Angle and History

The sesqui-quadrate belongs to the eightfold family of divisions and is among the aspects most confidently credited to Johannes Kepler by name, set out in the Harmonices Mundi of 1619 together with the quintile and biquintile as angles derived from consonant musical ratios rather than from the older twelvefold sign scheme. Its glyph reflects its composite definition directly: the semi-square's open-angle sign drawn combined with the square's sign, so that the mark itself states the arithmetic, a square's ninety degrees plus a semi-square's forty-five.

Of the minor aspects it is one of the few to have been given a dedicated place in the Unicode Standard, SESQUIQUADRATE at U+26BC in the Miscellaneous Symbols block, sitting in the character table immediately after the quincunx, itself one place removed in kind though not in origin, since the quincunx's admission as an aspect is credited to the later harmonic tradition rather than to Kepler specifically. The narrow orb customarily given to the sesqui-quadrate reflects its status as one of the weaker, later-recognised relations even within Kepler's own scheme. Kepler set out his reasoning for admitting it, and for the quintile and biquintile alongside it, in the fourth of the five books that make up the Harmonices Mundi, the volume most directly concerned with astrology among a work that ranges across geometry, music theory and what became his third law of planetary motion.