Martin Perry wrote:
I have an old Boxford lathe. The bed shows signs of wear but otherwise
it is in excellent working order. Everything works smoothly without
any undue slack.

I checked the accuracy the other day by turning an eight inch length
of two inch diameter bar. There was a slight taper on the finished
work. The bar was 0.001 inches smaller nearest to the chuck.

This is most likely due to workpiece deflection. Even a 2" bar,
especially mounted in a small chuck, will deflect some. A better
way is to use a bar of known and constant diameter, and mount it without
wobble, then read it with a dial test indicator.

..001" over 8 inches is no big surprise. If you need more accurate
diameter, you may need to turn between centers, and may also need to
grind to final diameter.

I then bored a two inch diameter hole in a four inch diameter bar and
found that the bore was 0.001 inches larger nearest to the chuck.

This is also no surprise. Again, workpiece deflection, but you are pushing
in the opposite direction.

These two errors seem to be in opposite directions. Can anybody
suggest an explanation please.
No, they are not opposite. Both are most likely caused by the cutting
forces deflecting the work.

As you have already figured out, it is not a misalignment between
the spindle and bed, as this would have caused the opposite pattern
in the ID boring job.

Jon