What is this dot appearing in the middle of proofs?

Question

In G.H. Hardy's 6th edition of Introduction to the Theory of Numbers, this dot appears in some proofs. For example, on page 2,

$$b\mid a \;.\; c\mid b \implies c\mid a$$

What is the dot on the left hand side of the arrow?

Answer 1

In the section REMARKS ON NOTATION right at the beginning of the book we find:

and \begin{align*} b\,|\,a \,.\,c\,|\,b\ \rightarrow\ c\,|\,a\qquad (p.1) \end{align*} means 'if $b$ divides $a$ and $c$ divides $b$ then $c$ divides $a$'.