A method for cube roots using fractions
Leonardo's fractional cube roots, cubic braccia and dice
On folio 200r Leonardo sets out a method for finding cube roots by means of fractions, which he claims is his own discovery. He writes the cube root of 3 as 3/9, of 4 as 4/16 and of 6 as 6/36, checking each by cubing the fraction to recover the whole number. He extends the idea to cubic measure, reading 8/4 as eight dice each holding a quarter of a cubic braccio, and explains that only the numerators are operative while the denominators name the parts used. Three small figures with numbers accompany the calculations.
On this page
Cube roots written as fractions
Leonardo gives the cube root of 3 as 3/9, of 4 as 4/16 and of 6 as 6/36, verifying each by squaring and then cubing the fraction to return the original whole number. He claims that by this method all cube roots can be found as rational fractions and that no one before him had done so.
Cube roots and cubic measure
Extending the method to solids, he multiplies fractions as with whole numbers, reading 2/4 cubed as 8/4, that is eight dice each holding a quarter of a cubic braccio. He notes that only the numerators above the line are operative, the denominators serving only to name the parts, so that '1/4' means a cube containing the fourth part of a greater cube.