diff --git a/eulerg/eulerg.tex b/eulerg/eulerg.tex
index 9f9a8f8..c7acdc7 100644
--- a/eulerg/eulerg.tex
+++ b/eulerg/eulerg.tex
@@ -40,8 +40,8 @@ The spaitial relationship between the curves defines the set theoretic relations
 
 The definitions above allow us to read an Euler diagram
 and write down set theory equations.
-The interest here, is to define relationships between the contours, that allow
-processing and parsing of the diagram without resorting to area operations in the concerete plane.
+The interest here though, is to define relationships between the contours, that allow
+processing and parsing of the diagram without resorting to extra area operations in the concerete plane.
 
 \section{Defining `pure intersection' and `enclosure'}
 \begin{figure}[h]
@@ -130,7 +130,7 @@ an {\em enclosure} relationship as a directed vertice and
 The diagram in figure \ref{fig:eulerg1} can now be represented as a graph thus:
 \begin{figure}[h]
  \centering
- \includegraphics[width=70pt,bb=0 0 128 108,keepaspectratio=true]{./eulerg/eulerg_g.jpg}
+ \includegraphics[width=50pt,bb=0 0 128 108,keepaspectratio=true]{./eulerg/eulerg_g.jpg}
  % eulerg_g.jpg: 128x108 pixel, 72dpi, 4.52x3.81 cm, bb=0 0 128 108
  \caption{Graph Representaion of figure \ref{fig:eulerg1}}
  \label{fig:eulerg1_g}
@@ -175,9 +175,9 @@ becomes part of the pure~intersection~chain. Hmmmm thats true but a better way t
 %The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships.
 
 
+\vbox{
 The diagram in figure \ref{fig:eulerg_pic} can be represented by the following relationships.
 
-
 $$ M {\pin} N $$
 $$ N {\pin} O $$
 $$ O {\pin} P $$
@@ -188,7 +188,7 @@ $$ A {\enc} N $$
 $$ A {\enc} O $$
 $$ A {\enc} P $$
 $$ A {\enc} Q $$
-
+}
 
 To form the {\pic} we can follow
 reachable pure intersection relationships.
@@ -258,13 +258,13 @@ see figure \ref{fig:eulerg_pic_g_a_unc}.
 \section{reduction of searches for available zones}
 
 
-Another property of any {\pic} $PIC$, is that
+Another property of any {\pic} $P$, is that
 the maximum number of euler zones within it is 
 
-$$ MaxZones =  2^{|PIC|}  $$
+$$ MaxZones =  2^{|P|}  $$
 
 Because no contours external to the {\pic} 
-bi-sect any in it, no extra zones can be formed.
+bisect any in it, no extra zones can be formed.
 By enclosing a {\pic} with 
 a contour, we change the nature of the zones within
 the {\pic}, but the number of zones contributed by the {\pic}
@@ -273,6 +273,8 @@ stays the same.
 A pure intersection chain has a maximum number of possible Euler zones, and exists as independent entities in the diagram. This
 allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones.
 \end{definition}
+This is to say, the the number of zones within a {\pic} is not affected by changes in the diagram 
+that do not alter the {\pic}.
 
 
 \vspace{40pt}