Geometric Algebra: the Ninth and Tenth Demonstrations
Segments divided into equal and unequal parts, with a triangle and figures proving the results.
This page carries two of Leonardo's 'Alcibra' (geometric-algebra) demonstrations, numbered the Ninth and the Tenth. For the Ninth he draws a line segment divided into equal and unequal parts (6 and 6, then 3 and 3) and a triangle used to demonstrate the theorem, labelled 6, 3, 3. The Tenth then divides a longer line into 12 = 6 + 6 + 4 and sets beside it a figure demonstrating the same relation with the parts 6, 6, 4. The work belongs to the Euclidean study of how a line, when divided and squared, yields a figure decomposable into the squares and rectangles of its parts.
On this page
Ninth demonstration: a segment split into equal and unequal parts
The Ninth demonstration opens with a line divided into equal and unequal parts, marked 6 and 6, then 3 and 3. It sets up the relation between a whole segment and the sub-lengths into which it is cut, in the manner of Euclid's propositions on divided lines.
Triangle proving the theorem
A triangle is drawn to demonstrate the theorem, its parts labelled 6, 3 and 3. The figure translates the numerical division of the line into a geometric proof.
Tenth demonstration: dividing 12 into 6 + 6 + 4
The Tenth demonstration divides a longer segment into 12 = 6 + 6 + 4 and places beside it a demonstrative figure carrying the same parts, 6, 6 and 4. The numbers track how the whole and its pieces combine.