The balance: dividing the arms among four weights
Rules and numerical schemes for spacing weights along the beam of a scale
The right-hand column sets out rules for finding the divisions of a balance's arms and the intervals between several weights, worked out arithmetically by subtracting, doubling and taking remainders, for a beam of four arms carrying four different weights. A column of small balance-beam diagrams runs down the sheet, each annotated with pairs of numbers giving the positions and sizes of the weights. A faint red-chalk sketch resembling foliage or a cloud sits in the lower centre, and some faint marginal jottings along the lower-left edge are not part of the transcription.
On this page
Finding the divisions of the balance's arms
To find the divisions of the arms of the balance one subtracts the smaller number from the greater; here, subtracting 2 from 5 leaves 3, which gives the spaces from the fulcrum a to the end of the beam c. The rule extends to a four-armed balance carrying four different weights, dividing each arm by the remainder left after the operation.
The method of doubling and subtracting
To place two weights on one side, the smaller weight is doubled (2 becomes 4) and subtracted from the greater (5), the remainder giving the space between them. Leonardo restates the procedure twice, seeking the clearest wording for the numerical rule.
Counterweighting the greater arm with a middle weight
The arm bearing the greater weight can be balanced by placing at its opposite extremity one of the two middle weights, whichever is chosen. The note treats the free interchange of the middle weights as counterpoise.