N I706 Philippe de Lahire (I640-I7I8) published a Traite des roulettes in which he showed that if a smaller circle rolls without slipping along the inside of a larger circle with diameter twice as great, then (i) the locus of a point on the circumference of the smaller circle is a straight line segment (a diameter of the larger circle); and (2) the locus of a point which is not on the circumference but which is fixed with respect to the smaller circle is an ellipse.' The first portion of this theorem he had enunciated some dozen years before in a Traite des epicycloides. 2 Many years later it was pointed out 3 that the first half had been given more than four hundred years before by Nasir Eddin (I20I-I274) and had reappeared subsequently in the works of Copernicus (I473-I543), Cardan (I50I-I576), and Tacquet (I6I2i66o). This portion of the theorem, therefore, no longer is ascribed to Lahire. However, the second part -characterized by Lahire as plus remarquable -continues to bear his name 4 in spite of the fact that it also was known to Copernicus. The quadricentennial edition of De revolutionibus includes a passage from the manuscript copy of Copernicus in which both portions of the Lahire theorem are stated (in the language of epicycles instead of that of roulettes), together with a demonstration of the first part.5 For some reason the second half of the theorem was deleted before publication in I543. The passage in question closes with the words sed de his alias, implying that the author had intended to pursue the matter further. There is every reason to suppose that Copernicus, an able trigonometer, was in a position to give a proof of the general proposition, yet there appears to be no further record in this connection. The theorem of Copernicus seems virtually to have disappeared for about a century. Cardan, writing in 1572, apparently was not acquainted with the second half, for he includes such epicyclic motions among those which are erratic. 6 One catches a glimpse of the theorem again in a brief note by Kepler (I57I-I630):