Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/32

 PROPOSITION 2. PROBLEM. From a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line: it is required to draw from the point A a straight line equal to BC. From the point A to B draw the straight line AB; and on it describe the equilateral triangle DAB, and produce the straight lines DA, DB to E and F. From the centre B, at the distance BC, describe the circle CGH, meeting DF at G. From the centre D, at the distance DG, describe the circle GKL, meeting DE at L. AL shall be equal to BC. Because the point B is the centre of the circle CGH, BC is equal to BG. And because the point D is the centre of the circle GKL, DL is equal to DG; and DA, DB parts of them are equal; therefore the remainder AL is equal to the remainder BG. But it has been shewn that BC is equal to BG; therefore AL and BC are each of them equal to BG. But things which are equal to the same thing are equal to one another. Therefore AL is equal to BC. Wherefore from the given point A a straight line AL has been drawn equal to the given straight line BC. PROPOSITION 3. PROBLEM. From the greater of two given straight lines to cut off a part equal to the less. Let AB and C be the two given straight lines, of which