Everything here turns on one halving
Two inches down to one and three quarters is 250 thou of stock, and 125 thou of tool movement. The tool sits on one side of the work and the work turns, so every thou the tool moves in takes two thou off the diameter. That is the entire difference and it is responsible for more undersized shafts than any other single mistake on a manual lathe.
With the default numbers, 20 thou is left on the diameter for finishing, which is 10 thou on the radius, so roughing has 115 thou of radius to remove. At 30 thou a pass that is 3.83 passes, which rounds to four, and the page spreads them evenly at 28.75 thou each rather than taking three at 30 and a fourth at 25. An even set of passes keeps the tool loaded the same way each time, and a thin final roughing pass is the one that rubs.
The dial is the second halving and nobody labels it
A direct reading cross slide dial moves the tool by what it says. A diameter reading dial is geared or marked so that the number on it is what comes off the diameter, which means the tool moves half of it. Set 28.75 thou on a diameter dial expecting a direct one and the pass takes 14.4 thou off the radius instead of 28.75 — half the cut, twice the passes, and no harm done. Do it the other way round and the part goes undersize on the first cut.
The page prints both the infeed and the dial reading in every row of the table, and the two are equal only when the dial reads direct. On a diameter dial the default roughing pass reads 57.5 thou.
What a spring pass is actually for
The tool, the toolpost, the cross slide and the work all bend under cutting load, and they bend away from the cut. That means the last pass left metal behind that the dial believes is gone. A spring pass takes the same setting again with nothing dialled in, and it cuts what the deflection hid. On a stiff short part it takes almost nothing. On a long slender shaft held between centres it can take a real chip, and a second spring pass will often still find something.
This is also why the finish allowance is not zero. A finish pass into 10 thou of clean stock deflects less than the roughing pass did, which is why the size it leaves is repeatable. A finish pass taken as an afterthought into the surface a heavy roughing pass left has both the deflection and the work hardened skin to deal with.
Boring runs the same arithmetic backwards
Set the operation to boring and the diameters swap roles: the starting bore is smaller and the finished bore is larger, the stock is still half the difference on the radius, and the diameter after each pass grows instead of shrinking. What does not carry over is the stiffness. A boring bar hanging out of the hole is far more flexible than a turning tool over the cross slide, the deflection is larger, and it is the reason boring usually wants more spring passes and a smaller finish allowance than turning does.
Where the paper plan stops
Backlash is absent from all of it. So is tool wear, which walks the size slowly across a long part. So is what the material does when it is cut: some alloys move as the stress in the bar is relieved, and a shaft that measured right in the chuck measures differently once it is out of it. The plan above is what the dial should read on each pass if the machine does exactly what the numbers say, and its value is in getting the halvings right and the pass count sensible before any metal moves. The micrometer settles the rest, and it is the only thing that does.
Questions people ask
How do I work out the depth of cut per pass on a lathe?
Take the difference between the starting and finished diameters, halve it to get the stock on the radius, subtract whatever you are leaving for finishing, and divide by the depth you want to take. Two inches down to 1.750 with 20 thou left on gives 115 thou of radius to rough; at 30 thou a pass that is four passes at 28.75 thou each once it is spread evenly.
Is depth of cut on the radius or the diameter?
The tool moves on the radius and the part loses twice that on the diameter. Which number your machine wants depends on how the cross slide dial is graduated: a direct dial moves the tool by what it says, a diameter reading dial takes what it says off the diameter and moves the tool half as far. The page prints both for every pass so the halving only has to be got right once.
What is a spring pass?
A pass taken with nothing dialled in, repeating the last setting. It exists because the tool and the work bend away from each other under cutting load, so the previous pass left metal behind that the dial thinks is gone. On a stiff part a spring pass takes almost nothing; on a long slender shaft it takes a real chip and a second one often still finds something.
How much stock should I leave for a finish pass?
That is a judgement about your machine, your tooling and the material, and this page takes it as an input rather than choosing for you. The reason to leave any is that a finish pass into clean stock deflects less and repeats better than one taken into the work hardened surface a heavy roughing cut left. The default of 20 thou on the diameter is a placeholder to replace, not a recommendation.
Does the pass plan account for backlash and tool wear?
No. It assumes the dial means what it says and that the tool does not change size during the job. Backlash means an infeed made after reversing direction is partly slack rather than metal, and wear walks the size slowly along a long part. Both are handled by measuring rather than by arithmetic, which is why the last word on size is always the micrometer.