are alſo known; and therefore, for brevity's ſake, I omit any farther demonſtration of them.
Lemma XXII.
To tranſform figures into other figures of the ſame kind. Pl. 10. Fig. 5.
Suppoſe that any figure figure, HGI is to be tranſformed. Draw, at pleaſure, two parallel lines AO, BL, cutting any third line AB given by psſition, in A and B, and from any point G of the figure, draw out any right line GD, parallel to OA, till it meet the right line AB. Then from any given point O in the line OA, draw to the point D the right line OD, meeting BL in d, and from the point of concourſe raiſe the right line dg containing any given angle with the right line BL, and having ſuch ratio to Od, as DG has to OD; and g will be the point in the new figure hgi, correſponding to the point G. And in like manner the ſeveral points of the firſt figure will give as many correſpondent points of the new figure. If we therefore conceive the point G to be carried along by a continual motion through all the points of the firſt figure, the point g will be likewiſe carried along by a continual motion through all the points of the new figure, and deſcribe the ſame. For diſtinction's ſake, let us call DG the firſt ordinate, dg the new ordinate, AD the firſt abſciſſa, ad the new abſciſſa; O the pole, OD the abſcinding radius, OA the firſt ordinate radius, and Oa (by which the parallelogram
