Meaning of correlation In probability books it is sometimes mentioned that correlation is a measure of linearity of a relationship between random variables. 
This claim is supported by the observation that $\left| \rho(X,Y)\right|=1 \iff X=b+aY$.
But let consider a less extreme case: $X,Y,Z$ are random variables such that $\rho(X,Y)=0.5$ and $\rho(Y,Z)=0.8$. Does it mean that the relationship between $X$ and $Y$ is "less" linear than between $Y$ and $Z$? How at all the degree of linearity in a relationship can be defined (especially given the fact, that a relationship between random variables can be non-functional, so one can not always use the second derivative)?    
Edit:
I am talking about Pearson correlation here.

correlation $=0.99$:

correlation $=0.95$:

correlation $=0.9$:

correlation $=0.7$:

correlation $=0.5$:

correlation $=0.3$:

correlation $=0$:

correlation $=-0.3$:

correlation $=-0.5$:

correlation $=-0.7$:

correlation $=-0.9$:

correlation $=-0.95$:

correlation $=-0.99$:

Appendix: Here is how I created these plots.
First I took a sample from a bivariate normal distribution with correlation $0$.  This of course gave me a scatterplot with non-zero correlation.
Then I replaced each $y$ value with the corresponding residual $y-\hat y$ where $\hat y$ is the fitted value, i.e. the $y$-coordinate of the least-squares line for predicting $y$ based on $x$.  This has two effects: $(1)$ It makes the correlation exactly $0$, and $(2)$ it makes the average of the $y$ values exactly $0$.
Then I divided the $y$ values by the standard deviation of the $y$ values.  I also subtracted the average $x$ value from all of them, and then divided them all by their standard deviation.  At this point I had a scatterplot in which the two averages were $0$, the two standard deviations were $1$, and the correlation was $0$.  That is the plot with correlation $0$ that you see above.
Then, to get a plot with correlation $\rho$, I replaced each data point $\begin{bmatrix} x \\ y \end{bmatrix}$ with
$$
\frac 1 2 \begin{bmatrix} \sqrt{1+\rho}+\sqrt{1-\rho}, & \sqrt{1+\rho}-\sqrt{1-\rho} \\ \sqrt{1+\rho}-\sqrt{1-\rho}, & \sqrt{1+\rho}+\sqrt{1-\rho} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}.
$$
That $2\times 2$ matrix is the positive-definite symmetric square root of the correlation matrix $\begin{bmatrix} 1, & \rho \\ \rho, & 1 \end{bmatrix}$.
