( 06I )


- . p.r. if the Exponent be , then 5: g-t i-i-; whence it mu
the Ratio is call'd Subfipertrpartiens quintas.   er a
Asto Names of Irratiomal Ratio's, no-body ever attempted  &'i
Soi, or Identic Ratio's, are thofe whofe Antecedents have an hai
equal refpec6t to their Confequents, i. e. whofe Antecedents di- of
Aided by their Confequents, give equal Exponents. See IDEN-
And hence may the Identity of Irrational Ratio's be concei- Bis
ved.
Hence, Fir]?, as oft as the Antecedent of one Ratio contains it i
its Confequent, or what ever Part it contains of its Confequent,
b oft, or fuch Part of the other Confequent does the Antece- is i
-dentof the other Ratio contain: Or, as oft as the Antecedent fre
of the one is contain'd in its Confequent, fo oft is the Antece- of
dent of the other contain'd in its Confequent.
Seco C: hi
D; on the contrary we call C:L) the lejjer, if C:D    <  ai
A:B.                                                     6
Hence, we exprefs a greater and lefs Ratio thus. E. gr. 6 to of
13 has a-greater Ratio, than 5 to 4; for, 6: 3 (=2) > S: 4
I       But 3 to 6 has a lefs Ratio than 4 to 5, for g- In
4A. TV(
I he Ratio is faid to be compounded of two or more other Ra- fu
ie's, which the Fadtum of the Antecedents of two or more li
Ratio's has to the Fa6tum of their Confequents.  Thus 6 to 7 is f
in a Ratio compounded of 2 to 6, and 3 to I2.
Particularly, if it be compounded of two, it is call'd a Dupli- K
{mte Ratio; if of three, a Triplicate; if of four, uadr#plicate; K
and in the general Multiplicate, if it be compofed of feveral fi- a]
Miilar Ratio's.  Thus 48: 3 is a duplicate Patio of 4  I and d-
a12 :3-
Properties of Ratio's of 5antditier.         h
ti
Eiirfl, Ratio's fnmilar to the fame third, are alfo fimilar to onef
4nother; and thofe fimilar to Similar, are alfo fimilar to one an-
other.
Secondly, If A:B-C:D; then, inversely, B: A = D: C.
Thirdly, Similar Parts P and p have the fame Ratio to Wholes t
T and t; and if the Wholes have the fame Ratio, the Parts are
fimilar.
Fourthly, If A: B=C: D; then, alternately, A: C = B: D.
And hence, A=C; hence, alfo, if A:B=C: D; and A: F        t
LYzC:G; we flhall have B: FzD:G.      Hence, again, if A:B  r
C: D; and F: A-G: C; we fhall have F :B _ G: D.
fthly, Thofe things which have the fame Ratio to the fame, t
or equal things, are equal: & 'vice verfa.
Sixthy, If you multiply any Quantities, as A and B. by the t
k hime or equal Quantities ; their Produats D and E will be to each I
other as A and B.
Sevent hly, If you divide any Quantities as A and B, by the
fiae or equal Quantities, the Quotients F and G will be to each
other as A and B.
Eigbthly, The Exponent of a compound Ratio is equal to the
Faium of the Exponents of the fimple Ratio.   See Expo-
;ENT.
Nintbly, If you divide either the Antecedents, or the Confe-
.Equtsof firnilar Ratio's, A: B and C: D by the fame E; in the
Mfarer Cafe, the Quotients F and G will have the fame Ratio
'to the Confequents B and D; in the latter, the Antecedents A
md B will have the fame Ratio to the Quotients H and K.
AtlY, If there be feveral Quantities in the fame continued
Ratio A, B. C, D, E, &c. the firfi, A is to the third, C, in a
c. .' icae Ratio, to the fourth, D, in a Triplicate, to the fifth E, in
a  uakruplicate, &c. Ratio of the Ratio of the firi, A, to
Condm B.
Elkventhl  If there be any Series of Quantities in the fame
Ratio, A, B, C, D, E, F, &c. the Ratio of the firfc, A) to the
i  laf F. is compounded of the intermediate Ratios A: B, B : C,
C: D, D: E, E: F, cb'.
2hwbthly, Ratio's compounded of Ratio's, whereof each is e-
1ala to another, are equal among themselves. Thus the Ratio's
90: 3 = 960: 32 are compounded of 6: 3 = 4: 2, and 3:
1= 12- 4X and 5 : I _= 20 -4-
Forother Properties of fimilar or equalRatio's, fee PROPORTION.
RATIO, in our Law Writers, is ufed for a Reafon, or Judg-
"lt given in a Caufe.  Hence, ponere ad ratioaw, is to Cite
Oft to appear in Judgment, Walfxgb. 88.
RATIO lft4W, RAGIONE deflate. See REASON of State.
RATIlOCINATION, the Aaion of REASONING. See
REASONING
RATK)N, in the Forces, a Pittance, or Proportion of Am-


RAT


nitions Bread, Drink, or Forrage, distributed to each Soldi-
md Seaman for his daily Subhftence.  See AMMUNITION,
rhe Rations of Bread are regulated by Weight.-The Officers
re feveral Rations according to their Quality, and the Number
Attendants they are obliged to keep.
The Horfe have Rations of Hay and Oats, when they cannot
out to Forrage.-The Ship's Crews have their Rations of
ket, Pulfe, and Water, proportioned according to their Stock.
When the Ration is augmented on Occafions of Rejoycingi
is call'd double Ration.
The ufual Ration at Sea, particularly among the Portugeze, &c.
a Pound and half of Bisket, a Pint of Wine, and a Quart of
ih Water per Day, And each Month an Arrobe or 3 I Pound
Salt Meat, with fome dried Fifli and Onions.
Some write Racion, and borrow the Word from the Spanill
scion. But they both come from the Latin Ratio; and in forne
Its of the Sea they call it Reafon.
RATIONABIL[ parte bonoruim, a Writ which lies for the
'ife, againfi the Executors ot her Husband denying her the
ird Part of her Husband's Goods, atter Debts and Funeral Ex-
nces paid. See GOODS.
Fitzherbort quotes Magna Charta and Glanville, to prove that
the Common Law of England, the Goods of the deceafed,
s Debts firifl paid, fhould be divided into three Parts; whereof
s Wife to have one, his Children a f cond, and the Executors
third. Adding, that this Writ lies as well for the Children,
Ic. as the Wife. But it feems only to obtain where the Cuftom
the Country makes for it.
RATIONABILES Expenfe, Reafonable Expences; the Coin-
nons in Parliament, as well as the Proaors of the Clergy in Con-
)cation, were antiently allow'd Rationabiles Expenfas; that is,
Lch Allowance as the King, confidering the Prices of all things,
Lall judge meet to impofe on the People, to pay for the Sub-
lance of their Reprelentatives. SeeREPRESENTATIvE, &c.
This in the I7th of Edward II. was 1o Groats per Day for
:nights, and 5 for Burgeffes. Afterwards, 4 Shillings a Day for
Knights, and 2 Shillings for Burgeffes; which was then deem'd
i ample Retribution both for Expences, for Labour, Atten-
ance, Neglca of their own Affairs, &c. See PARLIAMENT.
RATIONABILIBUS divfis, is a Writ that lies where two
Lords have the Seigneuries joining together, for him that finds
is Wafte encroached upon, within the Memory of Man, againft
he Encroacher; thereby to redify the Bounds of the Seigneu-
ies: in which R efpe& Fitzleerbert calls it, in its own Nature, a
Writ of Right.
RATIONAL, Reafonable. See REASON.
RATIONAL, or true Horizon, is that whofe Plane is conceived
o pafs through the Centro of the Earth; and which therefore
lividesthe Globe into two equal Portions., or Hemilpheres. See
H0RIZON.
'Tis call'd the Rational Horizon, becaufe only conceived by
he Underflanding; in opposition to the fenfible or aparent Ho-
,izon, which is vifible to the Eye. See SENSIBLE.
RATIONAL 6Uantlty or Number, a Quantity or Number Coin-
menfurable to Unity. See NUMBER and UNITY.
Suppofing any Quantity to be I, there are infinite other Quan-
ities, fome whereof are Commenfurable to it either fimnply, or in-
?ower; thefe Euclid calls, Rational uanxtities. See QUAN-
rITY.
The reft, that are Incommenfurable to i, he calls, Irrational
mantities, or Surds. See SURD.
RATIONAL Integer, or 'whole Number, is that whereof Unity
is an aliquot Part. See NUMBER and ALIQUOT Part.
RATIONAL frac!ion, or broken Number, is that equal to fome
aliquot Part or Parts of Unity. See FRACTION.
RATIONAL Mixt Number, is that confifting of an Integer and a
Fradtion, or of Unity, and a broken Number.
Commenfurable Quantities are defined by being to one ano-
ther, as one Rational Number to another.
For Unity is an aliquot Part of Unity; and a Fraeion has
Come aliquot Part common with Unity: In things, therefore, that
are as a Rational to a Rational Number, either the one is an ali-
quot Part of the other, or there is fome common aliquot Part
of both: Therefore they are Commenfurable. See COMMEN-
SURABLE.
Hence, if a Rational Number be divided by a Rational the
Quotient is a Rational
RATIONAL Ratio, is a Ratio whofe Terms are Rational
Quantities; or a Ratio which is as one RationalNumber to ano-
ther, e. gr. as 3 to 6. See RATIO.
Tho Exponent of a Rational Ratio is a Rational Quantity.
See EXPONENT.
RATIONALE, an Account, or Solution of fome Opinion,
Afion, Hypothefis, Phxnomenons or the like, on Principles of
Reafon.
Hence Rationale has become the Title of feveral Books; the
moft confiderable is the Rationale of the Di'une OffeF, by Guill.
Durardfs, a celebrated School Divine, Bifhop of Mede; finid
in 1286. as he hirafeif tells us.
RATIONALE, is alfo an antient facerdotal Vefiment, wore by
the lith-Prieft under the old Law; and cald by the Hebrews iM
hhocben ;


RAT


1W.M