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- ...of areas and volumes, including elements of the theory of limits. Euclid's Elements constitute a typical deductive system, containing the basic propositions of ...and 5) the whole is greater than any of its parts (in some editions of the Elements there are four additional axioms).9 KB (1,351 words) - 20:43, 26 November 2016
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- The set of prime numbers is infinite (Euclid's Elements, Book IX, Prop. 20). The [[Chebyshev theorems on prime numbers|Chebyshev th823 bytes (139 words) - 11:52, 14 February 2020
- ''Euclid's axiom of parallelism'' ...ersect $AA_1$ and lies in the plane containing $P$ and $AA_1$. In Euclid's Elements the fifth postulate is given in the following equivalent form: "If a strai4 KB (682 words) - 17:25, 24 April 2016
- ...measure of two intervals. It was described in geometrical form in Euclid's Elements (3rd century B.C.).2 KB (351 words) - 20:40, 16 November 2023
- I) Quantity in the sense of Euclid's Elements is equivalent to what is presently called positive scalar. It is a generali and 8) is the definition of homogeneous magnitudes (see Euclid's Elements, Book V, Definition 4).7 KB (1,159 words) - 08:08, 6 June 2020
- ...of areas and volumes, including elements of the theory of limits. Euclid's Elements constitute a typical deductive system, containing the basic propositions of ...and 5) the whole is greater than any of its parts (in some editions of the Elements there are four additional axioms).9 KB (1,351 words) - 20:43, 26 November 2016
- ...matically (though not sufficiently rigorous) in the [[Elements-of-Euclid|''Elements'' of Euclid]]. The space of Euclidean geometry is usually described as a se3 KB (380 words) - 20:59, 25 October 2014
- 2 KB (281 words) - 14:01, 12 November 2023
- ...ty. Its first appearance in the extant classical literature is in Euclid's Elements, Book II, 11 (3th century B.C.).2 KB (325 words) - 21:24, 3 April 2015
- ...ce on the development of mathematics of the problem of the independence of Euclid's [[Fifth postulate|fifth postulate]] in the axiom system for geometry. <table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Novikov, "Elements of mathematical logic" , Oliver & Boyd and Addison-Wesley (1964) (Tra3 KB (478 words) - 17:19, 7 February 2011
- ...d throughout Book 12 of Euclid's Elements as the principal deductive tool. Euclid's chain of reasoning may be written in modern form as follows: If all the rat ...qual base areas have equal volumes. Euclid overcame this difficulty in his Elements by using the method of exhaustion.22 KB (3,357 words) - 17:34, 1 January 2021
- ...unique up to the 19th century — was the geometric system known as Euclid's Elements (ca. 300 B.C.). At the time the problem of the description of the logical t ...ntary geometry by M. Pasch, G. Peano and D. Hilbert which, unlike Euclid's Elements is logically unobjectionable, and Peano's first attempt at the axiomatizati17 KB (2,707 words) - 10:02, 25 April 2020
- Real numbers form a non-empty totality of elements which contains more than one element and displays the following properties. A non-empty totality of elements which has all the above properties forms a totally ordered field (cf. [[Tot26 KB (4,086 words) - 09:51, 4 April 2020
- satisfying the three conditions below. Here, the elements of $ \mathfrak C $ are called chains and two different points (i.e., elements of $ P $)5 KB (775 words) - 16:43, 4 June 2020
- ...nd postulates which constitute the fundaments of geometry. In Euclid's $ Elements $( ...geometry]] on the basis of the same principles and concepts as in the $ Elements $25 KB (3,631 words) - 19:39, 5 June 2020
- ...F$ satisfying $\sigma ( x , x ) \neq 0$ for all $x \in X$, $x \neq 0$. The elements of $X$ are called points, and a set of points $p + F . v $ ($p , v \in X$, ...bed in a fairly elementary way: Let $P$ be a set (no stipulation about the elements of $P$ is made, except that they will be called points). Let $P _ { 2 }$ be12 KB (1,960 words) - 17:44, 1 July 2020
- ...angles with sides of integer lengths. Euclid (3rd century B.C.) in his $ Elements $ <table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian) {{MR|0010 KB (1,503 words) - 08:03, 6 June 2020
- ...s the notion of elementary methods is extended by bringing in the simplest elements of mathematical analysis. Traditionally, proofs are deemed to be non-elemen ...general theory of divisibility was created, in essence, by Euclid. In his Elements (3rd century B.C.), he introduces an algorithm for finding the greatest com13 KB (2,043 words) - 20:28, 13 October 2014
- ...ic theorems also appeared during that period. The [[Elements-of-Euclid| $ Elements $ ...in general, the choice of a suitable mathematical space, and bringing its elements into correspondence with the objects of the system under study, depends on20 KB (2,829 words) - 18:41, 11 December 2020
- Books 7–9 of Euclid's Elements (3rd century B.C.) deal exclusively with arithmetic in the sense in which t20 KB (3,091 words) - 18:48, 5 April 2020
- ...e sequence of natural numbers, possibly attributable to the Greeks. One of Euclid's theorems states: "There exist more than any given number of primes" . Also and if for any collection of elements of the system $ A $23 KB (3,482 words) - 08:03, 6 June 2020