To the fourth article you say, “_the description of those curvilineal figures is easy_.” True, to some men; and now that I have showed you the way, it is easy enough for you also. For the way you propound is wholly transcribed out of the figure of the second article, which article you had before rejected. For seeing the lines H F, G E, A B, &c. are equal to the lines C Q, C O, C D; and the lines Q F, O E, B D, equal to the lines C H, C G, C A; the proportion of D B to O E, will be triple (that is, triplicate) to the proportion of C O to G E; and the proportion of D B to Q F, triple to the proportion of C D to C Q; and consequently, because the complement B D C F E B is made by the decrease of A C in triple proportion to that of the decrease of C D, it will be (by the second article) a third part of the figure A B E F C A. So that it comes all to one pass, whether we take triple proportion in decreasing to make the complement, or triple proportion in increasing to make the figure; for the proportion of H F to B A, is triple to the proportion of C H to C A. Wherefore you have done no more but what you have seen first done, saving that from your construction you prove not the figure to be triple to the complement; perhaps because you have proved the contrary in your _Arithmetica Infinitorum_. But your way differs from mine, in that you call the proportion subtriplicate, which I call triplicate; as if the divers naming of the same thing made it differ from itself. You might as well have said briefly, the proposition is true, but ill proved, because I call the proportion of one or two triple, or triplicate of that of one to eight; which you say is false, and hath infected the fourth, fifth, ninth, tenth, eleventh, thirteenth, fourteenth, fifteenth, sixteenth, seventeenth, and nineteenth articles of the sixteenth chapter. But I say, and you know now, that it is true; and that all those articles are demonstrated. Lastly you add, “_Tu vero, in presente articulo, &c. id est, you bid find as many mean proportionals as one will, between two given lines; as if that could not be done by the geometry of planes, &c._” You might have left out _Tu vero_ to seek an _Ego quidem_. But tell me, do you think that you can find two mean proportionals (which is less than as many as one will) by the geometry of planes? We shall see anon how you go about it. I never said it was impossible, and if you look upon the places cited by you more attentively, you will find yourself mistaken. But I say, the way to do it has not been yet found out, and therefore it may prove a solid problem for anything you know. The fifth article you reject, because it citeth the corollary of the twenty-eighth article of the thirteenth chapter, where there is never a word to that purpose. But there is in the twenty-sixth article; which was my own fault, though you knew not but it might have been the printer’s.