An explication is offered for the notion of dimensionality both for tests and items. A set of n tests or of n binary items is unidimensional if and only if the tests or the items fit a common factor model, generally non-linear, with one common factor, that is, one latent trait. Both test scores and item responses in general contain stable specific factors as well as errors of retest measurement. The two-parameter normal ogive model can be obtained from a joint space which in general is of n + 1 dimensions. One of these is the latent trait continuum while the remaining n are dimensions of unique (specific and error) variation. If and only if the items fit the perfect scale the n + 1 dimensions collapse into one dimension. Proposals to regard coefficient alpha as a coefficient measuring homogeneity, internal consistency, or generalizability, do not appear to be well founded.