some improvements

Author: thomasWeise

bookbase

@@ -224,7 +224,7 @@
 \FloatBarrier%
 \endhsection%
 %
-\hsection{LIU Hui's Method and the Approximation of~$\numberPi$}%
+\hsection{LIU Hui's Method for the Approximation of~$\numberPi$}%
 \label{sec:approximatePiLiuHui}%
 %
 \begin{figure}%
@@ -248,16 +248,16 @@
 \def\liuhuistf{\ensuremath{{\color{liuhui-s24-color}s_{24}}}}%
 %
 Let us now come to a more serious example.
-I am not good at mathematics, but I really like mathematics anyway, so we will go with a mathematics example: approximating~\numberPi.
+I am not good at mathematics, but I still really like mathematics anyway, so we will go with a mathematics example: approximating~\numberPi.
 The number~\numberPi\ is the ratio of the circumference of a circle and its diameter.
 A we already mentioned before in \cref{sec:float}, it is transcendental, a never-ending and never-repeating sequence of digits.
 We can compute it to a certain precision, e.g., as the \pythonilIdx{float} constant \pythonilIdx{pi} with value \pythonil{3.141592653589793}.
 But we can never really write it down.
 
-Well, we I say \inQuotes{we can compute it}, then the question \inQuotes{How?} immediately arises.
-One particularly ingenious answer was given by the Chinese mathematician LIU Hui~(刘徽) somewhere in the third century~AD~\cite{OR2003LH} in his commentary to the famous Chinese mathematics book \emph{Jiu Zhang Suanshu}~(九章算术)~\cite{OR2003LH,SCL1999TNCOTMACAC,S1998LHATFGAOCM,D2010AALHOCAS,C2002LFLHADWTDM}.
-In \cref{fig:liuHuiCircle}, we show how~\numberPi, i.e., the ratio of the circumference and the diameter of a circle can be approximated.
-The idea of LIU Hui~(刘徽) was to inscribe regular~$e$\nobreakdashes-gons with an increasing number~$e$ of edges into a circlem such that the corners of the $e$\nobreakdashes-gons lie on the circle.
+Well, when I say \inQuotes{we can compute it}, then the question \inQuotes{How?} immediately arises.
+One particularly ingenious answer was given by the Chinese mathematician LIU Hui~(刘徽) somewhere in the third century~\pgls{CE}~\cite{OR2003LH,Y2024COACMMLHFHTIOMACE} in his commentary to the famous Chinese mathematics book \emph{Jiu Zhang Suanshu}~(九章算术)~\cite{OR2003LH,SCL1999TNCOTMACAC,S1998LHATFGAOCM,D2010AALHOCAS,C2002LFLHADWTDM}.
+In \cref{fig:liuHuiCircle}, we show how~\numberPi\ can be approximated based on the idea of LIU Hui~(刘徽):
+By inscribing regular~$e$\nobreakdashes-gons with an increasing number~$e$ of edges into a circlem such that the corners of the $e$\nobreakdashes-gons lie on the circle.
 
 We start with a hexagon~($e=6$) where the radius~\liuhuir\ is equal to the radius of the circle.
 All the $e$~edges~\liuhuiss\ of this hexagon then have length~\liuhuir\ as well.

text/main/basics/variables/assignment/assignment.tex

@@ -178,7 +178,7 @@
 %
 Finally, we confirm that \pythonil{gcd} and \pythonil{math_gcd} compute the same result for four test cases at the bottom of our program.
 Now that all is said and done, it should be mentioned that the Euclidean Algorithm has a particularly efficient binary variant which is faster than our implementation in \cref{lst:functions:def_gcd}.
-This binary variant may have been developed in China in the first century~CE~\cite{B1999FAOTBEA}.%
+This binary variant may have been developed in China in the first century~\pgls{CE}~\cite{B1999FAOTBEA}.%
 %
 \FloatBarrier%
 \endhsection%

text/main/controlFlow/functions/functions.tex

@@ -358,7 +358,7 @@
 \label{sec:whileLoop}%
 %
 Old clay tablets show that the Babylonians were able to approximate~$\sqrt{2}$ maybe as far back as 4000~years ago~\cite{FR1998SRAIOBMY7IC,S2011NA:NA}.
-The mathematician Hero(n) of Alexandria, who lived in the first century~CE, specified an abstract algorithm for computing the square root of numbers which, today, is known as Heron's Method~\cite{S2011NA:NA,K2009BMOCTSRJBOFTAOCC}.
+The mathematician Hero(n) of Alexandria, who lived in the first century~\pgls{CE}, specified an abstract algorithm for computing the square root of numbers which, today, is known as Heron's Method~\cite{S2011NA:NA,K2009BMOCTSRJBOFTAOCC}.
 
 Given a number~$a$ and aiming to find the square root~$\sqrt{a}$ of~$a$, this algorithm starts with a guess~$x_0$, let's say~$x_0=1$.
 In each iteration~$i$, it will compute a new guess as follows~\cite{S2011NA:NA,K2009BMOCTSRJBOFTAOCC}:%