Newton on the equiangular spiral: An addendum to Erlichson's account

In a recent article Herman Erlichson called attention to a flaw in Newton's proof of Proposition IX of Book I of the Principia. How did Newton fall into this error? A valid proof was near to hand, by an easy addition to Lemma III of Book II; but evidently Newton wished to attempt a different line of argument. The figure for Proposition IX in the first two editions of the Principia differs from that in the third edition, and does not involve the quadrilateral “given in kind” that Erlichson rightly objects to. But the basic error remains: the assumption without proof of the similarity of all segments of the spiral with the same central angle. By 1671 Newton had proved this assumption by an integration, establishing the logarithmic property of the equiangular spiral. In Proposition IX it would thus appear that he either simply presented as known an analytic result now familiar to himself, or hoped that his readers would consider it as valid for infinitesimal segments