A brief note on Renaissance algebra

While investigating the background to Cardano's use of ‘capitulum’ to refer to a type or category of algebraic equation, I came across an interesting passage from a letter written by Regiomontanus in 1471:

Sunt enim qui se iactant ampliorem habere artem algebricam quam in sex capitulis vulgatissimis traditur. Sed ipsi profecto ignorant hanc artem ad cubos, census censuum, atque ulteriores potentias extendi non posse nisi prius geometria solidorum equipollentium edatur. Quemadmodum enim tria capitula composita superficierum equipollentiis nituntur, ita novum artis additamentum ex commutatione solidorum hauriatur [? l. hauriri] necesse est.

Menso Folkerts, to whom all historians of medieval and Renaissance mathematics must be grateful, translates this as follows (1996):

Many flatter themselves that they understand the higher (ampliorem) algebra from the six standard forms. But they completely ignore the fact that this art cannot be extended to cubes or to fourth and higher powers, unless the geometry of solids of equal volume is first treated. Just as the three composed forms (of quadratic equations) are proved by means of figures of equal area, so the new extension of the art must be based upon the transformation of solids.

Which is puzzling, because the Latin is conspicuously rather different:

For there are those who boast that they have a more extensive algebraic art than is handed down in the six most commonly known capitula. But these people clearly do not know that this art cannot be extended to cubes, squares of squares, and further powers unless the geometry of equivalent solids is provided first. For just as the three compound capitula rely on equivalences between surfaces, so the new addition to the art must be drawn from the transformation of solids.

At any rate, the sex capitula in question are from al-Khwārizmī's Algebra, and it might be helpful to list them here in modern notation:

Simple	Compound
ax² = bx	ax² + bx = c
ax² = c	ax² + c = bx
bx = c	bx + c = ax²

Codex on Aristotle's Doctrine of the Mean

When I first studied the Nicomachean Ethics, I did so with the help of Urmson's wonderful book Aristotle's Ethics (1988), which surprised me with its strictures against interpreting the doctrine of the mean as a simple thesis of moderation.  Surely, I thought, no one could have read that into II.6.  Now, six and a half years later, I've had a glimpse of what Urmson meant – in the pages of an Inspector Morse novel:

“Morse skipped his way along [the report].  ‘… would suggest a period of between 72–120 hours before the body was discovered.  Any greater precision about these time limits is precluded in this case…’  As in all cases you ever have, muttered Morse.  He had never ceased to wonder why, with the staggering advances in medical science, all pronouncements concerning times of death remained so disconcertingly vague.  For that was the real question: when had Quinn died?  If Aristotle could be believed (why not?) the truth would probably lie somewhere in the middle: 94 [sic] hours, say.”

By coincidence, I was only reading this because of a sloppy piece of academia.  In Twentieth-Century Crime Fiction (2005), Lee Horsley, comparing the relationship between the authors and readers of detective fiction to that between the setters and solvers of cryptic crosswords, promised me that “Colin Dexter also gestures towards parallels between the mystery story and the cryptic crossword in The Silent World of Nicholas Quinn (1977), which involves Inspector Morse with a suspect who is a crossword-setter called Daedalus.”  Alas, this turns out to be an embellishment from Julian Mitchell's TV adaptation (1987), which made Ogleby a more interesting character.