I'm not sure if I'm correct but here it goes...
Instead of calling the radius R, call it x as it's probably something we're both familiar with.
Lets only imagine the top left corner of the box and cut it out so we have box of sides of length of the radius x. Also in the box is top left quarter of our circle of radius x so it's not touching the top left corner.
Now we should find the length of the diagonal that goes from the centre of the big circle to the top left corner of the box:
How do we get the radius of the smaller circle?
Well we get the length from the edge of the big circle to the top left corner leaving just the smaller circle within it. To do this it is just the diagonal length minus the radius of the big circle:
Which can not be canceled down further.
HOWEVER, this is the diagonal of the line from where the big circle touches the smaller one so really we have the diameter of the circle plus the bit in the corner. So in plain terms, we have the diameter plus top left corner minus the bottom right corner (PS. to visualise this, draw it all out with the diameter always drawn as the diagonal through the box).
So since the corners cancel out, we actually had the diameter all along! And in order to get the radius, just halve it. Which gives:
And since you want it in terms of R....
I hope this is alright for you to understand! I was really working it out as I was going along so you have my thoughts here too and this is my second helping post I think too!
I think a good tip would be to draw out each stage as you read it!
Also you should note that you can keep working out the radiuses of each smaller circle like this an infinite number of times as it is impossible to reach the corner with circles!