Studies of the balance: centres of gravity and arm ratios

Centre of gravity of a balance arm lies at its midpoint

A principle repeated across the sheet: the centre of weight of each arm of a balance — equal or unequal, whatever its length — always falls at the middle of its length. Leonardo states it several times, including in two struck-out notes at the top of the leaf.

Load a short arm sheds onto the long arm

Whatever proportion the shorter arm bears to the longer, such is the weight the shorter discharges onto the longer. This ties the transferred load directly to the ratio of the arm lengths.

Simple beam d e lifting at b and a

For the two beams lettered e d c b a, the short beam d e balances the whole of c d beyond the pole, so it lifts a weight equal to itself. Because b is twice, and a three times, as far from the pole as d e, the same d e lifts one half of itself at b c and one third of itself at a b.

Worked balance s a r f t with weights 6 and 8

On the beam s a r f t, the value 4 at r s (its midpoint being half that of t r) lifts two at t r, leaving 6 there; placing 6 at s makes it stand level with the 6 at f, and indeed level with the whole of r t, which totals 8.

Counterweights proportional to distances from the pole

The counterweights of the two arms stand in the same proportion as the distances between the pole and the centres of weight of the arms. Likewise the part taken from the longer beam relates to the whole shorter beam as the shorter beam relates to its opposite longer one.

In alto, due note biffate:

Figura di bilancia:

Due aste di bilancia, con:

Asta di bilancia con:

Asta di bilancia con:

Asta di bilancia:

Nel margine destro, aste di bilancia composita.

Colonna sinistra.

Figura colle lettere:

At the top, two struck-out notes:

Figure of a balance:

Two balance beams, with:

Balance beam with:

Balance beam with:

Balance beam:

In the right margin, beams of a compound balance.

Left column.

Figure with the letters: