Page:Cyclopaedia, Chambers - Volume 2.djvu/577

 RAT

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RAT

Ratines are chiefly manufactured in Trance, Holland, and Ita- ly; and are moftlv ufed in Linings.

The Frize is a coarle Ratine ; the Drugget zRatine half Thread, half Wool.

RATIFICATION, an A&, approving of, and confirming fomething done by another, in our Name.

A Treaty of Peace is never fare till the Princes have ratified it. See Treaty.

All Procuration imports a promife of Ratifying and approving what is done by the Proxy or Procurator. After treating with a Procurator, Agent, Facfor, ef c. A Ratification is frequently ne- ceffary on the Part of his Principal.

Ratification is particularly ufed in our Laws for the Confir- mation of a Clerk in a Benefice, Prebend, &c. formerly given him by the Bifhop, ajre. where the right of Patronage is doubted to be in the King.

Ratification is alio ufed for an Aift confirming fomething we our felves have done in our own Name. An Execution, by a Major, of an Act pafs'd in his Minority, is equivalent to a Rati- fication.

RATIO, Reason, in Arithmetick and Geometry, that Re- lation of homogeneous things which determines the Quantity of one from the Quantity of another, without the intervention of any third. See Relation.

The homogeneous things thus compared, we call the Terms of the Ratio ; particularly, that referr'd to the other, we call the Antecedent; and that to which the other is referr'd, the Confe- quent. See Term, &c.

Thus, when we conlider one Quantity, by comparing it with another, to fee what Magnitude it has in Cotnparifon ot that o- ther ; the Magnitude this Quantity is found to have in Compari- fon thereof is call'd the Ratio, Reafin, of this Quantity to that; whichfome think would be better exprefs'd by the Word Compari- fon. See Comparison.

Euclid defines Ratio by the Habitude or Relation of Magnitudes of the fame Kind in refpecj of Quantity — But this Definition is found defe6Hve; there being other Relations of Magnitudes which are conftanr, yet are not included in the Number of Ra- tio's; fuch as that of the Right Sine, to the Sine of the Com- plement in Trigonometry.

Hobbs endeavour'd to amend Euclid's Definition of Ratio, but unhappily ; for in defining it, as he does, by the Relation of Mag- nitude to Magnitude ; his Definition has not only the fame Detect with Euclid's, in not determining the particular Kind of Relati- on; but has this further, that it does not exprefs the Kind of Magnitudes, which may have a Ratio to one another.

Ratio is frequendy confounded with Proportion; yet ought they by all means to be diftinguilhed, as very different things. Pro- portion, in effea, is an Identity, or Similitude of two Ratio's. See Proportion.

Thus, if the Quantity A be triple the Quantity B; the Re- lation of A to B, ;'. t. of i to l, is call'd the Ratio of A to B. If two other Quantities, C,D, have the fame Ratio to one ano- ther that A and B have, i. e. be triple one another ; this fame- nefs of Ratio conftitutes Proportion: and the four Quantiiies A : B : : C : D, are in Proportion, or Proportional to one ano- ther.

So that Ratio exifh between two Terms, Proportion requres more.

There is a twofold Comparifon of Numbers : By rhe firft, we find how much they differ, i. e. by how many Units the Ante- cedent exceeds, or comes fhort of, the Confequent.

This Difference is call'd, the Arithmetical Ratio, or Exponent of the Arithmetical Relation or Habitude of the two Numbers. Thus if 5 and 7 be compared, their Aritlimetical Ratio is 2.

By the i'econd Comparifon, we find how oft the Antece- dent contains, or is contain'd in the Confequent ; i. e. as before, what Part ot the greater is equal to the lefs.

This Ratio, being common to all Quantity, may be call'd Ra- tio in the General, or, by way of Eminence. But it is ufually called Geometrical Ratio; becaufe exprefs'd, in Geometry, by a Line, though it cannot be exprefs'd by any Number.

mifius, better diftinguifhes Ratio, with regard to Quantity in the general, into Rational, and Irrational.

Rational Ratio, is that which is as one rational Number to a- nother. e. gr. as 3 to 4. See Number.

Irrational Ratio, is that which cannot be exprefs'd by Ratio- nal Numbers. r '

Suppofe, for an Illuftrarion, two Quantities A and B; and let A be lefs than B If A be fubftrafted as often as it can be, from B, c. gr five times, there will either be left nothing or fomething. In the former Cafe A will be to B, as 1 to ; ; that is, A is contain d in B five times ; or Azr ■ ' JJ. The Ratio, here, therefore, is rational.

■ H u he J at i et Cafc • d *7 therc » &I "= Part, which be- ing fubtracled certain times from A, e. g r . three times, and like- wife from B, e. gr. 7 times leaves nothing; or tnere is no fuch

Part, if the former : A will be to B,

1 3 £ o 7> or A=4B, and

therefore the Ratio, Rational. If the latter, the Ratio of A to B, >. e. what Part A is of B, cannot be exprefs'd by rational Numbers ; nor any other way than either by Lines, or by infinite approaching Series. See Series. ' e

The Exponent of a Geometrical Ratio is the Quotient arifin" from the Divifion of the Antecedent by the Confequent : Thus the Exponent of rhe Ratio of 3 to ±, is 1 £; that of the Rath of 2 to 3, is 4-; for when the lefs Term is the Antecedent, the Ratio, or rather the Exponent, is an improper Fraction. Hence the Fraction $ = 3 : 4. If the Confequent be Unity, the Ante- cedent itfelf is the Exponent of the Ratio : Thus the Exponent of 4 to 1 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 referr'd 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 determines how often the lels is contain'd in the greater, or how often the greater contains the lefs, i. e. to what Part of the greater, the lefs is equal.

The Ratio the greater Term has to the lefs, e. gr. 6 to 3, is called the Ratio of the greater Inequality. The Ratio the lefs Term has to the greater, e. gr. 3 to 5, is called the Ratio of the

lefs Inequality.

This Ratio correfponds to Quantity in the General, or is ad- mitted of by all Kinds of Quantities, difcrete or continued, Com- menfurable, or Incommenfurable. Difcrete Quantity, or Num- ber does likewife admit of another Ratio.

If the lefs Term of a Ratio be an aliquot Part of the greater, the Ratio of the greater Inequality is faid to be Multiplex, Mul- tiple: And the Ratio of the lefs Inequality, Suhmultiple. See Multiple.

Particularly, in the firft Cafe, if the Exponent be 2, the Ra- tio is call'd duple; if 3, triple, Sec. In the fecond Cafe, if the Exponent be i, the Ratio is call'd Subduple ; if i, Subtriple, &c.

E. gr. 6" to 2 is in a triple Ratio ; becaufe 6 contains two thrice. On the contrary, 2 to 6 is in a Subtriple Ratio, becaufe 2 is the third Part of 6. See Duple, Subduple, &c.

It the greater Term contain the lefs once; and over and a- bove, an aliquot Part of the fame ; the Ratio of the greater Ine- quality is call'd Superparticularis; and the Ratio of the lefs Sub- fuperparticularis.

Particularly, in the firft Cafe, if the Exponent be 1 h it is call'd Sefquialterate ; if 3 A, Sefquitertia, 6cc. In the other, if the Exponent be f, the Ratio is call'd Subfrfquialtera; if J, Sub- fefquitertia, &c.

E. gr. 3 to 2 is in a Sefquialterate Ratio; 2 to 3 in a Subfef- quialterate.

If the greater Term contain the lefs once, and over and above feveral aliquot Parts; the Ratio of the greater Inequality is call'd Superpartiens; that of the lefs Inequality, Subfuperpartiens.

Particularly, in the former Cafe, if tie Exponent be I f, the Ratio is call'd Superbipartiem tertias ; if the Exponent be 1 4, Su- pertripartiens quart as; if 1 f, Superquadripartiens feptimas, &C. In the latter Cafe, if the Exponent be f, the Ratio is call'd Sub- fuperbipar liens tertias ; if 4, Subfuperlriparliexsquartas; if -? T, Sub- fuperquadripartiens feptimas.

E. gr. the Ratio o£ 5 to 3 is Superbipartiens tertias; that of 3 to 5, Subfperbipartient tertias.

If the greater Term contain the lefs feveral times ; and, be- fides, fonie quota Part of the fame ; the Ratio of the greater In- equality is call'd Multiplex Superparticularis; and the Ratio of the lels Inequality, Submultiplexftbfitperparticularis.

Particularly, in the former Cale, if the Exponent be 2 £, the Ratio is call'd, Duplaffquiahera; if 3 -J, triple Sefquiquarta, &C. In the latter Cafe, if the Exponent be \, the Ratio is call'd Sub- dupla fubffquialtera; if -f\, Subtriple fubffquiquarta, &ZC.

E. gr. the Ratio of 16 to 5 is triple Sefquiquinta; that of 4 to ej, Subdup/a fubffquiqarta.

Lafily, if the greater Term contain the lefs feveral times, and feveral aliquot Parts thereof belides; the Ratio of the greater In- equality is call'd Multiplex fiperpartiens ; that of the lefs Inequali- ty, Submultiplex Subfuperpartiens. Particularly, in the former Cafe, if the Exponent be 2 \, the

Ratio is call'd, dttpla Superbipartiens tertias; if 3 }, tripla Superbi- quadripartiens feptimas, &c. In the latter Cafe, if the Exponent be y, the Ratio is call'd Subdupla fubfiperbipartiens tertias; if ^ T> Subtripla fulfuperquadripartiens feptimas, &c.

E. gr. the Ratio of 25 to 7 is tripla fuperquadripartiens fepti- mas; that of 3 to 8, jlibdupla Subfuperbipartiens tertias.

Thefe are the various Kinds of Rational Ratio's ; the Names whereof, though they occur but rarely among the modern Wri- ters, (for in lieu thereof they ufe the Imalleft Terms of the Ra- tio's, e. gr. for duple, 2 : I, for ffquialterate, 3 : 2) yet are they ahfolutely neceilary to fuch as converfe with the antient Au- thors.

Clavks obferves, that the Exponents denominate the Ratio's of the greater Inequality, both in Deed andName; but xheRa- tio's of the lefs Inequality, only in Deed, not in Name. But 'tis eafy finding the Name in thefe; if you divide the Denominator of the Exponent, by the Numerator.

M. t r.