Archimedes employed an infinite process which led to a geometric series of areas under the parabolic segment.
M is the midpoint of AD, and the segment MT is 3/4 of the parallel segment BD, according to the quadratic property of parabolas. The ratio is proved by combining trapezoid MTBD with triangle ATM, and comparing them with ABD. The process is iterated by inscribing new triangles in the unmeasured areas (outside of ATBD). Using midpoints of the intervals along AD, we get the series
which amounts to four-thirds of ADB [Maor, pp. 43-45].