to one, is a _double_ proportion, but that every number in that progression is _double_ to the number next before it; and yet he does not call it _analogia dupla_, but _duplicate_. This distinction in proportions between _double_ and _duplicate_, proceeded long after from want of knowledge that the proportion of one to two is _double_ to the proportion of one to four; and this from ignorance of the different nature of proportions of _excess_, and proportions of _defect_. And you that have nothing but by tradition saw not the absurdities that did hang thereon. In the second article I make E K, (fig. 1) the third part of L K, which you say is false; and consequently the proposition undemonstrated. And thus you prove it false: “_Let A C be to G C, or G K to G L, as eight to one_ (_for seeing the point G is taken arbitrarily, we may place it where we will, &c._)” and upon this placing of G arbitrarily, you prove well enough that E K is not a third part of L K. But you did not then observe, that I make _the altitude A G, less than any quantity given_, and by consequence E K to differ from a third part by a less difference than any quantity that can be given. Therefore as yet the demonstration proceedeth well enough. But perceiving your oversight, you thought fit (though before, you thought this confutation sufficient) to endeavour to confute it another way; but with much more evidence of ignorance. For when I come to say, _the proportion therefore between A C and G C is triple, in arithmetical proportion, to the proportion between G K and G E, &c._ you say, “_the proportion of A C to G C is the proportion of identity, as also that of G K to G E.”_ But why? Does my construction make it so? Do not I make G C less than A C, though with less difference than any quantity that can be assigned? And then where I say, _therefore E K is the third part of L K_, you come in, by parenthesis, with (_or a fourth, or a fifth, &c._). Upon what ground? Because you think it will pass for current, without proof, that a point is nothing. Which if it do, geometry also shall pass for nothing, as having no ground nor beginning but in nothing. But I have already in a former lesson sufficiently showed you the consequence of that opinion. To which I may add, that it destroys the method of _indivisibles_, invented by Bonaventura; and upon which, not well understood, you have grounded all your scurvy book of _Arithmetica Infinitorum_; where your indivisibles have nothing to do, but as they are supposed to have quantity, that is to say, to be _divisibles_. You allow, it seems, your own nothings to be somethings, and yet will not allow my somethings to be considered as nothing. The rest of your objections having no other ground than this, “_that a point is nothing_,” my whole demonstration standeth firm; and so do the demonstrations of all such geometricians, ancient and modern, as have inferred any thing in the manner following, viz. _If it be not