I R R 



pliet Elijah, t Kings, xviii. 27, by Solomon, F.cclef. xi. 9, 

 and by our Saviour himfelf, Mark, \ii. 9. 



IROQUOIS. See Six Nations. 



Iroquois, National Mufic. Claude Perrault. an un- 

 believer in the liarmony or counterpoint of, the Tincicnts, 

 tells us, in his " Phyfical EIFavB," that Louis XIV., near 

 the end of his reign, when fome of tlie Iroquois nation were 

 brought into France, wirtiing to ii'-ar tht-ra fnig, that he 

 might form fome idea of their mniic, many of them fang 

 their wild melodies in unifons and oclaves, ivhile others ac- 

 companied them in Li'rimting like pigs ; regularly, however, 

 marking the meafure by a violent jolt. And thus they at- 

 tempered the acute voices by the mixture of the grave 

 orunts and rhythmical pulfations of the others. 



Perrault imagines, from the defcription which CnHlodorus 

 gives of harmony, or finging together, which tiie ancients 

 called j^iw^/ix/nj, that Roman harmony refembled that ot the 

 Iroquois. 



IRRADIATION, fignifiesan emanation or fliooting out 

 of rays, or fubtle effluvia, from any body. See 1L.M.\.S'.\- 

 TION, R.'iY, Eppi rviA, and Qi^ality. 



IRRATIONAL Nu.mbers, the fame ?& furJ numbm. 

 See Surd and Numbkks. 



Irr.\tional Quantities. See Rational Quarititia. 



Irr.\tionai, Sou!. See Soul. 



IRREDUCIBLE C.\se, in y!!-ebra, h an expreffion 

 ariling from the folutiou of certain equations of the third 

 degree, which always appears under an imaginary form, 

 notwithftanding it is, in facft, a real quantity, but the reduc- 

 tion of it to a rational, or irrational finite expreffion, has at 

 prefent refifled the united efforts of many of the moll cele- 

 brated mathematicians of Europe. Every cubic equation 

 may be reduced to the form .»' -\- ax = b; and then, ac- 

 cording to the common rule. 



See Equations. 



Now, when a is negative, - - is alfo recrative ; and, there- 

 27 



R R 



when - is crreater than — , but wlien — exceeds -, you 



4 ° .27 27 4 ^ 



cannot refolve the equation ; and, therefore, you rcqueft me 

 to fend you the folution of the equation x' — ^x = 10. 

 To which I reply, that you have not ufed a good method 

 iu that cafe, and tlrat your whole procefs is entirely falfe. As 

 to refolving you the equation you h.ave fent, I muil fay that 

 I am v»ry forry th:it I have already given you fo much as 

 I have done ; for I have been informed by a credible per- 

 fon, that you are about to publifli another algebraical work, 

 and that you have been boalting through Milan of having 

 difeovered fome new rules in algebra. But take notice, 

 that if you break your faith with me, I fhall certamly keep 

 my word with you, nay, I even affure you to do more than 

 I promifcd." (Dr. Hutton's Mathematical Dictionary, ar- 

 ticle Algebra.) Tartalea, however, notwithtlanding what he 

 fays iu this letter, was himfelf well aware of the difficulty in 

 qneftion, as appears from fome of his private memoranda : 

 and from that time to the prefent, which is near 300 years, 

 the fame impediaient remains, notwithllanding the repeated 

 attempts of many very dillinguillied mathematicians ; in 

 faft, there is great reafon to fuppofe, independently of the 

 failure of fo many ingenious attempts, that the formula is in- 

 expreffible in any other finite form, than that under which 

 it naturally arifes by the folutiou. See Equations. 



Notxvithftanding, however, that no analytical folution can 

 be given to the cafe in queftion, every equation of this form 

 has three real roots, which are obtainable by other methods, 

 fuch as by means of a table of fines and tangents, infinite 

 feries, continued fraftions, and a new method, lately publilhed 

 by Mr. Barlow in the Mathematical Repofitory, wliich fcems 

 by far the readied and moft accurate of any that has been at 

 prefent difeovered ; the rationale of which is as follows ; 



I. Every cubic equation may be reduced to the form i-' 

 + n .r = + b, by the known rules in algebra ; but when 

 the equation is in the irreducible cafe, this ambiguous form 

 ceafes, and the equation becomes x' — a x =^ + b ; the 

 folution of which, by the following method, is the fame for 

 either fign of b ; only when b is pofitive, the root firft found 

 will he pofitive, and when b is negative, the root will be 

 negative alfo ; it will, therefore, be fufficient to confider the 



fore, when — < — , the quantity 

 4 27 



idcr the inferior radical, ^.^^^ 



b, 





ihich — < — , as we have before 

 4 27 



IS imaginary j 



bee 



ve- cannot ex- 



traft the fquare root of a negative quantity ; and this is what 

 eonftitutes that which is g«nerally called the irreducible 

 cafe. This difficulty foon prefented itfelf to Cardan, after 

 Tartalea had communicated to him his method for the folu- 

 tion of cubic equations, which rule is now con;monly, though 

 very improperly, attributed to the former. Cardan informs 

 Tartalea, m a letter dated Auguft 4th, 1^39, that he un- 

 cerftood the folution of the equation *' -j- <; v = b, and 



alfo oi x' — a X = b, when - > — ; but when - < - 

 4 27 4 27' 



his attempts always failed ; and he therefore begged of Tar- 

 talea to clear up his difficulty, by fending him the folution 

 of the equation x' — qv — 10. Tartalea was himfelf per- 

 feilly aware of this difficulty, but he was by no means fa- 

 tisfied with Cardan's conduct, wliom he at that time fufpetled 

 to be about puhlifhing as his own the rules he had taught 

 iiiro ; and, therefore, inllead of giving him an explicit 

 anfwer, he writes to him in the following terms — " M. 

 Kieronime, I h.ivt- received your letter, in which you write, 

 iia: you uaderiland the rule for the cafe .b' — a.v = b, 



Now, every equation of the form x'' ~ a x = b, may be 

 transformed to anot!>er dependent equation, in which tiic 

 CO. efficient of the fecond term (hall be unity ; that is, to 



anotherof the form v' —V =r r. For make jr=-, then the 



equation becomes — 



w take ax'^ — i , or c = 

 ba\ . 



and, coiifequently, ^z' 



ce we have _y' — ^ = — - ; or putting - 



= c, it becomes 



jr' — y =z r, as required. 

 And the value of ^ being found in thi.s reduced equation, 

 we immediately obtain that of .v, in the original one, by 



means of the expreffion x ■=■ — , or .v = j» .J a. 



Now, the original equation being by hypothefis of the 

 irreducible form, the transformed ec^uationmutl DectlTarily be 



fo 



