Abstract: An apparatus and a method for aligning a vertical shaft or multiple axially coupled vertical shafts in a hydroelectric turbine generator or a similar vertical-shaft system, and for providing precise plumb inclination alignment of a vertical rotating shaft. Precision inclinometers attached to the vertical shaft measure plumb inclination. Proximity probe displacement devices mounted externally of the vertical shaft measure radial movement, throw, or run out at various shaft elevations as the shaft is rotated relative to a fixed point. Data acquisition devices and communication devices accumulate and transmit alignment data to a micro-computer which receives and processes such data. Methods of defining shaft plumb inclination in a static single rotational position and defining plumb inclination of the virtual centerline of a shaft's rotational throw position. A method of swinging, tilting, or adjusting a vertical shaft to a corrected or different plumb position relative to the earth's gravity.