It seems a number of people are confused by the X,
Y and Z axes. It is better to start without thinking of any machine. The X and Y
axes are at right angle to each other and cross the axis of the workholding
device. The Z axis is in line with, towards and away from the workholder. Thus,
the outline that Peter wrote is correct for a vertical milling machine. For a
lathe, the X axis is the cross slide and the Z axis is the carriage and any
movement of the tailstock quill. There is no Y axis on a lathe, unless you mount
a vertical slide on the carriage. It is easier to consider all movements in
relation to the workholding device.
HTH
???????
Ellis