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The factorization 3-D reconstruction method requires that all feature points must occur in all images in a sequence. A long sequence has to be divided into multiple subsequences for partial reconstructions. This paper proposes an algorithm for dividing a long sequence for factorization-based Structure and Motion (SaM). First, we propose an Algorithm for Detecting a few Critical Configurations (ADCC) where Euclidean reconstruction degenerates. The critical configurations include: (1) coplanar 3-D points, (2) pure rotation, (3) rotation around two camera centers, and (4) presence of excessive noise and outliers in the measurements. The configurations in cases of (1), (2) and (4) will affect the rank of the scaled measurement matrix (SMM). The number of camera centers in case of (3) will affect the number of independent rows of the SMM. By examining the rank and the row space of the SMM, we detect the above-mentioned critical configurations. With the proposed ADCC algorithm, we are able to divide a long sequence into subsequences such that a successful 3-D reconstruction can be obtained on each subsequence with a high confidence. Experimental results on both synthetic and real sequences demonstrate the effectiveness of the proposed algorithm for an automatic 3-D reconstruction using the factorization method.