Two halvings and a ratio
Setting the tailstock over tilts the axis of the work relative to the path the carriage travels, so the tool cuts a cone. How much cone you get depends on two things: how far the tailstock moved, and how much of the work is being cut. The formula is half the diameter difference multiplied by the distance between centres over the length of the taper.
With the defaults — 1.000 down to 0.900 over 8 in of taper on work 12 in between centres — that is 0.050 times 1.5, or 0.075 in of setover. Notice how easily two different wrong answers appear. Forget the halving and you get 0.150. Forget the ratio and you get 0.050. Both look like plausible setover numbers and neither cuts the part on the drawing.
Why it is half
Move the tailstock centre sideways by one thou and the axis of the work tilts. At the far end of the taper the tool is now one thou deeper on one side and one thou shallower on the other, so the diameter has changed by two thou. Diameters always move at twice the rate of a sideways displacement, which is the same reason a cross slide infeed of one thou takes two off the diameter. It is the same halving wearing different clothes, and it appears on every manual lathe control that moves a tool or an axis rather than a diameter.
The tidy result that falls out of it
Rearrange the formula and the taper length cancels: taper per foot equals 24 times the setover divided by the distance between centres. On 12 in between centres, an inch of setover would give 2 in per foot, so a thou of setover is worth 2 thou per foot of taper, whatever length of the work is actually tapered. That is a useful thing to carry around, because it turns the setover dial straight into the units taper gauges and drawings use, with no reference to the job at all.
It also explains why short work between centres is hard to set a fine taper on. Halve the distance between centres and every graduation of setover is worth twice as much taper.
Where the arithmetic and the metal part company
The formula treats the centre holes as points. They are not — they are 60 degree cones, and when the axis is tilted the centre no longer sits square in them. The bearing moves to one side, the contact area shrinks, and the effective pivot is not quite where the geometry assumes. On a gentle taper the difference is small enough to be swallowed by a trial cut and a correction. On a steep one the centre holes wear visibly, the work is poorly supported, and the finish tells you before the gauge does.
This is the reason the method survives for long shallow tapers and is abandoned for anything steep, where a taper attachment or the compound rest does the job instead. It is not that the arithmetic gets worse. It is that the setup does.
Included angle, half angle, and which machine control wants which
A drawing that calls out an angle for a taper normally means the included angle, measured across both flanks. The compound rest is set to half of that, measured from the axis. The setover is neither of those — it is a length. The default taper here is 0.150 in per foot, which is 0.7162 degrees included and 0.3581 degrees from the axis, and 0.075 in of setover. Three numbers describing the same cone, and putting any one of them into the control that wanted a different one is a scrapped part rather than a slightly wrong one.
Questions people ask
What is the tailstock offset formula?
Half the difference between the large and small diameters, multiplied by the distance between the centres divided by the length of the taper. For 1.000 down to 0.900 over 8 in of taper on work 12 in between centres, that is 0.050 times 1.5, which is 0.075 in of setover.
Why is the tailstock offset half the diameter difference?
Because moving the centre sideways changes the radius, and a diameter changes at twice the rate of a radius. One thou of setover puts the tool one thou deeper on one side and one thou shallower on the other, which is two thou on the diameter. It is the same factor of two that makes a one thou cross slide infeed take two thou off a turned diameter.
How does taper length affect the setover?
Through the ratio of the distance between centres to the tapered length. The setover acts over the whole length between the centre holes, but the taper only occupies part of it, so a taper confined to 8 in of a 12 in shaft needs the setover multiplied by 1.5. When the taper runs the entire length the ratio is 1 and it disappears, which is the case most people remember and the reason the ratio gets left out on the jobs where it matters.
How do I convert a setover to taper per foot?
Taper per foot equals 24 times the setover divided by the distance between centres. The tapered length cancels out. On work 12 in between centres, a thou of setover is worth 2 thou per foot of taper regardless of how much of the shaft is being tapered, which makes a setover dial directly readable in the units a taper gauge uses.
Is the offset method as accurate as a taper attachment?
This page will not rank them, and the honest answer is that they fail differently. Offsetting swings the centre holes off the work axis so the centres bear on one side, which wears the holes and supports the work worse as the setover grows. That is a setup problem rather than an arithmetic one, and it is why the method is normally kept for long shallow tapers. Whether the result is within the tolerance on your print is settled by a gauge, not here.