Squares are what we call those little bits of cloth showing (in a hardback) around the edges of the endpaper where the endpaper is glued into the case holding the book in the binding. In the picture below, the squares are the blue bits showing around the green endpaper. In an ideal world the squares will be an even all around, about 1/8″ in a 6″ x 9″ book. As a book comes off the binding line the glue is still wet and the book block can be squished around to even up the squares, but because this costs it’s not much done now.

In this photo you can see that the squares are not exactly even, which is a little embarrassing because this is Volume 2 of Elizabeth Eisenstein’s The printing press as an agent of change. Still, not too bad, and let’s assume all the others were perfect. Book loving production people will also make sure that a slightly faulty book doesn’t get into stock. This they do by selflessly taking the “faulty” book home to read. I am embarrassed to confess that I have indeed heard a colleague offer to hurt a book for me. (I’d usually say I’ll take it uninjured!)

You can also make out by the shadows the edges of the cloth where it’s folded over the board forming the case. Along the spine you can also discern by its shadow the mull liner which adds strength to this binding — it is glued to the spine of the book block and held between case and the endpapers at front and back. There’s a useful diagram at my earlier post Hardcover parts.

Very rarely you’ll see a big fat book bound with almost no square at the bottom, so that the bookblock isn’t hanging within the case, tempting it to tip forward towards the fore-edge, weakening the bond at the top of the spine. Not sure whether there’s a mechanical reason why we don’t do this, or whether it’s just habit (or more darkly a desire that the book should fall apart, necessitating replacement with a new copy).

Like so many things we used to spend our working days caring about evenness of squares is hardly a matter of concern these days. The book market, certainly the trade book market, is price sensitive, so little quality touches tend to fall away in an attempt to preserve margins.