Quadrature of curvilinear figures by superimposed segments
Squaring triangle a b using a quarter- and an eighth-circle from doubled circles
A geometry page developing Leonardo's quadrature procedure. He describes seeking a circular portion equal to a figure of two straight lines and one curve, so that the remainder of the sickle-shaped falcata is left squared and can be laid over the quadrature of the whole falcata. In the worked case he squares triangle a b by superimposing two equal surfaces, a quarter-circle and an eighth-circle, equal because their names are converse to the subduple circles they come from, and concludes that if a quarter of one circle equals an eighth of another the circles are double, and this halving continues to infinity.
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Method of quadrature by superimposed circular portions
Leonardo seeks a portion of a circle equal to a figure of two straight lines and one curve, provable equal, so the remainder of the falcata is left squared and can be placed over the quadrature of the whole falcata. What remains uncovered of that square equals the curvilinear portion first removed; halving that uncovered remainder gives one half equal to the portion and the other to the curvilinear figure.
Squaring triangle a b with a quarter- and an eighth-circle
To square triangle a b, two equal surfaces are superimposed: a b c and c d, one a quarter of a circle (d c) and the other an eighth. They are equal because their denominations are converse to the subduple circles from which they derive; with c their common contact, a b and d remain equal, and the squared triangle d is laid on the falcata.
Converse denominations: a quarter equals an eighth of the double circle
If the 4th of one circle equals the 8th of the other, the circles from which the triangles derive are double one another. It follows that the 16th of one equals the 8th of the other, and so on infinitely, always dividing each part in half.