A method and apparatus is disclosed for measuring motion signals of gyroscopes in downhole instruments used to determine the heading of a borehole. An illustrative embodiment of the invention includes a measuring-while-drilling system which may experience motion even while the drill string is suspended in rotary table slips when the heading of the drill string is being determined. Accelerometer and magnetometer data along three orthogonal axes of a measurement sub are used to obtain unit gravitational vectors g at a first time and at a second time and unit magnetic vectors h at the first time and the second time. The difference between the two unit gravitational vectors at the different times, .DELTA.g, and the difference between the two unit magnetic vectors at the different times, .DELTA.h, are used along with the unit vectors g and h and the difference in time .DELTA.t to determine the rotation vector of the probe .OMEGA..sup.p which has occurred during such time difference. The vector representing the rotation of the earth, .OMEGA..sup.e is then determined by subtracting .OMEGA..sup.p from the vector .OMEGA..sup.g from three gyroscope instruments placed along the axes of the measurement sub. The heading of the drill string is determined from the gravitational vector and the earth rotation vector.