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	Euclid's Elements, Book X
		The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number will also have their sides commensurable in length.
		To find two medial straight lines commensurable in square only, containing a rational rectangle, and such that the square on the greater is greater than the square on the less by the square on a straight line commensurable in length with the greater.
		To find two medial straight lines commensurable in square only, containing a medial rectangle, and such that the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater.
	www.claymath.org /library/historical/euclid/book10.html (2698 words)

	PlanetMath: commensurable numbers
		The commensurability is an equivalence relation in the set
		The equivalence classes of the commensurability are of the form
		This is version 6 of commensurable numbers, born on 2008-07-08, modified 2008-07-09.
	planetmath.org /encyclopedia/Incommensurable.html (162 words)

	Euclid's Elements, Book X, Definitions I
		In particular X.5 and X.6 state that two magnitudes are commensurable if and only if their ratio is the ratio of a number to a number.
		Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.
		They are commensurable in square since the square on B is twice the square on A, by I.47.
	aleph0.clarku.edu /~djoyce/java/elements/bookX/defX.I.html (504 words)

	Chapter Commendation <i>to</i> Commission of C by Webster's Dictionary (1913 Edition)
		Having a common measure; commensurable; reducible to a common measure; as, commensurate quantities.
		In a commensurate manner; so as to be equal or proportionate; adequately.
		The act of commensurating; the state of being commensurate.
	www.bibliomania.com /2/3/257/1194/22233/2.html (256 words)

	incommensurables
		Two quantities A and B are commensurable when there is a quantity Q such that Q is contained in A and B respectively n and m times.
		For example, a foot and a yard are commensurable because there is a quantity, the inch, which is contained in them respectively 12 and 36 times.
		Suppose AC and AB are commensurable; let a/b be their ratio expressed in the smallest possible numbers.
	www.siue.edu /~evailat/incommensurables.html (559 words)

	Aristotle looks at motion.
		Now if they are all commensurable and if two things to have the same velocity must accomplish an equal motion in an equal time, then we may have a circumference equal to a straight line, or, of course, the one may be greater or less than the other.
		Moreover it does not as a matter of fact make any difference to the argument to say that the one motion must inevitably be quicker or slower than the other: for then the circumference can be greater or less than the straight line; and if so it is possible for the two to be equal.
		Thus horse and dog are so commensurable that we may say which is the whiter, since that which primarily contains the whiteness is the same in both, viz.
	www.humboldt.edu /~mef2/Aristo.htm (2572 words)

	Euclid's Elements, Book X
		The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number also have their sides commensurable in length.
		To find two rational straight lines commensurable in square only such that the square on the greater is greater than the square on the less by the square on a straight line incommensurable in length with the greater.
		But if neither is commensurable in length with the rational straight line set out, and the square on the whole is greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a third apotome.
	aleph0.clarku.edu /~djoyce/java/elements/bookX/bookX.html (3490 words)

	Merriam-Webster Online
		"Commensurable" means "having a common measure" or "corresponding in size, extent, amount, or degree." Its antonym "incommensurable" generally refers to things that are unlike and incompatible, sharing no common ground (as in the "incommensurable theories" of the example sentence), or to things that are very disproportionate, often to the point of defying comparison ("incommensurable crimes").
		Both words entered English in the 1500s and were originally used (as they still can be) for numbers that have or don't have a common divisor.
		They came to English by way of Middle French and Late Latin, ultimately deriving from the Latin noun "mensura," meaning "measure." "Mensura" is also an ancestor of "commensurate" (meaning "coextensive" or "proportionate") and "incommensurate" ("disproportionate" or "insufficient"), which overlap in meaning with "commensurable" and "incommensurable" but are not exact synonyms.
	leopard.eb.com /cgi-bin/mwwodarch.pl?Jun.01 (172 words)

	Henri Poincare: Science and Hypothesis: Chapter 2: Mathematical Magnitude and Experiment (Site not responding. Last check: )
		Commensurable numbers may be divided into classes in an infinite number of ways, subject to the condition that any number whatever of the first class is greater than any number of the second.
		From Kronecker's point of view, the incommensurable number 2 is nothing but the symbol of this particular method of division of commensurable numbers; and to each mode of repartition corresponds in this way a number, commensurable or not, which serves as a symbol.
		It is clear, in fact, that if the points whose co-ordinates are commensurable were alone regarded as real, the in-circle of a square and the diagonal of the square would not intersect, since the co-ordinates of the point of intersection are incommensurable.
	spartan.ac.brocku.ca /~lward/Poincare/Poincare_1905_03.html (3684 words)

	COMMENSURABLE - Definition
		[adj] able to be measured by a common standard; "hours and minutes are commensurable"
		{Commensurable numbers} or {quantities} (Math.), those that can be exactly expressed by some common unit; thus a foot and yard are commensurable, since both can be expressed in terms of an inch, one being 12 inches, the other 36 inches.
		{Numbers}, or {Quantities}, {commensurable in power}, those whose squares are commensurable.
	www.hyperdictionary.com /dictionary/commensurable (82 words)

	Golden Ratio is made from irrational & rational numbers. Great Pyramid has Golden Proportions inside & out. No ...
		Commensurables are naturally finite numbers that are also called rational or real or exact or absolute numbers
		Another way of saying 'commensurable' is 'exactly proportional.' When you say 'rational,' you are also saying (and you should know) that the ratio deals with two whole numbers.
		Saying 'commensurable' is then superior to saying 'rational' because 'commensurable' deals with an exact measurement (using a standard) or exact proportion (using two things or whole numbers).
	www.hyperflight.com /golden_numbers-proportion.htm (8512 words)

	numbers - definition by dict.die.net
		Commeasurable.] Having a common measure; capable of being exactly measured by the same number, quantity, or measure.
		Commensurable numbers or quantities (Math.), those that can be exactly expressed by some common unit; thus a foot and yard are commensurable, since both can be expressed in terms of an inch, one being 12 inches, the other 36 inches.
		Numbers, or Quantities, commensurable in power, those whose squares are commensurable.
	dict.die.net /numbers (287 words)

	Commensurability effects on the spectra of integrable systems
		The asymptotic constant is determined for all sets of aspect ratios for the AHO and is estimated semi-analytically for several sets of aspect ratios for the cuboid.
		The crossover from incommensurable to commensurable behaviour is studied by following a sequence of rational aspect ratios which approaches an irrational limit.
		The distribution of energy level spacings in the commensurable regime is determined for the AHO and a sequence of cuboids.
	www.iop.org /EJ/abstract/0305-4470/24/18/017 (357 words)

	Commensurability (mathematics) - Wikipedia, the free encyclopedia
		A time measured in weeks and a time measured in minutes are commensurable because a week is a constant number of minutes (10080), so that one can convert between the two units by multiplying or dividing by 10080.
		This gives rise to a general notion of commensurable subgroups: two subgroups A and B of a group are commensurable when their intersection has finite index in each of them.
		Sometimes in fact this relation is called commensurate, and to be commensurable requires only to be conjugate to a commensurate subgroup.
	en.wikipedia.org /wiki/Commensurability_(mathematics) (431 words)

	Course-History of Mathematics;Professor-Edwards;Lecture1-Lesson14;Format-text
		What are commensurable segments, well if you have two segments ab and cd they are said to be commensurable if you can find a smaller segment ef that goes evenly into both of these segments.
		This is equivalent to saying that two segments are commensurable if their ratio is a fraction taking what we just had for ab and cd forming a fraction, canceling out the length of ef we see that the ratio of the measures of those two segments would be the fraction p divided by q.
		Since our assumption that the diagonal and the side of a square are commensurable led to a contradiction it must be the case that they are not commensurable.
	blog.wku.edu /podcasts/Edwards_409_DocPart15.php (876 words)

	Journal of Theoretics - Nonprofit peer-reviewed Journal of scientifically credible theories from all disciplines. ...
		Among others, the most common characteristic of the performed experiments lies in the fact that the “source” masses of the measured gravitational field are much larger then the “test” masses.
		About six years ago our Hungarian research group started an extended study on the problem of what kind of technical solution would be the most suitable for the experimental study of gravity between equal, or near equal masses.
		We initially concluded that the torsion balance of gravity measurement was not a proper method for the investigation of gravity between commensurable masses.
	www.journaloftheoretics.com /Articles/3-6/Grav-pub.htm (2010 words)

	A GEOMETRIC PROOF THAT THE SQUARE ROOT OF 2 IS IRRATIONAL
		Thus a line segment of length 3/8 and a line segment of length 5/6 were commensurable since they were both integer multiples of a line segment of length 1/24.
		They assumed all pairs of line segments were commensurable (which is the same as assuming all numbers are rational).
		Then the side and diagonal of this square are commensurable with AB and AC as well.
	web.aurora.edu /~bdillon/square2.htm (721 words)

	Uncategorized « Low Dimensional Topology
		Two groups are called commensurable if one has a finite index subgroup that is isomorphic to a finite index subgroup of the other.
		Thus if two 3-manifolds are commensurable then their fundamental groups are commensurable.
		Walsh and Reid prove their result by showing that in the commensurability class for a knot complement, there is a unique minimal element (i.e.
	ldtopology.wordpress.com /category/uncategorized (1821 words)

	Untitled Document (Site not responding. Last check: )
		The term commensurability is widely used in astronomy and its use contradicts to the mathematical concept of commensurability (it was pointed out by V.B. Braginsky).
		we offer to widen the concept of commensurability for the case of unequal values accepting that the values concerned are commensurable as integers if their ratio is integer or the values concerned are rationally commensurable if their ratio is expressed through the natural numbers.
		To avoid the mistakes the maximum value for discrete commensurability is accepted to be 39.
	www.friends-partners.org /partners/russeds/unknown/astrochem/e3_2.htm (374 words)

	commensurate - Definitions from Dictionary.com
		corresponding in size or degree or extent; "pay should be commensurate with the time worked" [ant: incommensurate]
		Commeasurable.] Having a common measure; capable of being exactly measured by the same number, quantity, or measure.
		commensurate was Word of the Day on June 4, 2001.
	dictionary.reference.com /browse/commensurate (253 words)

	Physics Book 7 Chapter 4 (Site not responding. Last check: )
		A difficulty may be raised as to whether every motion is commensurable with every other or not.
		But these are not commensurable: and so the corresponding motions are not commensurable either.
		Similarly in the case of motion: two things are of the same velocity if they occupy an equal time in accomplishing a certain equal amount of motion.
	www.aristoteles-heute.de /SeinAlsGanzesBewegt/Physics/Physics0704.htm (1569 words)

	Real numbers 1
		Theodorus was writing out for us something about roots, such as the sides of squares whose area was 3 or 5 units, showing that the sides are incommensurable with the unit: he took the examples up to 17, but there for some reason he stopped.
		We suppose that the discovery that √2 was not commensurable with 1 came earlier which is why Theodorus started with √3.
		Definition X.1 Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.
	www-history.mcs.st-andrews.ac.uk /HistTopics/Real_numbers_1.html (2733 words)

	FIZIKA SARKADI DEZSÕ
		No report on the determination of G for interaction between commensurable masses can be found so far.
		Keywords: experimental gravity, dynamical gravity, gravity of commensurable masses, enforced resonance method, extension of the Newtonian Law of Gravity.
		Abstract – According to generally accepted physical model, the synthesis of the heavy elements may happen in the supernova explosions at a very high temperature.
	www.geocities.com /fhunman/mgrav/mgrav.html (394 words)

	The Harmony of the World
		Kepler devised a system for classifying both commensurable and incommensurable magnitudes into a system of "degrees of knowability," using the circle as a relative unit of measure.
		Magnitudes of the second degree of knowability are commensurable with the circle's diameter.
		Magnitudes of the fourth degree are not commensurable with the diagonal as squares but are commensurable as rectangles.
	www.keplersdiscovery.com /Harmonies.html (1513 words)

	Springer Online Reference Works
		One of the principal problems in the theory of discrete subgroups of Lie groups is the classification of such subgroups up to commensurability.
		If one of two commensurable subgroups of a locally compact topological group is a discrete subgroup (or a lattice, or a uniform discrete subgroup), so is the other.
		Up to the middle of the 20th century one basically studied individual classes of discrete subgroups of Lie groups occurring in arithmetic, function theory and physics.
	eom.springer.de /d/d033150.htm (1915 words)

	commensurable - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "commensurable" is defined.
		commensurable : The American Heritage® Dictionary of the English Language [home, info]
		Phrases that include commensurable: commensurable in power, commensurable numbers
	www.onelook.com /?w=commensurable (170 words)

	Math Seminars: Alan Reid (Site not responding. Last check: )
		In previous work we showed that arithmetic hyperbolic 2-manifolds that are isospectral are commensurable.
		We had previously shown that if arithmetic hyperbolic 3-manifolds are complex iso-length spectral they are commensurable.
		What we will actually prove here is that arithmetic hyperbolic 3-manifolds that are iso-length spectral are commensurable.
	www.math.ias.edu /abstract.php?event=10402 (76 words)

	Union College Math Department (Thesis/2003-04): Thesis Topics - J. Barbanel
		Ancient Greek mathematicians are credited with the discovery of irrational numbers, but their approach and presentation are very different from anything that you have probably seen before.
		It focussed on the notion of commensurable line segments: two line segments are commensurable if and only if each is a (whole number) multiple of some common line segment.
		The Greeks showed that there are line segments that are not commensurable.
	www.math.union.edu /courses/thesis/2003-04/barbanel1.html (274 words)

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