Dividing a Cube into Pyramids and Their Weights
Proving the weight of pyramids is proportional to their bases
Folio 127r reasons about solid figures whose weight is treated as proportional to their bases. In red chalk and pen Leonardo sketches prisms, single pyramids and cubes cut into pyramids, and argues that a cube can be divided into five square-based pyramids whose apexes meet at the centre. Because the axis of the tallest pyramid is double that of the other four, it counts for two of them, and by fractions of the whole (1/8, 3/8, 5/8) he balances the smaller pyramid against the larger to make them equal in weight.
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Cube divided into five square-based pyramids
The cube has six sides and is divided into five pyramids of square base, whose bases are the sides of the cube and whose apexes all meet at the middle of its base, at point e. Because the axis of the larger pyramid is double the height of the other four, it is worth two of those pyramids.
Weight of a pyramid follows the ratio of its base
In pyramids of equal height and sides, the proportion of weight is as the proportion of their bases. The pyramid a b c d e, having a base four times the upper base, is therefore four times that upper pyramid, which had already been found to be one eighth of the whole.
Balancing the two pyramids to equal weight
The parts contain five eighths of the whole, so three eighths remain, embracing the larger pyramid; added to the smaller these make it equal in weight to the larger, which is the required result.
Five pyramids of doubled base
Here are five pyramids drawn in red chalk of equal height but of double base, the uppermost carrying the base of the other four.