I'm trying to find the origin of the integration process. To do that I'm studying "De Analysi" by Newton. I would like to know the process that lead Newton to the Rule I $\ref{Rule I}$ below, and if it is linked with the quadrature problem.
This is the first page of the original book digitalized by Google. The latin is quite simple this is a "user friendly" translation:
General method to measure the "quantity" [area] of a curve by a series of infinte terms that I've found once, you [will] have it demonstrate in the following short explainations rather than an accurate demostration.
Draw $BD$ perpendicular to the base $AB$ of the $AD$ curve: we call $AB=x$, $BD=y$. $a,b,c,e$ are note quantities and $m,n$ integer. Then:
Here comes the question:
Curvarum Simplicium Quadratura
In latin means "the quadature of a simple curve" and then Newton doesn't explain how he founds the classic rule, but simply explains that works.
$y=ax^{m/n};$ it shall be $(\frac{an}{m+n})x^{\frac{m+n}{n}} =$Area ABD $\tag{Rule I}$
Quadrature was a well know ancient Greek problem to build a square with the same area of another shape. The first thing I thought is to compare the area of a quarter of circle to the integration of the equation of the curve related but I'm not sure is right. Anybody has already faced this problem?