RATIO. One of the most frequent mathematical terms has no other name in our language than a Latin word which is but a bad translation from the Greek of Euclid. The older English writers introduced the word reason, as a translation of ratio, which completed the confusion ; for it is easier to attach any meaning we please to ' a word in a dead language than to the literal translation of it in our own.
The word ratio is the translation of 74.6yos, as used in the third definition of the fifth book of Euclid, which is Abyos ?crrl 5tio peyercia, 4uryf way i? eceris IrnAtud-nrra irpbs 1taxssaa seta (rxials. This has been translated by English writers, "Ratio is a mutual habitude of two magnitudes of the same kind with respect to quantity." By Gregory, in his translation which accompanies the Greek," Ratio eat duarum magnitudinum ejusdem generic secundum quantuplicitalcm mutua qu:edam habitude." The common translation is unmeaning; and it will be worth while to offer some remarks on the probable meaning of Euclid. In the first place, let it be observed that be never attempts this vague sort of definition except when, dealing with a well-known term of common life, he wishes to bring it into geometry with something like an expressed meaning, which may aid the conception of the thing, even though it does not furnish a perfect-criterion. Thua, when in speaking of a straight lino, he says that it is the line which lies evenly &Vast, usirat) between its extreme pointe, he merely calls the reader's attention to the well known term €50B7a way.,ui, tries how far he can present the conception which accompanies it in other words, and trusts for the correct use of the term to the axioms which the universal conception of a straight line makes self-evident. Let us suppose him doing the same thing here, and we shall find that the definition before us, con sidered with reference to the place it is in, and the subsequent purpose which it serves, is as clear as the translation of it is confused.
The term X67os contains (asy, ?toy), a root the original meaning of which seems to have contained the idea of collection or bringing together.
It is certain however that the secondary sense which it obtained in common usage was that of speaking ; so that the first sense in which Ad-yos appears in writings is that of speech. Subsequently, speech being the dis tinctive character of reasoning beings, and their mode of communication, the word was applied to every sort of communication, not only with reference to the mode of communication, but also to its subject ; thus explanation, defence, apology, teaching, assignment of cause or reason, &c., are among the recognised uses of the word. The Latin translators have taken the geometrical word as being properly translated by ratio, a word which may very well signify the technical meaning of NeS7or, but has no reference to its primary meaning. For ratio, in its primitive sense, means rather computation or reckoning than reason.
But what has speech to do with the sense of ratio in geometry I Robert Recordo answers this question [7;umEnavox], when he reduces his pupil to silence by forbidding him the use of number, and asking him questions. Numbers are but certain ratios, and ratio is a generalised idea of number. Our gift of speech with reference to magnitudes would be altogether annihilated if we did not consider a certain habi tude or mode of existence which they have, or morn correctly a certain conception of our own, which always accompanies the presence of two magnitudes, and prompts us to inquire how many times one is contained in the other. A foot being known, speech can carry a correct knowledge of other lengths all over the world ; but let it be attempted to describe a foot in words without reference nark imaged rsrra to some other inagnitude, and all the powers of language utterly fail. We conceive then that in this definition Euclid simply conveys the fact that the mode of expressing quantity in terms of quantity, is entirely based upon the notion of gnantuplicity, or that relation of which we take cognizance when we find how many times one is con tained in the other.