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Chambers, Ephraim, 1680 (ca.)-1740 / Cyclopædia, or, An universal dictionary of arts and sciences : containing the definitions of the terms, and accounts of the things signify'd thereby, in the several arts, both liberal and mechanical, and the several sciences, human and divine : the figures, kinds, properties, productions, preparations, and uses, of things natural and artificial : the rise, progress, and state of things ecclesiastical, civil, military, and commercial : with the several systems, sects, opinions, &c : among philosophers, divines, mathematicians, physicians, antiquaries, criticks, &c : the whole intended as a course of antient and modern learning

(1728)

R - rectification, pp. 951-966 PDF (18.2 MB)

Page 960

RAT ( 966 ) RAT Ratines are chiefly tranufa6tured in *anwc, HRoll and Ita- ly; and are moftlv ufed in Linings. The Frize is a coarfe Ratine the Drugget aRatine half Thread, half Wool. RATIFICATION, an Af, approving of, and confirming Something done by another, in our Name. A Treaty of Peace is never fure till the Princes have rat lied 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, Fa&or, &c. A Ratjfcatiox is frequently ne- ceffiary 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 Bifop, &c. where the right of Patronage is doubted to be in the King. RATIFICATION is alfo ufed for an A& confirming Something we our felves have done in our own Name. An Execution, by a Major, of an A&1: paf'd in his Minority, is equivalent to a Rati- ficat am. 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 ,Antecedmnt; and that to which the other is reterrd, the Confi. juext. See TERM, &c. Thus, when we confider one Quantity, by comparing it with another, to fee what Magnitude it has in Comparifon of that o- ther; the Magnitude this Quantity is found to have in Compari- fon thereof is call'd the Ratio, Reafon, of this Quantity to that; -which Come think would be better exprefs'd by the Word Compari- fox. See COMPARISON. Euclid defines Ratio by the Habitude or Relation of Magnitudes of the fame Kind in refpel of fyantity.-But this Definition is found defe6tive; there being other Relations of Magnitudes which are confiant, 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 Eucld's Definition of Ratio, but unhappily; for in defining it, as he does, by the Relation of Mag- xitade to Magnitude; his Definition has not only the fame Defe& with Euc.d'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 frequently confounded with Proportion; yet ought they by all means to be difidnguithed, as very different things. Pro- portion, in effet, is an Identity, or Similitude of two Ratios. See PROPORTION. Thus, if the Quiantity A be triple the Quantity B; the Re- lation of A to B. i. c. of toiT is calrd the Ratio of A to B. If two other Quantities, CD, have the fame Ratio to one ano- ther that A and B have, i. e. be triple one another; this fame- nefs of Ratio conflitutes Proportion: and the four Quantities A: B :: C : D, are in Proportion, or Proportional to one ano- ther. So that Ratio exifls between two Terms, Proportion requres more. There is a twofold Comparifon of Numbers: By the firft, we find how much they differ, i. e. by how many Units the Ante- cedent exceeds, or comes (hort 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 Arithmetical Ratio is 2. By the fecond Comparifon, we find how oft the Antece- dent contains, or is contain'd in the Confequent; i. e. as before, what Part of 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 exprel'd, in Geometry, by a Line, though it cannot be exprefisd by any Number. Wolfius, better diinguifhes 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. Suppofe, for an Illultration, two Quantities A and B; and let A be lef than B. If A be fubftraft&ed as often as it can be, from B, e. gr. five times, there will either be left nothing or Something. In the former Cafe A will be to R, as I to 5; that is, A is contain'd in B five times; or A=f B. The Ratio, here, therefore, is rational. In the latter Cafe, either there is Come Part, which be- ing fubtramed certain times from A, e. gr. three times, and like- wife from B, e. gr. 7 times leaves nothing ; or there is no fuch Part, if the former: A will be to B, as 3 to 7, or A=41B, and therefore the Ratio, Rational. If the latter, the Ratio of A to D, i. e. what Part A is of B, cannot be expres'd by rational Numbers; nor any other way than either by Liues, or by infinite ppwrching Series. Sot SRiEas. The Exponent of a Geometrical Ratio is the Quotient ariing from the Divifion of the Antecedent by the Confequent: Thus the Exponent of rhe Ratio of 3 to 2, is I A; that of the RatiD of 2 to 3, is T; for when the lefs Term is the Antecedent, the Ratio, or rather the Exponent, is an improper Fraaion. Hence the Fraftion - 3: 4. If the Confequent be Unity, the Ante- cedent itfelf is the Exponent of the Ratio: Thus the Exponent of to I is , 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 le~s: 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 lefsi is contain'd in the greater, or how often the greater contains the lefs, i. e. to what Part of the greater, the legs 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 6, is called the Ratio of the lefi Inequality. This Ratio correfponds to Quantity in the General, or is ad- mitted of by all Kinds of Quantities, difcrete or continued, Coin- menfurable, or Incommenfurable. Dilerete 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 Multplex, May- tiple: And the Aatio of the lefs Inequality, Submultiple. See MULTIPLE. Particularly, in the firfi Cafe, if the Exponent be 2, the Ra- tio is call'd duple; if 3, tripk, &c. In the fecond Cafe, if the Exponent be I, the Ratio is cali'd SUbduple; if 4, Subtreiple &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, &e. 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 Superparticularif; and the Ratio of the lefs Sub- fuperparticularis. Particularly, in the firft Cafe, if the Exponent be I I1, it is call'd Sefrqialterate; if 3 a., Sefluitertia, &c. In the other, if the Exponent be .3 the Ratio is call'd Sbfeqauialtera; if 4. SW fefquitertia, &C. E. gr. 3 to 2 is in a Se/quialterate 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, Subfuperpartes. Particularly, in the former Cafe, if the Exponent be i am the Ratio is call'd Superbipartiens tertias; if the Exponent be I j, Su pertripartiens g artas; if x 4, Superquadripartiens feptimas, &c. In the latter Cafe, if the Exponent be 4, the Ratio is call'd Sub- fuperbpartiens tertias; if 4, Sufiupertripartiew quartas; if .-,I, Sub fuperquadipartiens feptimas. E. gr. the Ratio of 5 to 3 is Superbiparties tertias; that of 3 to I, SublJperbipartiens tertias. If the greater Term contain the lefs feveral times; and, be- fides, fome quota Part of the fame; the Ratio of the greater In- equality is cali'd Multiplex Sperparticularis; and the Ratio of the le's Inequality, Submaltiplexfibfuperparticularis. Particularly, in the former Cale, if the Exponent be 2 id the Ratio is call'd, Duplaffquialtera; if 3 I, triple Sefquiquarta, &c. In the latter Cafe, if the Exponent be T, the Ratio is cald Sub.& dupla fubfefquialtera; if 4 Subtriple fub/frjuiquarta, &c. E. gr. the Ratio of I6 to 5 is triple Sefquiquixta; that of 4to 9, Subdupla fubfeJluiqarta. La~jly, if the greater Term contain the lefs feveral times, and feveral aliquot Parts thereof befides; the Ratioof the greater In- equality is call'd MultiplexfJperpartie s; that of the lefa Inequali- ty, Submultiplex Sutbfuperpartiens. Particularly, in the former Cafe, if the Exponent be 2 A, the Ratio is call'd, dupla Superbipartiens tertias; if 3 4, trpla Superbi- quadripartiens feptimas, &c. In the latter Cafe, if the Exponent be TV. the Ratio is call'd Subdupla fubfuperbipartiens teruiasi if -,7r Smbtriplafubfuperquadripartiens feptimas, &c. E. gr. the Ratio of 25 to 7 is tripltfuperqguadripartiens Jepti- mas; that of 3 to 8, fubdupla 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 fmalleft Terms of the Ra- tio's, e. gr. for duple, 2 : I, for fefquialterate, 3: a) yet are they absolutely neceflary to fuch as converfe with the antient Au- thors. Clavius obferves, that the Exponents denominate the Rat's of the greater Inequality, both in Deed and Name; but the Ra- tio's of the lefs Inequality, only in Deed, not in Name. But 'tie eafy finding the Name in thefe; if you divide the Denoouiator of the Exponent, by the Numerator. ~~~~~~~~z v.

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