It needs a back.
That would do it. But I don't want one (don't want the weight, the expense, or the block for a fairly beautiful wall), so I am hoping for another solution.
Can you entertain a diagonal brace? You need something that is neither horizontal or vertical.
Sure. Can you or someone advise me on how to put that on so that it is straight, as opposed to simply preventing it from deviating from its current crookedness?
(And making the next bookshelf start straight? I've got 3 to go!)
Given its current structure, you should be able to force it into a square shape.
Put one nail (only one) in diagonal brace, force to square, then put a second nail on on the other end of the brace. It should now be stable and then you can finish securing the brace. Just use a carpenters square to make sure you are dead on.
I can't see the other side in the picture. I assume the two vertical sides are parallel? If not, you have another problem and need to check the lengths of the shelves to make sure they are all the same.
Sheeting: you mean plywood? Something else? I may go with Kit's suggestion below and just use it behind the bottom shelf.
One option that would give you the strength, stability, and squareness you want would be to put a back on just the bottom shelf or two. That gives you the structural benefits of a back without covering the wall anywhere near eye level.
Hey, I like that. What would you use? How thick?
Hardboard should suffice.
I agree with GE - 1/4" hardboard would be ideal.
I did this. I took one of the shelves and made it into a back for the bottom shelf. Two crosspieces at the bottom and one at the top, and it feels stable.
I think the main things I did to make it work this time were:
* loosen the crosspiece at the bottom that I think made it lean to begin with
* made damned sure all crosspieces had right angles at the end,
* ...and were exactly the right size, so they could just fit in their respective holes
* clamped a spirit level to the side and made sure the case stayed level as I put all those crosspieces in.
Since you only braced the bottom, I suggest you put some weight on the upper half and see if you are still satisfied with its stability before you make the other three shelves.