Geometry and arithmetic exercises with wave studies

Proving a right angle by Euclid

Beside a triangle Leonardo writes that the angle n is to be proved a right angle by the penultimate proposition of the first book, and that where squares are equal the sides are equal by the eighth. The columns are labelled 'Arithmetic' and 'Geometry'.

Right-triangle numbers and the gnomon

Triangles are annotated with the sets 5 4 3 and 6 5 - 12, and a separate figure is labelled 'Gnomon'. A square carries the equality a n will be equal to m S, tied to references to the twenty-second and twenty-eighth propositions.

Multiplication reaching square numbers

A worked column adds and multiplies: 6 and 6, add 4, four times sixteen makes 64, six times six makes 36, together 100; ten times ten also makes 100. Elsewhere 12 yields 144 through a chain of 64, 32 and 16.

Water that falls less obliquely hollows more

Below the wave and vortex sketch: every motion follows its beginning, whether straight or oblique. Water that falls along a less oblique line hollows out more, and the motion of water within water behaves like the motion of air within air.

Foglio capovolto, da destra a sinistra.

Triangolo con:

Quarta figura:

Colonna successiva.

Triangolo con:

Idem con:

Triangolo con:

Colonna successiva.

Quadrati con:

Figura di:

Colonna centrale.

Terza figura con:

Quinta figura con:

A destra, figura con:

In basso:

A sinistra.

Linea con:

Linea divisa in:

Linea ampliata con:

A destra delle figure:

Figura con:

Foglio ricapovolto, a destra. Figura mutilata con:

Linea divisa in:

A matita, lungo il margine:

A sinistra del precedente:

A sinistra della successiva figura:

Foglio capovolto, in basso a destra. Figura di onda e vortici:

Figura di onda:

Sheet turned upside down, from right to left.

Triangle with:

Fourth figure:

Next column.

Triangle with:

The same with:

Triangle with:

Next column.

Squares with:

Figure of:

Central column.

Third figure with:

Fifth figure with:

On the right, figure with:

Below:

On the left.

Line with:

Line divided into:

Line extended with:

To the right of the figures:

Figure with:

Sheet turned back over, on the right. Mutilated figure with:

Line divided into:

In pencil, along the margin:

To the left of the preceding:

To the left of the next figure:

Sheet turned upside down, lower right. Figure of a wave and vortices:

Figure of a wave:

On this page

Proving a right angle by Euclid

Right-triangle numbers and the gnomon

Multiplication reaching square numbers

Water that falls less obliquely hollows more