The books cover plane and solid euclidean geometry. Note that euclid takes both m and n to be 3 in his proof. If in a circle a straight line cuts a straight line into two equal parts and at right angles. This is the third proposition in euclids first book of the elements. Take the center g of the circle abdc and the center h of ebfd. But c also equals ad, therefore each of the straight lines ae and c equals ad. About half the proofs in book iii and several of those in book iv begin with taking the center of a given circle, but in plane geometry, it isnt necessary to invoke this proposition iii. Built on proposition 2, which in turn is built on proposition 1. Use of proposition 3 this proposition begins the geometric arithmetic of lines. Draw a straight line ab through it at random, and bisect it at the point d. Euclid s elements, book iii department of mathematics.
Euclids elements book i, proposition 1 trim a line to be the same as another line. In later books cutandpaste operations will be applied to other kinds of magnitudes such as solid figures and arcs of circles. It is a collection of definitions, postulates, propositions theorems and constructions, and mathematical proofs of the propositions. Draw dc from d at right angles to ab, and draw it through to e. The proof starts with two given lines, each of different lengths, and shows. Use of proposition 35 this proposition is used in the next two propositions and in xi. Paraphrase of euclid book 3 proposition 16 a a straight line ae drawn perpendicular to the diameter of a circle will fall outside the circle.
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