Swell is thread, counted
Every sewn section puts a thread inside its fold, and those threads stack at the spine and nowhere else. Sixteen sections sewn with a 0.020 in thread pile up 0.320 in of raw swell. Press and hammer 30 percent of that out and 0.224 in is left, which on a block measuring 1.100 in at the fore edge makes the spine 1.324 in — about a fifth thicker at one edge than the other.
That is a very large difference to have arrived at without meaning to, and it is why swell is worth calculating rather than discovering. The two inputs that move it are the number of sections and the thread diameter, and both are settled long before anybody picks up a hammer.
The round is a consequence, not a decision
Rounding a spine does not make the paper shorter. The spine still measures 1.324 in around the curve after rounding, because bending a stack of folded sheets does not compress them. What changes is the span: the two edges of that arc are pulled together until they sit the same distance apart as the fore edge, which has no thread in it and is stuck at 1.100 in.
So the geometry is fixed the moment those two numbers are known. An arc of length 1.324 spanning a chord of 1.100 subtends 118.6 degrees, on a radius of 0.640 in, with the middle of the spine standing 0.313 in above the line joining its edges. That is close to the shape often described as about a third of a circle, and it got there without anybody aiming at it — it fell out of sixteen sections and a 0.020 in thread.
Halve the thread and the round nearly disappears
Run the same sixteen sections with a 0.010 in thread and the swell halves to 0.112 in. The spine goes to 1.212 in, the ratio to 1.102, and the arc the swell supports drops from 118.6 degrees to 86.6. The book is not a third of a circle any more; it is a shallow curve with less than half the crown.
The page prints half-swell, actual and double-swell side by side for exactly this reason. The relationship is not linear and it is not intuitive, and a small change at the sewing bench moves the finished shape a long way.
Where the arithmetic stops
It treats the swell as evenly spread along the spine. It is not — thread piles at the stations and thins between them, which is what makes a freshly sewn block look lumpy and what makes a hurried round come out uneven. It treats the spine as a circular arc, which is an approximation of a shape that is really set by how the sheets slide against each other.
And it says nothing at all about whether a given round will last, whether the paper will take it, or whether the structure is sound. Those depend on the paper, the thread, the adhesive and the hands, none of which are numbers on a form.
Questions people ask
What is swell in bookbinding?
The thread stacking up in the folds at the spine. It only happens at the spine because that is the only place the thread is, so a sewn block is thicker there than at the fore edge. Sixteen sections sewn with a 0.020 in thread build 0.320 in of raw swell, and about 0.224 in survives pressing at 30 percent compression. That is the number the rest of the binding has to accommodate.
How much swell do I need to round a spine?
Enough that the spine arc is longer than the fore edge it has to span, and how much longer sets the angle. The page solves it the other way round, which is more useful: it takes the swell you actually have and tells you what arc it supports. At the values in the form, 0.224 in of swell on a 1.100 in fore edge gives a 118.6 degree round with a 0.313 in crown.
What if I have too much swell?
Then the arithmetic says you would need an arc tighter than a spine sensibly takes, and the page says so rather than giving you a number. The four terms in the sum are the section count, the thread diameter, how much you press out, and the thickness of the block itself. Which one to move is a question about the book, and this page has no view on it.
Does hammering change the round?
It changes how much swell survives to be rounded, which changes the arc. That is the compression field. What it does not do is create a round out of nothing: if the thread was thin and the block short, there is not enough extra spine length to bend, and no amount of hammering will produce one. The shape was decided at the sewing bench.
How high should the backing shoulder be?
The page prints your board caliper next to the crown for comparison and stops there. The shoulder has to hold the board, so it is normally worked to the board thickness, but the figure is yours and depends on the board you actually have rather than on the one named in a catalogue. This page states no specification and makes no judgement about whether a joint will hold.