THE DELPHIC PROBLEM, OR DUPLICATION OF THE CUBE.

_A._ Have you seen a printed paper sent from Paris, containing the duplication of the cube, written in French?

_B._ Yes. It was I that writ it, and sent it thither to be printed, on purpose to see what objections would be made to it by our professors of algebra here.

_A._ Then you have also seen the confutations of it by algebra.

_B._ I have seen some of them; and have one by me. For there was but one that was rightly calculated, and that is it which I have kept.

_A._ Your demonstration then is confuted though but by one.

_B._ That does not follow. For though an arithmetical calculation be true in numbers, yet the same may be, or rather must be false, if the units be not constantly the same.

_A._ Is their calculation so inconstant, or rather so foolish as you make it?

_B._ Yes. For the same number is sometimes so many lines, sometimes so many planes, and sometimes so many solids; as you shall plainly see, if you will take the pains to examine first a demonstration I have to prove the said duplication, and after that, the algebraic calculation which is pretended to confute it. And not only that this one is false, but also any other arithmetical account used in geometry, unless the numbers be always so many lines, or always so many superficies, or always so many solids.

_A._ Let me see the geometrical demonstration.

_B._ There it is. Read it.

TO FIND A CUBE DOUBLE TO A CUBE GIVEN:

Let the side of the cube given be V D. Produce V D to A, till A D be double to D V. Then make the square of A D, namely A B C D. Divide A B and C D in the middle at E and F. Draw E F. Draw also A C cutting E F in I. Then in the sides B C and A D take B R and A S, each of them equal to A I or I C.

Lastly, divide S D in the middle at T, and upon the centre T, with the distance T V, describe a semi-circle cutting A D in Y, and D C in X.

I say the cube of D X is double to the cube of D V. For the three lines D Y, D X, D V are in continual proportion. And continuing the semi-circle V X Y till it cut the line R S, drawn and produced in Z, the line S Z will be equal to D X. And drawing X Z it will pass through T. And the four lines T V, T X, T Y and T Z will be equal. And therefore joining Y X and Y Z, the figure V X Y Z will be a rectangle.

_Delphic Problem.