Forty five degrees is half of ninety, and the semi-square is the eighth harmonic's defining division. Its relation to the square is exact and arithmetical, which is the source of both its plausibility and the objection to it. If a quarter of the circle is a real configuration, the argument runs, an eighth ought to be as well. The counter-argument is that the square is not important because it is a quarter but because it joins signs of a shared modality, and forty five degrees joins nothing at all, since eight does not divide twelve.
Like the quintile it comes from Kepler and the search for aspects among the regular figures, and like the quintile it was taken up seriously only in the last century and a half. Its usual companion is the sesqui-quadrate at a hundred and thirty five degrees, which is three eighths of the circle and belongs to the same harmonic; the two are often treated as one family and read together, and some writers group them with the square under the heading of the hard aspects.
Practice gives the semi-square a narrow orb, commonly two degrees, and a reading of persistent low-level irritation, friction that never rises to the level of crisis and never quite goes away. There is no traditional testimony to support that description and it should not be presented as though there were. What can be said without reservation is the geometry: forty five degrees, an eighth of the circle, no relation to the signs, and a documented origin in the seventeenth century rather than in antiquity.