Isaac Newton

“ Draw the right lines FG, GH, HI, FI; and (by lem. 27.) describe a trapezium fghi that may be similar to the trapezium FGHI, and whose angles f, g, h, i, may touch the right lines given by position, AB, AD, BD, CE, severally according to their order. And then about this trapezium describe a trajectory, that trajectory will be similar to the curve line FGHI.
Scholium. ”

“ For if A and P are supposed to be the points of contact situated any where else in the tangents, and through any of the points H, I, K, L, as I, situated in either tangent HI a right line IT is drawn, parallel to the other tangent KL, and meeting the curve in X and I and in that right line there be taken IZ equal to a mean proportional between IX and IT ”

“ Conceive the point of contact of any tangent removed in infinitum, and the tangent will degenerate into an asymptote, and the constructions of the preceding problem will be changed into the constructions of those problems wherein the asymptote is given. ”

“ Bisect in M and N, AF, BE the diagonals of the quadrilateral figure ABFE contained under any four of them; and (by cor. 3. lem. 25) the right line MN drawn through the points of bisection will pass through the centre of the trajectory. Again. bisect in P and Q the diagonals (if I may so call them) BD, GF of the quadrilateral figure BGDF contained under any other four tangents, and the right line PQ drawn through the points of bisection will pass through the centre of the trajectory. And therefore the centre will be given in the concourse of the bisecting lines. ”