The set of prime numbers is infinite (Euclid's Elements, Book IX, Prop. 20). The [[Chebyshev theorems on prime numbers|Chebyshev th
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''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 strai
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...measure of two intervals. It was described in geometrical form in Euclid's Elements (3rd century B.C.).
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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).
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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).
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...matically (though not sufficiently rigorous) in the [[Elements-of-Euclid|''Elements'' of Euclid]]. The space of Euclidean geometry is usually described as a se
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...ty. Its first appearance in the extant classical literature is in Euclid's Elements, Book II, 11 (3th century B.C.).
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...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) (Tra
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...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.
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...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 axiomatizati
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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. [[Tot
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satisfying the three conditions below. Here, the elements of $ \mathfrak C $
are called chains and two different points (i.e., elements of $ P $)
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...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 $
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...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 $P _ { 2 }$ be
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...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|00
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...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 com
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...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 on
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Books 7–9 of Euclid's Elements (3rd century B.C.) deal exclusively with arithmetic in the sense in which t
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...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 $
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arithmetic and the first book of Euclid's Elements to the calculus
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...itely many primes. Let $\pi(x)$ be the number of primes not exceeding $x$. Euclid's theorem can then be formulated as follows: $\pi(x)\to +\infty$ as $x\to\inf
|valign="top"|{{Ref|Vi}}||valign="top"| I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
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...the moment when the concept of convergence has been introduced on a set of elements.
...convergence of elements of a set can be applied to one and the same set of elements, depending on the problem under consideration. The concept of convergence p
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the first six books of Euclid's elements. He had even tried his hands on the
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...by a rectangle with sides representing the respective factors. In Euclid's Elements (3th century B.C.), quantities are denoted by two letters, the initial and
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...field|skew-field]] is an associative ring in which the set of all non-zero elements is a multiplicative group. A [[Field|field]] is a skew-field in which multi
...}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' ,
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mathematics. It would appear that he had read the "Elements'' of
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...les of 1-1 correspondences between the elements of an infinite set and the elements of a proper subset
...s, the elements of an infinite set could be put in 1-1 correspondence with elements of one of its proper subsets.
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...al structures" rather than as "intuitively given magnitudes inherited from Euclid's geometry."<ref>Boyer p. 605</ref>
For example, Cauchy's definition was constructed using only these three elements
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