ROOT. 



necelTary wherever there is a number of cows or other forts of 

 cattle to be fupported on roots of the carrot, parinep, turnip, 

 and potatoe kinds, as well as for cabbages, as without them it 

 would not be only inconvenient, but in many cafes, in fevere 

 weather, impoflible to provide them for the daily fupply of 

 fuch (lock. The cabbages mould not, however, ever be 

 kepi long in thefe houfes, as they are very apt to take on 

 the putrid fermentation, and become ufelefs. The farmer 

 ihould be careful that the yard-man conftantly keeps fuch 

 places perfectly clean and fweet, in order that the roots may 

 contract no bad fmell, as cattle are in many cafes extremely 

 nice in their feeding, and when once difgutted with any fort 

 of food of this kind, feldom take to it again in a proper 

 manner. 



Root-houfes are always the bell and fweeteft when laid 

 over on the infides with a coarfe plailer, or boarded with 

 lome rough common boarding material. They may in many 

 cafes be divided for different forts of roots, with great ad- 

 vantage and convenience. The doors of them fhould for the 

 mod part be large, fo that the carts may be backed and 

 readily emptied in at them without any difficulty. 



Root, in Arithmetic and Algebra, denotes a quantity 

 which, being multiplied by ltfelf, produces fome higher 

 power, and it is called the 2d, 3d, 4th, &c. root, according 

 to the number of times that the multiplication by itfelf is 

 performed, that number being always one lefs than the deno- 

 mination of the root : thus, if a number is multiplied once 

 by itfelf, it is called the fquare root, or 2d root of the pro- 

 duct ; if twice, it is called the cube, or 3d root ; if three 

 times, the biquadrate, or 4th root ; and fo on to other roots, 

 which, beyond the 4th, are commonly denoted by the jth, 

 6th, &c. root ; though ancient authors, and even fome mo- 

 dern ones, ufe particular denominations for all higher roots, 

 as we do for the fquare root and cube root. Thas, the 

 2d root, is called the fquare root. 



3d root 



4th root 



5th root 



6th root 



&c. 



cube root. 

 f quadrato-quadratum, or biqua- 

 \ dratic root. 



furfolid root. 



quadrato-cubo. 

 &c. 



But fuch diilin&ions are ufelefs, and are, therefore, now 

 commonly omitted. 



For the method of extracting the roots of numbers, fee 

 the articles AprROxiMATON and Extraction ; and for a 

 table of the fquare roots and cube roots of numbers, fee 

 the conclulion of this article. 



Roots of Equations, in Algebra, denote fuch a number 

 or quantity as, when fubftituted for the unknown quantity, 

 will produce an equality between both tides of the equa- 

 tion : thus, in the equation 



x 1 — 6x~ -f- 11 x = 6, or a- 5 — 6: ** + 1 1 .v — 6=0; 



if we fubftitute 1 inllead of .v, we have 1 — 6+11 = 6; 

 therefore 1 is a root of that equation : if we fubftitute 2 

 inftead of .r, we have 2 ' — 6.2'+ 11.2 = 6; therefore 



2 is alfo a root of the fame equation : and if we fubftitute 



3 for x, then 3 ' — 6 . 3 1 4- 11.3=6; and, therefore, 

 3 is likewife a root of the fame equation : hence the roots 

 are I, 2, and 3 ; that is, there are three diftincl numbers, 

 which, when fubftituted for .v, will produce the equality 

 required. And it is the fame in all equations, viz. it has 

 always as many roots, real or imaginary, as there are units in 

 the index of the higheft power of the unknown quantity. 

 This property has place in equations of the mod fimple forms, 

 as X 2 = i,x' = 1, x* = 1, #' = 1, &c. each of thefe 



having as many roots as there are units in the exponent of 

 the power : thus, the 



two fquare roots of I, are -f 1 and — 1 ; 



three cube roots of 1, are 1, - I, + i j — 3, and - §, 



- hs - 3 ; 



four 4th roots of 1, are I, — J, + ,/ — i.and — J — 1 ; 



fn\'5throot5 

 of 1, 



are 1,- 



1 



and 



- 1 + 

 4 



^ ± v{(=i±^/ 



i 



the two latter forms containing each two roots, in confe- 

 quence of the ambiguous lign, +, which enters into their 

 compofition. 



The doctrine of the roots of equations is one of the moft 

 intricate, but at the fame time moft interefting, of any 

 branch of algebra. The method of finding the roots of 

 quadratic equations is found in the earlieft algebraic authors; 

 it is even given, though fomewhat different in form, in the 

 Bija Ganita, a Sanfcrit algebra, written about the latter 

 end of the 1 2th, or the beginning of the 13th century, tranf- 

 lated into Perfian in 1634, and lately into Englifh by 

 Mr. Strachey, of the Eall India Company's Bengal civil 

 eftablifhment. The folution of cubic equations was firft 

 publiihed by Cardan about 1540, though it is clear that 

 he was not the inventor of the method, having received it 

 from Tartaglia, who is commonly confidered as the real 

 author ; however, Lagrange attributes the general invefti- 

 gation to Hudde, a celebrated Dutch mathematician, a 

 contemporary of Defcartes and Format. Equations of the 

 4th degree were firft folved by Ferrari, a pupil of Cardan's, 

 and publifhed by the latter in 1540 ; fince which time no 

 further extent has been given to the fubjefrt, the 5th, and 

 all higher equations having refilled the whole accumulated 

 power of the modern analyfis. Still, however, many important 

 properties of the roots of equations have been difcovered ; 

 the whole theory has been reduced to one uniform prin- 

 ciple of operation ; and approximations have been made in 

 all thofe cafes where direct methods of folution were un- 

 attainable. We cannot, of courfe, enter upon this fubject 

 at any great length ; but a fummary of the moft interefting 

 particulars, though not accompanied, in all cafes, with their 

 demonitration, will not, we prefume, be unacceptable to the 

 general reader ; in the enumeration of which we fhall avail 

 ourfelves of the Introduction to Barlow's Mathematical 

 Tables. 



General Properties of the Roots of Equations. 



1. Every equation of the general form 



*■» + A X " ■ ' + B.v™ ' + C «— ' + &c. + K = c 



has m roots real or imaginary (fee I.m iginary Roots) ; and 

 may be fuppofed to be formed by the continued producl ot 

 m factors, 



(.r - a) (x - 9) [x - , ) (x - J) &C. = O, 

 where x, /J, -., J, &c. are the routs of the equation. 



2. The imaginary roots of an equation always enter in 

 pairs, and if a + b v / — 1 be one of thofi roots, a — b 

 ■v/ — I is another of them ; fo that the fum of every pair 

 of them is a real quantity, and the fquare of their difference 

 a real negative quantity ; and an equation can have no 

 imaginary root but i< reducible to the above form. Thefe 



properties of the imaginary roots of equation are generally 

 attributed to d'Alembi 1 1 , 



J T 2 *. Sin. ■• 



