My demonstration consisteth of three cases: the first is when both the proportions are of defect, which is then when the first quantity is the least; as in these three quantities, A B, A C, A D. The first case I demonstrated thus: (A B C D)/(a) Let it be supposed that the point A were moved uniformly through the whole line A D. The proportions, therefore, of A B to A C, and of A C to A D, are determined by the difference of the times in which they are described. And the proportion also of A B to A D, is that which is determined by the difference of the times in which they are described; but the difference of the times in which A B and A C are described, together with the difference of the times wherein A C and A D are described, is the same with the difference of the times wherein are described A B and A D. The same cause, therefore, which determines both the proportions of A B to A C, and of A C to A D, determines also the proportion of A B to A D. Wherefore, by the definition of _the same proportion_, article six, the proportion of A B to A C, together with the proportion of A C to A D, is the same with the proportion of A B to A D. Consider now your argumentation against it. “_Let there be taken_,” say you, “_between A and B the point_ a; and then in your own words, I argue thus: _The difference of the times wherein are described A B and A C, together with the difference of the times wherein are described A C and A D, is the same with the difference of the times in which are described_ a _B and_ a _C (namely, B D, or B C + C D_); wherefore, the same cause which determines the two proportions of A B to A C, and of A C to A D, determines also the proportion of a _B to_ a _D_.” Let me ask you here whether you suppose the motion from _a_ to B, or from _a_ to D, to have the same swiftness with the motion from A to B, or from A to D? If you do not, then you deny the supposition. If you do, then B C, which is the difference of the times A B and A C, cannot be the difference of the times in which are described _a_ B and _a_ C, except A B and _a_ B are equal. Let any man judge now whether there be any paralogism in Orontius that can equal this. And whether all that follows in the rest of this, and the next two whole pages, be not all a kind of raving upon the ignorance of what is the meaning of proportion, which you also make more ill-favoured by writing it; not in language, but in _gambols_; I mean in the symbols, which have made you call those demonstrations short, which put into words so many as a true demonstration requires, would be longer than any of those of Clavius upon the twelfth Element of Euclid.