RATIO. 



admitted of by all kinds of qnantities, difcrete or continued, 

 commenfurable or incommeiifurable : but difcrete quantity, 

 or number, does likevvife admit of another ratio. 



If the Icfs term of a ratio be an aliquot part of the greater, 

 the ratio of the greater inequality is faid to be mulliplex, mul- 

 tiple ; and the ratio of the L-fs inequality, fubmultiple. 



Particularly, in the firft cafe, if the exponent be 2, the 

 ratio is called rt'a;>/f ; if 3, tripU, Sec. In the fecond cafe, if 

 the exponent be 4, the ratio is called fuiilupie ; \f 'jjfubtriplc, 

 &c. 



E.gr. 6 to 2 is in a triple ratio ; becaufe 6 contains two 

 thrice. On the contrary, z to 6 is a fubtriple ratio ; becaufe 



2 is the third part of 6. 



If the greater term contains the lefs once, and over and 

 above an aliquot part of the fame ; the ratio of the greater 

 inequality is called fuperparticularis, and the ratio of the lefs 

 fub-fuperpariicularu. 



Particularly, in the firft cafe, if the exponent be i-j, it is 

 C3\l<::d fefquialtcrate ; if 7i\, fefquitertial, &c. In the other, 

 if the exponent be -?» the ratio is aWeA fubfefqulalterate ; if 

 ^, fubfejquitert'ial, &c. 



E. gr. 3 to 2 is in a fefquialterate ratio ; 2 to 3 in a fub- 

 fefquiaiterate. 



If the greater term contains the lefs once, and over and 

 above feveral aliquot parts ; the ratio of the greater inequa- 

 lity is called fuperfiartiens ; that of the lefs inequality isjitb- 

 fuperparliens. 



Particularly, in the former cafe, if the exponent be 1 1, 

 the ratio is called fuperb'ipartiens terlias ; if the exponent be 

 14, fiipertripartitns quarlas ; \i l\-, fuperquadrtpartiens fepti- 

 mas, &c. In the latter cafe, if the exponent be I, the 

 ratio is c^Udf^d fubjuperb'tpartiens tertias ; if *., fubfupertripar- 

 tiens qtiartas ; if x"r> fubfupcrqiiadnparl'iens feptimas. 



E. gr. the ratio of 5 to 3 \% fuperh'ipartiens tertias ; that of 



3 to 5, fubfuperb'ipartiens tertias. 



If the greater term contains the leis feveral times, and, 

 befides, fome quota part of the fame ; the ratio of the 

 greater inequality is called multiplex fuperparticularis ; and the 

 ratio of the lefs inequality, fubmultiplex fubfuperparticu- 

 laris . 



Particularly, in the former cafe, if the exponent be 24, 

 the ratio is called dupla fefquialtera ; if 34, tripla fefqui- 

 quc.rta. Sec. In the latter cafe, if the exponent be -J, the ratio 

 is called fubdupla fubfefquialtera ; if -rV> fubtripla fubfefqiti- 

 quarta, Sic. 



E.gr. the ratio of 16 to 5 is tripla fefquiquinta ; that of 



4 to 9, fuldupla fulfefquiquarta. 



LalUy, if the greater term contains the lefs feveral timei:, 

 and feveral aliquot parts of it befides ; the ratio of the 

 greater inequality is called multiplex fuperpartiens ; that of the 

 lefs inequality, fubmultipkx fubfuperpartiens. 



Particularly, in the former ca(e, if the exponent be ^, the 

 ratio is called diipla fiiperbiparliens terlias ; if 3^, triplnfu- 

 perbiquadraparliens feptimas, &c. In the latter cafe, if 

 the exponent be J, the ratio is called fubdupla fubfuperbi- 

 partiens tertias ; if 4st fubtripla fubfuperquadripartient feptimas , 

 Sec. 



E. gr. the ratio of 25 to 7 is tripla fuperquadrtpartiens fep- 

 timas ; that <if 3 to 8, fubdupla fuhfupcrlipartleru tertias. 



Thofe are the various kinds of rational ratios ; the names 

 of which, though they occur but rarely among the modern 

 writers (for in lieu of them they ufe the fmalleil terms of the 

 ratios, e. gr. (oi- duple 2:1, fox fefquialterate ■}, : 2) ; yet are 

 they abfolutely neceffary to fucli as convcrfe with the ancient 

 authors. 



Clavius obferves, that the exponents denominate the ra- 

 tios of the greater inequality, both in deed and name ; but 



Vol. XXIX. 



the ratios of the lefs inequality, only in deed, not in name : 

 but it is eafy finding the name in thcfe, if you divide the de- 

 nominator of the exponent by the numerator. 



E. gr. if the exponent be ^, then 5:8^1?; whence it 

 appears, the ratio is called fubfupertripartiens quinlas. As 

 to the names of irrational ratios, nobody ever attempted 

 them. 



Same, or identic ratios, are thofe whofe antecedents have 

 an equal refpeft to their confequents, ;. e. whofe antece- 

 dents divided by their confequents, give equal exponents. 

 And hence may the identity of irrational ratios be con- 

 ceived. 



Hence, firft, as oft as the antecedent of one ratio con- 

 tains its confequent, or whatever part it contains of its con- 

 fequent, fo oft, or luch part of the other confequent does 

 the antecedent of the other ratio contain : or, as oft as 

 the antecedent of the one is contained in its confequent, 

 fo oft is the antecedent of the other contained in its con- 

 fequent. 



Secondly, if A be to B as C to D ; then will A : B :; 

 C : D ; or A : B = C : D. The former of which is the 

 ufual manner of reprefenting the identity of ratios, the latter 

 is that of the excellent Wolfius ; which has the advantage of 

 the former, in that the middle charafter, =, which denotes 

 the laraenefs, is fcientifical ; i. e. it exprefles the relation of 

 the thing reprefented, which the other, ::, does not. See 

 Character. 



Two equal ratios, c . ^r. B : C = D : E, we have already 

 obferved, conftitute a proportion : of two unequal ratios, 

 e. gr. A : B and C : D, we call A : B the greater, if A : B 

 > C : D ; on the contrary, we call C : D the leffer, if C 

 : D > A :B. 



Hence, we exprefs a greater and lefs ratio thus : e. gr. 6 to 



3 has a greater ratio than 5 to 4 ; for, 6:3(=2)>5:4 

 (= ij). But 3 to 6 has a lefs ratio than 4 to 5 ; for ^ ;= 



4 > ni. Compound ratio is that made up of two or more 

 other ratios, which the faftum of the antecedents of two 

 or more ratios has to the faftum of their confequents. Thus, 

 6 to 72 is in a ratio compounded of 2 to 6, and 3 

 to 12. 



Particularly if it be compounded of two, it is called a 

 duplicate ratio ; if of three, a triplicate; if of four, quadru- 

 plicate ; and, in the general, multipUcate, if it be compofed of 

 feveral fimilar ratios. Thus 48 : 3 is a duplicate ratio of 

 4 : I and 12:3. 



Ratio, Additive. See Additive. 



Ratio, Alternate. See Alternate. 



Ratio, Ordinate. See Ordinate. 



Ratio Modularis and Modulus, were terms introduced 

 into ufe by Cotes, but more modern authors do not ufe them 

 always in the fame fenfe : according to Cotes, the modulus 

 in logarithms is that number which connefts any fyftem of 

 logarithms with the hyperbolic fyftem, or that number by 

 which the hyperbolic number of a logarithm mult be multi- 

 plied, or by the reciprocal of which it muft be divided, in 

 order to transform it to another fyftem ; and this modulus 

 is, therefore, always the reciprocal of the hyperbolic loga- 

 rithm of the radix of that fyftem to which the modulus 

 belongs. 



This IS what Cotes calls the modulus, to whom we owe 

 the introduftion of the term ; and the reciprocal of it he 

 calls the ratio modularis : but fome modern authors, as La- 

 grange, &c. ufe the term modulus to denote the ratio modu- 

 laris of Cotes. 



The modulus of the hyperbohc fyftem is I, this being 



the reciprocal of the hyperbolis logarithm of 2.71828182, 



the radix of this fyftem ; and the modulus of the common 



3 M logarithmic 



