Dividing lines, gnomons and decomposed squares

A line divided into equal and unequal parts

A horizontal segment is split into two equal halves marked 6 and 6, and a companion segment into unequal parts 8 and 4. A further line combines both, divided 10 and 2, then 6, 4, 2. These divisions are the base on which the rectangles below are constructed.

Rectangle and square applied to the same line

A rectangle is applied to the given line, and beside it a rectangle and a square are set on the same base. The parts carry the measures 16, 8, 4 and 6, 4, 2. The figures test how areas relate to the divided segment beneath them.

Construction and enlargement of the gnomon

A gnomon (an L-shaped carpenter's-square figure) is constructed from squares marked 12 and 12, then shown enlarged with the labels 12 and 4. A decomposed square marked 16, 8 and 8 accompanies it. The sequence studies how a square can be extended into a gnomon.

Two similar but differently decomposed figures

Two figures that are similar in outline are broken up into different sets of parts, and an analogous figure carries the labels 20 and 4. A note in the left column reads 'geometric proof.' The comparison is meant to show equal figures reassembled in unlike ways.

Numerical measures of the divided segments

Each construction is annotated with the arithmetic of its parts: 6 and 6, 8 and 4, 10 and 2 summing to 20, and 16, 8, 4. The numbers turn the geometric divisions into a running check of lengths and areas. They accompany rather than replace the drawn figures.

Linea divisa in parti uguali, con:

Linea divisa in parti ineguali, con:

Linea divisa in parti uguali e ineguali, con:

Rettangolo applicato alla linea data.

Rettangolo e quadrato applicati alla stessa, con:

Costruzione dello gnomone, con:

Colonna sinistra.

Due figure simili ma diversamente scomposte.

Figura analoga, con:

Gnomone e suo ampliamento, con:

Quadrato scomposto, con:

Line divided into equal parts, with:

Line divided into unequal parts, with:

Line divided into equal and unequal parts, with:

Rectangle applied to the given line.

Rectangle and square applied to the same, with:

Construction of the gnomon, with:

Left column.

Two similar but differently decomposed figures.

Analogous figure, with:

Gnomon and its enlargement, with:

Decomposed square, with: