If two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines. Summary of the proof euclid begins by assuming that the sum of a number of powers of 2 the sum beginning with 1 is a prime number. This is the thirty sixth proposition in euclid s first book of the elements. To place at a given point as an extremity a straight line equal to a given straight line. If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is square, then all the rest are also square. Quinary code construction of the leech lattice ozeki, michio, nihonkai mathematical journal, 1991. Noneuclid hyperbolic geometry article and javascript software. This proof shows that if you have two parallelograms that have equal bases and end on the same parallel, then they will.
If as many numbers as we please beginning from a unit are set out. If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if. If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect. Euclid, elements, book i, proposition 36 heath, 1908. This proof shows that if you have two parallelograms that have equal. Noneuclid supports two different models of the hyperbolic plane. This is the thirty sixth proposition in euclids first book of the elements. On a given finite straight line to construct an equilateral triangle. Checklist for planning a great block party two months before. Euclid, book iii, proposition 9 proposition 9 of book iii of euclid s elements is to be considered.
Let p be the number of powers of 2, and let s be their sum which is prime. Hyperbolic geometry used in einsteins general theory of relativity and curved hyperspace. It must be a neighborhood where your close friends can gather, but. Euclid, book iii, proposition 36 proposition 36 of book iii of euclid s elements is to be considered. Let abcd, efgh be parallelograms which are on equal bases bc, fg and in the same parallels ah, bg.
The statements and proofs of this proposition in heaths edition and caseys edition are to be compared. In the first proposition, proposition 1, book i, euclid shows that, using only the. Proposition 36 if a point is taken outside a circle and two straight lines fall from it on the circle, and if one of them cuts the circle and the other touches it, then the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference. Translation group and modular automorphisms for local regions borchers, h. Book iii, propositions 16,17,18, and book iii, propositions 36 and 37. Scribd is the worlds largest social reading and publishing site. Jan 16, 2002 in all of this, euclids descriptions are all in terms of lengths of lines, rather than in terms of operations on numbers. The theory of the circle in book iii of euclids elements. Noneuclid is java software for interactively creating straightedge and collapsible compass constructions in both the poincare disk model of hyperbolic geometry for use in high school and undergraduate education. Proposition 36 book 9 is euclids a great numbertheoretical achieve ment because he gave a sufficient condition for even numbers to be. The theta functions of sublattices of the leech lattice kondo, takeshi and tasaka, takashi, nagoya mathematical journal, 1986.
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