Sunday, 26 May 2013

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