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Synopsis
Excerpt from The Thirteen Books of Euclid''s Elements, Vol. 2: Books III-IX
Many editors have held that this should not have been included among definitions. Some, e.g. Tartaglia, would call it a postulate; others, e.g. Borelli and Playfair, would call it an axiom; others again, as Billingsley and Clavius, while admitting it as a definition, add explanations based on the mode of constructing a circle; Simson and P?eiderer hold that it is a t/zeorem. I think however that Euclid would have maintained that it is a definition in the proper sense of the term and certainly it satisfies Aristotle''s requirement that a definitional statement (optonxbs Aéyos) should not only state the fact (76 511) but should indicate the cause as well (de anima 11. 2, 413 a The equality of circles with equal radii can of course be proved by superposition, but, as we have seen, Euclid avoided this method wherever he could, and there is nothing technically wrong in saying By equal circles I mean circles with equal radii. No ?aw is thereby introduced into the system of the Elements; for the definition could only be objected to if it could be proved that the equality predicated of the two circles in the definition was not the same thing as the equality predicated of other equal figures in the Elements on the basis of the congruence-axiom, and, needless to say, this cannot be proved because it is not true. The existence of equal circles (in the sense of the definition) follows from the existence of equal straight lines and 1. Post. 3.
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Equal circles are those the diameters of which are equal, or the radii of which are equal. 2. ,A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle. 3. Circles are said to touch one another which, meeting one another, donot cut one another. 4. In a circle straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal. 5. And that straight line is said to be at a greater distance on which the greater perpendicular falls. 6. A segment of a circle is the figure contained by a straight line and a circumference of a circle. 7. An angle of a segment is that contained by a straight line and a circumference of a circle. 8. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined. 9. A nd, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference. H. E. II.
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Paperback. Etat : New. Print on Demand. This book contains Euclid's foundational text on plane geometry. Composed in the 3rd century BC, it is widely considered one of the most influential works in the history of mathematics. Euclid builds up abstract mathematical concepts through definitions, postulates and theorems. Beginning with elementary propositions he advances to sophisticated theorems, solidifying the structure of Euclidean geometry. Basic concepts such as the relationship between angles, lines, and circles are defined, while advancing to complex propositions that explore the properties of more complex geometric constructions, including polygons and circles. Euclid's work forms the basis for much of modern geometric theory and is essential reading for those wanting to understand the history and development of mathematics. The author's logical, systematic approach has significantly impacted mathematical and scientific inquiry for centuries. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. N° de réf. du vendeur 9781330231722_0
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Etat : New. KlappentextrnrnExcerpt from The Thirteen Books of Euclid s Elements, Vol. 2: Books III-IXMany editors have held that this should not have been included among definitions. Some, e.g. Tartaglia, would call it a postulate others, e.g. Bore. N° de réf. du vendeur 2147741299
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