There’s likely a more elegant way of doing this, but this was my approach. I’ve rotated the figure and put in place a coordinate system with the origin at the centre of the lower left circle. Without loss of generality I’m using unit circles. By Pythagoras the top right circle is at (10,0) as part of a 3,4,5 triangle. Since the distance between adjacent circles is 2, and we know the triangles formed with the horizontal are in the ratio 3,4,5, we can easily find the coordinates of the circles next to the origin circle, and by extension, any circle we wish. The red line is the critical line that goes from (0,1) to (10,1). The first two circles it passes through are centred 0.6 off the line. The third circle is indeed tangent as appears from the figure. The other two circles it passes through are centred 0.2 off the line. Using Pythagoras we can find the lengths of the chords to be 1.6 and 0.8*sqrt(6) respectively. The overall proportion then is 4/25(2+sqrt(6)) or just under 0.72.