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 quise iactantamplioremhabere artem algebricamquamin sexcapitulisvulgatissimis traditur. Sed ipsi profectoignoranthanc artem ad cubos, census censuum, atque ulteriores potentias extendi non posse nisi prius geometria solidorum equipollentium edatur. Quemadmodum enim triacapitulacomposita superficierum equipollentiisnituntur, 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):

Manyflatter themselves that they understand thehigher(ampliorem) algebrafromthe six standardforms. But they completelyignorethe 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 composedforms(of quadratic equations)are proved by means offigures 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 whoboast that they have amore extensivealgebraic artthanis handed down in the six most commonly known capitula. But these people clearlydo not knowthat 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 capitularely onequivalences 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 |

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