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where the numerator of the sequence terms is the nth square pyramidal number P<sub>n</sub>.
But (4a) state that: the area (measured in green triangles) of the circumscribed figure is one-third the area of the triangle ABC, at the limit of n = infinity. ''End of proof''
This proof is very beautiful! Notice its three fundamental steps:
# Choice of ''equivalent'' triangles for measuring areas.
# With this choice, the area of triangle ABC measure n<sup>3</sup> triangles.
# Counting the number of triangles in the saw-tooth figure that encloses the parabolic segment and ''discovery'' that, for each number n of divisions, this number is the ''square pyramidal number'' !
The rest came by itself.
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