Remark

We can define a homomorphism of $\mathbf{C}^*$ onto $\mathfrak{A}_0$ by mapping each complex number $z \neq 0$ to the cross which the line through o and $z$ makes with the real axis. Clearly this homomorphism is the composition of $\varphi$ and the homomorphism $z \to \mathrm{Am}\,(z)$ of $\mathbf{C}^*$ onto $\mathfrak{A}$; it is therefore a strict morphism of the topological group $\mathbf{C}^*$ onto the topological group $\mathfrak{A}_0$, and the associated bijective homomorphism is an isomorphism of the quotient group $\mathbf{C}^*/\mathbf{R}^*$ onto $\mathfrak{A}_0$.

We know that if D denotes a line making a cross $\theta_0$ with the real axis and if $(a, b)$ is a pair of direction ratios of D, then the tangent of the cross $\theta_0$ (denoted by $\tan\,\theta_0$) is the element $b/a$ of $\tilde{\mathbf{R}}$ ($= \infty$ if $a = 0$), which is also called the slope of the line D. If $\theta$ and $\theta + \omega$ are the two angles whose image under $\varphi$ is $\theta_0$, then we have $\tan\,\theta_0 = \tan\,\theta = \tan(\theta + \omega)$.

The mapping $\theta_0 \to \tan\,\theta_0$ is a homeomorphism of $\mathfrak{A}_0$ onto $\tilde{\mathbf{R}}$, for the topological space $\mathbf{C}^*/\mathbf{R}^*$ is just the real projective line $\mathbf{P}_1$, and from Chapter VI, § 3, no. 3, the mapping of a line (considered as a point of $\mathbf{P}_1$) to its slope is a homeomorphism of $\mathbf{P}_1$ onto $\tilde{\mathbf{R}}$. If now we transport to $\tilde{\mathbf{R}}$ the group structure of $\mathfrak{A}_0$ by means of the mapping $\theta_0 \to \tan\,\theta_0$, we have defined on $\tilde{\mathbf{R}}$ the structure of an abelian topological group, in which the product of two elements $t_1, t_2$ is $\frac{t_1 + t_2}{1 - t_1 t_2}$ whenever $t_1, t_2$ belong to $\mathbf{R}$ and $t_1 t_2 \neq 1$; for pairs $(t_1, t_2)$ which do not satisfy these conditions, the product of $t_1$ and $t_2$ is obtained by extending the function $\frac{x + y}{1 - xy}$ by continuity to $\tilde{\mathbf{R}} \times \tilde{\mathbf{R}}$, and is still denoted by $\frac{t_1 + t_2}{1 - t_1 t_2}$.