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Measurement Without a Protractor
Place the part to be measured into the headstock chuck. Put a tool bit in a holder with the flat, un-ground end facing the collet. The tool bit should be positioned so that the top of it is at the lathe center height. Now, adjust the carriage and cross-slide to bring the edge of the tool bit up against the finished, outside diameter of the chuck. Use the chuck as a straight edge to help adjust the tool bit so that it is in line with the longitudinal axis of the lathe. When you think you have it in position, tighten the tool post screw. After tightening, bring the tool up to the edge of the chuck and make sure that it makes even contact along the entire length of the tool, adjust if necessary.
Next, loosen up the bolts that lock the compound rest, just enough to allow it to be easily turned. By the use of the carriage and cross-feed, the bit may now be brought right up to the collet. By moving the compound rest, you can easily get the bit to exactly contact the taper for its entire length. Hold a piece of white paper below the bit so that you can easily see any gap. You can now read the angle directly off the compound rest. This will be the angle of the taper with reference to the centerline of the lathe.
The same principle may also be used, if required, on very short internal tapers. While this may not be the most accurate way to measure angles, I think you will find that it is sufficient for the majority of situations.
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