1. Newton method
1.A Estimate the square root and enter it via the input sliders.
1.B Calculate the square of the estimate.
1.C Mentally double the estimate and adjust the sliders appropriately.
1.D Add and subtract (with appropriate shifts) until the original number appears in the total register on top.
1.E The number in the turns counter register on top is an improved approximation to the square root.
1.F Repeat steps A-E until no improvement in the estimate occurs on successive iterations.
2. Dibble-dabble method
2.A Shift the carriage full CCW and zeroize.
2.B Enter "1" in the most significant input slider (IS).
2.C Add ONCE. If the value in the total register exceeds the target number, then subtract out the last number added and go to step F.
2.D Increase the value in the currently active IS by mentally adding "2". If adding "2" to "9" change the "9" to "1" and add "1" the the next left input slider (not relevant first time through).
2.E Repeat steps C & D.
2.F Reduce the setting of the current IS by "1" and shift the carriage CW one step. If the limit of carriage travel is reached, go to step I.
2.G Enter "1" in the next most significant IS (the one to the right of the one previously used).
2.H Repeat steps C through G.
2.I The number in the "turns counter" register is the true square root.
I purchased my Curta I in Vaduz in 1958 for $70 US. Both these algorithms were printed in the instruction book and described much more clearly than I have recited them here. This is reproduced from my memory, so if one of them doesn't quite work for you, consider that I may have slightly misremembered a step.
The principle involved in method #2 is that the sum of a series of odd numbers is always a perfect square. Of course, this principle requires some adaptation when the number being rooted is larger than 81 (9^2) or does not have an integer root.
Dan Massey email@example.com