RAT 



11AV 



ei s than a. X a. \ therefore, ft 1 -f- 2 a c : tC- t 

 or a -f- 2 x : a v/ill nearly express the ra- 

 tio of a 1 -f- 2 a.T ~|-.r 2 :V. 



Thus the ratio of the square of 1001 to 

 the square of 1000 is nearly 1002 : 1000 ; 

 the real ratio is 1002.001 : 1000, in which 

 the antecedent differs from its approxi- 

 mate value, only by one thousandth part 

 of an unit. 



Hence the ratio of the square root of 

 a -f- 2 a: to the square root of a is the ra- 

 tio a X x : a, nearly, that is, if the differ- 

 ence of two quantities be small with res- 

 pect to either of them, the ratio of their 

 square roots is nearly obtained by halving 

 their difference. 



In the same manner, a -j- 3 x : a ; 

 -f- 4 x : a ; a -f- m x : a ; are nearly equal 

 to the ratios a-j-a? 3 -. o3 ; a -f- x* : a* ; 

 a -}- ^ m a ; if true be small when com- 

 pared with a. 



Or we may treat the subject different- 

 ly, thus; ratio is that relation of homoge- 

 neous things which determines the 

 quantity of one from the quantity of 

 another, without the intervention of a 

 third. Two numbers, lines, or quantities, 

 A and B, being- proposed, their relations 

 one to another may be considered under 

 one of these two heads : 1. How much 

 A exceeds B, or B exceeds A ; and this 

 is found by taking A from B, or B from 

 A, and is called arithmetic reason, or 

 ratio. 2. Or how many times and parts 

 of a time, A contains B, or B contains A ; 

 and this is called geometric reason, or 

 ratio ; (or, as Euclid defines it, it is the 

 mutual habitude, or respect, of two mag- 

 nitudes of the same kind, according to 

 quantity ; that is, as to how often the one 

 contains, or is contained, in the other;) 

 and is found by dividing A by B, or B by 

 A ; and here note, that the quantity 

 which is referred to another quantity, is 

 called the antecedent of the ratio ; and 

 that to which the other is referred is 

 called the consequent of the ratio ; as, in 

 the ratio of A to B, A is the antecedent, 

 and B the consequent. Therefore any 

 quantity, as antecedent, divided by any 

 quantity as a consequent, gives the ratio 

 of that antecedent to the consequent. 



Thus the ratio of A to B is-, but the 

 B 



Tl 



ratio of B to A is ; and, in numbers, the 



12 



ratio of 12 to 4 is = 



4 



the ratio of 4 to 12 is 

 triple. 



3, or triple ; but 



4 1 



,-x = '^, or sub- 



The quantities, thus compared, must 

 be of the s-mie kind ; that is, such which, 

 by multiplication, :r.vy be made to ex- 

 ceed one the other, or us these quantities 

 are said to have a ratio between them, 

 which, being- multiplied, may be made to 

 exceed one another. Thus ;i line, how 

 short soever, nuy be multiplied, that is, 

 produced so long as to exceed in length 

 any given right line, and consequently 

 these may be compared together, and 

 the ratio expressed; but as a line can 

 never, by any multiplication whatever, 

 be made to have breadth, that is, to be 

 made equal to a superficies, how small 

 soever ; these can therefore never be 

 compared together, and consequently 

 have no ratio or respect one to another, 

 according to quantity ; that is, as to how 

 often the one contains, or is contained in 

 the other. 



RATION, in the army, a portion of 

 ammunition, bread, drink, and forage, 

 distributed to each soldier in the army, 

 for his daily subsistence, Sec. The horse 

 have rations of hay and oats when they 

 cannot go out to forage. The rations of 

 bread are regulated by weight. The 

 ordinafy ration of a foot soldier is a pound 

 and a half of bread per day. The officers 

 have several rations, according to their 

 quality .and the number of attendants 

 that they are obliged to keep. When 

 the ration is augmented on occasions of 

 rejoicing, it is called a double ration. 

 The ships' crews have also their rations 

 or allowances of biscuit, pulse, and 

 water, proportioned according to their 

 stock. 



RATIONAL, is a word applied to in- 

 tegral, fractional, and mixed numbers : 

 thus we say rational fraction, rational 

 integer, and rational mixed number ; for 

 the explanation and doctrine of which, 

 see NUMBER and FHACTIOX. 



Rational is applied to the true horizon, 

 in opposition to the sensible or apparent 

 one. See HORIZON. 



RATIONALE, a solution, or account 

 of the principles of some opinion, action, 

 hypothesis, phenomenon, or the like. 



RATLINES, or, as the seamen call 

 them, RATLIXS, those lines which make 

 the ladder steps to get up the shrouds 

 and puttocks, hence called the ratlins of 

 the shrouds. 



RATTLE snake. See CROTALTJS. 



RAVELIN, in fortification, was anci- 

 ently a flat bastion, placed in the middle 

 of a curtain ; but now a detached work, 

 composed only of two faces, which make 

 a ssdiant angle, without any flanks, and 



