To the fourteenth article you say that I “_commit a circle in that I require in the fourth article the finding of two mean proportionals, and come not till now to show how it is to be done_.” Nor now neither. But in the mean time you commit two mistakes in saying so. The place cited by you in the fourth article is, in the Latin, p. 215, line 26, in the English, p. 255, line 24. Let any reader judge whether that be a requiring it, or a supposing it to be done; this is your first mistake. The second is, that in this place the proportion itself, which is, “_If these deficient figures could be described in a parallelogram exquisitely, there might be found thereby between any two lines given, as many mean proportionals as one would_,” is a theorem, upon supposition of these crooked lines exquisitely drawn; but you take it for a problem. And proceeding in that error, you undertake the invention of two mean proportionals, using therein my first figure, which is of the same construction with the eighth that belongeth to this fourteenth article. Your construction is, “_Let there be taken in the diameter C A, (fig. 1) the two given lines, or two others proportional to them, as C H, C G, and their ordinate lines H F, G E (which by construction are in subtriplicate proportion of the intercepted diameters). These lines will show the proportions which those four proportionals are to have._” But how will you find the length of H F or G E, the ordinate lines? Will you not do it by so drawing the crooked line C F E, as it may pass through both the points F and E? You may make it pass through one of them, but to make it pass through the other, you must find two mean proportionals between G K and G L, or between H I and H P; which you cannot do, unless the crooked line be exactly drawn; which it cannot be by the geometry of planes. Go shew this demonstration of yours to Orontius, and see what he will say to it. I am now come to an end of your objections to the seventeenth chapter, where you have an epiphonema not to be passed over in silence. But because you pretend to the demonstration of some of these propositions by another method in your _Arithmetica Infinitorum_, I shall first try whether you be able to defend those demonstrations as well as I have done these of mine by the method of motion. The first proposition of your _Arithmetica Infinitorum_ is this lemma: “_In a series, or row of quantities, arithmetically proportional, beginning at a point or cypher, as 0, 1, 2, 3, 4, &c. to find the proportion of the aggregate of them all, to the aggregate of so many times the greatest, as there are terms_.” This is to be done by multiplying the greatest into half the number of the terms.