The fourth is, Ἀναλογία δέ ὲστιν ῆ των λόγων ὁμοιότης. “_Proportio vero est rationum similitudo._” Here we have no one word by which to render Ἀναλογία; for our word _proportion_ is already bestowed upon the rendering of λόγος. Nevertheless the Greek may be translated into English thus, _iterated proportions_. But iterated proportion is the same with _eadem ratio_. To what purpose then serveth the sixth definition, which is of _eadem ratio_? For Ἀναλογία and _eadem ratio_ and _similitudo rationum_, are the same thing, as appeareth by Euclid himself, where he defines those quantities, that are in the same proportion by ἀνάλογον. Therefore the sixth definition is but a _lemma_, and assumed without demonstration. The fourteenth, “_Compositio rationis est sumptio antecedentis cum consequente, ceu unius, ad ipsum consequentem_,” _To compound proportion, is to take both antecedent and consequent together as one magnitude, and compare it to the consequent_, is good; though he might have compared it as well with the antecedent; for both ways it had been a composition of proportion. We are to note here, that the composition defined in this place by Euclid is not adding together of proportions, but of two quantities that have proportion. And therefore it is not the same composition which he defineth in the fourth place before the sixth element, for there he defineth the addition of one proportion to another proportion in this manner: λόγος ἐκ λόγων συγκεῖσθαι λέγεται, &c. _A proportion is said to be compounded of proportions, when their quantities multiplied into one another make a proportion_; as when we would compound or add together the proportions of three to two, and of four to five, we must multiply three and four, which maketh twelve, and two and five, which maketh ten. And then the proportion of twelve to ten is the sum of the proportions of three to two, and of four to five, which is true, but not a definition; for it may and ought to be demonstrated. For to define what is addition of two proportions (which are always in four quantities, though sometimes one of them be twice named) we are to say, that they are then added together when we make the second to another in the same proportion, which the third hath to the fourth. And thus much of the definitions; of which some, very few, you see are faulty; the rest either accurate, or good enough if well interpreted. For the rest of the elements all are accurate, notwithstanding that you allow not for good any definition in geometry that hath in it the word _motion_, of which there be divers before the eleventh Element. But I must here put you in mind, that geometry being a science, and all science proceeding from a precognition of causes, the definition of a sphere, and also of a circle, by the generation of it, that is to say, by motion, is better than by the equality of distance from a point within.