\paragraph{10.} Each of the letters composing a term is called a \textbf{dimension}
of the term, and the number of letters involved is called
the \textbf{degree} of the term. Thus the product $abc$ is said to be \emph{of
three dimensions}, or \emph{of the third degree}; and $ax^4$ is said to be \emph{of
five dimensions}, or \emph{of the fifth degree}.

A numerical coefficient is not counted. Thus $8a^2b^5$ and $a^2b^5$
are each of \emph{seven} dimensions.

\paragraph{11.} But it is sometimes useful to speak of the dimensions
of an expression with regard to any one of the letters it involves.
For instance, the expression $8a^3b^4c$, which is of eight dimensions,
may be said to be of three dimensions in~$a$, of four dimensions
in~$b$, and of one dimension in~$c$.

\paragraph{12.} A compound expression is said to be \textbf{homogeneous} when
all its terms are of the same dimensions. Thus $8a^6 - a^4b^2 + 9ab^5$
is a \emph{homogeneous expression of six dimensions}.

\paragraph{13.} In dealing with Algebraical expressions, where the letters
denote numerical quantities, we may make use of the principles
with which the student is familiar in Arithmetic. Thus
$ab$ and $ba$ each denote the product of the two quantities represented
by the letters $a$ and~$b$, and have therefore the same value.
Again, the expressions $abc$, $acb$, $bac$, $bca$, $cab$, $cba$ have the same
value, each denoting the product of the three quantities $a$, $b$, $c$. It
is immaterial in what order the factors of a product are written;
it is usual, however, to arrange them in alphabetical order.

\textit{Example}~1.\quad If $x = 5$, $y = 3$, find the value of $4x^2y^3$.
\begin{align*}
4x^2y^3 &= 4 × 5^2 × 3^3, \\
        &= 4 × 25 × 27,   \\
        &= 2700.
\end{align*}
%% Also fine, though it does not match the scan:
%\[
%4x^2y^3 = 4 × 5^2 × 3^3,
%        = 4 × 25 × 27,
%        = 2700.
%\]

\textit{Example}~2.\quad If $a = 4$, $b = 9$, $x = 6$, find the value of $\dfrac{8bx^2}{27a^3}$.
\begin{align*}
\frac{8bx^2}{27a^3} % Replacing the letter x with the symbol × several times
  &= \frac{8 × 9 × 6^2}{27 × 4^3}, \\
  &= \frac{8 × 9 × 36}{27 × 64},   \\
  &= \tfrac{3}{2}. \\ % [** F1: Comma?]
  &= 1\tfrac{1}{2}.
\end{align*}

\paragraph{14.} If one factor of a product is equal to~$0$, the whole product
must be equal to~$0$, \emph{whatever values the other factors may have}.
A factor~$0$ is sometimes called a ``zero factor.''
