Oblique balances, compasses, and the squaring of lunes

Why an oblique balance wavers

The inequality of weight and length the balance-arms acquire on becoming oblique is said to cause the balance's wavering, generating an impetuous motion that transfers the heaviness of the upper arm to the lower. Leonardo attributes this transfer to the thickness of the pivot. Three figures of tilted balance-beams accompany the text.

Valleys rise as rivers carry earth to the sea

A note above the figures states that valleys continually become higher in surface the nearer they lie to the sea. Leonardo grounds this in an earlier proposition that rivers ceaselessly carry earth from the mountains down to the sea and carry nothing back from sea to mountains.

A lune and a triangle shown equal

Working from two opposed and divided semicircles, Leonardo argues that the whole lune a b c d equals the whole sector, while the portions a b equal the single portion d. Taking these away, he is left with c d f equal to the whole lune and the whole triangle. The reasoning repeatedly applies the rule that equals removed from equals leave equals.

Comparing circles by doubling

Lettered semicircles are compared on the principle that b c d e derives from a circle double the one giving b c, so that a b c equals b c d e and b c equals a. This 'doubling' relation between circles underlies the equalities drawn across the lower half of the sheet.

Compasses and a drawing instrument

Several compasses are drawn and labelled, one near the tear noted as 'half a pivot and compass-head.' A detail of an instrument for drawing is described as having 'one fixed and two movable' parts. These tool sketches sit among the mechanical figures of the upper half.

Metà superiore, girato il foglio sul lato sinistro. Tre figure di aste di bilancia oblique:

Sopra le precedenti figure:

Compasso segnato:

Altro compasso, con:

Altro compasso, con:

Presso la lacerazione, mezza testa di compasso:

Operazioni aritmetiche:

Metà inferiore, riquadro destro. Parte di semicerchi colle lettere:

Semicerchi con:

Doppia figura di frammenti di cerchi:

Due semicerchi contrapposti e divisi con:

Cerchio su semicerchio, con:

Figure collegate con porzioni:

Margine sinistro; elementi rettilinei e curvilinei fra parallele, con:

Riquadro sinistro. Figura con:

Particolare di strumento per disegnare:

Al centro del riquadro, foglio ruotato di 90°.Tre piccole figure, con:

Upper half, the sheet turned onto its left side. Three figures of oblique balance-beams:

Above the preceding figures:

A compass marked:

Another compass, with:

Another compass, with:

Near the tear, half a compass-head:

Arithmetical operations:

Lower half, right panel. Part of semicircles with the letters:

Semicircles with:

Double figure of fragments of circles:

Two opposed and divided semicircles, with:

Circle upon semicircle, with:

Figures joined with portions:

Left margin; rectilinear and curvilinear elements between parallels, with:

Left panel. Figure with:

Detail of an instrument for drawing:

At the centre of the panel, the sheet rotated 90°. Three small figures, with:

On this page

Why an oblique balance wavers

Valleys rise as rivers carry earth to the sea

A lune and a triangle shown equal

Comparing circles by doubling

Compasses and a drawing instrument