Geometry of triangles and circles, with balance notes

Triangles on an equal base are proportional to their lengths

Of triangles of various lengths made on an equal base, the proportion from triangle to triangle is as from length to length. With triangle r a d double the height of triangle r o d, the larger is double the smaller, so its excess equals the whole of the smaller. Triangle a o d is therefore equal to triangle o c d, being on equal bases a o and o c between the parallel lines a e and c d.

Equal remainders from unequal circles

Two concentric circles are labelled 8 and 9: take one from the 8 and two from the 9 and equal amounts remain. Generalised, from unequal bodies, by removing unequal parts, it is possible that the remainder be equal.

Rebound of swinging weights increased by a blow

Beside a set of suspended, oscillating weights Leonardo notes that here the reflected (rebounding) motion is increased by the blow of the opponent.

Excess of superimposed double surfaces

When two double surfaces are laid wholly one upon the other, all the excess of the larger is equal to the whole of the smaller surface. The claim is illustrated by concentric semicircles in the ratio 1 : 2.

Reminder about a proclamation concerning the vineyard

Among the geometry Leonardo wrote a note to himself: go and see whether another proclamation has been issued at the Provvisione concerning your vineyard.

Columns of calculation

Short columns of arithmetic are set out in the margins, including figures such as 366, 73, 100 and 55 arranged for addition or subtraction.

Girato il foglio sul lato sinistro. Facciata sinistra. Due cerchi, con:

Due semicerchi, con:

Pesi sospesi e oscillanti:

Operazioni aritmetiche:

Foglio capovolto. A matita presso il margine destro. Triangolo rettangolo, con:

Torre(?), con:

A penna, cerchi concentrici, con:

Facciata ora a sinistra.

Semicerchi concentrici, proporzione 1 : 2:

Quattro strisce sovrapposte in croce:

Foglio riportato nella sua posizione. A destra, due triangoli entro rettangolo, con:

Tra varie figure geometriche:

13Più in basso.

Piccolo cerchio, con:

Sheet turned to the left side. Left face. Two circles, with:

Two semicircles, with:

Suspended and swinging weights:

Arithmetical operations:

Sheet upside down. In pencil near the right margin. Right triangle, with:

Tower(?), with:

In pen, concentric circles, with:

Face now to the left.

Concentric semicircles, ratio 1 : 2:

Four strips overlaid in a cross:

Sheet returned to its position. On the right, two triangles within a rectangle, with:

Among various geometric figures:

Lower down.

Small circle, with: