Ml KATIO. 



the amount of the rate ; and they have authority to negative the 

 imposition of a rate altogether ; the only mode of compelling them to 

 impose a rate being by ecclesiastical censured and laying the pariah 

 under interdict. The existing poor-rate of the pariah is generally 

 taken an the criterion for the impoidtion of the church-rate. All pro- 

 perty in the parish U liable except the glebe-land of that parish and 

 the poBeanoD* of the crown. The ecclesiastical courts have the exclu- 

 sive authority of deciding on the validity of a rate, and the liability of 

 a party to pay it ; but a rate-payer cannot by an original proceeding in 

 those court* raise objections to a rate for the purpose of quashing it 

 altogether. If he wishes to dispute it, he ought to attend at the vestry, 

 and there state his objections ; if they are not removed, he may enter 

 a caveat against the confirmation of the rate, or refuse to pay his 

 assessment. In the latter case, if proceeded against in the ecclesiastical 

 court, he may in his defence show either that the rate is generally 

 invalid, or that he in unfairly assessed. The consequence of entering a 

 caveat is an appeal to the ecclesiastical judge, who will see that right 

 is done. Previously to 53 Oeo. III., c. 127, the only mode of recover- 

 ing church-rates from parties refusing to pay was by suit in the eccle- 

 siastical court for subtraction of rate. By that statute, where the sum 

 to be recovered is under 10/.,and there U no question as to the validity 

 of the rate, or the liability of the party assessed, any justice of the 

 county where the church is situated may, on complaint of the church- 

 warden, inquire into the merits of the case, and order the payment. 

 Against his decision there is an appeal to the quarter-sessions. As to 

 other rates, see SEWEBS ; SHIRK: \\'AY. 



(Lyndwood; John de Athon : Selden's I/ixturi/ of Tithei ; Gibson's 

 Codex ; Burn's Ecclesiastical Law ; Rogers's Ecclesiastical Law, 1840.) 



RATIQ. 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 retuua, 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 \6yot, as used in the third 

 definition of the fifth book of Euclid, which is AAyos ta?\ Svo /ttyfffuv 

 tluTYfvav fi KOTO JTjAunJTTiTa irpbs fiAATjAa JTOIO <rx"'<j. 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 est duaruin 

 niagnitudinuiii ejusdem generis secundum ijuantujjlicitatcm mutua 

 quaxlam 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 he 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. Thus, when in speaking of a 

 straight line, he says that it is the line which lies evenly (4(l<rou KC?TOI) 

 between its extreme points, he merely calls the reader's attention to 

 the well known term cufcm 7po,u/i1j, 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 \6yos contains (Ary, \oy), 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 \Ayos 

 appears in writings is that of speech. Subsequently, speech beingthe 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 \Ayos, 

 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 ? Robert 

 Recorde answers this question [NUMERATION], 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 wo did not consider a certain habi- 

 tude or mode of existence which they have, or more 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 nthcr. A f<iot 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 KOTO mjAiii- 

 TTfru to some other magnitude, 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, 



RATIO. i'.i 



is entirely based upon the notion of rjHaiitujJirity, or that relation of 

 which we toko cognizance when we find how many times one U con- 

 tained in the other. 



The word ngAuc^rqi has been translated " quantity," by many editors, 

 which makes nonsense of the whole ; for magnitude has hardly a 

 different meaning from quantity, and a relation of magnitudes with 

 respect to quantity may give clear ideas to those who want a word to 

 convey a notion of architecture with respect to building, or of battles 

 with respect to fighting ; and to no others. Wallis, we believe, restored 

 the true meaning of the word, and was followed by Gregory, as seen 

 above : and Euclid himself, or some very old commentator, in aii"tl -r 

 place, shows in what sense he used it. In the fifth definition of the 

 sixth book (omitted by many editors), he says that a ratio is com- 

 pounded of two other ratios when the mjAcfTrr of the lattrr 

 Mul/i/flieil ^*oAAowAo<na<r0r<TaO together, make the former. Now. tlii 

 would be unmeaning if the Greek word meant simply qu.-n 

 unless they were quantities represented by numbers (though Gregory 

 has here forgotten his own previous correction, and writes quantitas 

 instead of quantuplicitas). The lexicographers generally give " quan- 

 titas ; " but they are not adepts in the mathematical use of terms 

 implying relations of magnitude. 



The first and rough notion of ratio being thus given, we may find a 

 synonyme for the word in the more intelligible term relative ma'jn 

 Six feet, though greater than three feet, is, relatively to four feet, a 

 less magnitude than three feet is, relatively to one foot : the number 

 of times which six feet contains four feet is less than tin 1 miml*. r <>! 

 times which three feet contains one foot. The relative magnitude of 

 six to four is less than the relative magnitude of three to one; or tin 

 ratio of six to four is less than the ratio of three to one. 



Given two magnitudes, how are we to find the means of expressing 

 the first in terms of the second ? Euclid answers this question, when 

 it can be answered, in the tenth book, by giving the rule for finding 

 the greatest common measure of two magnitudes, in which he employs 

 a process exactly the same as that of the arithmetical rule in common 

 use. Of two magnitudes which are multiples of the same third 

 magnitude, each is a definite arithmetical fraction of the other. 



But it may happen that the magnitudes have no common measure 

 [INCOMMENSUUABLK], in which case the means of expression would 

 fail. Vi'e can describe the diagonal of a square as a part of a certain 

 figure, and the description is perfect ; but if we attempted a description 

 tecundmn guantuplicilatem, we should never succeed ; for no possible 

 line exists of which it can be said that the diagonal of a square and its 

 side both contain that line exactly. Such quantities are called by 

 Euclid oAo-yo, irrational, or having no ratio; and in the primitive 

 meaning of the term this is correct, for there is no quanluplicitative 

 mode of expressing one by the other. But the term ratio, both in 

 Euclid and all other writers, immediately acquires another sense ; and 

 it is this new sense in which we proceed to speak of ratio. Since the 

 relative magnitude of two quantities is always shown by the quantu- 

 plicitative mode of expression, when that is possible, and since propor- 

 tional quantities (pairs which have the same relative magnitude) are 

 pairs which have the same mode (if possible) of expression by means of 

 each other ; in all such cases sameness of relative magnitude leads to 

 sameness of mode of expression ; or proportion is sameness of ratios 

 (in the primitive sense). But sameness of relative magnitude may 

 exist where quantuplicitative expression is impossible ; thus the diagonal 

 of a larger square is the same compared with its side as the diagonal of 

 a smaller square compared with lit side. It is an easy transition to 

 speak of sameness of ratio even in this case ; that is, to use the term 

 ratio in the sense of relative magnitude, that word having originally 

 only a reference to the mode of expressing relative magnitude, in cases 

 which allow of a particular mode of expression. The word irrational 

 does not make any corresponding change, but continues to have its 

 primitive meaning, namely, incapable of quantuplicitative expression. 

 And it is worth noting that this of itself shows that the original 

 meaning of \6yot referred to expression, not to the thing expressed ; 

 for &Ao7o (not having a ratio) would have been absurd as applied to 

 incommensurable quantities, if the primitive mathematical meaning of 

 the first word had coincided with its modern one. 



The idea of relative magnitude is one which strikes us in all < 

 which we compare the parts of an original with the corresponding 

 parts of any model or imitation. It does not closely connect itself 

 with any mode of expression or measurement : if a part of the model 

 were only in a slight degree too large or too small, the detection of the 

 error might require a formal measurement, but anything which is very 

 much out would be rejected by one glance of the eye. Let us suppose 

 now that the formal measurement is attempted. The first and simplest 

 notions of relative magnitude are gained from repetition ; and the ideas 

 of two, three, four, etc., originally used in their simple cumulative 

 sense, soon become the representatives of those simple relative magni- 

 tudes which are suggested by pairs in which one is quantuple of the 

 other. The next step is to those magnitudes in which neither is 

 quantuple of the other, but both are quantuple of a third : from 

 which we learn how, admitting aliquot parts, to extend the mode of 

 expression. Thus, of the magnitudes 10 K and 7 R, we see that every 

 ivl itinn of quantuplicity can be derived from the simple numbers 1 

 and 7 : the first number is 1J of the second, a mode of expression 

 which equally applies to the magnitudes 10 n and 7 it. 



