Perpendicular bisector of two points and the definition of the circle
Compass construction of the locus equidistant from two points; lunes drawn on the facing leaf
Folio 219r sets a pure geometry problem: to draw a line every part of which is equally distant from two given points a and b. Leonardo strikes equal circles centred on each point, joins their two intersections with the line m n, and shows that any third point on it, such as c, is equidistant from a and b, appealing repeatedly to the definition of the circle. The facing folio 222v carries no transcribed text but is filled with lettered figures, including rows of crescent lunes and rectangles crossed by diagonals from his studies of transforming areas. The British Museum crown stamp appears on the left page.
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Constructing the locus equidistant from two points
To find a line every part of which is equally distant from two given points a and b, Leonardo strikes equal circles o m p and q n r centred on a and b and joins their two intersections by the line m n extended to f. Any third point taken on this line, such as c, is equidistant from a and b, giving the perpendicular bisector of the pair.
The construction rests on the definition of the circle
For the points a b the line c d is shown to lie midway between them by the definition of the circle, since equal radii make every point of c d equidistant from a and b. A marginal note names the governing principle simply the definition of the circle.
Lunes and divided rectangles on the facing leaf
The left half of the opening (folio 222v) carries a series of crescent lunes and rectangles crossed by diagonals, drawn without accompanying transcribed text. They belong to Leonardo's studies of transforming curved and straight-sided areas into one another.