## Download PDF by Robert Simson: The Elements of Euclid: Viz. the First Six Books together

By Robert Simson

ISBN-10: 1402151497

ISBN-13: 9781402151491

This Elibron Classics booklet is a facsimile reprint of a 1830 variation by way of C.J.G. Rivington; etc., London.

**Read or Download The Elements of Euclid: Viz. the First Six Books together with the Eleventh and Twelfth. Also the Book of Euclid's Data to Which are Added the Elements of Plane and Spherical Trigonometry PDF**

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**Download e-book for iPad: The Elements of Euclid: Viz. the First Six Books together by Robert Simson**

This Elibron Classics publication is a facsimile reprint of a 1830 version through C. J. G. Rivington; and so on. , London.

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**Extra resources for The Elements of Euclid: Viz. the First Six Books together with the Eleventh and Twelfth. Also the Book of Euclid's Data to Which are Added the Elements of Plane and Spherical Trigonometry**

**Sample text**

1). From the results obtained above, and the methods used in their proof, there now follow these results: Every ring which is isotopic to a quasi-field is itself a quasi-field. In fact, let there be given on the additive Abelian group G a ring with multiplication a o b and a quasi-field isotopic to it with the multiplication a-b. Zn];)x = 0, and because the zero of the group G remains fixed under the iso morphism x> then ay-bty = 0. e. the given ring does not contain divisors of GROUPS AND RINGS 57 zero.

We note that the ring which we have constructed is in no way the unique (or minimal ring) with identity containing (in the sense of isomorphic embedding) a given ring R—thus the ring R itself may possess an identity. 5. Every groapoid can be isomorphically embedded in the multi plicative groupoid of some ring, where an associative or commu tative groupoid can be embedded in a ring possessing the same property. To prove this we consider all sums of the form £ Ka, (3) aeG where a varies through all the elements of the given groupoid G9 and the coefficients ka are integers, and where not more than a finite number of these coefficients are distinct from zero.

This enables us to consider equation (1) as the definition of multiplication of classes of equal fractions. The associativity and commutativity of this multiplication are obvious, and there fore we have made G into an Abelian semigroup. All fractions of the form z/z, z e S, are equal to each other. e. after cancellation by z, a = x. Fractions of the form z/z thus form a separate class. This class plays the part of the identity in the semigroup G. e. b — ay. Finally, if ax _ 6y x y ' then axy = byx, whence a = b.

### The Elements of Euclid: Viz. the First Six Books together with the Eleventh and Twelfth. Also the Book of Euclid's Data to Which are Added the Elements of Plane and Spherical Trigonometry by Robert Simson

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