How to calculate a curve length in gnuplot?

I have a data file containing the following data:

x    z
0.1  0
0.12 0.1
0.12 0.14
0.13 0.15
0.15 0.15
...   ...

I would like to calculate the curve length

sum_{n=0->end of file}(sqrt((x_n+1-x_n)^2+(z_n+1-z_n)^2))

I guess I have to use a recursive function.But don't know how to use it with a data file. How could I do that in gnuplot script?

You certainly could define a recursive function, but it is not necessary. You simply sum up all the path lengths between two consecutive points. For the first line, i.e. if the pseudocolumn 0 (basically, line index zero-based, check help pseudocolumns) is equal to 0, initialize your variables x1 and y1 and L. For each line assign x0=x1 and x1=$1. With this, x0 holds the previous x-value and x1 the current x-value. Same for y0=y1 and y1=$2. Calculate the length via sqrt((x1-x0)**2 + (y1-y0)**2) and sum all pieces up in L.

The function sumUpSegments(colX,colY) takes the x- and the y-column (here: 1 and 2) as argument.

Furthermore, the construct '+' u (x1):(y1) ... every ::0::0 is one way of many ways to plot a single data point.

Script:

### determine path length
reset session

$Data <<EOD
 0.1   0
 0.12  0.1
 0.12  0.14
 0.13  0.15
 0.15  0.15
 0.17  0.10
 0.14  0.04
 0.13  0.08
 0.13  0.11
 0.15  0.12
 0.16  0.10
 0.14  0.07
EOD

set key noautotitle
sumUpSegments(colX,colY) = ($0==0 ? (L=0, x1=column(colX), y1=column(colY)):0, \
                            x0=x1, y0=y1, x1=column(colX), y1=column(colY), \
                            L=L+sqrt((x1-x0)**2 + (y1-y0)**2), x1)

plot $Data u (sumUpSegments(1,2)):2 w l lc "red", \
       '+' u (x1):(y1):(sprintf("L=%.3g",L)) every ::0::0 w labels offset 0,-0.5
### end of script

Result: