R A T 



miles S.E. of Breflau. N. lat. 50'. E. long. iS'-- 5'. The 

 principality is bounded N. by the principality of Oppeln, 

 on the E. by Poland, on the S. by Tefcheii, and on the 

 W. by the principality of Jagendorf. Its foil is better than 

 that of Oppeln, as it produces a fufiicient fupply of wheat, 

 rye, and barley, with fruits ; and befides, it has alfo good 

 pafture grounds. Its only river is the Oder, which paiies 

 through its wefteru part ; but it is abuirdantly watered in 

 all its parts with llreams, ponds, and lakes. It contains 

 only three cities, and the inhabitants are univerfally Poli(h. 

 It became a principality in 1288, and about zoo years 

 afterward it was united to Oppeln, from which it has never 



been feparated. 



RATIFICATION, Ratificatio, an aft, approving 

 of, and confirming, fomething done by another, in our 



name. ,- ' i, > • . 



A treaty of peace is never fecure till the princes have 



ratified it. -r r -r ■ j 



All procuration imports a promife ot ratifying and ap- 

 proving what is done by the proxy, or procurator : after 

 treating with a procurator, agent, faftor, &c. a ratifica- 

 tion is frequently neccffary on the part of his principal. 



Ratification is particularly ufed, in our Laws, for the 

 confirmation of a clerk in a benefice, prebend, &c. formerly 

 given him by the biihop, &c. where the right of patronage 

 is doubted to be in the king. 



Ratification is alfo ui^d for an aft confirming fome- 

 thing we ourfelves have done in our own name. 



An execution, by a major, of an aft paffed in his mino- 

 rity, is equivalent to a ratification. 



RATING. See Raiting. 



RATINGEN, or Rattingen, in Geography, a town 

 of the duchy of Berg ; 4 miles N.E. of Duffeldorp. N. 

 lat. 51' 15'. E. long. 6^47'. 



RATING, a town of Naples, in the county of Mohfe ; 

 6 miles S.E. of Molife. 



RATIO, in Arithmetic and Geometry, that relation of ho- 

 mogeneous things, wliich determines the quantity of one 

 from the quantity of another, without the intervention of 

 any third. 



The homogeneous things, thus compared, we call the 

 terms of the ratio ; particularly that referred to the other, 

 we call the antecedent ; and that to which the other is referred, 

 the confequent. 



Thus, when we confider one quantity by comparing it 

 with another, to fee what magnitude it has in comparifon of 

 that other ; the magnitude this quantity is found to have in 

 comparifon with it, is called the ratio of this quantity to 

 that : which fome think would be better expreffed by the 

 word comparifon. 



Euclid defines ratio by a mutual relation of two magnitudes 

 of the fame kind in refped of quantity. But this definition is 

 found defeftive ; there being other relations of magnitudes 

 which are conftant, yet are not included in the number of 

 ratios : fuch as that of the right fine, to the fine of the 

 complement in trigonometry. 



Hobbes endeavoured to improve Euclid's definition of 

 ratio, but without fuccefs : for in defining it, as he does, by 

 the relation of magnitude to magnitude, his definition has not 

 only the fame defeft with Euclid's, in not determining the par- 

 ticular kind of relation ; but it has this farther, that it does 

 not exprefs the kind of magnitudes which may have a ratio 

 to one another. 



Ratio is frequently confounded, though very improperly, 

 with proportion. Proportion, in effeft, is an identity or fimi- 

 litude of two ratios. 



Thus, if the quantity A be triple the quantity B ; the re- 



RAT 



lation ©f A to B, i.e. of 3 to i, is called the ratio of A to 

 B. If two other quantities, C, D, have the fame ratio to 

 one another that A and B have, i. e. be triple one another, 

 this equality of ratio conltitutes proportion ; and the four 

 quantities A : B :: C : D, are in proportion, or propor- 

 tional to one another. 



So that ratio exills between two terms ; proportion re- 

 quires more. 



There is a twofold comparifon of numbers : by the firft, we ' 

 find how much they differ, i. e. by how many units the ante- 

 cedent exceeds, or comes fliort of, the confequent. 



Tliis difference is called the arithmetical ratio, or exponent 

 of the arithmetical relation or habitude of the two num- 

 bers. Thus, if 5 and 7 be compared, their arithmetical ra- 

 tio is 2. 



By the fecond comparifon, we find how oft the antecedent 

 contains, or is contained in, the confequent ; /'. e. as before, 

 what part of the greater is equal to the Icfs. 



This ratio, being common to all quantity, may be called 

 ratio in the general, or by way of eminence : but is ufually 

 aWei geometrical ratio ; becaufe expreffed, in geometry, by a 

 line, though it cannot be expreffed by any number. 



Modern authors dift;ingui/h ratio, witli regard to quantity 

 in the general, into rational and irrational. 



Ratio, Rational, is that which is as one rational number 

 to another ; e. gr. as 3 to 4. 



Ratio, Irrational, is that which cannot be expreffed by 

 rational numbers. 



Suppofe, for an illuftration, two quantities, A and B ; 

 and let A be lefs than B. If A be fubtrafted as often as it 

 can be from B, e. gr. five times, there will either be left no- 

 thing, or fomething. In the former cafe, A will be to B, 

 as 1 to 5 ; that is, A is contained in B five times ; or 

 A ^ ^ B. The ratio here, therefore, is rational. 



In the latter cafe, either there is fome part, which, being 

 fubtrafted certain times from A, e. gr. 3 times, and like- 

 wife from B, e. gr. 7 times, leaves nothing ; or there is no 

 fuch part : if the former, A will be to B as 3 to 7, or 

 A = 4- B ; and therefore the ratio, rational. If the latter, 

 the ratio of A to B, i. e. what part A is of B, cannot be 

 expreffed by rational numbers ; nor any other way than 

 either by lines, or by infinite approaching feries. 



The exponent of a geometrical ratio is the quotient arifing 

 from the divifion of the antecedent by the confequent. Thus, 

 the exponent of the ratio of 3 to 2, is li ; that of the ra- 

 tio of 2 to 3, is 4 : for when the lefs terra is the antecedent, 

 the ratio, or rather the exponent, is a proper fraftion. 

 Hence the fraftion f = 3 -^ 4. If the confequent be 

 unity, the antecedent itfelf is the exponent of the ratio : 

 thus, the exponent of 4 to I, is 4. See Exponent. 



If two quantities be compared, without the intervention 

 of a third ; either the one is equal to the other, or unequal : 

 hence, the ratio is either of equality or inequality. If the 

 terms of the ratio be unequal, either the lefs is referred to 

 the greater, or the greater to the lefs : that is, either the lefs 

 to the greater, as a part to the whole, or the greater to the 

 lefs, as the whole to a part. The ratio, therefore, deter- 

 mines how often the lefs is contained in the greater, or how 

 often the greater contains the lefs ; i. e. to what part of the 

 greater the lefs is equal. 



The following dillinftions of ratios are fometijnes found in 

 early authors. 



The ratio which the greater term has to the lefs, e. gr. 6 to 

 3, is called the ratio of the grecUer inequality : the ratio which 

 the lefs term has to the greater, e.gr. 3 to 6, is caHed the 

 ratio of the lejfer inequality. 



This ratio correfponds to quantity in the general, or is 



admitted 



