_A._ There is I see some little difference between this arithmetical and your geometrical demonstration. And though it be insensible, yet if his calculation be true, yours must needs be false, which I am sure cannot be. _B._ His calculation is so true, that there is never a proposition in it false, till he come to the conclusion, that the cube of D Y is equal to 45, want the square root of 1682. But that, and the rest, is false. _A._ I shall easily see that A D is certainly 2, whereof D V is 1, and A V is certainly 3, whereof D V is 1. _B._ Right. _A._ And B R is without doubt the square root of 2. _B._ Why, what is 2? _A._ 2 is the line A D as being double to D V which is 1. _B._ And so, the line B R is the square root of the line A D. _A._ Out upon it, it is absurd. Why do you grant it to be true in arithmetic? _B._ In arithmetic the numbers consist of so many units, and are never considered there as nothings. And therefore every one line has some latitude, and if you allow to B I, the semi-diagonal, the same latitude you do to A B, or to B R, you will quickly see the square of half the diagonal to be equal to twice the square of half A B. _A._ Well, but then your demonstration is not confuted; for the point Y will have latitude enough to take in that little difference which is between the root of 1681 and the root of 1682. This putting off an unit sometimes for one line, sometimes for one square, must needs mar the reckoning. Again he says, the cube of A B is equal to 8; but seeing A B is 2, the cube of A B must be just equal to four of its own sides; so that the unit which was before sometimes a line, sometimes a square, is now a cube. _B._ It can be no otherwise when you so apply arithmetic to geometry, as to number the lines of a plane, or the planes of a cube. _A._ In the next place, I find that the cube of D Y is equal to 45, want the square root of 1682. What is that 45? Lines, or squares, or cubes? _B._ Cubes; cubes of D V. _A._ Then if you add to 45 cubes of D V the square root of 1682, the sum will be 45 cubes of D V; and if you add to the cube of D Y the same root of 1682, the sum will be the cube of D Y, plus the square root of 1682, and these two sums must be equal. _B._ They must so. _A._ But the square root of 1682, being a line, adds nothing to a cube; therefore the cube alone of D Y, which he says is equal almost to 4 cubes of D V, is equal to 45 cubes of the same D V. _B._ All these impossibilities do necessarily follow the confounding of arithmetic and geometry. _A._ I pray you let me see the operation by which the cube of D Y (that is, the cube of 3, want the root of 2) is found equal to 45, want the square root of 1682. _B._ Here it is. A DETECTION OF THE ABSURD USE OF ARITHMETIC AS IT IS NOW APPLIED TO GEOMETRY.