Newton. Let be a line and a point not on . Let be a right angle which moves so that equals the distance from to and remains on , while the other leg slides along . Then the midpoint of describes the curve. To see this, let the distance between and be . By translation and rotation, take and the line . Let and let be the angle between and the -axis; this is equal to the angle between and . By construction, , so the distance from to is . In other words . Also, is the -coordinate of if it is rotated by angle , so . After simplification, this produces parametric equations Change parameters by replacing with its complement to get or, applying double angle formulas, But this is polar equation given above with . Note that, as with the double projection construction, this can be adapted to produce a mechanical device that generates the curve. Delian problem The Greek geometer Diocles used the cissoid to obtain two mean proportionals to a given ratio. This means that given lengths and , the curve can be used to find and so that is to as is to as is to , i.e. , as discovered by Hippocrates of Chios. As a special case, this can be used to solve the Delian problem: how much must the length of a cube be increased in order to double its volume? Specifically, if is the side of a cube, and , then the volume of a cube of side is so is the side of a cube with double the volume of the original cube. Note however that this solution does not fall within the rules of compass and straightedge construction since it relies on the existence of the cissoid. Let and be given. It is required to find so that , giving and as the mean proportionals. Let the cissoid be constructed as above, with the origin, the point , and the line , also as given above. Let be the point of intersection of with . From the given length , mark on so that . Draw and let be the point where it intersects the cissoid. Draw and let it intersect at . Then is the required length. To see this, rewrite the equation of the curve as and let , so is the perpendicular to through .