Let the first series be of three terms subscribed with the greatest (0. 1. 8.)/(8. 8. 8.); the sum of the cubes is nine; the sum of all the greatest is 24; a quarter whereof is 6. But 9 is greater than 6 by three unities. An unity is something. Let it be therefore A. Therefore the row of cubes is greater than a quarter of three times eight, by three A. Again, let the series have four terms, as (0. 1. 8. 27)/(27. 27. 27. 27); the sum of the cubes is 36; a quarter of the sum of all the greatest is twenty-seven. But thirty-six is greater than twenty-seven by nine, that is, by 9 A. The excess therefore of the sum of the cubes above the fourth part of the sum of all the greatest, is increased by the increase of the number of terms. Again, let the terms be five, as (0. 1. 8. 27. 64)/(64. 64. 64. 64. 64), the sum of the cubes is one hundred; the sum of all the greatest three hundred and twenty; a quarter whereof is eighty. But one hundred is greater than eighty by twenty, that is, by 20 A. So you see that this lemma also is false. And yet there is grounded upon it all that which you have of comparing parabolas and paraboloeides with the parallelograms wherein they are accommodated. And therefore though it be true, that the parabola is (2)/(3) and the cubical paraboloeides (3)/(4) of their parallelograms respectively, yet it is more than you were certain of when you referred me, for the learning of geometry, to this book of yours. Besides, any man may perceive that without these two lemmas (which are mingled with all your compounded series with their excesses) there is nothing demonstrated to the end of your book: which to prosecute particularly, were but a vain expense of time. Truly, were it not that I must defend my reputation, I should not have showed the world how little there is of sound doctrine in any of your books. For when I think how dejected you will be for the future, and how the grief of so much time irrecoverably lost, together with the conscience of taking so great a stipend, for mis-teaching the young men of the University, and the consideration of how much your friends will be ashamed of you, will accompany you for the rest of your life, I have more compassion for you than you have deserved. Your treatise of the _Angle of Contact_ , I have before confuted in a very few leaves. And for that of your _Conic Sections_ , it is so covered over with the scab of symbols, that I had not the patience to examine whether it be well or ill demonstrated. Yet I observed thus much, that you find a tangent to a point given in the section by a diameter given; and in the next chapter after, you teach the finding of a diameter, which is not artificially done.