Euclid's book 10 prop 36

Dependence of propositions in book i of euclids elements. The other pa rt, proposition 21b, stating that if j is a p oint inside a triangle ab c, then. In other words, 367272 isosceles triangles are characterized by this property. 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 equals the square on the tangent. 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 the sum multiplied into the last makes some number, then the product is perfect. Thus, it is one short step from this proposition to the construction of a regular decagon inscribed in a circle. The 72, 72, 36 degree measure isosceles triangle constructed in iv. 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. In the first proposition, proposition 1, book i, euclid shows that, using only the. The triangle abdconstructed in this proposition is one of ten sectors of a regular decagon 10gon. 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. Published on apr 12, 2017 this is the thirty sixth proposition in euclids first book of the elements.

This proof shows that if you have two parallelograms that have equal bases and end on the same parallel, then they will. Let abcd and efgh be parallelograms which are on the equal bases bc and fg and in the same parallels ah and bg. All that follows is what i think hes driving at, but expressed in somewhat more modern terms. Hide browse bar your current position in the text is marked in blue. This is the thirty sixth proposition in euclids first book of the elements. To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line. But we use the golden ratio in this proof as a special. Problem understanding euclid book 10 proposition 1 mathoverflow. Proposition 37, which ends the book, is the converse of proposition 36. Book iii of euclids elements concerns the basic properties of circles, for example. For the love of physics walter lewin may 16, 2011 duration. No, the golden ratio wasnt really used in prop 36 or prop 37 of book 3. This proof shows that if you have two parallelograms that have equal bases and end on the same.

An invitation to read book x of euclids elements core. Click anywhere in the line to jump to another position. Let abcdand efghbe parallelograms which are on the equal bases bcand fgand in the same parallels ahand bg. Proposition 36 parallelograms which are on equal bases and in the same parallels equal one another. Propositions 36 to 72 of book x describe properties of certain sums of pairs of. If a straight line be bisected and a straight line be added to it in a.