Successive halving and the vessel of wine and water
A bisection table carried to 127/128, and propositions that a fed-and-drained vessel never wholly changes its liquid
The upper part of the sheet is a table headed 'always add 1/2', halving repeatedly and adding a further half at each step so that the running fractions climb 3/4, 7/8, 15/16, 31/32, 63/64, 127/128; a branching diagram of fractions and a marginal column repeat the sequence. Below and in two columns Leonardo works out a related thought-experiment: if a vessel full of wine is repeatedly half-emptied and topped up with water, the water can never become pure and free of wine, however often it is done. Parallel propositions labelled 'Conceptione' argue that a vessel which continually receives as much water above as it pours out below can never be entirely stripped of the liquid that first filled it.
On this page
Table of successive halving ('always add 1/2')
A stepped table halves a quantity and adds half again at each stage: 1/2, then 1/4 and 1/4 making 3/4; 1/8 and 3/8; 1/16 and 7/16 making 15/16; on down through 1/32, 1/64 and 1/128, the remainders approaching unity as 31/32, 63/64, 127/128. It sets out an unending geometric bisection.
Marginal column of fractions
A short column in the left margin repeats the halving sequence in stacked form, from whole numbers down through 1/4, 1/8 and 3/8 to 1/16 and 7/16, echoing the main table.
The vessel of wine and water
If a vessel is full of wine and you empty half and add as much water, then again empty half and refill with water, you can repeat this diminution and addition infinitely, yet the water will never remain simple and without wine. A vessel that continually pours out below as much as it receives above still cannot exchange enough to lose all its own liquid.
Propositions ('Conceptione') on inexhaustible mixture
A full vessel cannot receive more water than it pours out; and though it continually takes in above and lets out below, it can never be wholly deprived of the liquid that first filled it. Even if half were removed and replaced infinitely many times, some part of the first water would always remain.
