  In this article, we will introduce methods of non-standard analysis into
projective geometry. Especially, we will analyze the properties of a projective
space over a non-Archimedean field. Non-Archimedean fields contain numbers that
are smaller than every real number: the so called "infinitesimal" numbers. The
theory is well known from non-standard analysis. This enables us to define
projective objects that deviate only infinitesimally (in an appropriate metric)
from each other. And we show that in most cases operations involving such
objects that deviate infinitesimally also experience only infinitesimal change.
Another focus will be where this property does not hold true and show that this
usually involves discontinuity or degeneration.
  Furthermore, we will explore common projective concepts like projective
transformations, cross-ratios and conics in a non-standard setting.
