[medium wave]] radio frequency, 300&nbsp;kHz
|-
|1609 m
|1 [[mile|international mile]]
|-
|1852 m
|1 [[nautical mile]]<br>&nbsp;
|-
|8848 m
|height of the highest mountain on earth, [[Mount Everest]]
|-
|rowspan=3|[[1 myriametre|10<sup>4</sup>]]<!-- Please consider renaming the obsolete term myriametre to [[10 kilometres (order of magnitude)]] -->
|rowspan=3|10&nbsp;km
|10.911&nbsp;km
|depth of deepest part of the ocean, [[Mariana Trench]]
|-
|13&nbsp;km
|narrowest width of the [[Strait of Gibraltar]], separating Europe and Africa
|-
|90&nbsp;km
|width of the [[Bering Strait]]
|-
|rowspan=3|[[100 kilometres|10<sup>5</sup>]]
|rowspan=3|100&nbsp;km
|111&nbsp;km
|distance covered by one degree of [[latitude]] on Earth's surface
|-
|163&nbsp;km
|length of the [[Suez Canal]]
|-
|974.6&nbsp;km
|greatest diameter<ref>{{cite journal|first=P. C.|last=Thomas|author2=Parker, J. Wm. |author3=McFadden, L. A. |author4= et al. |title=Differentiation of the asteroid Ceres as revealed by its shape|year=2005|journal=Nature|volume=437|pages=224&ndash;226|doi=10.1038/nature03938| bibcode=2005Natur.437..224T|pmid=16148926|issue=7056}}</ref> of the dwarf planet<ref name="Asteroid-planet?" group=note>The exact [[Solar System#Terminology|category]] (asteroid, dwarf planet or planet) to which particular solar system objects belong, has been subject to some revision since the discovery of [[extrasolar planet]]s and [[Trans-Neptunian object|trans-Neptunian]] objects</ref> [[Ceres (dwarf planet)|Ceres]]
|}


<blockquote>The first of these two misguided visionaries filled me with a great ambition to do a feat I have never heard of as accomplished by man, namely to convince a circle squarer of his error! The value my friend selected for Pi was 3.2: the enormous error tempted me with the idea that it could be easily demonstrated to BE an error. More than a score of letters were interchanged before I became sadly convinced that I had no chance.
===Astronomical===
</blockquote>
{| class="wikitable" border=1 cellpadding=3px
!Factor ([[metre|m]])
!Multiple
!Value
!Item
<div id="1E6"/>
|-
|rowspan=7|[[1 megametre|10<sup>6</sup>]]
|rowspan=7|1,000&nbsp;km = 1 [[megameter]] (Mm)
|2,390&nbsp;km
|diameter of dwarf planet [[Pluto]], formerly the smallest [[planet]] category<ref name="Asteroid-planet?" group=note/> of our solar system
|-
|3,480&nbsp;km
|diameter of the [[Moon]]
|-
|5,200&nbsp;km
|typical distance covered by the winner of the [[24 Hours of Le Mans]] automobile endurance race
|-
|6,400&nbsp;km
|length of the [[Great Wall of China]]
|-
|6,600&nbsp;km
|approximate length of the two longest rivers, the [[Nile]] and the [[Amazon River|Amazon]]
|-
|7,821&nbsp;km
|length of the [[Trans-Canada Highway]]
|-
|9,288&nbsp;km
|length of the [[Trans-Siberian Railway]], longest in the world
<div id="1E7"/>
|-
|rowspan=2|[[10 megametres|10<sup>7</sup>]]
|rowspan=2|10,000&nbsp;km
|12,756&nbsp;km
|equatorial diameter of the Earth
|-
|40,075&nbsp;km
|length of the Earth's [[equator]]
<div id="1E8"/>
|-
|rowspan=3|[[100 megametres|10<sup>8</sup>]]
|rowspan=3|100,000&nbsp;km
|142,984&nbsp;km
|diameter of [[Jupiter]]
|-
|299,792.458&nbsp;km
|distance travelled by light in one second
|-
|384,000&nbsp;km = 384 Mm
|[[Moon]]'s orbital distance from Earth
<div id="1E9"/>
|-
|rowspan=2|[[1 gigametre|10<sup>9</sup>]]
|rowspan=2|1 million km = 1 [[gigameter]] (Gm)
|1,390,000&nbsp;km = 1.39&nbsp;Gm
|diameter of the [[Sun]]
|-
|4,200,000 &nbsp;km = 4.2&nbsp;Gm
|greatest mileage ever recorded by a car (A 1966 [[Volvo P1800|Volvo P-1800S]], still driving)
<div id="1E10"/>
|-
| [[10 gigametres|10<sup>10</sup>]]
| 10 million km
|18 million km
|approximately one [[light-minute]]
<div id="1E11"/>
|-
| rowspan=2 | [[100 gigametres|10<sup>11</sup>]]
| rowspan=2 | 100 million km
|150 million&nbsp;km = 150&nbsp;Gm
|1 [[astronomical unit]] (AU); mean distance between Earth and Sun
|-
|~ 900 Gm
|optical diameter of [[Betelgeuse]] (~600 × Sun)
<div id="1E12"/>
|-
|rowspan=6|[[1 terametre|10<sup>12</sup>]]
|rowspan=6|1000 million km = 1 [[terameter]] (Tm)
|1.4 {{E|9}}&nbsp;km
|orbital distance of [[Saturn]] from Sun
|-
|1.96 {{E|9}}&nbsp;km
|estimated optical diameter of [[VY Canis Majoris]] (1420 × Sun)
|-
|2.3 {{E|9}}&nbsp;km
|estimated optical diameter of [[NML Cygni]] (1650 × Sun)
|-
|2.37 {{E|9}}&nbsp;km
|median point of the optical diameter of [[UY Scuti]], as of 2014 the [[List of largest known stars|largest known star]]
|-
|5.9 {{E|9}}&nbsp;km = 5.9&nbsp;Tm
|orbital distance of Pluto from Sun
|-
|~ 7.5 {{E|9}}&nbsp;km = 7.5&nbsp;Tm
|outer boundary of the [[Kuiper belt]], inner boundary of the [[Oort cloud]] (~ 50 AU)
<div id="1E13"/>
|-
|rowspan=3|[[10 terametres|10<sup>13</sup>]]
|rowspan=3|10 Tm
|
|diameter of our [[Solar System]] as a whole<ref name="Burgess_and_Quevedo"/>
|-
|16.25{{E|9}}&nbsp;km = 16.25&nbsp;Tm
|distance of the [[Voyager 1]] spacecraft from Sun ({{As of|2009|alt=as of Feb 2009}}), the farthest man-made object so far<ref>[http://heavens-above.com/solar-escape.asp Spacecraft escaping the Solar System<!-- Bot generated title -->]</ref>
|-
|62.03{{E|9}}&nbsp;km = 62.03&nbsp;Tm
|estimated radius of the [[event horizon]] of the [[supermassive black hole]] in [[NGC 4889]], the largest known black hole to date
<div id="1E14"/>
|-
|[[100 terametres|10<sup>14</sup>]]
|100 Tm
|1.8{{E|11}}&nbsp;km = 180&nbsp;Tm
|size of the [[debris disk]] around the star [[51 Pegasi]] <ref>[http://esciencenews.com/articles/2009/09/24/twin.keck.telescopes.probe.dual.dust.disks Twin Keck telescopes probe dual dust disks]</ref> <div id="1E15"/>
|-
|rowspan=2|[[1 petametre|10<sup>15</sup>]]
|rowspan=2|1 [[petameter]] (Pm)
|~ 7.5 {{E|12}}&nbsp;km = 7.5&nbsp;Pm
|supposed outer boundary of the Oort cloud (~ 50,000 AU)
|-
|9.46{{E|12}}&nbsp;km = 9.46&nbsp;Pm<br> = 1 [[light year]]
|distance traveled by light in one year; at its current speed, Voyager 1 would need 17,500 years to travel this distance
<div id="1E16"/>
|-
|rowspan=3|[[10 petametres|10<sup>16</sup>]]
|rowspan=3|10 Pm
|3.2616 light-years<br>(3.0857{{E|13}}&nbsp;km = 30.857&nbsp;Pm)
|1 [[parsec]]
|-
|4.22 light-years = 39.9&nbsp;Pm
|distance to nearest star ([[Proxima Centauri]])
|-
|4.37 light-years = 41.3&nbsp;Pm
|as of March 2013, distance to nearest discovered [[extrasolar planet]] ([[Alpha Centauri Bb]])
<div id="1E17"/>
|-
|rowspan=2|[[100 petametres|10<sup>17</sup>]]
|rowspan=2|100 Pm
|20.4 light-years = 193&nbsp;Pm
|as of October 2010, distance to nearest discovered extrasolar planet with potential to support life as we know it ([[Gliese 581 d]])
|-
|65 light-years = 6.15{{E|17}}&nbsp;m = 615&nbsp;Pm
|approximate radius of humanity's radio bubble, caused by high-power TV broadcasts leaking through the atmosphere into outer space
<div id="1E18"/>
|-
|rowspan=1|[[1 exametre|10<sup>18</sup>]]
|rowspan=1|1 [[exameter]] (Em)
|200 light-years = 1.9&nbsp;Em
|distance to nearby [[solar twin]] ([[HIP 56948]]), a star with properties virtually identical to our Sun <ref>{{cite web | last = Shiga | first = David | url = http://space.newscientist.com/article/dn12725-suns-twin-an-ideal-hunting-ground-for-alien-life.html | title = Sun's 'twin' an ideal hunting ground for alien life | publisher = New Scientist | accessdate = 2007-10-03 }}</ref>
<div id="1E19"/>
|-
|[[10 exametres|10<sup>19</sup>]]
|10 Em
|1,000 light-years = 9.46 Em or 9.46 × 10<sup>15</sup> km
|average thickness of [[Milky Way Galaxy]]<ref>{{cite web |last=Christian |first=Eric |last2=Samar |first2=Safi-Harb |title=How large is the Milky Way? |url=http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/980317b.html |accessdate=2008-11-14}}</ref> (1000 to 3000 ly by [[Hydrogen line|21&nbsp;cm observations]]<ref>{{cite web |last=Duncan |first=Martin |title=Physics 216 – Introduction to Astrophysics |chapter=16 |url=http://www.physics.queensu.ca/~phys216/ch16B.pdf |year=2008 |accessdate=2008-11-14}}</ref>)
<div id="1E20"/>
|-
|rowspan=2|[[100 exametres|10<sup>20</sup>]]
|100 Em
|12,000 light-years = 113.5 Em or 1.135 × 10<sup>17</sup> km
|thickness of [[Milky Way Galaxy]]'s gaseous disk<ref>{{cite web|url=http://news.smh.com.au/milky-way-fatter-than-first-thought/20080220-1tbv.html |title=Milky Way fatter than first thought |accessdate=2008-11-14 |date=2008-02-20 |work=[[The Sydney Morning Herald]] |publisher=[[Australian Associated Press]]}}</ref>
|-
|950 Em
|100,000 light-years
|diameter of galactic disk of [[Milky Way Galaxy]]<ref name="Burgess_and_Quevedo"/>
<div id="1E21"/>
|-
|rowspan=5|[[1 zettametre|10<sup>21</sup>]]
|rowspan=5|1 [[zettameter]] (Zm)
|-
|50 [[kiloparsecs]]
|distance to [[Supernova 1987a|SN 1987A]], the most recent naked eye supernova
|-
|52 kiloparsecs = 1.62{{E|21}}&nbsp;m = 1.62&nbsp;Zm
|distance to the [[Large Magellanic Cloud]] (a [[dwarf galaxy]] [[orbit]]ing the [[Milky Way]])
|-
| 54 kiloparsecs = 1.66&nbsp;Zm
|distance to the [[Small Magellanic Cloud]] (another dwarf galaxy orbiting the Milky Way)
|-
| 200 kiloparsecs = 6.15&nbsp;Zm
|diameter of the low surface brightness disc halo of the giant spiral galaxy [[Malin 1]]
<div id="1E22"/>
|-
|rowspan=4|[[10 zettametres|10<sup>22</sup>]]
|rowspan=4|10 Zm
| 13.25&nbsp;Zm = 1.4 million light years<br>= 600 kiloparsecs
| radius of the diffuse stellar halo of [[IC 1101]], one of the largest known galaxies
|-
| 24&nbsp;Zm = 2.5 million light-years<br>= 770 kiloparsecs
|distance to [[Andromeda Galaxy]]
|-
|3.26 million light-years<br>=30.8&nbsp;Zm = 1 megaparsec
|1 [[megaparsec]]
|-
|50&nbsp;Zm (1.6&nbsp;Mpc)
|diameter of [[Local Group]] of [[galaxy|galaxies]]
<div id="1E23"/>
|-
|[[100 zettametres|10<sup>23</sup>]]
|100 Zm
|300–600&nbsp;Zm = 10–20 [[megaparsecs]]
|distance to [[Virgo cluster]] of [[galaxy|galaxies]]
<div id="1E24"/>
|-
|rowspan=4|[[1 yottametre|10<sup>24</sup>]]
|rowspan=4|1 [[yottameter]] (Ym)
|200 million light-years<br>= 1.9&nbsp;Ym = 61 megaparsecs
|diameter of the [[Local Supercluster]] and the largest voids and filaments.
|-
|300 million light-years<br>= 2.8&nbsp;Ym = 100 megaparsecs
|[[Observable universe#End of Greatness|End of Greatness]]
|-
|550 million light-years<br>~170 megaparsecs ~5&nbsp;Ym
|diameter of the enormous [[Horologium Supercluster]] <ref>http://www.atlasoftheuniverse.com/superc/hor.html The Horologium Supercluster</ref>
<div id="1E25"/>
|-
|1 billion light-years<br>= 9.46&nbsp;Ym =306 megaparsecs
|diameter of the [[Pisces–Cetus Supercluster Complex]], the supercluster complex where we live.
|-
|rowspan=3|[[10 yottametres|10<sup>25</sup>]]
|rowspan=3|10 Ym
|1.37 billion light years<br>= 1.3{{E|25}}&nbsp;m = 13&nbsp;Ym
|Length of the [[Sloan Great Wall]], a giant wall of galaxies ([[galactic filament]]).<ref>[http://arxiv.org/abs/astro-ph/0310571 J. R. Gott III ''et al.'', ''Astrophys. J.'', '''624''', 463 (2005).] Figure 8 &ndash; "Logarithmic Maps of the Universe" &ndash; is available as a poster from [http://www.astro.princeton.edu/~mjuric/universe/ the homepage of Mario Juric.]</ref>
|-
|3.26 billion light years<br>=30.8&nbsp;Ym = 1 gigaparsec
|1 [[gigaparsec]]
|-
|4 billion light years<br>=37.84&nbsp;Ym
|Length of the [[Huge-LQG]], a group of 73 [[quasar]]s
|-
<div id="1E26/>
|-
|rowspan=3|[[100 yottametres|10<sup>26</sup>]]
|rowspan=3|100 Ym
|1{{E|10}} light-years<br>= 9.5{{E|25}}&nbsp;m = 95&nbsp;Ym
|estimated [[Distance measures (cosmology)|light travel distance]] to certain [[quasars]]<br>


Perhaps the most famous and effective ridiculing of circle squaring appears in [[Augustus de Morgan]]'s [[A Budget of Paradoxes]] published posthumously by his widow in 1872. Originally published as a series of articles in the ''Athenæum'', he was revising them for publication at the time of his death. Circle squaring was very popular in the nineteenth century, but hardly anyone indulges in it today and it is believed that de Morgan's work helped bring this about.<ref>{{citation|first=Underwood|last=Dudley|title=A Budget of Trisections|publisher=Springer-Verlag|year=1987|isbn=0-387-96568-8|pages=xi-xii}} Reprinted as ''The Trisectors''.</ref>
length of the [[Hercules-Corona Borealis Great Wall]], a colossal wall of galaxies, the largest and the most massive structure in the observable universe as of 2014
|-
|13.42 billion light years<br>=1.27{{E|26}}&nbsp;m = 127&nbsp;Ym
|Estimated [[Distance measures (cosmology)|light travel distance]] to [[UDFj-39546284]], the [[List of the most distant astronomical objects|most distant object]] ever observed
|-
|9.2{{E|10}} light years<br>= 8.7{{E|26}}&nbsp;m = 870&nbsp;Ym
|approx. diameter ([[comoving distance]]) of the [[visible universe]]<ref name="Burgess_and_Quevedo"/>
|-
<div id="1E27/>
|-
|[[1000 yottametres|10<sup>27</sup>]]
|1000 Ym
<!-- previous version of this entry reported diameter as radius -->
|130 billion light years<br>= 1.2{{E|27}}&nbsp;m = 1200&nbsp;Ym
|lower bound of the (possibly infinite) radius of the universe, if it is a [[3-sphere]], according to one estimate using the [[WMAP]] data at 95% confidence.<ref>http://arxiv.org/abs/astro-ph/0605709v2 How Many Universes Do There Need To Be?</ref> It equivalently implies that there are at minimum 21 [[particle horizon]]-sized volumes in the universe.
<!-- The article this entry is based on has been withdrawn by the author, specifically due to an error in the lower-bound estimate itself
|-
|~400 billion light years<br>= 3.8{{E|27}}&nbsp;m = 3800&nbsp;Ym
|Lower bound of the homogeneous universe derived from the [[Planck spacecraft]] <ref>[http://arxiv.org/abs/1304.1181 data]</ref>
-->
|-
|<math>10^{10^{115}}</math><ref name="exponents" group=note>10<sup>115</sup> is 1 followed by 115 zeroes, or a [[googol]] multiplied by a quadrillion. 10<sup>10<sup>115</sup></sup> is 1 followed by a quadrillion googol zeroes. 10<sup>10<sup>10<sup>122</sup></sup></sup>is 1 followed by 10<sup>10<sup>122</sup></sup> (a [[googolplex]]<sup>10 sextillion</SUP>) zeroes. These numbers are so vast that they are essentially the same in whatever units we could use to list them.</ref>
|<math>10^{10^{115}}</math> Ym
|<math>10^{10^{115}}</math> megaparsecs<br>= <math>10^{10^{115}}</math>&nbsp;m<br>= <math>10^{10^{115}}</math>Ym
|According to the laws of probability, the distance one must travel until one encounters a volume of space identical to our [[observable universe]] with conditions identical to our own.<ref name="TegmarkPUstaple">"Parallel universes. Not just a staple of science fiction, other universes are a direct implication of cosmological observations.", Tegmark M., Sci Am. 2003 May;288(5):40-51.</ref><ref>{{cite journal |author1=Max Tegmark |journal=In "Science and Ultimate Reality: from Quantum to Cosmos", honoring John Wheeler's 90th birthday. J. D. Barrow, P.C.W. Davies, & C.L. Harper eds. Cambridge University Press (2003) |title=Parallel Universes |year=2003 |arxiv=astro-ph/0302131|bibcode = 2003astro.ph..2131T }}</ref>
|-
|<math>10^{10^{10^{122}}}</math><ref name="exponents" group=note/>
|<math>10^{10^{10^{122}}}</math> Ym
|<math>10^{10^{10^{122}}}</math> Mpc<br>= <math>10^{10^{10^{122}}}</math>&nbsp;m<br>= <math>10^{10^{10^{122}}}</math>Ym
|size of universe after [[cosmological inflation]], implied by one resolution of the No-Boundary Proposal<ref>http://arxiv.org/abs/hep-th/0610199 "Susskind's Challenge to the Hartle-Hawking No-Boundary Proposal and Possible Resolutions "</ref>
|}


==Impossibility==
==See also==
The solution of the problem of squaring the circle by compass and straightedge demands construction of the number <math>\scriptstyle \sqrt{\pi}</math>, and the impossibility of this undertaking follows from the fact that pi is a [[transcendental number|transcendental]]
* [[List of examples of lengths]]
([[algebraic number|non-algebraic]] and therefore [[constructible number|non-constructible]]) number. If the problem of the quadrature of the circle is solved using only compass and straightedge, then an algebraic value of pi would be found, which is impossible.  [[Johann Heinrich Lambert]] conjectured that pi was transcendental in 1768 in the same paper in which he proved its irrationality, even before the existence of transcendental numbers was proven. It was not until 1882 that [[Ferdinand von Lindemann]] proved its transcendence.
* [[Metric_prefix#List_of_SI_prefixes|List of metric prefixes]]
 
* [[List of semiconductor scale examples]]
The transcendence of pi implies the impossibility of exactly "circling" the square, as well as of squaring the circle.
* [[Earth's location in the universe]]
 
* ''[[Powers of Ten (film)|Powers of Ten]]'', a 1968 short documentary film which depicts the relative scale of the Universe in factors of ten.
It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of pi, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations.
 
Bending the rules by allowing an infinite number of compass-and-straightedge operations or by performing the operations on certain [[non-Euclidean space]]s also makes squaring the circle possible. For example, although the circle cannot be squared in [[Euclidean space]], it can be in [[Gauss–Bolyai–Lobachevsky space]]. Indeed, even the preceding phrase is  overoptimistic.<ref>{{Cite journal| doi = 10.1007/BF03024895  | last = Jagy  | first = William C. | title = Squaring circles in the hyperbolic plane | journal = Mathematical Intelligencer  | volume = 17  | issue = 2  | pages = 31–36  | year = 1995  | url = http://zakuski.math.utsa.edu/~jagy/papers/Intelligencer_1995.pdf  | format = [[Portable Document Format|PDF]]  | postscript = <!--None--> }}</ref><ref>{{Cite book
  | last = Greenberg
  | first = Marvin Jay
  | title =  Euclidean and Non-Euclidean Geometries
  | publisher = W H Freeman
  | year = 2008
  | pages = 520–528
  | edition = Fourth
  | isbn =  0-7167-9948-0
  | postscript = <!--None--> }}</ref>  There are no squares as such in the hyperbolic plane, although there are regular quadrilaterals, meaning quadrilaterals with all sides congruent and all angles congruent (but these angles are strictly smaller than right angles).
There exist, in the hyperbolic plane, (countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area.
However, there is no method for starting with a regular quadrilateral and constructing the circle of equal area, and there is no method for starting with a circle and constructing a regular quadrilateral of equal area (even when the circle has small enough radius such that a regular quadrilateral of equal area exists).
 
==Modern approximative constructions==
Though squaring the circle is an impossible problem using only compass and straightedge, approximations to squaring the circle can be given by constructing lengths close to pi.
It takes only minimal knowledge of elementary geometry to convert any given rational approximation of pi into a corresponding compass-and-straightedge construction, but constructions made in this way tend to be very long-winded in comparison to the accuracy they achieve. After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding elegant approximations to squaring the circle, defined roughly and informally as constructions that are particularly simple among other imaginable constructions that give similar precision.
 
Among the modern approximate constructions was one by [[E. W. Hobson]] in 1913.<ref>{{Cite book|last=Hobson|first= Ernest William |year=1913|title=Squaring the Circle: A History of the Problem|publisher= Cambridge University Press}} Reprinted by Merchant Books in 2007.</ref> This was a fairly accurate construction which was based on constructing the approximate value of 3.14164079..., which is accurate to 4 decimals (i.e. it differs from pi by about {{Val|4.8|e=-5}}).
 
Indian mathematician [[Srinivasa Ramanujan]] in 1913, [[C. D. Olds]] in 1963, [[Martin Gardner]] in 1966, and [[Benjamin Bold]] in 1982 all gave geometric constructions for


:<math>\tfrac{355}{113} = 3.1415929203539823008\dots</math>
==Notes==
 
{{reflist|group=note}}
which is accurate to six decimal places of pi.
 
[[Image:Kochanski-1.svg|right|thumb|[[Adam Adamandy Kochański|Kochański's]] approximate construction]]
Srinivasa Ramanujan in 1914 gave a ruler-and-compass construction which was equivalent to taking the approximate value for pi to be
 
:<math>\left(9^2 + \frac{19^2}{22}\right)^{1/4} = \sqrt[4]{\frac{2143}{22}} = 3.1415926525826461252\dots</math>
 
giving a remarkable eight decimal places of pi.
 
In 1991, [[Robert Dixon (mathematician)|Robert Dixon]] gave constructions for
 
:<math>\frac{6}{5} (1 + \varphi)\text{ and }\sqrt{{40 \over 3} - 2 \sqrt{3}\  }</math>
 
([[Kochański's approximation]]), though these were only accurate to four decimal places of pi.
 
==Squaring or quadrature as integration==
The problem of finding the area under a curve, known as [[Integral|integration]] in [[calculus]], or [[numerical quadrature|quadrature]] in [[numerical analysis]], was known as ''squaring'' before the invention of calculus.  Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge.  For example [[Isaac Newton|Newton]] wrote to [[Henry Oldenberg|Oldenberg]] in 1676 "I believe M. Leibnitz will not dislike the Theorem towards the beginning of my letter pag. 4 for '''squaring Curve lines''' Geometrically" (emphasis added).<ref>{{cite book|url=http://books.google.com/?id=OVPJ6c9_kKgC&pg=PA259&vq=squaring&dq=newton+squaring-curves+date:0-1923|title=Correspondence of Sir Isaac Newton and Professor Cotes: Including letters of other eminent men|author1=Cotes|first1=Roger|year=1850}}</ref>  After Newton and [[Gottfried Leibniz|Leibniz]] invented calculus, they still referred to this integration problem as squaring a curve.
 
==Claims of circle squaring==
===Connection with the longitude problem===
The mathematical proof that the [[numerical integration|quadrature]] of the circle is impossible using only compass and straightedge has not proved to be a hindrance to the many people who have invested years in this problem anyway.  Having squared the circle is a famous [[crank (person)|crank]] assertion. (''See also'' [[pseudomathematics]].) In his old age, the English philosopher [[Thomas Hobbes]] convinced himself that he had succeeded in squaring the circle.
 
During the 18th and 19th century, the notion that the problem of squaring the circle was somehow related to the [[Longitude prize|longitude problem]] seems to have become prevalent among would-be circle squarers. Using "cyclometer" for circle-squarer,  [[Augustus de Morgan]] wrote in 1872:
 
<blockquote>
[[Jean Etienne Montucla|Montucla]] says, speaking of France, that he finds three notions prevalent among cyclometers: 1. That there is a large reward offered for success; 2. That the longitude problem depends on that success; 3. That the solution is the great end and object of geometry. The same three notions are equally prevalent among the same class in England. No reward has ever been offered by the government of either country.<ref>{{cite book|author=[[Augustus de Morgan]] |year=1872|title=[[A Budget of Paradoxes]]|page= 96}}</ref></blockquote>
 
Although from 1714 to 1828 the British government did indeed sponsor a £20,000 prize for finding a solution to the longitude problem, exactly why the connection was made to squaring the circle is not clear; especially since two non-geometric methods (the astronomical [[Lunar distance (navigation)|method of lunar distances]] and the mechanical [[marine chronometer|chronometer]]) had been found by the late 1760s. De Morgan goes on to say that "[t]he longitude problem in no way depends upon perfect solution; existing approximations are sufficient to a point of accuracy far beyond what can be wanted." In his book, de Morgan also mentions receiving many threatening letters from would-be circle squarers, accusing him of trying to "cheat them out of their prize".
 
===Other modern claims===
Even after it had been proved impossible, in 1894, amateur mathematician Edwin J. Goodwin claimed that he had developed a method to square the circle. The technique he developed did not accurately square the circle, and provided an incorrect area of the circle which essentially redefined pi as equal to 3.2. Goodwin then proposed the [[Indiana Pi Bill]] in the Indiana state legislature allowing the state to use his method in education without paying royalties to him. The bill passed with no objections in the state house, but the bill was tabled and never voted on in the Senate, amid increasing ridicule from the press.
 
==In literature==
The problem of squaring the circle has been mentioned by poets such as [[Dante]] and [[Alexander Pope]], with varied [[metaphor]]ical meanings.
 
The character [[Meton of Athens]] in the play [[The Birds (play)|The Birds]] by [[Aristophanes]] (first performed in 414 BC) mentions squaring the circle.<ref>{{citation
| last = Amati | first = Matthew
| issue = 3
| journal = [[The Classical Journal]]
| jstor = 10.5184/classicalj.105.3.213
| pages = 213–222
| title = Meton's star-city: Geometry and utopia in Aristophanes' ''Birds''
| volume = 105
| year = 2010}}.</ref>
 
Dante's ''Paradise'' canto XXXIII lines 133–135 contain the verses:
<poem style="margin-left: 2em">
As the geometer his mind applies
To square the circle, nor for all his wit
Finds the right formula, howe'er he tries</poem>
For Dante, squaring the circle represents a task beyond human comprehension, which he compares to his own inability to comprehend Paradise.<ref>{{citation
| last1 = Herzman | first1 = Ronald B.
| last2 = Towsley | first2 = Gary B.
| journal = Traditio
| jstor = 27831895
| pages = 95–125
| title = Squaring the circle: ''Paradiso'' 33 and the poetics of geometry
| volume = 49
| year = 1994}}.</ref>
 
By 1742, when [[Alexander Pope]] published the fourth book of his [[Dunciad]], attempts at circle-squaring had come to be seen as "wild and fruitless":<ref>{{citation
| last = Schepler | first = Herman C.
| journal = [[Mathematics Magazine]]
| jstor = 3029832
| mr = 0037596
| pages = 165–170, 216–228, 279–283
| title = The chronology of pi
| volume = 23
| year = 1950}}.</ref>
<poem style="margin-left: 2em">Mad Mathesis alone was unconfined,
Too mad for mere material chains to bind,
Now to pure space lifts her ecstatic stare,
Now, running round the circle, finds it square.</poem>
 
Similarly, the [[Gilbert and Sullivan]] comic opera ''[[Princess Ida]]'' features a song which satirically lists the impossible goals of the women's university run by the title character, such as finding [[perpetual motion]].  One of these goals is "And the circle – they will square it/Some fine day."<ref>{{citation
| last = Dolid | first = William A.
| issue = 2
| journal = The Shaw Review
| jstor = 40682600
| pages = 52–56
| title = Vivie Warren and the Tripos
| volume = 23
| year = 1980}}. Dolid contrasts Vivie Warren, a fictional female mathematics student in ''[[Mrs. Warren's Profession]]'' by [[George Bernard Shaw]], with the satire of college women presented by Gilbert and Sullivan. He writes that "Vivie naturally knew better than to try to square circles."</ref>
 
The [[sestina]], a poetic form first used in the 12th century by [[Arnaut Daniel]], has been said to square the circle in its use of a square number of lines (six stanzas of six lines each) with a circular scheme of six repeated words. {{harvtxt|Spanos|1978}} writes that this form invokes a symbolic meaning in which the circle stands for heaven and the square stands for the earth.<ref>{{citation|title=The Sestina: An Exploration of the Dynamics of Poetic Structure|first=Margaret|last=Spanos|journal=Speculum|volume=53|issue=3|year=1978|pages=545–557|jstor=2855144}}.</ref>
A similar metaphor was used in "Squaring The Circle", a 1908 short story by [[O. Henry]], about a long-running family feud. In the title of this story, the circle represents the natural world, while the square represents the city, the world of man.<ref>{{citation|title=Twentieth-century American literature|first=Harold|last=Bloom|authorlink=Harold Bloom|publisher=Chelsea House Publishers|year=1987|isbn=9780877548034|page=1848|quotation=Similarly, the story "Squaring the Circle" is permeated with the integrating image: nature is a circle, the city a square.}}</ref>
 
In [[James Joyce|James Joyce's]] novel ''[[Ulysses_(novel)|Ulysses]]'', Leopold Bloom dreams of becoming wealthy by squaring the circle, unaware that the quadrature of the circle had been proved impossible 22 years earlier and that the British government had never offered a reward for its solution.<ref>{{citation
| last = Pendrick | first = Gerard
| issue = 1
| journal = [[James Joyce Quarterly]]
| jstor = 25473619
| pages = 105–107
| title = Two notes on "Ulysses"
| volume = 32
| year = 1994}}.</ref>
 
==See also==
* The two other classical problems of antiquity were [[doubling the cube]] and [[trisecting the angle]], described in the [[compass and straightedge]] article. Unlike squaring the circle, these two problems can be solved by the slightly more powerful construction method of [[origami]], as described at [[mathematics of paper folding]].
* For a more modern related problem, see [[Tarski's circle-squaring problem]].
* The [[Indiana Pi Bill]], an 1897 attempt in the [[Indiana state legislature]] to dictate a solution to the problem by legislative fiat.
* [[Squircle]], a mathematical shape with properties between those of a square and those of a circle.
* [[Squared circle]], a [[professional wrestling]] ring.


==References==
==References==
{{Reflist|2}}
{{reflist|30em}}


==External links==
==External links==
{{Wikisource}}
* [http://www.vendian.org/howbig/ How Big Are Things?] displays orders of magnitude in successively larger rooms
* ''[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html Squaring the circle]'' at the [[MacTutor History of Mathematics archive]]
* [http://microcosm.web.cern.ch/Microcosm/P10/english/welcome.html Powers of Ten] Travel across the Universe. Altering perspective by changing scale by just a few powers of ten (interactive) {{dead link|date=May 2013}}
* ''[http://www.cut-the-knot.org/impossible/sq_circle.shtml Squaring the Circle]'' at [[cut-the-knot]]
* [http://www.shekpvar.net/~dna/Publications/Cosmos/cosmos.html Cosmos &ndash; an Illustrated Dimensional Journey from microcosmos to macrocosmos] &ndash; from Digital Nature Agency {{dead link|date=May 2013}}
* ''[http://mathworld.wolfram.com/CircleSquaring.html Circle Squaring]'' at [[MathWorld]], includes information on procedures based on various approximations of pi
* [http://htwins.net/scale2/ Scale of the universe]- interactive guide to length magnitudes
* "[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1207&bodyId=1357 Squaring the Circle]" at "[http://mathdl.maa.org/convergence/1/ Convergence]"
* [http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1203&bodyId=1593 The Quadrature of the Circle and Hippocrates' Lunes] at [http://mathdl.maa.org/convergence/1/ Convergence]
* [http://www.song-of-songs.net/Squaring_the_Circle.html ''How to Unroll a Circle''] Pi accurate to eight decimal places, using straightedge and compass.
* [http://www.gresham.ac.uk/event.asp?PageId=45&EventId=624 ''Squaring the Circle and Other Impossibilities''], lecture by [[Robin Wilson (mathematician)|Robin Wilson]], at [[Gresham College]], 16 January 2008 (available for download as text, audio or video file).
* {{cite web|last=Grime|first=James|title=Squaring the Circle|url=http://www.numberphile.com/videos/squaring_circle.html|work=Numberphile|publisher=[[Brady Haran]]}}


{{Greek mathematics}}
{{Orders of magnitude wide}}
{{Use dmy dates|date=September 2010}}
{{Units of length used in Astronomy}}


{{DEFAULTSORT:Squaring The Circle}}
[[Category:Length]]
[[Category:Pi]]
[[Category:Orders of magnitude|Length]]
[[Category:Euclidean plane geometry]]
[[Category:Orders of magnitude (length)|*]]
[[Category:Mathematical problems]]
[[Category:Lists by length]]
[[Category:History of geometry]]
[[Category:Compass and straightedge constructions]]

Revision as of 15:51, 10 August 2014

The following are examples of orders of magnitude for different lengths.

Section Range (m) Unit Example Items
≥ <
Planck area ？ 10−35 Quantum foam
Subatomic 10−35 10−15 am electron, quark, string
Atomic and cellular 10−15 10−12 fm atomic nucleus, proton, neutron
10−12 10−9 pm wavelength of gamma rays and X-rays, hydrogen atom
10−9 10−6 nm DNA helix, virus, wavelength of optical spectrum
Human scale 10−6 10−3 µm bacterium, fog water droplet, human hair[1]
10−3 100 mm mosquito, golf ball, football
100 103 m human being, football field, Eiffel Tower
103 106 km Mount Everest, length of Panama Canal, larger asteroid
Astronomical 106 109 Mm the Moon, Earth, one light-second
109 1012 Gm Sun, one light-minute, Earth's orbit
1012 1015 Tm orbits of outer planets, Solar System
1015 1018 Pm one light-year; distance to Proxima Centauri
1018 1021 Em galactic arm
1021 1024 Zm Milky Way, distance to Andromeda Galaxy
1024 101010122[note 1] Ym Huge-LQG, Hercules-Corona Borealis Great Wall, visible universe

Detailed list

To help compare different orders of magnitude, the following list describes various lengths between 1.6Template:E meters and 101010122meters.

Subatomic

Factor (m) Multiple Value Item
10−35 1 Planck Length 0.0000000000162 ym (1.62Template:E m) Planck length; typical scale of hypothetical loop quantum gravity or size of a hypothetical string and of branes; according to string theory lengths smaller than this do not make any physical sense.[2] Quantum foam is thought to exist at this level.
10−24 1 yoctometer (ym) 20 ym (2 × 10−23 meters) effective cross section radius of 1 MeV neutrinos[3]
10−21 1 zeptometer (zm) Preons, hypothetical particles proposed as subcomponents of quarks and leptons; the upper bound for the width of a cosmic string in string theory.
7 zm (7 × 10−21 meters) effective cross section radius of high energy neutrinos[4]
310 zm (3.10 × 10−19 meters) de Broglie wavelength of protons at the Large Hadron Collider (4 TeV as of 2012)
10−18 1 attometer (am) upper limit for the size of quarks and electrons
sensitivity of the LIGO detector for gravitational waves
upper bound of the typical size range for "fundamental strings"[2]
10−17 10 am range of the weak force

Atomic and cellular

Factor (m) Multiple Value Item
10−15 1 femtometer (fm) 1.5 fm size of an 11 MeV proton[5]
2.81794 fm classical electron radius[6]
scale of the atomic nucleus[2][7]
10−14 10 fm
10−13 100 fm
10−12 1 picometer (pm) ... longest wavelength of gamma rays
2.4 pm Compton wavelength of electron
5 pm wavelength of shortest X-rays
10−11 10 pm 25 pm radius of hydrogen atom
31 pm radius of helium atom
53 pm Bohr radius
10−10 100 pm 100 pm (0.1 nm) 1 Ångström (also covalent radius of sulfur atom[8])
154 pm (0.154 nm) length of a typical covalent bond (C–C).
500 pm (0.50 nm) width of protein α helix
10−9 1 nanometer (nm) 1 nm diameter of a carbon nanotube[9]
2.5 nm Smallest microprocessor transistor gate oxide thickness (as of Jan 2007)
6–10 nm thickness of cell membrane
10−8 10 nm 10 nm thickness of cell wall in gram-negative bacteriaPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
40 nm extreme ultraviolet wavelength
90 nm Human immunodeficiency virus (HIV) (generally, viruses range in size from 20 nm to 450 nm)
10−7 100 nm 121.6 nm wavelength of the Lyman-alpha line[10]
380–435 nm wavelength of violet light—see color and optical spectrum[11]
625–740 nm wavelength of red light[11]

Human scale

Factor (m) Multiple Value Item
10−6 1 micrometer (µm) 1 µm also called one micron
1–3 µm particle size that a surgical mask removes at 80–95% efficiencyPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
6-8 µm diameter of a red blood cell[12]
10−5 10 µm 10 µm typical size of a fog, mist or cloud water droplet. Chip 10 µm process in 1971.
12 µm width of acrylic fibre
25.4 µm 1/1000 inch, commonly referred to as one thou or one mil
10−4 100 µm 100 µm average width of a strand of human hair[1]
200 µm typical length of Paramecium caudatum, a ciliate protist
750 µm maximum diameter of Thiomargarita namibiensis, the largest bacterium ever discovered
10−3 1 millimeter (mm) 2.54 mm 1/10th inch; distance between pins in DIP (dual-inline-package) electronic components
5 mm length of average red ant
7.62 mm common military ammunition size
10−2 1 centimeter (cm) 1.5 cm length of a large mosquito
2.54 cm 1 inch
4.267 cm diameter of a golf ball
10−1 1 decimeter (dm) 10 cm wavelength of the highest UHF radio frequency, 3 GHz
30.48 cm 1 foot
91.44 cm 1 yard
100 1 meter 1 m wavelength of the lowest UHF and highest VHF radio frequency, 300 MHz
1.7 m (5 feet 7 inches) average height of a human
8.38 m The length of a London Bus (Routemaster)
101 1 decameter (dam) 10 m wavelength of the lowest VHF and highest shortwave radio frequency, 30 MHz
33 m length of longest blue whale measured, the largest animal[13]
93.47 m height of the Statue of Liberty (foundation of pedestal to torch)
102 1 hectometer (hm) 100 m wavelength of the lowest shortwave radio frequency and highest medium wave radio frequency, 3 MHz
137 m (147 m) height (present and original) of the Great Pyramid of Giza
979 m height of the Salto Angel, the world's highest free-falling waterfall (Venezuela)
103 1 kilometer (km) 1 km wavelength of the lowest medium wave radio frequency, 300 kHz
1609 m 1 international mile
1852 m 1 nautical mile
 
8848 m height of the highest mountain on earth, Mount Everest
104 10 km 10.911 km depth of deepest part of the ocean, Mariana Trench
13 km narrowest width of the Strait of Gibraltar, separating Europe and Africa
90 km width of the Bering Strait
105 100 km 111 km distance covered by one degree of latitude on Earth's surface
163 km length of the Suez Canal
974.6 km greatest diameter[14] of the dwarf planet[note 2] Ceres

Astronomical

Factor (m) Multiple Value Item
106 1,000 km = 1 megameter (Mm) 2,390 km diameter of dwarf planet Pluto, formerly the smallest planet category[note 2] of our solar system
3,480 km diameter of the Moon
5,200 km typical distance covered by the winner of the 24 Hours of Le Mans automobile endurance race
6,400 km length of the Great Wall of China
6,600 km approximate length of the two longest rivers, the Nile and the Amazon
7,821 km length of the Trans-Canada Highway
9,288 km length of the Trans-Siberian Railway, longest in the world
107 10,000 km 12,756 km equatorial diameter of the Earth
40,075 km length of the Earth's equator
108 100,000 km 142,984 km diameter of Jupiter
299,792.458 km distance travelled by light in one second
384,000 km = 384 Mm Moon's orbital distance from Earth
109 1 million km = 1 gigameter (Gm) 1,390,000 km = 1.39 Gm diameter of the Sun
4,200,000  km = 4.2 Gm greatest mileage ever recorded by a car (A 1966 Volvo P-1800S, still driving)
1010 10 million km 18 million km approximately one light-minute
1011 100 million km 150 million km = 150 Gm 1 astronomical unit (AU); mean distance between Earth and Sun
~ 900 Gm optical diameter of Betelgeuse (~600 × Sun)
1012 1000 million km = 1 terameter (Tm) 1.4 Template:E km orbital distance of Saturn from Sun
1.96 Template:E km estimated optical diameter of VY Canis Majoris (1420 × Sun)
2.3 Template:E km estimated optical diameter of NML Cygni (1650 × Sun)
2.37 Template:E km median point of the optical diameter of UY Scuti, as of 2014 the largest known star
5.9 Template:E km = 5.9 Tm orbital distance of Pluto from Sun
~ 7.5 Template:E km = 7.5 Tm outer boundary of the Kuiper belt, inner boundary of the Oort cloud (~ 50 AU)
1013 10 Tm diameter of our Solar System as a whole[2]
16.25Template:E km = 16.25 Tm distance of the Voyager 1 spacecraft from Sun (Template:As of), the farthest man-made object so far[15]
62.03Template:E km = 62.03 Tm estimated radius of the event horizon of the supermassive black hole in NGC 4889, the largest known black hole to date
1014 100 Tm 1.8Template:E km = 180 Tm size of the debris disk around the star 51 Pegasi [16]
1015 1 petameter (Pm) ~ 7.5 Template:E km = 7.5 Pm supposed outer boundary of the Oort cloud (~ 50,000 AU)
9.46Template:E km = 9.46 Pm
= 1 light year 		distance traveled by light in one year; at its current speed, Voyager 1 would need 17,500 years to travel this distance
1016 10 Pm 3.2616 light-years
(3.0857Template:E km = 30.857 Pm) 		1 parsec
4.22 light-years = 39.9 Pm distance to nearest star (Proxima Centauri)
4.37 light-years = 41.3 Pm as of March 2013, distance to nearest discovered extrasolar planet (Alpha Centauri Bb)
1017 100 Pm 20.4 light-years = 193 Pm as of October 2010, distance to nearest discovered extrasolar planet with potential to support life as we know it (Gliese 581 d)
65 light-years = 6.15Template:E m = 615 Pm approximate radius of humanity's radio bubble, caused by high-power TV broadcasts leaking through the atmosphere into outer space
1018 1 exameter (Em) 200 light-years = 1.9 Em distance to nearby solar twin (HIP 56948), a star with properties virtually identical to our Sun [17]
1019 10 Em 1,000 light-years = 9.46 Em or 9.46 × 1015 km average thickness of Milky Way Galaxy[18] (1000 to 3000 ly by 21 cm observations[19])
1020 100 Em 12,000 light-years = 113.5 Em or 1.135 × 1017 km thickness of Milky Way Galaxy's gaseous disk[20]
950 Em 100,000 light-years diameter of galactic disk of Milky Way Galaxy[2]
1021 1 zettameter (Zm)
50 kiloparsecs distance to SN 1987A, the most recent naked eye supernova
52 kiloparsecs = 1.62Template:E m = 1.62 Zm distance to the Large Magellanic Cloud (a dwarf galaxy orbiting the Milky Way)
54 kiloparsecs = 1.66 Zm distance to the Small Magellanic Cloud (another dwarf galaxy orbiting the Milky Way)
200 kiloparsecs = 6.15 Zm diameter of the low surface brightness disc halo of the giant spiral galaxy Malin 1
1022 10 Zm 13.25 Zm = 1.4 million light years
= 600 kiloparsecs 		radius of the diffuse stellar halo of IC 1101, one of the largest known galaxies
24 Zm = 2.5 million light-years
= 770 kiloparsecs 		distance to Andromeda Galaxy
3.26 million light-years
=30.8 Zm = 1 megaparsec 		1 megaparsec
50 Zm (1.6 Mpc) diameter of Local Group of galaxies
1023 100 Zm 300–600 Zm = 10–20 megaparsecs distance to Virgo cluster of galaxies
1024 1 yottameter (Ym) 200 million light-years
= 1.9 Ym = 61 megaparsecs 		diameter of the Local Supercluster and the largest voids and filaments.
300 million light-years
= 2.8 Ym = 100 megaparsecs 		End of Greatness
550 million light-years
~170 megaparsecs ~5 Ym 		diameter of the enormous Horologium Supercluster [21]
1 billion light-years
= 9.46 Ym =306 megaparsecs 		diameter of the Pisces–Cetus Supercluster Complex, the supercluster complex where we live.
1025 10 Ym 1.37 billion light years
= 1.3Template:E m = 13 Ym 		Length of the Sloan Great Wall, a giant wall of galaxies (galactic filament).[22]
3.26 billion light years
=30.8 Ym = 1 gigaparsec 		1 gigaparsec
4 billion light years
=37.84 Ym 		Length of the Huge-LQG, a group of 73 quasars
1026 100 Ym 1Template:E light-years
= 9.5Template:E m = 95 Ym 		estimated light travel distance to certain quasars

length of the Hercules-Corona Borealis Great Wall, a colossal wall of galaxies, the largest and the most massive structure in the observable universe as of 2014

13.42 billion light years
=1.27Template:E m = 127 Ym 		Estimated light travel distance to UDFj-39546284, the most distant object ever observed
9.2Template:E light years
= 8.7Template:E m = 870 Ym 		approx. diameter (comoving distance) of the visible universe[2]
1027 1000 Ym 130 billion light years
= 1.2Template:E m = 1200 Ym 		lower bound of the (possibly infinite) radius of the universe, if it is a 3-sphere, according to one estimate using the WMAP data at 95% confidence.[23] It equivalently implies that there are at minimum 21 particle horizon-sized volumes in the universe.
1010115[note 1] 1010115 Ym 1010115 megaparsecs
= 1010115 m
= 1010115Ym 		According to the laws of probability, the distance one must travel until one encounters a volume of space identical to our observable universe with conditions identical to our own.[24][25]
101010122[note 1] 101010122 Ym 101010122 Mpc
= 101010122 m
= 101010122Ym 		size of universe after cosmological inflation, implied by one resolution of the No-Boundary Proposal[26]

See also

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

Template:Orders of magnitude wide Template:New Testament manuscript infobox

Codex Augiensis, designated by Fp or 010 (in the Gregory-Aland numbering), α 1029 (von Soden) is a 9th-century diglot uncial manuscript of the Pauline Epistles in double parallel columns of Greek and Latin on the same page.[27]

Description

The codex contains 136 parchment leaves (Template:×), with some gaps in the Greek (Romans 1:1-3:19, 1 Corinthians 3:8-16, 6:7-14, Colossans 2:1-8, Philemon 21-25, Hebrews). Hebrews is given in Latin only.[28] It is written in two columns per page, 28 lines per page.[27]

Text

Textual character

The Greek text of this codex is a representative of the Western text-type. According to Kurt and Barbara Aland it agrees with the Byzantine standard text 43 times, and 11 times with the Byzantine when it has the same reading as the original text. It agrees 89 times with the original text against the Byzantine. It has 70 independent or distinctive readings. Alands placed it in Category II.[27]

Textual features

In Romans 12:11 it reads καιρω for κυριω, the reading of the manuscript is supported by Codex Claromontanus*, Codex Boernerianus 5 it d,g, Origenlat.[29]

In 1 Corinthians 2:4 the Latin text supports reading πειθοι σοφιας (plausible wisdom), as 35 and Codex Boernerianus (Latin text).[30]

In 1 Corinthians 7:5 it reads τη προσευχη (prayer) along with P11, P46, א*, A, B, C, D, G, P, Ψ, 6, 33, 81, 104, 181, 629, 630, 1739, 1877, 1881, 1962, it vg, cop, arm, eth. Other manuscripts read τη νηστεια και τη προσευχη (fasting and prayer) or τη προσευχη και νηστεια (prayer and fasting) – 330, 451, John of Damascus.[31][32]

The section 1 Cor 14:34-35 is placed after 1 Cor 14:40, like other manuscripts of the Western text-type (Claromontanus, Boernerianus, 88, itd, g, and some manuscripts of Vulgate).[33][34]

Relationship to Codex Boernerianus

The Greek text of both manuscripts is almost the same; the Latin text differs. Also lacunae omissions are paralleled to the sister manuscript Codex Boernerianus. According to Griesbach Augiensis was recopied from Boernerianus. According to Tischendorf two codices were recopied from the same manuscript. Scrivener enumerated 1982 differences between these two codices. Among textual scholars, there is a tendency to prefer Augiensis above Boernerianus. The codex is also similar to Codex Claromontanus, and again scholars favour the readings in Augiensis above those in Claromontanus.

History

Codex Augiensis is named after the monastery of Augia Dives in Lake Constance.[35] In 1718 Richard Bentley (1662–1742) was its owner. The Greek text of the codex was edited by Scrivener in 1859.[28] It was examined, described and collated by Tischendorf.[36] E. M. Thompson edited a facsimile.[37]

The codex today is located in the library of Trinity College (Cat. number: B. XVII. 1) in Cambridge.[27][38]

See also

Sportspersons Hyslop from Nicolet, usually spends time with pastimes for example martial arts, property developers condominium in singapore singapore and hot rods. Maintains a trip site and has lots to write about after touring Gulf of Porto: Calanche of Piana.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

External links

Template:Link GA

  1. ↑ 1.0 1.1 According to The Physics Factbook, the diameter of human hair ranges from 17 to 181 µm. Template:Cite web
  2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 Template:Cite news
  3. ↑ Template:Cite web (6.3 × 10−44 cm2, which gives an effective radius of about 2 × 10−23 m)
  4. ↑ Template:Cite web (area for 20 GeV about 10 × 10−42 m2 gives effective radius of about 2 × 10−21 m; for 250 GeV about 150 × 10−42 m2 gives effective radius of about 7 × 10−21 m)
  5. ↑ Template:Cite web (diameter of the Scattering Cross Section of an 11 MeV proton with a target proton)
  6. ↑ NIST. CODATA Value: classical electron radius. Retrieved 2009-02-10
  7. ↑ Template:Cite web
  8. ↑ Template:Cite web
  9. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  10. ↑ Cohn, J. University of California, Berkeley Lyman alpha systems and cosmology. Retrieved 2009-02-21
  11. ↑ 11.0 11.1 Color
  12. ↑ Template:Cite web
  13. ↑ Template:Cite web
  14. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  15. ↑ Spacecraft escaping the Solar System
  16. ↑ Twin Keck telescopes probe dual dust disks
  17. ↑ Template:Cite web
  18. ↑ Template:Cite web
  19. ↑ Template:Cite web
  20. ↑ Template:Cite web
  21. ↑ http://www.atlasoftheuniverse.com/superc/hor.html The Horologium Supercluster
  22. ↑ J. R. Gott III et al., Astrophys. J., 624, 463 (2005). Figure 8 – "Logarithmic Maps of the Universe" – is available as a poster from the homepage of Mario Juric.
  23. ↑ http://arxiv.org/abs/astro-ph/0605709v2 How Many Universes Do There Need To Be?
  24. ↑ "Parallel universes. Not just a staple of science fiction, other universes are a direct implication of cosmological observations.", Tegmark M., Sci Am. 2003 May;288(5):40-51.
  25. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  26. ↑ http://arxiv.org/abs/hep-th/0610199 "Susskind's Challenge to the Hartle-Hawking No-Boundary Proposal and Possible Resolutions "
  27. ↑ 27.0 27.1 27.2 27.3 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  28. ↑ 28.0 28.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  29. ↑ UBS3, p. 564.
  30. ↑ UBS3, p. 581.
  31. ↑ NA26, p. 450.
  32. ↑ UBS3, p. 591.
  33. ↑ NA26, p. 466.
  34. ↑ Bruce M. Metzger, A Textual Commentary on the Greek New Testament (Deutsche Bibelgesellschaft: Stuttgart, 2001), pp. 499-500.
  35. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  36. ↑ K. Tischendorf, Anecdota sacra et profana ex oriente et occidente allata sive notitia, Lipsiae 1861, pp. 209-216.
  37. ↑ Facsimiles of Manuscripts and Inscriptions, ed. E. A. Bond, E. M. Thompson and others I (London, 1873-1883), 127.
  38. ↑ Template:Cite web


Cite error: <ref> tags exist for a group named "note", but no corresponding <references group="note"/> tag was found

Retrieved from "https://en.formulasearchengine.com/index.php?title=Main_Page&oldid=38673"