Dave –
You could be onto something with that drawing, but surely it would need a plunger not lever type indicator?
(The jig shown by Hopper would also need a plunger indicator, and its drawing seems to give that.)
My thought though was perhaps for measuring the actual diameter to fine limits readily than by using a rather unwieldy caliper, using chord geometry to measure the radius.
The only fly-in-the suds though is that there appears in the photo to be no method for accurately measuring the ball-to-DTI centres. Unless that was done in some way on a surface-table or against a reference gauge, and then the method becomes one of change from that set datum.
So I looked back at your drawing….
… and played Whatif.
IF the handle stays in the position shown in Hans' photo, as you have drawn it, and the plate between it and the rest of the device has a ground outer face hidden in the picture….
Also, note there are two "shiny things" on the device, one each side, not just one.
Place the instrument as you have drawn, on the surface-plate with the appropriate ball pair for the size-range.
Use parallels and slips to set the ball axis at the lathe's centre-height, but from where the device will stand on the bed or saddle; not necessarily the lathe's nominal centre-height.
We already know the height to the indicator plunger, as that is constant.
Now set the balls and DTI to touch an angle-plate with the indicator reading 0.
Transfer the instrument to its predetermined place on the lathe, where with the balls just touching the work, the radius is a function of the half-chord length given by the ball to indicator centres, and the DTI deflection.
Having two spheres symmetrical about the axis automatically aligns it correctly perpendicular to the work axis.
Geometrically, the deflection is the chord's distance from its diameter intersection to the circumference.
.
Thinking on again, if this is all so, it use need not just be on the lathe but also on some other machine-tool types or a surface table, as long as the centre-distance from the datum plane is known. In some situations, e.g. a plano-mill, it might need clamping to an angle-plate or some other vertical, true surface.
It might also measure a large-radius sphere, since the two reference-balls will always place the instrument's axis on a diametral plane.
Given that it has a lever indicator it might have been intended as a comparator against a reference radius rather than direct measuring tool.
.
I don't know if this the right answer, nor if what I suggest would work: the lever DTI shown may not be the right type, and it does rely on full concentricity so is not for initial setting-up.
Worth a thought though….
…. and I'm still intrigued by that round thingamyjig with its strange numbers we saw back in, err, 2019?! (A calibrated Tuit?)
vice