By Isaac Asimov
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Voila! By that act, you have;,in effect, squared the circle. But no. I'm afraid you're still out of the club. You have made use of a rolling circle with a mark on it and that comes under the heading of an instrument other than the straightedge and compass. The point is that there are any number of ways of squaring the circle, but the Greeks were unable to find any way of doing it with straightedge and compass alone in a finite number of steps. (They spent I don't know how many man-hours of time searching for a method, and looking back on it, it might all seem an exercise in futility now, but it wasn't.
And on and on without end. 414214 . . times as long as another when you cannot possibly ever know exactly how many times as long you want it to be. Actually, it's easy. Imagine a given line from point A to point B. (I can do this without a diagram, I think, but if you feel the need you can sketch the lines as you read. ) Let this line, AB, represent L. Next, construct a line at B, perpendicular to AB. Now you have two lines forming a right angle. Use the compass to draw a circle with its center at B, where the two lines meet, and passing through A.
In order to achieve consistent results, mathematicians have decided that the multiplication of a negative number by a positive number yields a negative product, while the multiplication of a negative number by a negative number yields a positive product. If, in the equation x2 + Ax — 5 = 0, —5 is substituted for x, then x2 becomes —5 times —5, or +25, while 4x becomes +4 times —5, or —20. The equation becomes 25 — 20 — 5 = 0, which is true. We would say, then, that there are two solutions to this equation, +1 and -5.