Can someone explain Newton's Lemma 2 and 3 from Book I Section 1 of Principia Mathematica in English?

Answer 1

Lemma 2 says that if the curved area is approximated by inscribed and also by exscribed rectangles, where all rectanlges are of equal width, then the ratio of the exscribed areas summed up, to the curved area will be 1 as the number of rectangles tends to infinity. Also, the ratio of the sum of the area of the inscribed rectangles to the curved area to the curved area will be equal to 1 as the number of rectangles tends to infinity. In less technical language, the rectangles approximate the area under the curve, and as we use more and more (and thus thinner and thinner) rectangles, the approximation will get better and better; in fact we can make it as accurate as we want to. Lemma 3 says the same as the above, but drops the requirement that the rectangles all be of equal width.