**Archimedes,**__Quadrature of the Parabola__Prop. 23- translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 22

Prop. 24

23. If magnitudes are placed successively in a ratio of four-times, all the magnitudes and yet the third part of the least composed into the same magnitude will be a third-again the largest.

(diagram 1) And so let there be magnitudes, however many, placed successively, A, B, G, D, E each four-times its follower, and let A be largest and let Z be a third of B, H of G, Q of D, and I of E. (diagram 2) And so, since Z is a third part of B, and B is a fourth part of A, both together, B, Z, are a third part of A. (diagram 3) And for the same reasons, in fact, H, G is a third part of B, and Q, D of G, and I, E of D. (diagram 4) In fact, B, G, D, E, Z, H, Q, I altogether are a third part of A, B, G, D altogether. But Z, H, Q are themselves also a third part of B, G, D. (diagram 5) Therefore, the remainders B, G, D, E, I are also a third part of A. (diagram 6) And so it is clear that A, B, G, D, E, and I, that is, the third of E, altogether are a third again of A.

A : B = B : C = C : D = D : E = 4 : 1

let B :Z = G : H = D : G = E : I = 3 : 1

Hence,

B + Z = 1/3*A

G + H = 1/3*B

D + Q = 1/3*G

E + I = 1/3*D

Hence, B + G + D + E + Z + H + Q + I = 1/3*(A+B+G+D)

Also, Z + H + Q = 1/3 (B + G + D)

Hence, (subtracting equals from equals) B + G + D + E + I = 1/3*A

Adding equals, A + B + G + D + E + I = 4/3 *A

Or A + B + G + D + E + 1/3*E = 4/3 *A