Squares on a divided line and tangents to a circle
Binomial (irrational) magnitudes, and tangent constructions from Euclid's third book
The recto pairs two lines of study. At the top, squares are raised on the two unequal parts of a line and combined into a rectangle, with the note that the compound 'h b' is a binomial and therefore irrational. Below, a circle is drawn with a tangent, two perpendicular radii, and the chord joining their ends, the margin citing the 'fifteenth' and 'sixteenth of the third' — the tangent propositions of Euclid's Book III. A vertical segment a-b is then treated in turn as the diameter of a circle, as the radius of a fourfold circle, and reversed with two diameters in place of the radii.
On this page
Squares on unequal parts: a binomial irrational
Two squares are built on the two unequal parts of a line and joined by a rectangle. The compound labelled h b is called a binomial and, as such, irrational.
Tangent to a circle with orthogonal radii
A circle is drawn with a tangent, two radii meeting at a right angle, and the line joining their ends; a variant brings the radii close with a line in the curvilinear angle. The margin cites the fifteenth and sixteenth of the third book, Euclid's tangent propositions.
A segment as diameter and as radius of a fourfold circle
The vertical segment a-b is used first as the diameter of a circle and then as the radius of a fourfold circle, the same figure being reversed with two diameters in place of the two radii.