Centres of Gravity of Pyramids, Prisms and Cubes

Centre of gravity of a truncated pyramid at 7/16

Beside a triangular prism carrying a balance scheme Leonardo notes that the centre of gravity of the half-pyramid lies at 7/16 of the way toward the base. He calls this the method for finding the centres of truncated pyramids (piramide scapezze). It is a general rule extracted from the numbered figures around it.

Three principal centres of every heavy body

In the upper-left corner he asserts that we have three principal centres of the mass of each heavy body, and that the others are infinite but all correspond to these three. The claim frames the whole page of centre-of-gravity studies. It is tested against the balance schemes drawn below.

Proportion of equal squares equals that of their heights

Repeatedly Leonardo argues that when the sides of several squares are equal among themselves, the proportion between those squares equals the proportion of their heights. He restates the rule in the lower-left corner for equilateral squares arranged so that four sides of one equal four sides of each of the others. The figures of cubes divided into numbered squares illustrate the relation.

Decorum to be observed in figures

At the centre of the upper margin Leonardo sets down a precept of painting: in figures a certain decorum must be observed according to their rank, their quality, their age, their kind and their dignity. It is an art-theory aphorism inserted among the mechanical studies. The rule governs how painted figures should be characterised.

Marginal animal and hand studies

Small sketches in the lower-left margin include what appears to be an animal study and a drawn hand, set apart from the geometric figures. They are faint and not transcribed in the bundle. Such marginal life studies recur among Leonardo's technical sheets.

Al centro del margine superiore:

Nell'angolo superiore sinistro:

Sotto la precedente nota, disegno di tronco di piramide adagiata su un fianco, segnata con crocetta, colla base divisa in quadretti numerati:

Prova della bilancia coi numeri:

Al centro del foglio figura analoga, segnata con crocetta, coi numeri:

Sotto la precedente, analoga figura, segnata con croce, e coi numeri:

Prova della bilancia coi numeri:

Sul lato destro, dall'alto: figura di prisma triangolare con schema di bilancia nel centro e i numeri:

Analoga figura sotto la precedente:

A sinistra della penultima:

Sotto la precedente, figura di parallelepipedo:

In basso:

A sinistra della adiacente figura colla prova della bilancia:

Più in alto, figure di parallelepipedo e prismi coi numeri:

A sinistra, figure di prismi, uno dei quali colle lettere:

Lungo il margine sinistro:

Figura di prisma triangolare troncato della parte superiore, coi numeri:

Figura analoga più in basso coi numeri:

Nell'angolo inferiore sinistro:

At the centre of the upper margin:

In the upper-left corner:

Below the preceding note, a drawing of a truncated pyramid lying on one side, marked with a little cross, its base divided into numbered squares:

Proof of the balance with numbers:

At the centre of the sheet, a similar figure marked with a little cross, with the numbers:

Below the preceding, a similar figure marked with a cross, and with the numbers:

Proof of the balance with numbers:

On the right side, from the top: a triangular prism with a balance scheme in the centre, and the numbers:

A similar figure below the preceding:

To the left of the second-to-last:

Below the preceding, a figure of a parallelepiped:

At the bottom:

To the left of the adjacent figure with the proof of the balance:

Higher up, figures of a parallelepiped and prisms with numbers:

On the left, figures of prisms, one of them with the letters:

Along the left margin:

A triangular prism truncated at its upper part, with numbers:

A similar figure lower down, with numbers:

In the lower-left corner:

On this page

Centre of gravity of a truncated pyramid at 7/16

Three principal centres of every heavy body

Proportion of equal squares equals that of their heights

Decorum to be observed in figures

Marginal animal and hand studies