Centres of gravity, a waterfall, hexagons and proportions

One centre of gravity at the crossing of three medians

The right column restates in four near-identical proofs that a triangle has three medians, each dividing it into two equal parts and each carrying the centre of gravity, so three apparent centres reduce to one at their intersection. A closing sentence generalises: where the lower central lines of the weights of a body's equated parts intersect, there is the centre of gravity of the whole body.

Falling water rebounds and scours the bed

Water descending at a slight slant strikes the bottom and would rise at a similar slant, were it not blocked by the water where it strikes. That water pours it back, and it falls again on the succeeding water, penetrating to the bottom, where it wears away much of it. A drawing accompanies the note.

An inscribed hexagon is half the larger hexagon

Triangle a b c is 1/6 of the hexagon a b c g d f e and half the triangle a b d, which is 1/6 of the larger hexagon g h i K b d g. Therefore the smaller hexagon is half of the larger. A drawing shows the inscribed hexagon and its extension.

Rule of three and the proportion of squares

Under the heading 'Rule of three', columns of numbers accompany the claim that equilateral squares stand in the same proportion as their heights when their sides are equal. Leonardo warns the proposition seems to give only equality but in fact yields proportions of infinite variety.

Squaring circle-sectors of an eighth and a sixteenth

Having 1/4 and 1/16 of the same circle (a b and o p), Leonardo removes o from each piece, leaving p squared; since a b was three times o p, he transposes p onto b so that the remainder of b a equals p with its addition. Overlapping sectors are labelled eighth, sixteenth, and 'double' or 'equal' circles.

Refutation: restoring equality between doubled quantities

Taking 4 and 8 (a double ratio) and subtracting 2 from each leaves 2 and 6, now a triple ratio; drawing 2 from 6 and adding it to 2 restores 4 and 4, the ratio of equality. Leonardo tests the method on 8 and 16 and on 6 and 12, heading it 'Refutation of the opponent'.

Foglio 526 a Verso

Margine destro, verticalmente:

Colonna destra.

Colonna centrale.

Figura di una caduta d'acqua:

Esagono iscritto e suo prolungamento, con:

A destra del precedente:

Regola del tre:

Foglio 526 b Verso

Colonna centrale. Semicerchio con:

Triangolo in semicerchio, con:

Settori sovrapposti:

Idem con:

Sul margine inferiore, settori sovrapposti, con:

Sopra la precedente figura:

Margine superiore sinistro. Settori sovrapposti con:

Al centro della pagina, gruppo di proporzioni:

In basso a destra, altri gruppi di proporzioni:

Al centro:

A sinistra:

Sheet 526a Verso

Right margin, vertically:

Right-hand column.

Central column.

A figure of a waterfall:

An inscribed hexagon and its extension, with:

To the right of the preceding:

Rule of three:

Sheet 526b Verso

Central column. A semicircle with:

A triangle in a semicircle, with:

Overlapping sectors:

The same, with:

On the lower margin, overlapping sectors, with:

Above the preceding figure:

Upper-left margin. Overlapping sectors with:

In the centre of the page, a group of proportions:

Lower right, other groups of proportions:

In the centre:

On the left: