In 1669 manuscript De Analysi, Newton adopts three rules: (I) The area under the curve \(y(x)=x^{\frac{m}{n}}\) equals \(\frac{n}{m+n}x^{\frac{m+n}{n}}\) , (II) The area under finitely or infinitely many curves equals the sum of areas under each curve, (III) Shows how to expand into a power series functions such as \(\frac{a^2}{b+x}\) or \(\sqrt{a^2+x^2}\) . Newton proves the converse of Rule I, namely: If the area under the curve y is given by the function \(z= \tfrac{n}{m+n}x^{\tfrac{m+n}{n}}\) , then \(y(x)=x^{\tfrac{m}{n}}\) . His approach relies on indivisibles and summing up infinitesimal area moments, although these concepts are left undefined. In this paper, we interpret De Analysi with techniques of nonstandard analysis. We represent Newton’s arguments on a hyperfinite grid, define the area under a curve as a hyperfinite sum and provide a rigorous proof of Rule I.
