Euclid book 3 proposition 31

T he next two propositions depend on the fundamental theorems of parallel lines. Definitions heath, 1908 postulates heath, 1908 axioms heath, 1908 proposition 1 heath, 1908. Book iii of euclids elements concerns the basic properties of circles. Book 11 deals with the fundamental propositions of threedimensional geometry. For euclid, an angle is formed by two rays which are not part of the same line see book i definition 8. The parallel line ef constructed in this proposition is the only one passing through the point a. How to draw a straight line through a given point, parallel to another given line.

This is the thirty first proposition in euclid s first book of the elements. Euclids elements book i, proposition 1 trim a line to be the same as another line. This edition of euclids elements presents the definitive greek texti. With links to the complete edition of euclid with pictures in java by david joyce, and the well known comments from heaths edition at the. The part of this proposition which says that an angle inscribed in a semicircle is a right angle is often called thales theorem. Use of proposition 31 this construction is frequently used in the remainder of book i starting with the next proposition.

The national science foundation provided support for entering this text. In euclids elements book 1 proposition 24, after he establishes that again, since df equals dg, therefore the angle dgf equals the angle dfg. It is also frequently used in books ii, iv, vi, xi, xii, and xiii. If on the circumference of a circle two points be taken at random, the straight line joining the points will fall within the circle. Propostion 27 and its converse, proposition 29 here again is. So, to euclid, a straight angle is not an angle at all, and so proposition 31 is not a special case of proposition 20 since proposition 20 only applies when you have an angle at the center. Built on proposition 2, which in turn is built on proposition 1. A right line is said to touch a circle when it meets the circle, and being produced does not cut it. This construction proof shows how to build a line through a given point that is parallel to a given line.