RAT 
roots is nearly obtained by halving their 
difference. 
In the same manner, « + 3 X : a ; 
a; a-^-mx : a-, are nearly equal 
to the ratios a : a’ ; a : a* ; 
a -{- xl’" '■ a'" J if mx be small when com- 
pared with a. 
Or we may treat the subject differently, 
thus, ratio is that relation of homogeneous 
things which determines the quantity of one 
from the quantity of another, without the in- 
tervention 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 ration 
(or, as Euclid defines it, it is the mutual 
habitude, or respect, of two magnitudes 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. There- 
fore any quantity, as antecedent, divided by 
any quantity as a consequent, gives the ratio 
of that antecedent to the consequent. 
4 
Thus the ratio of A to B is g, but the 
jg 
ratio pf B to A is ~ ; and, in numbers, the 
12 
ratio of 12 to 4 is = 3, or triple ; but 
4 1 
the ratio of 4 to 12 is — = g, or subtriple. 
The quantities, thus compared, must be 
of the same kind ; that is, such which, by 
multiplication, may be made to exceed one 
the other, or as these quantities are said to 
have a ratio between them, which, being 
multiplied, may be made to exceed one an- 
other. Thus a line, how short soever, may 
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, 
RAU 
how small soever; these can tlrerefore 
never be compared together, and conse- 
quently have no ratio or respect one to an- 
other, 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 am- 
munition, bread, drink, and forage, distri- 
buted to each soldier in the army, for his 
daily subsistence, &c. The horse have ra- 
tions of hay and oats when they cannot go 
out to forage. The rations of bread are 
regulated by weight. The ordinary 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 expla- 
nation and doctrine of which, see Number 
and Fraction. 
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, hypo- 
thesis, phenomenon, or the like. 
RATLINES, or, as the seamen call them. 
Ratlins, those lines which make tlie lad- 
der steps to get up the shrouds and put- 
tocks, hence called the ratlins of the 
shrouds. 
RATTLE snake. See Crotalus. 
RAVELIN, in fortification, was anciently 
a flat bastion, placed in the middle of a cur- 
tain ; hut now a detached work, composed 
only of two faces, which make a saliant an- 
gle, without any flanks, and raised before 
the curtain on the counterscarp of the place, 
A ravelin is a triangular work resembling 
the point of a bastion, with the flanks cut 
off. Its use before a curtain is to cover 
the opposite flanks of the two next bas- 
tions. It is used also to cover a bridge, or- 
a gate, and is always placed without the 
moat. There are also double ravelins, that 
serve to cover each other : they are said 
to be double when they are joined by a 
curtain. 
RAVEN. See Corvds. 
RAUWOLFIA, in botany, so named iit 
