Mean proportionals, roots and the quadrature of circles

Finding a mean proportional with a right triangle

As the ratio of c d to d e, so is a b to that same e d. A line is drawn from a to the angle e and continued straight to where it cuts the line o S and then d S at the point r; there the lesser mean proportional is found, the greater lying in a b.

Graphic extraction of the cube root of 27

In the large triangle Leonardo writes that he seeks the cube root of 27, placed at a S, and that he will increase it by one, S e, to make 28. The letters r r, c S, b, a mark the construction used to read the root off the figure.

Quarters and halves of doubled circles

Working with the figure a b c, he treats a b as a quarter of a circle double the one below and b c as half the lower circle, then reasons that the half of the smaller doubled circle equals the quarter of the greater, concluding a, b and c are equal in quantity.

Octagon dissected into circle-portions

An octagon is drawn with the eight portions of its circumscribed circle separated out, and below it a square joined to a large circular portion. He argues that a equals b and holds portion e, eightfold one portion of a taken from a circle eightfold circle a, so that b equals the whole circle a.

Girato il foglio sul lato destro, si veda il triangolo rettangolo distinto dalle lettere:

Nota biffata distinta dalla lettera:

Capovolto il foglio:

Rigirato il foglio, grande triangolo colle lettere:

Al centro in colonna:

Piccolo triangolo colle lettere:

Triangolo senza lettere alfabetiche:

Riportato il foglio in posizione normale, figura colle lettere:

Ottagono colle otto porzioni del cerchio circoscritto separate, con sotto un quadrato unito a una porzione grande di cerchio, distinti colle lettere:

In basso a sinistra, figura colle lettere:

Turn the sheet to the right to see the right triangle marked by the letters:

A note struck through, marked by the letter:

The sheet turned upside down:

The sheet turned again, a large triangle with the letters:

At the centre, in a column:

A small triangle with the letters:

A triangle without alphabetical letters:

The sheet returned to its normal position, a figure with the letters:

An octagon with the eight portions of the circumscribed circle separated, and below a square joined to a large portion of a circle, marked with the letters:

Lower left, a figure with the letters:

Mean proportionals, roots and the quadrature of circles

On this page

Finding a mean proportional with a right triangle

Graphic extraction of the cube root of 27

Quarters and halves of doubled circles

Octagon dissected into circle-portions