5. That the nature of an angle consisteth in that which lies between the lines that comprehend the angle in the very point of their concourse, that is to say, an angle is the superficies which lies between the two points which touch, or, as they understand a point, the superficies that lies between the two nothings which touch. 6. That the quotient is the proportion of the division to the dividend. Upon these and some such other principles is grounded all that Dr. Wallis has said, not only in his _Elenchus_ of my geometry, but also in his treatises of the _Angle of Contact_, and in his _Arithmetica Infinitorum_; which two last I have here in two or three leaves wholly and clearly confuted. And I verily believe that since the beginning of the world, there has not been, nor ever shall be, so much absurdity written in geometry, as is to be found in those books of his; with which there is so much presumption joined, that an ἀποκατάϛασις of the like conjunction cannot be expected in less than a Platonic year. The cause whereof I imagine to be this, that he mistook the study of _symbols_ for the study of _geometry_, and thought symbolical writing to be a new kind of method, and other men’s demonstrations set down in symbols new demonstrations. The way of analysis by squares, cubes, &c., is very ancient, and useful for the finding out whatsoever is contained in the nature and generation of rectangled planes, which also may be found without it, and was at the highest in Vieta; but I never saw anything added thereby to the science of geometry, as being a way wherein men go round from the equality of rectangled planes to the equality of proportion, and thence again to the equality of rectangled planes, wherein the symbols serve only to make men go faster about, as greater wind to a windmill.