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Archimedes
Quadrature of the Parabola
Proposition 23 [quoting]
Proposition 23.
P23. Given a series of areas A, B, C, D, ... Z, of which A is the greatest, and each is equal to four times the next in order, then
A + B + C + ... + Z + 1/3Z = 3/4A.
Take areas b, c, d, … such that
b = 1/3B
c = 1/3C,
d = 1/3D, and so on.
Then, since b = 1/3B,
and B = 1/4A,
B + b = 1/3A.
Similarly C + c = 1/3B.
Therefore
B + C + D + ... + Z + b + c + d+ ... + z = 1/3(A + B + C + ... + Y).
But b + c + d + ... +y = 1/3(B + C + D + ... + Y). 
Therefore, by subtraction,
B + C + D + … + Z + z = 1/3A
A + B + C + … + Z + 1/3Z = 4/3A.
Archimedes. “Quadrature of the Parabola.” In The Works of Archimedes. Ed. T.L. Heath. Cambridge UP, 1897. Obtained at
https://archive.org/details/worksofarchimede00arch
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