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	Euclid - Crystalinks
		Euclid quickly called to his slave to give the boy a coin because "he must make gain out of what he learns." Another story relates that Ptolemy asked the mathematician if there was some easier way to learn geometry than by learning all the theorems.
		Euclid was unable to prove this statement and needing it for his proofs, so he assumed it as true.
		Euclid's Phaenomena is a tract on sphaeric, the study of spherical geometry for the purpose of explaining planetary motions.
	www.crystalinks.com /euclid.html (1856 words)

	What's Special About This Number?
		is the number of planar partitions of 10.
		is the number of planar partitions of 11.
		is the number of planar partitions of 12.
	www.stetson.edu /~efriedma/numbers.html (7292 words)

	Euclid
		Euclid is one of the world's most famous mathematicians, yet very little is known of his life, except that he taught at Ptolemy’s university at Alexandria, Egypt.
		Euclid's Elements are remarkable for the clarity with which the theorems and problems are selected and ordered.
		Euclid is not known to have made any original discoveries, and the Elements is based on the work of the people before him, like Exodus, Thales, Hippocrates, and Pythagoras.
	library.thinkquest.org /4116/History/euclid.htm (292 words)

	Euclid's Elements Summary
		Euclid's Elements (Greek: Στοιχεῖα) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC.
		Euclid based his work in Book I on 23 definitions, such as point, line and surface, five postulates and five "common notions" (both of which are today called axioms).
		Euclid probably chose to describe results in number theory in terms of geometry because he couldn't develop a constructible approach to arithmetic.
	www.bookrags.com /Euclid's_Elements (3047 words)

	Prime number - ExampleProblems.com
		A natural number that is greater than one and is not a prime is called a composite number.
		Prime numbers are of fundamental importance in number theory.
		The prime number theorem says that the proportion of primes less than x is asymptotic to 1/ln x (in other words, as x gets very large, the likelihood that a number less than x is prime is inversely proportional to the number of digits in x).
	www.exampleproblems.com /wiki/index.php/Prime_number (3311 words)

	EUCLID, The Elements
		Euclid is known to almost every high school student as the author of The Elements, the long studied text on geometry and number theory.
		Euclid concerns himself in several other propositions of Book VIII with determining the conditions for inserting mean proportional numbers between given numbers of various types.
		If as many numbers as we please are in continued proportion, and there is subtracted from the second and the last numbers equal to the first, then, as the excess of the second is to the first, so will the excess of the last be to all those before it.
	www.math.tamu.edu /~dallen/history/euclid/euclid.html (2784 words)

	The origins of proof
		Euclid was born about 365 BC in Alexandria, Egypt, and died about 300 BC.
		Euclid's definitions are neither true nor false: they simply act as a kind of dictionary, explaining what is meant by the various terms he will be using.
		Euclid goes on in his Elements to present various geometric propositions, and shows them to be true using deductive inference within his axiomatic system.
	www.plus.maths.org.uk /issue7/features/proof1/index.html#euclid (2379 words)

	Types
		Integers are the numbers..., -2, -1, 0, 1, 2,...
		Rational numbers are numbers which can be expressed as the ratio of two integers.
		Euclid supports simple algebraic numbers of the form M * N ^ (1 / R), where M is rational and N and R are integers.
	kevingong.com /Euclid/Types.html (403 words)

	Euclid's Proof of the Infinitude of Primes (c. 300 BC)
		Euclid's Proof of the Infinitude of Primes (c.
		Euclid may have been the first to give a proof that there are infinitely many primes.
		Finally, Euclid sometimes wrote his "proofs" in a style which would be unacceptable today--giving an example rather than handling the general case.
	primes.utm.edu /notes/proofs/infinite/euclids.html (545 words)

	EUCLID (Site not responding. Last check: 2007-11-04)
		Euclid wrote the classical book 'The Elements', a collection of geometrical theorems which became a standard work for over 2000 years.
		Euclid lived in Alexandria where he founded the first school of mathematics.
		Euclid proved the number of primes, numbers which are divisible only by themselves and one, to be infinite.
	www.hyperhistory.com /online_n2/people_n2/persons2_n2/euclid.html (74 words)

	References for Euclid
		H L L Busard, The Latin translation of the Arabic version of Euclid's 'Elements' commonly ascribed to Gerard of Cremona (Leiden, 1984).
		A C Bowen, Euclid's 'Sectio canonis' and the history of Pythagoreanism, in Science and philosophy in classical Greece (New York, 1991), 167-187.
		H Guggenheimer, The axioms of betweenness in Euclid, Dialectica 31 (1-2) (1977), 187-192.
	www-groups.dcs.st-and.ac.uk /~history/References/Euclid.html (1349 words)

	10.8. Euclid (330?-275? B.C.)
		Euclid is one of the most influential and best read mathematician of all time.
		Archimedes in the invention of integral calculus, and the proof that the set of all prime numbers is infinite.
		In his time, many of his peers attacked him for being too thorough and including self-evident proofs, such as one side of a triangle cannot be longer than the sum of the other two sides.
	web01.shu.edu /projects/reals/history/euclid.html (805 words)

	Euclid's Algorithm
		The generalization of the Corollary to an arbitrary field is known as Bézout's identity or Bézout's Lemma after the French mathematician Éttiene Bézout (1730-1783), so it often happens that the result stated in the Corollary is also often referred to as Bézout's identity or Bézout's Lemma.
		If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers.
		Since, by definition, a number is composite if it has factors other than 1 and itself, and these factors are bound to be smaller than the number, we can keep extracting the factors until only prime factors remain.
	www.cut-the-knot.org /blue/Euclid.shtml (656 words)

	Euclid's Elements - Wikipedia, the free encyclopedia
		Euclid's Elements (Greek: Στοιχεῖα) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC.
		Euclid's Book 1 begins with 23 definitions — such as point, line, and surface — followed by five postulates and five "common notions" (both of which are today called axioms).
		In the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points.
	en.wikipedia.org /wiki/Euclid's_Elements (1977 words)

	HMCo College Algebra Exercises, Chapter 1 (Site not responding. Last check: 2007-11-04)
		Thus Euclid reasoned that the set of prime numbers cannot be finite.
		Some people would have conjectured that every number of this type would be a prime number.
		Now we have a contradiction, because by the fundamental theorem of arithmetic, it is impossible for a number to have an odd number of prime factors and an even number of prime factors.
	college.hmco.com /mathematics/aufmann/collalg/chapter1.html (625 words)

	Euclid Theorem:
		Since these are all possible prime numbers, M is not divisible by any prime number, and therefore M is not divisible by any number.
		That means that M is also a prime number.
		But clearly M > N, which is impossible, because N was supposed to be the largest possible prime number.
	web01.shu.edu /projects/reals/logic/proofs/euclidth.html (172 words)

	Read This: Euclid: The Creation of Mathematics
		Interspersed among the sections dealing with the contents of the Elements are sections which use Euclid's text as a starting-point to investigate "the origins of mathematics": the axiomatic method, the role of definitions, the importance of generalization, the nature of infinity, and incommensurability.
		Of course, it is a purely factual question whether Euclid's "magnitudes" have the structure of a group (Weil stated specifically that he was not attributing the abstract concept of a group to Euclid); but, for Dauben, such an observation is not "history".
		A well-known topic in Euclid is that of a "construction" with straightedge and compasses.
	www.maa.org /reviews/artmann.html (2231 words)

	The golden number: nature seems to have a sense of proportion Natural History - Find Articles
		Turn the crank, and the number that solves the equation for x is equal to the never-ending, never-repeating number 1.6180339887..., commonly denoted by the Greek letter phi, or [phi].
		The number phi would have remained in the relative obscurity of pure mathematics were it not for its propensity to pop up where least expected.
		The number of clockwise spirals and the number of counterclockwise spirals vary, depending on the size of the sunflower.
	www.findarticles.com /p/articles/mi_m1134/is_2_112/ai_98254967 (809 words)

	Prime number - Wikipedia, the free encyclopedia
		In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors.
		The number 6 would be built with a side-by-side pair of 3's (with each 3 being built as a row of three unit squares).
		The number 8 would be a cube built with two 2's stacked on top of another pair of 2's (with each 2 being built as a row of two unit cubes).
	en.wikipedia.org /wiki/Prime_number (4806 words)

	Euclid's Elements, Book V, Proposition 8 (Site not responding. Last check: 2007-11-04)
		We can also have variables for numbers, instead of having to choose a specific number as Euclid does when he takes N to be 4D.
		Euclid carefully proved distributivity of multiplication by numbers over addition of magnitudes in V.1, which is used in this proof.
		In order to be as correct as Euclid, we should verify the rules of algebra and be aware when we use them.
	aleph0.clarku.edu /~djoyce/java/elements/bookV/propV8.html (1161 words)

	Amazon.com: Euclid - The Creation of Mathematics: Books: Benno Artmann,B. Artmann (Site not responding. Last check: 2007-11-04)
		The material in Euclid's Elements may be divided into four categories of very different degrees of interest for modern readers.
		Euclid's construction draws on all previous books, in accordance with his aim to hide his masterplan and unveil it in a flash of brilliance just as we though he was getting lost in a mass of technicalities.
		Euclid brings this up in connection with the marvellous constructions of the regular polyhedra in book XIII --- the culmination of the entire Elements.
	www.amazon.com /Euclid-Creation-Mathematics-Benno-Artmann/dp/0387984232 (2241 words)

	The Euclidean Algorithm
		For example, data encryption is based on theories, conjectures, and algorithms that arise from number theory; it is used every time you send your credit card number over the internet.
		Much of the utility of number theory is derived from the fact that there is a simple and fast algorithm to find the greatest common divisor known as the Euclidean Algorithm.
		In general, for many large numbers, the Euclidean Algorithm is conjectured to be a million (or more) times faster than any algorithm that tries to find all the factors of each number individually.
	www.rpi.edu /~mitchj/math1900/topics/euclideanalgo (478 words)

	EHS Complex Homepage
		Euclid High School is now six small schools.
		The Euclid tradition of sustained excellence was founded on the premises of continuous improvement and life-long learning.
		It is our vision for the future to continue this quest as we, the Euclid Academy of the Arts, pursue excellence in academics, visual arts and performing arts.
	www.euclid.k12.oh.us /HS/index.html (415 words)

	Central Middle School
		When calling, speak clearly and state your child's first name, last name, reason for absence, who you are, and a contact number.
		It is the mission of Central Middle School to provide a safe environment, which nurtures disciplined independent learners and fosters a cooperative effort among staff, parents, and the community to develop high achieving, productive citizens.
		Our mission is to provide students with the opportunity to acquire the technological knowledge, skills and experiences necessary to function successfully in an information/communication age society and to provide teachers with the knowledge, experience and tools to successfully integrate technology into the teaching/learning curriculum.
	www.euclid.k12.oh.us /CE (293 words)

	An Index to Hardy & Wright's The Theory of Numbers
		Hardy and Wright's The Theory of Numbers was published in 1938 and is now in its fifth edition (1979).
		The authors admitted that there were large gaps in their book and that the topics were presented with very little depth.
		It has always seemed, to us, that this had to be an oversight on the part of Hardy and Wright or their publishers.
	primes.utm.edu /notes/hw_index.html (290 words)

	EUCLID'S PROOF (Site not responding. Last check: 2007-11-04)
		Suppose there are only a finite number of primes, say n primes.
		This is where it gets exciting: Euclid asked himself to consider the interesting number which we will call 'a' for brevity.
		Now 'a' is either a new prime or it's the product of prime numbers none of which are in the list p1,p2,...,pn.
	www-personal.umich.edu /~gomez/euclid.html (159 words)

	Getopt::Euclid - Executable Uniform Command-Line Interface Descriptions - search.cpan.org
		=for Euclid: h.type: integer >; 0 w.type: number <= 100
		=for Euclid: h.type: integer, h > 0 && h < 100 h.type.error: <h> must be between 0 and 100 (not h) w.type: number, Math::is_prime(w)
		You can also specify a default value for any placeholders that aren't given values on the command-line (either because their argument isn't provided at all, or because the placeholder is optional within the argument).
	search.cpan.org /perldoc?Getopt::Euclid (2430 words)

	Index to Hardy and Wright (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-04)
		Higher-quality versions of this index are available either as a dvi file or as a PostScript file.
		There, the authors write that their own personal interests dictated the material to be included and chose topics that they considered "congenial".
		The page numbers are the same in both the softcover and hardcover editions.
	www.mathpropress.com.cob-web.org:8888 /HardyAndWright.html (288 words)

	Euclid Public Library
		Euclid Municipal Court - 555 E. 222nd St., 44123
		Euclid Central Middle School, 20701 Euclid Ave., 44117
		Euclid Cooperative Preschool 21000 Lake Shore blvd. 44123
	www.euclid.lib.oh.us /public/community.asp (391 words)

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