If on the circumference of a circle two points be taken at random, the straight line joining the points will fall within the circle. Euclids elements, book iii, proposition 35 proposition 35 if in a circle two straight lines cut one another, then the rectangle contained by the segments of the one equals the rectangle contained by the segments of the other. Now it is clear that the purpose of proposition 2 is to effect the construction in this proposition. Euclid often supplies a proof for only one case, although occasionally he gives proofs for two or three cases. For the love of physics walter lewin may 16, 2011 duration. The inner lines from a point within the circle are larger the closer they are to the centre of the circle. Definitions from book iii byrnes edition definitions 1, 2, 3, 4. There are other cases to consider, for instance, when e lies between a and d.
Euclid s proof specifically treats the case when the point d lies between a and e in which case subtraction of a triangle is necessary. If in a circle two straight lines cut one another, then the rectangle contained by the segments of the one equals. In equal circles equal circumferences are subtended by equal straight. That is, ag makes with ab, at the given point a, an anle equal to the given angle.
Leon and theudius also wrote versions before euclid fl. Book 11 deals with the fundamental propositions of threedimensional geometry. It is now 35 years since the publication of 40, and meantime, the technology of proof. 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. Propositions from euclids elements of geometry book iii tl heaths. Proposition by proposition 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 perseus collection of greek classics. W e shall see however from euclids proof of proposition 35, that two figures. Built on proposition 2, which in turn is built on proposition 1. In that case the point g is irrelevant and the trapezium bced may be added to the congruent triangles abe and dcf to derive the conclusion.