Nested squares and circles in geometric ratio
Infinitely halved squares; a circle that fits nine or four times in another
Three constructions explore how squares and their inscribed circles scale. Leonardo halves the base of triangle m n o repeatedly to inscribe an endless series of squares, each one half of the one before; shows that a circle S inscribed in one ninth of a square enters nine times into the circle f inscribed in the whole; and proves, through a chain of equal triangles, that a circle set in a 45-degree rotated square (centre m) enters four times into the circle around it. Letter labels a, b, c, d, e, f, m, n, r, s, t track the three figures.
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An infinite series of halved squares (triangle m n o)
Repeatedly dividing into half the remainders left along the base of triangle m n o produces an infinite number of equal-sided squares, each one half of the preceding. The upper edges of the shrinking squares are labelled a, b, c, d, e, f between the shared vertices m, n, o.
Circle S enters nine times into circle f
Circle S, inscribed in the lower-middle of nine equal squares, enters nine times into circle f inscribed in the whole square a r. The proof is simply that the small square m n enters nine times into the large square a r.
A nested circle that enters four times into the larger
In a square inscribed with a rotated square and a further inscribed square holding a circle (centre m), the small circle enters four times into the circle around it. This follows because square c d e g goes twice into a b h r and because triangle a m b equals square f n s t, giving the small square a quarter of the large.