For first, from the seventh chapter of my book _De Corpore_, to the thirteenth, I have rectified and explained the principles of the science; _id est_, I have done that business for which Dr. Wallis receives the wages. In the seventh, I have exhibited and demonstrated the proportion of the parabola and parabolasters to the parallelograms of the same height and base; which, though some of the propositions were extant without that demonstration, were never before demonstrated, nor are by any other than this method demonstrable. In the eighteenth, as it is now in English, I have demonstrated, for anything I yet perceive, equation between the crooked line of a parabola or any parabolaster and a straight line. In the twenty-third I have exhibited the centre of gravity of any sector of a sphere. Lastly, the twenty-fourth, which is of the nature of refraction and reflection, is almost all new. But your Lordship will ask me what I have done in the twentieth, about the quadrature of the circle. Truly, my Lord, not much more than before. I have let stand there that which I did before condemn, not that I think it exact, but partly because the division of angles may be more exactly performed by it than by any organical way whatsoever; and I have attempted the same by another method, which seemeth to me very natural, but of calculation difficult and slippery. I call them only aggressions, retaining nevertheless the formal manner of assertion used in demonstration. For I dare not use such a doubtful word as _videtur_, because the professors are presently ready to oppose me with a _videtur quod non_. Nor am I willing to leave those aggressions out, but rather to try if it may be made pass for lawful, (in spite of them that seek honour, not from their own performances, but from other men’s failings), amongst many difficult undertakings carried through at once to leave one and the greatest for a time behind; and partly because the method is such as may hereafter give farther light to the finding out of the exact truth. But the principles of the professors that reprehend these of mine, are some of them so void of sense, that a man at the first hearing, whether geometrician or not geometrician, must abhor them. As for example: 1. That two equal proportions are not double to one of the same proportions. 2. That a proportion is double, triple, &c. of a number, but not of a proportion. 3. That the same body, without adding to it, or taking from it, is sometimes greater, and sometimes less. 4. That a quantity may grow less and less eternally, so as at last to be equal to another quantity; or, which is all one, that there is a last in eternity.