"Direct solutions by means of conics. Pappus gives two solutions of the trisection problem in which conics are applied directly without any preliminary reduction of the problem to a νϵῡσɩς. ...The passage of Pappus from which this solution is taken is remarkable as being one of three passages in Greek mathematical works still extant (two being in Pappus and one in a fragment of Anthemius on burning mirrors) which refer to the focus-and-directrix property of conics. The second passage in Pappus comes under the heading of Lemmas to the Surface-Loci of Euclid . Pappus there gives a complete proof of the theorem that, if the distance of a point from a fixed point is in a given ratio to its distance from a fixed line, the locus of the point is a conic section which is an ellipse, a parabola, or a hyperbola according as the given ratio is less than, equal to, or greater than, unity. The importance of these passages lies in the fact that the Lemma was required for the understanding of Euclid's treatise. We can hardly avoid the conclusion that the property was used by Euclid in his Surface-Loci, but was assumed as well known. It was, therefore, probably taken from some treatise current in Euclid's time, perhaps from Aristaeus's work on Solid Loci."