To supply this want, I thought it necessary to seek out some way, whereby the proportion of two lines, commensurable or incommensurable, might be continued perpetually the same. And this I found might be done by the proportion of two lines described by some uniform motion, as by an efficient cause both of the said lines, and also of their proportions; which motions continuing, the proportions must needs be all the way the same. And therefore I defined those magnitudes to have the same geometrical proportion, _when some cause producing in equal times equal effects, did determine both the proportions_. This, you say, needs an Œdipus to make it understood. You are, I see, no Œdipus; but I do not see any difficulty, neither in the definition nor in the demonstration. That which you call perplexity in the explication, is your prejudice, arising from the symbols in your fancy. For men that pretend no less to natural philosophy than to geometry, to find fault with bringing motion and time into a definition, when there is no effect in nature which is not produced in time by motion, is a shame. But you swim upon other men’s bladders in the superficies of geometry, without being able to endure diving, which is no fault of mine; and therefore I shall, without your leave, be bold to say, I am the first that hath made the grounds of geometry firm and coherent. Whether I have added anything to the edifice or not, I leave to be judged by the readers. You see, you that profess with the pricking of bladders the letting out of their vapour, how much you are deceived. You make them swell more than ever. For the corollaries that follow this sixth article, you say they contain nothing new. Which is not true. For the ninth is new, and the demonstrations of all the rest are new, being grounded upon a new definition of proportion; and the corollaries themselves, for want of a good definition of proportion, were never before exactly demonstrated. For the truth of the sixth definition of the fifth Element of Euclid cannot be known but by this definition of mine; because it requires a trial in all numbers possible, that is to say, an infinite time of trial, whether the quimultiples of the first and third, and of the second and fourth, in all multiplications, do together exceed, together come short, and are together equal; which trial is impossible.