Notation question on the colon symbol I'm reading through the first chapter of Ahlfors's Complex Analysis book, and during the section on stereographic projections, he says that we can map any $z = x+iy \in \mathbb{C}$ onto the unit sphere in three dimensions injectively using the equation $z= \frac{x_1+ix_2}{1-x_3}$. 
He then says that $x:y:-1= x_1:x_2:x_3-1$, which implies that $(x,y,0)$, $(x_1,x_2,x_3)$, and $(0,0,1)$ all lie on the same line. 
My question is what the colon symbol means in this context. I only remember it being used in the context of sets as a replacement for the "|" symbol. 
 A: It is a symbol that represents ratios; $A:B$ means "the ratio of $A$ to $B$". (Shows up all the time in Euclid; e.g., Book V.) When we write "$A:B=C:D$", we mean the ratio of $A$ to $B$ is the same as the ratio of $C$ to $D$. 
Here he is talking about three ratios; the fact that $x:y:-1 = x_1:x_2:x_3$ means that the vectors determined by $(x,y,-1)$ and by $(x_1,x_2,x_3)$ are parallel. Note that $(x,y,-1) = (x,y,0) - (0,0,1)$ is the vector with starting point at $(0,0,1)$ and end point at $(x,y,0)$; and $(x_1,x_2,x_3-1) = (x_1,x_2,x_3) - (0,0,1)$ is the vector with starting point at $(0,0,1)$ and endpoint at $(x_1,x_2,x_3)$. Since $x:y:-1 = x_1:x_2:x_3$, they are parallel; since they both start at the same point, that means that the line through $(0,0,1)$ and $(x,y,0)$ (determined by the first vector) and the line through $(0,0,1)$ and $(x_1,x_2,x_3)$ (determined by the second vector) are the same, since they are parallel and they both go through $(0,0,1)$. 
A: The colon notation $\rm\ (x\::\:y\::\:z)\ $ denotes a point in projective space (vs. comma notation $\rm\ (x\:,\:y\:,\:z)\ $ for points in affine space).$\ $ The notation highlights the fact that $\rm\  (x\::\:y\::\:z)\ =\ (\lambda\:x\::\:\lambda\:y\::\:\lambda\:z)\:,\ $ which explains the equality stated by Ahlfors. See for example Kedlaya: Projective Geometry.
A: The notation $a:b = c:d$ usually means that the proportions between $a,b$ and $c,d$ are the same, e.g. $a/b=c/d$ or more generally $ad = bc$. In this case we have a triple proportion.
