Archimedes’ [P23] ‘Quadrature of the Parabola’, Proposition 23

by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Archimedes Entry Directory]

[Archimedes, Quadrature of the Parabola, Entry Directory]

 

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