 A locus, plural loci, is a figure that satisfies a certain relationship. Some common loci, given a point O and some distance r, the locus of all points in the plane distance r from O is a circle. Or given two points P and Q, the locus of all points X, where XP equals XQ, is a straight line perpendicular to PQ. Plaintian authors noted that the conic section solves certain types of locus problems. For example, given a point and a line, the locus of all points equidistant from the point and the line is a parabola. On the other hand, given a line, the locus of points where the square of the line drawns ordinal wise are equal to the rectangle of the line and the abscissa is also a parabola. And this raises an important question. How can you tell if an existing and well-known curve solves a locus problem? More generally, how can we solve an arbitrary geometric construction problem? For that, we turn to a mercenary. Rene Descartes fought as a mercenary in the wars of the 17th century. After he retired, he studied philosophy. And among his philosophical contributions are what we now call the anthropic principle. In modern descriptions, the anthropic principle is the following, the universe is conducive to life because if it wasn't, we wouldn't be here to ask why it isn't. Now Descartes had his own take on it, and this sounds better in Latin, cogito ergo sum. I think, therefore, I am. Descartes had the following insight. Every problem in geometry can easily be reduced to such terms that a knowledge of the length of certain straight lines is sufficient for its construction. What Descartes meant was this. Suppose you had a geometry problem. You could reframe the problem as an algebra problem, solve the algebra problem, then translate the algebraic solution into a geometric one. Now this only works if we know how to interpret an arithmetic expression as a geometric one. So addition and subtraction pose no problems, nor does division or multiplication buy a whole number. For example, suppose a, b equals a, and b, c equals b, we can show a plus b, 3a, and b over 2. So a plus b would be a line equal to a, b, and b, c joined together, 3a, well that would be 3a's joined together, and b over 2, well we could take b, c and cut it in half. However, products are a problem. Traditionally, products of two lines were interpreted as the areas of rectangles. And products of three lines were interpreted as volumes of solids. So products of four terms don't make sense. There's another problem. The product of two or more numbers can be interpreted in two different ways. So if I have the product 2 times 3, I could view this as the area of a rectangle with sides of length 2 and 3, but it can also be two lines of length 3 joined together, two 3s. And Descartes realized the theory of ratio and proportion could offer a solution. The addition and subtraction of lines didn't change, but to multiply or divide, we need to introduce a new line, the unit line. This would mean that lines corresponding to specific numbers would have specific lengths. So how can we multiply two lines? To multiply two lines, bd equals a by bc equals b, we'll set the two lines down at any convenient angle and mark a b equal to 1, the unit. We'll join ac and draw de parallel to ac, then be is our product ab. And this result follows from the theory of ratio and proportion. It's worth observing that we can reverse the construction. We'll set down two lines, be, bd, again at any convenient angle, and join de. Again we'll mark ab equal to 1, our unit, and draw ac parallel to de. Then bc is be divided by bd. We can also find square roots this way. To find the square root of gh equal to a, we'll extend gh by fg equal to 1, our unit. Then we'll draw the circle with diameter fh and construct our perpendicular gi. Then gi is the square root of a. And this follows from the properties of circles. So for example, let ab equal to 1, bc equal to a, let's construct a line equal to a squared plus the square root of a. So we'll set down bc and bd of length a, again at any convenient angle, and then mark ab equal to 1, the unit. Join ad, then draw ce parallel to ad, and be is going to be a squared. Next, to construct square root of a, so extend be by bf equal to 1, our unit. Draw the circle, construct the perpendicular, and the perpendicular will be square root of a. And now we'll join the lines together. So if we extend our line be, which is a squared by bh equal to the square root of a, we obtain he a squared plus square root of a.