I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an example of a nonstandard differentiation:

The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity... the typical method from Newton through the 19th century would have been simply to discard the $dx^2$ term.

I've never heard anything like this before, and really find it fascinating that Newton's method was to define the relation $dx^2 = 0$. If we actually formalize the above structure by taking $\mathbb{R}$ and adjoining an element $dx^2 = 0$ to it, we get the "dual numbers," isomorphic to the quotient ring $\mathbb{R}[x]/x^2$. I'd seen some things about how this algebra plays into automated differentiation algorithms for some computer software systems, but I've never heard anything about Newton directly working in this algebra. So I have a few questions:

- Does anyone have more historical information on the way that Newton performed differentiation, and its relation to the dual numbers?
- Does anyone know how effectively real analysis can be formalized with the dual numbers? Does the resulting system play nice enough to develop all of the important modern results?
- If we start with $\mathbb{C}[x]/x^2$ instead, can we likewise develop complex analysis?

Since this idea is so simple, I'm very curious how powerful it is. I'm also curious if it has any major drawbacks too, since I'm not sure why anyone would mess with the foundational baggage involved in defining the hyperreals if this simple 2-dimensional real algebra could really do the trick.