To the fifth article, though your discourse be long, you object but two things. One is, that “_Whereas the spiral of Archimedes is made of two motions, one straight, the other circular, both uniform, I taking the motion compounded of them both for one of those that are compounded, conclude falsely, that the generation of the spiral is like to the generation of the parabola_.” What heed you use to take in your reprehensions, appears most manifestly in this objection. For I say in that demonstration of mine, that _the velocity of the point A in describing the spiral increaseth continually in proportion to the times_. For seeing it goes on uniformly in the semidiameter, it is impossible it should not pass into greater and greater circles, proportionally to the times, and consequently it must have a swifter and swifter motion circular, to be compounded with the uniform motion in every point of the radius as it turneth about. This objection therefore is nothing but an effect of a will, without cause, to contradict. The other objection is, that “_Granting all to be true hitherto, yet because it depends upon the finding of a straight line equal to a parabolical line in the eighteenth chapter, where I was deceived, I am also deceived here_.” True. But because in the eighteenth chapter of this English edition I have found a straight line equal to the spiral line of Archimedes. I must here put you in mind that by these words in your objections to the fifth article at your number two, _Quatenus verum est, etc._, _we have demonstrated prop. 10, 11, 13_, _Arithmetica Infinitorum_; you make it appear that you thought your spiral (made of arches or circles) was the true spiral of Archimedes; which is fully as absurd as the quadrature of Joseph Scaliger, whose geometry you so much despise. To the sixth article, which is a digression concerning the analytics of geometricians, you deny _that the efficient cause of the construction ought to be contained in the demonstration_. As if any problem could be known to be truly done, otherwise than by knowing first how, that is to say, by what efficient cause, and in what manner, it is to be done. Whatsoever is done without that knowledge, cannot be demonstrated to be done; as you see in your computation of the parabola, and paraboloeides, in your _Arithmetica Infinitorum_.