Describe a square A B C D, and in it a quadrant D C A. In the side D C take D T two-fifths of D C, and between D C and D T a mean proportional D R, and describe the quadrantal arcs R S, T V. I say the arc R S is equal to the straight line D C. For seeing the proportion of D C to D T is duplicate of the proportion of D C to D R, it will be also duplicate of the proportion of the arc C A to the arc R S, and likewise duplicate of the proportion of the arc R S to the arc T V. Suppose some other arc, less or greater than the arc R S, to be equal to D C, as for example _r s_: then the proportion of the arc _r s_ to the straight line D T will be duplicate of the proportion of R S to T V, or D R to D T. Which is absurd; because D _r_ is by construction greater or less than D R. Therefore the arc R S is equal to the side D C, which was to be demonstrated. COROL. [Illustration] Hence it follows that D R is equal to two-fifths of the arc C A. For R S, T V, D T, being continually proportional, and the arc T V being described by D T, the arc R S will be described by a straight line equal to T V. But R S is described by the straight line D R. Therefore D R is equal to T V, that is to two-fifths of C A. And your said servant most humbly prayeth you to consider, if the demonstration be true and evident, whether the way of objecting against it by square root, used by Dr. Wallis; and whether all his geometry, as being built upon it, and upon his supposition of an infinite number, be not false. CONSIDERATIONS UPON THE ANSWER OF DOCTOR WALLIS TO THE THREE PAPERS OF MR. HOBBES. Dr. Wallis says, all that is affirmed, is but _if we_ SUPPOSE _that, this will follow_. But it seemeth to me, that if the supposition be impossible, then that which follows will either be false, or at least undemonstrated. First, this proposition being founded upon his _Arithmetica Infinitorum_, if there he affirm an absolute infiniteness, he must here also be understood to affirm the same. But in his thirty-ninth proposition he saith thus: “_Seeing that the number of terms increasing, the excess above sub-quadruple is perpetually diminished, so at last it becomes less than any proportion that can be assigned; if it proceed in infinitum it must utterly vanish. And therefore if there be propounded an infinite row of quantities in triplicate proportion of quantities arithmetically proportioned (that is, according to the row of cubical numbers) beginning from a point or 0; that row shall be to a row of as many, equal to the greater, as 1 to 4._” It is therefore manifest that he affirms, that in an infinite row of quantities the last is given; and he knows well enough that this is but a shift.