Let p be the number of powers of 2, and let s be their sum which is prime. For some people, the dayin, dayout of an ordinary life makes. Begin sequence its about time for me to let you browse on your own. 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. 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. 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. 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. A similar remark can be made about euclids proof in book ix, proposition 20, that there are infinitely many prime numbers which is one of the most famous proofs in the whole of mathematics.
Book iii, propositions 16,17,18, and book iii, propositions 36 and 37. A given figure can be viewed in either model by checking either disk or upper halfplane in the model command of the view menu. Scribd is the worlds largest social reading and publishing site. 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. It must be a neighborhood where your close friends can gather, but. Euclid, elements, book i, proposition 36 heath, 1908. This proof shows that if you have two parallelograms that have equal. Checklist for planning a great block party two months before. 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. In the first proposition, proposition 1, book i, euclid shows that, using only the.
Noneuclid hyperbolic geometry article and javascript software. The 72, 72, 36 degree measure isosceles triangle constructed in iv. Noneuclid supports two different models of the hyperbolic plane. The theta functions of sublattices of the leech lattice kondo, takeshi and tasaka, takashi, nagoya mathematical journal, 1986. 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. This is the thirty sixth proposition in euclid s first book of the elements. Let abcd, efgh be parallelograms which are on equal bases bc, fg and in the same parallels ah, bg. Im relatively new to jeuclid and im using it to convert some mathml content to pngs for inclusion in html content. 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. Hyperbolic geometry used in einsteins general theory of relativity and curved hyperspace. I say that the parallelogram abcd is equal to efgh.
This proof shows that if you have two parallelograms that have equal bases and end on the same parallel, then they will. This is the thirty sixth proposition in euclids first book of the elements. 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. The statements and proofs of this proposition in heaths edition and caseys edition are to be compared. On a given finite straight line to construct an equilateral triangle. Translation group and modular automorphisms for local regions borchers, h. Parallelograms which have the equal base and equal height are equal in area. Proposition 36 book 9 is euclids a great numbertheoretical achieve ment because he gave a sufficient condition for even numbers to be. Euclid, book iii, proposition 36 proposition 36 of book iii of euclid s elements is to be considered.