Postulate 3. A circle may be described from any centre, and at any distance from that centre. Postulate 1. A straight line may be drawn from any one point to any other. Definition 15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. Axiom 1. Magnitudes which are equal to the same magnitude are equal to one another. Definition 24. Of three-sided figures, an equilateral triangle is that which has three equal sides. PROPOSITION II. PROBLEM. From a given point to draw a straight line equal to a given straight line. GIVEN that A is a point, and BC a straight line; IT IS REQUIRED TO to BC. DRAW from the point A, a straight line equal H K Join AB. Upon AB describe the equilateral triangle ABD. Produce DA to E, and DB to F. Post. 1. Post. 2. From the centre B, at the distance BC, describe the circle CGH; Post. 3. and let the circumference of it cut DF at G. h. From the centre D, at the distance DG, describe the circle GKL, Post. 3. and let the circumference of it cut AE at L. h. To be proved. Constr. Def. 15. Constr. Def. 15. Proved. Constr. Axiom. 3. Proved. Axiom. 1. Because AL and BC are each equal to BG; therefore AL is equal to BC. Therefore, from the given point A, a straight line AL has been drawn equal to the given straight line BC. Q. E. F. Postulate 1. A straight line may be drawn from any one point to any other point. Proposition I. An equilateral triangle may be described upon a given finite straight line. Postulate 2. A terminated straight line may be produced to any length in a straight line. Postulate 3. A circle may be described from any centre, and at any distance from that centre. Definition 15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines, drawn from a certain point within the figure to the circumference are equal to one another. Axiom 3. If equals be taken from equals, the remainders are equal. Axiom 1. Magnitudes which are equal to the same magnitude are equal to one another. PROPOSITION III. PROBLEM. From the greater of two given straight lines to cut off a part equal to the less. GIVEN that AB and CG are two straight lines, of which AB is the greater; IT IS REQUIRED TO CUT OFF from AB a part equal to CG. A B E From A draw AD equal to CG. I. 2. From the centre A, at the distance AD, describe the circle DEF, Post. 3. and let the circumference of it cut AB at E. h. To be proved. Constr. Def. 15. Proved. Constr. Axiom 1. Therefore, from AB, the greater of two given straight lines, a part AE has been cut off, equal to CG the less. Q. E. F. Proposition II. A straight line may be drawn from a given point equal to a given straight line. Postulate 3. A circle may be described from any centre and at any distance from that centre. Definition 15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. Axiom 1. Magnitudes which are equal to the same magnitude are equal to one another. |