ratio 
atoms belonging to the different groups of acidic or basic 
compounds in the composition of a mineral. The oxygen 
ratio of silica, sesquioxid, and protoxid in garnet is 2 : 1 : 1. 
Pedal ratio, in anc. pros., the proportion of the num- 
ber of times in the arsis to that in the thesis, or vice versa. 
The pedal ratio (Aoyos iro$i6) is usually either equal or 
isorrhythmic ratio (1 : 1), diplasic or double ratio (1 : 2), 
or hemiolic ratio (2:8 = 1: 1J). Besides these three, the 
ordinary pedal ratios, two others were anciently recog- 
niZL'd the triplasic or triple ratio (1 :3), and the epitritic 
ratiu (3 : 4 = 1 : 1J). The dochmius, regarded as a sin- 
gle foot^ had a pedal ratio different from all these (3 : f, ; 
w I -* ^ ). Isorrhythmic, diplasic, hemiolic, triplasic. 
epitritic, and dochmiac feet are feet having thepedal ratios 
just named. See foot, 11, irrational, rhythm. Prime and 
ultimate ratios, phrases first introduced, at least in a 
system, by Newton, who preferred them to the terms 
suggested by his own method of fluxions. The method 
of prime and ultimate ratios is a method of calculation 
which may be considered as an extension of the ancient 
method of exhaustions. It may be thus explained : let 
there be two variable quantities constantly approaching 
each other in value, so that their ratio or quotient con- 
tinually approaches to unity, and at last differs from 
unity by less than any assignable quantity; the ultimate 
ratio of these two quantities is said to be a ratio of equal- 
ity. In general, when different variable quantities re- 
spectively and simultaneously approach other quantities, 
considered as invariable, BO that the differences between 
the variable and the invariable quantities become at the 
same time less than any assignable quantity, the ultimate 
ratios of the variables are the ratios of the invariable 
quantities or limits to which they continually and simul- 
taneously approach. They are called prime ratios or ulti- 
mate ratios according as the ratios of the variables are 
considered as receding from or approaching to the ratios 
of the limits. The first section of Newton's "Principia" 
contains the development of prune and ultimate ratios, 
with various propositions. Progression with n ra- 
tios. See progression. Quadruple ratio, the ratio of 4 
to 1. Quadruplicate ratio, a ratio of fourth powers. 
Quintuple ratio, the ratio of 6 to i. Ratio cogno- 
scendl (L.), a reason. Ratio decidendi (L.\ in law, 
the ground or reason on which a judicial decision is con- 
ceived as proceeding. The effect of such a decision as a 
precedent or evidence of the law is largely dependent on 
the ratio decidendi, which is usually indicated in the opin- 
ions of the court, but often obscurely or with conflict; 
hence what was the ratio decidendi is often a question for 
commentators and text-writers. Ratio essendi (L.\ a 
cause. Rational ratio, a ratio between rational quanti- 
ties. Ratio of equality. See equality. Ratio of ex- 
change, in polit. econ., the proportion in which a given 
quantity of one commodity may be exchanged for a given 
quantity of another, especially when the commodities cor- 
respond in form and mode of measurement: as, the ratio 
of exchange between gold and silver, or between wheat and 
barley. 
When I proposed in the first edition of this book to use 
Ratio of Exchange instead of the word value, the expres- 
sion had been so little if at all employed by English Econ- 
omists that it amounted to an innovation. . . . Yet ratio 
is unquestionably the correct scientific term, and the only 
term which is strictly and entirely correct. 
IT. S. Jevons, Theory of Polit. Econ., p. 89. 
Ratio of greater (or lesser) inequality, the ratio of a 
greater quantity to a lesser one (or of a lesser to a great- 
er). Ratio of similitude, in geom., the ratio between 
corresponding dimensions of similar figures. See homo- 
thetic. Ratio sufflciens (L.). Same as sufficient reason 
(which see, under reason). Reciprocal ratio. Same as 
inverse ratio. Simple ratio, (a) A ratio between first 
powers. (b) A ratio not compound. Subduple ratio. 
See duple. Subduplicate ratio, an inverse ratio of 
squares (sub in all names of ratio indicating the inver- 
sion of the ratio) : as, the gravity of two equal masses is 
in the subduplicate ratio of their distances from the grav- 
itating center. Submultiple ratio, the ratio which 
exists between an aliquot part of any number or quantity 
and the number or quantity itself : thus, the ratio of 3 to 
21 is submultiple, 21 being a multiple of 3. To cut a 
line in extreme and mean ratio. See extreme. 
Triple ratio, the ratio of 3 to 1. 
ratiocinant (rash-i-os'i-nant), a. [< L. ratio- 
cinan(t-)s, ppr.of ratiocinari, reason: see ratio- 
cinate.] Reasoning Ratiocinant reason. See 
reason. 
ratiocinate (rash-i-os'i-nat), v. 1 ; pret. and 
pp. ratiocinated, ppr. ratiocinating. [< L. ratio- 
cinate, pp. of ratiocinari ( > It. raziocinare = Sp. 
Pg. raciocinar = F. ratioeiner), reckon, compute, 
calculate, consider, deliberate, meditate, rea- 
son, argue (cf. ratiocinium, a reckoning, a com- 
putation, > It. raziocinio = Sp. Pg. raciocinio, 
reasoning), < ratio(n-), reckoning, reason : see 
ratio, reason.] To reason; from two judg- 
ments to infer a third. The word usually im- 
plies an elaborate deductive operation. 
ratiocinate (rash-i-os'i-nat), a. [< L. ratioci- 
natus, pp. of ratiocinari, reason: see the verb.] 
Reasoned about. Ratiocinate reason. See rea- 
son. 
ratiocination (rash-i-os-i-na'shon), n. [< F. 
ratiocination = Pr. raciocinacio = Sp. raciocina- 
eion = Pg. raciociitaqa'o (cf. It. raziocinamento, 
raziocinio, reasoning), < L. ratiocinatio(n-), rea- 
soning, argumentation, a syllogism, < ratioci- 
nari, pp. ratiocinatus, reason: see ratiocinate.'] 
1. The mental process of passing from the cog- 
nition of premises to the cognition of the con- 
clusion ; reasoning. Most writers make ratiocination 
synonymous with reasoning. J. S. Mill and others hold 
that the word is usually limited to necessary reasoning. 
The Latin word is especially applied by Cicero to proba- 
ble reasoning. 
4971 
The great instrument that this work [spiritual medita- 
tion] is done by is ratiocination, reasoning the case with 
yourselves, discourse of mind, cogitation, or thinking ; or, 
if you will, call it consideration. 
liaxti'r, Saints' Rest, iv. 8. 
The schoolmen make a third act of the mind, which they 
call ratiocination, and we may stile it the generation of a 
judgement from others actually in our understanding. 
A. Tucker, Light of Nature, I. i. 11. 
Ratiocination is the great principle of order in thinking ; 
it reduces a chaos into harmony ; it catalogues the ac- 
cumulations of knowledge ; it maps out for us the rela- 
tions of its separate departments ; it puts us in the way to 
correct its own mistakes. 
J. H. Newman, Gram, of Assent, p. 273. 
2. A mental product and object consisting of 
premises and a conclusion drawn from them ; 
inference ; an argumentation. 
Can any kind of ratiocination allow Christ all the marks 
of the Messiah, and yet deny him to be the Messiah ? 
South. 
Ratiocination denotes properly the process, but, improp- 
erly, also the product of reasoning. 
Sir W. Hamilton, Logic, xv. 
= Syn. Reasoning, etc. See inference. 
ratiocinative (rash-i-os'i-na-tiv), a. [< F. ra- 
tiocinatif, < L. ratiocinaticus, of or belonging to 
reasoning, syllogistic, argumentative, < ratioci- 
nari, reason: see ratiocinate.] Of the nature 
of reasoning; pertaining to or connected with 
the act of reasoning. The word is misused by 
some modern writers. See ratiocination, 2. 
The conclusion is attained quasi per saltum, and with- 
out any thing of ratiocinative process. 
Sir It. Hale, Orig. of Mankind, p. 51. 
The ratiocinative meditativeness of his character. 
Coleridge. 
Again, itnotunfrequently happens that, while the keen- 
ness of the ratiocinative faculty enables a man to see the 
ultimate result of a complicated problem in a moment, it 
takes years for him to embrace it as a truth, and to recog- 
nize it as an item in the circle of his knowledge. 
J. H. Newman, Gram, of Assent, p. 159. 
ratiocinatory (rash-i-os'i-na-to-ri), a. [< ra- 
tiocinate + -ory.] S&mea&raiiocinatire. [Rare.] 
ration (ra'shon or rash'on), . [< F. ration = 
Sp. radon = Pg. racSo, reqao = It. razione, a 
ration, a rate or allowance, < L. ratio(n-), a 
calculation, reckoning, hence in ML. a com- 
puted share or allowance of food : see ratio, rea- 
son (which are doublets of ration), and cf . rate 2 .] 
1. An allowance of means of subsistence for a 
fixed period of time ; specifically, in the army 
and navy, an allotment or apportionment of 
provisions for daily consumption to each offi- 
cer and man, or of forage for each horse. Offi- 
cers' rations are generally commuted for a money pay- 
ment at a prescribed rate ; and soldiers' and sailors' rations 
may be partly or wholly commuted under some circum- 
stances. 
2. Any stated or fixed amount or quantity dealt 
out ; an allowance or allotment. 
At this rate [two years and a half for three vowels], to 
master the whole alphabet, consonants and all, would be 
a task fitter for the centurial adolescence of Methuselah 
than for our less liberal ration of years. 
Lowell, Harvard Anniversary. 
ration (ra'shon or rash'on), v. t. [< ration, 
n.] 1. To supply with rations ; provision. 
It had now become evident that the army could not be 
rationed by a wagon train over the single narrow and almost 
impassable road between Milliken's Bend and Perkins' 
plantation, U. S. Grant, Personal Memoirs, I. 471. 
2. To divide into rations ; distribute or appor- 
tion in rations. [Rare.] 
The presence of hunger began ; they began to ration out 
the bread. The Nation, March 9, 1871, p. 160. 
rationability (ragb/on-a-biri-ti), n. [= Sp. 
racionabilidad = Pg. racionabilidade = It. ra- 
zionabilita,<.'LL.rationabiUta(t-)s,<,rational}ilis, 
reasonable : see rationable.] The possession of 
reason, as the distinctive attribute of man. 
Rationability, being but a faculty or specifical quality, 
is a substantial part of a man, because it is a part of his 
definition, or his essential difference. 
Bramhall, ii. 24. (Dailies.) 
rationable (rash'on-a-bl), a. [= OF. rationable 
= Sp. rationable = Pg. racionavel = It. razion- 
abile, < LL. rationabilis, reasonable, rational, < 
L. ratio(n-), reason: see reason.] Reasonable, 
as an agent or an act. 
She was, I take it, on this matter not quite rationable. 
Miss Edgeworth, Belinda, xxvi. 
rational (rash'on-al), a. and re. [I. a. < OF. 
rationel, rational, F\ rationnel = Pr. Sp. Pg. ra- 
tional = It. razionale, < L. rationalis, of or be- 
longing to reason, rational, reasonable, < ra- 
tio(n-), reason: see ratio, ration, reason. II. n. 
< OF. rational, < ML. rationale, a pontifical stole, 
a pallium, an ornament worn over the chasuble, 
neut. of L. rationalis, rational: see I.] I. a. 1. 
Of, pertaining to, or springing from the reason, 
in the sense of the highest faculty of cognition. 
rational 
He confesses a rational sovrantie of soule, and freedom 
of will in every man. Milton, Eikonoklastes, vi. 
Devout from constitution rather than from rational con- 
viction. Macaulay, Kssays, liistory, p. 394. 
Contradiction . . . must be absurd when it is regarded 
as fixed, and rational when it is regarded as superable. 
Veiteh, Introd. to Descartes's Method, p. clxxviii. 
2. Endowed with reason, in the sense of that 
faculty which distinguishes man from the 
brutes : as, man is a rational animal. 
It is our glory and happiness to have a rational nature. 
Law. 
Are these men rational, or are not the apes of Borneo 
more wise? Goldsmith, Citizen of the World, let x. 
He [man] is rational and moral according to the organic 
internal conformation of his mind. 
Swedenborg, Christian Psychol. (tr. by Gorman), p. 72. 
There has been an idea of good, suggested by the con- 
sciousness of unfulfilled possibilities of the rational nature 
common to all men. 
T. H. Green, Prolegomena to Ethics, 207. 
3. Conformable to the precepts of reason, es- 
pecially of the practical reason; reasonable; 
wise. 
You are one 
Of the deepest politics I ever met, 
And the most subtly rational. 
B. Jonson, Magnetick Lady, iii. 4. 
He had his Humour as other Men, but certainly he was 
a solid rational Man. Howell, Letters, I. vi. 17. 
His bounties are more rational and moderate than be- 
fore. Goldsmith, Vicar, iii. 
4. In aritli. and alg. : (a) Expressible in finite 
terms: applied to expressions in which no ex- 
traction of a root is left, or, at least, none such 
indicated which cannot be actually performed 
by known processes. The contraries of these are 
called turd or irrational quantities. Thus 2, 12}, -9, are ra- 
tional quantities, and y 1 ^ VT, etc., are irrational or surd 
quantities, because their values can only be approximately 
and not accurately assigned, (ft) In Euclid's "Ele- 
ments" and commentaries, etc., on that work, 
commensurable with a given line. In senses (a) 
and (b) rational (Latin rationalis) translates Greek p>jT6, 
expressible. It may be remarked that some inconvenience 
arises from the fact that words derived from Latin ratio, 
originally signifying an account, are used to translate 
words connected with Greek Aoyoc, whose original mean- 
ing (a word) is entirely different. 
5. In anc. pros., capable of measurement in 
terms of the metrical unit (semeion or mora). 
A rational time (xpovw PTT<K) is a time divisible by this 
unit without remainder. Thus, disemic times (times of 
two semeia) are rational, while irrational times (XP<J"<H 
aAoyot) can be expressed only by fractions (as f, 1J, 2$, 2J0 
of a semeion. Geometrically rational, algebraic. 
Rational and Integral function. See function. 
Rational certainty, cognition, cosmology. See the 
nouns. Rational class of functions, a class which is 
relative to a group of operations produced by combina- 
tions of additions, subtractions, multiplications, and divi- 
sions. Rational composition, in logic: (a) The compo- 
sition of elements which only differ as viewed by the mind, 
and not as they exist, as the composition of essence and 
existence, of being and relation, etc. (b) The union of 
several objects so far as they are brought together into or 
under one concept. Rational derivative. See deriva- 
tive. Rational formula. See chemical formula, under 
chemical. Rational fraction, function. See the nouns. 
Rational horizon, (a) The astronomical horizon. 
(6) The limits of rational knowledge. Rational infer- 
encet, a ratiocinative inference or syllogism. Rational 
instinct, an innate idea, or natural belief. Rational 
knowledge, (a) Knowledge of an object through its 
cause or causes. 
The knowledge why or how a thing is is termed the 
knowledge of the cause ; philosophical, scientific, rational 
knowledge. Sir W. Hamilton, Metaph., iii. 
(b) Knowledge springing directly or indirectly from rea- 
son, and not from experience. Rational mechanics, 
the science which establishes and puts into shape the 
laws of motion. Rational number, a number expressi- 
ble as an ordinary fraction, in contradistinction to a 
continued fraction. Rational power, proposition, 
ratio, etc. See the nouns. Rational psychology. 
See psychology. Rational theology, theology so far 
as drawn from a priori ideas. Rational transforma- 
tion, the transformation of a geometrical continuum 
into another, so as to make a one-to-one correspondence 
between the points of the two, except for a finite num- 
ber of exceptional points. =Syn. Rational, Reasonable, 
sensible, enlightened, discreet, intelligent, sane, sound. 
The first two words ore somewhat different, according 
as they refer to persons or things. As to persons, ration- 
al is the more speculative, reasonable the more practi- 
cal term ; rational means possessing the faculty of reason, 
while reasonable means exercising reason in its broader 
sense, in opposition to unreasonable that is, guided by 
prejudice, fancy, etc. In fever the patient may become ir- 
rational and give irrational answers ; when he is rational 
he may through weakness and fretfulness make unreason- 
able demands of his physician. As to things, the distinc- 
tion continues between the narrower and the broader 
senses : a rational proposition is one that might proceed 
from a rational mind ; a reasonable proposition is one that 
is marked by common sense and fairness. It is irrational 
to look for a coal-mine in a granite-ledge ; It Is unreason- 
able to expect good work for poor pay. See absurd. 
II. n. 1. A quiddity; a universal; a nature. 
Thus, in the first quotation "the world of rationale" is the 
rational world, the system of general or possible entities. 
The conception is Platonic. 
He, the great Father, kindled at one flame 
The world of rational*. Young, Night Thoughts, iv. 
