T ii i: 





ii i: 



the uni of the rooU of the equation in x being -/, : a,, 

 that of the equation in y will be 0. 



1.1. To umiup'v iill the rout* of an equation by in, inul- 



.. t!u- I'..-!.. 



.'i- ol ..ii cqua- 



tion by i, multiply all the ten. .lining 



from the lowest. N.B. Term* ai<i- n.< in an 



.on must neve. -r-3j: 



1 = ought to be writ:- 



This caution is of the utmost importance : in fact no 



process ought to be applied to nny equation without a 



momi-nfs tlioiiRht an to whether all the terms are formally 



n down, and if not, whether the process about to be 



applied will not require it. 



16. To change the signs of all the roots of an equation, 

 change the signs of the coefficients of all the odd powers, 

 or of all the even powers, as most convenient. 



17. To change an equation into another whose roots 

 shall be reciprocals of the former roots, for every power 

 of x write its complement to the hiirlic-t iliimn~io:i. 

 Thus in an equation of the seventh deirrec. for .r" w i . 



write a*. for af write .r 1 , and so on; lastly, 

 write af. N.B. Consider the independent term of the 

 equation as affected by ./:". From the reciprocal equation 

 can be found the sums of the negative powers of the roots 

 of the original. 



18. The' old methods of finding limits to the magnitude 

 of the positive and negative roots of an equation are so 

 rapid that they can hardly be said to be superseded by 

 those of Sturm or Fourier. In enunciating them we 

 speak of coefficients absolutely, without their siiriis, when 

 mentioning any increase or decrease they are to receive. 



if A be the greatest of all the quotients made by divid- 

 ing the co-efficients by the first co-efficient, no root. posi- 

 ti\c or ucsrativ, is numerically so great as A+l. And if 

 B be the greatest of all the quotients made by dividing the 

 co-efficients by the last co-efficient, no root, positive or 

 negative, is numerically so small as 1:(B+1). Better 

 thus : if L be the first co-efficient, M the greatest, and N 

 the last, signs not considered, then all the roots, numeri- 

 cally speaking, lie between 



M+L N 



-IT and M+N 



19. If L be the first co-efficient, and M the greatest co- 

 efficient which has a different sign from that of L, no 

 positive root is so great as (M+Li : L. And it I, be the 

 last co-efficient and M the u'rcalcst which has a different 

 sign, no positive root is so small as L : (M+L). And to 

 apply this to the negative roots, change the signs of all 



ots of the original (J 1C), and find limits to tlu 

 the roots of the new one. 



20. If L be the first co-efficient, M the greatext which 

 has a different sign, and if the jirxt which has a different 

 "itrn be in the <th place from the first term exclusive, or 

 bekng to the (wi + l)th term ; then no positive root is so 

 great as 



the original. It such a qua he readily found, the 



1 be 

 greatly diminished, and, pi .. 



would be gained m numerical solution. \Yhat is 

 \vanted to add to both Fourier's and Horuer's method, is a 

 ready mode of finding out when two luoU are nearly 



I-agrange's mode of approximation is as folio 

 Having found that a root of an equation lies between the 

 s .1 and 'i-f-1, diminish all the roots of that equation 

 ind take the reciprocal equation to the result. Find 

 of thela.it lying between the integers b and 4+1, 

 diminish all the roots by t>. and take the reciprocal equa- 

 tion of the result. Find a root of this last between c and 

 <+!. and proceed in the same way. Then the continued 

 fraction 



is a root of the original. 



26. When an equation has equal roots, those roots can 

 be found by an equation depending entirely on the u 



ts of equal roots. If fr have m roots equal to , j'x 

 has /,;-! of tliem, $"x has m 2 of them, and so on; 



finally, f x has one of them. If then f j- and $'x be 

 found to have a common measure, every root of that com- 

 mon measure enters in f .r one time more than in the com- 

 mon i: -elf. 



27. \Vhen an equation has an integer root, which must 

 he one of the divisors of the hist it, it may be dis- 

 covered by successive trial, as follows : Suppose d^j-'-f-d, 

 x'+a, a"+a, a?+a 4 =0, n . tic. being integers. Let A be 

 a di\ isor of a,, and let a, : /t=l, an integer. Then if A be 

 a root, we have u,J< i +ii l h*+u,ti+ti 1 +/=0, and a,+l is 

 divisible by /t, giving in. an integer. Hence aJP+a I k + 

 u -)-;=(), and ,-f-' divided by /r gives an integer, HIV . 

 Hence a,,A-r-<',-|-"=0, and a,+n divided by A gives o . 

 If all these conditions be fulfilled, A is a root. All the 

 divisors of o 4 being tried in this manner, settle thequ* 



of the integer roots entirely. 



28. If the co-efficients of an equation read backwards 

 and forwards the .-umc, both in sign and magnitude, every 

 root has it.s reciprocal also among the roots. By reducing 

 it to the form 



'-efficient which differs in sign from the 

 first term. ! by the sum of all which preci 



agree with >n itself included ', the 



ImHeM resulting fraction, inerensed by unity, is greater 

 than any posi 1 : ^nation. 



rr 1han the 

 ilion now merges in 



Fourier's theorem. It consists in finding a hyinsp 

 and trial, so that 4*1, <ft'ti, <j>''a, &c. shall all be posit. 



mode of ascertaining a limit greater than the 

 e root of an equation may l.e thus t 

 > riprocal equation ( 17), and tb. 

 i the legist posi 1 



iginal. Apply both to the equation of roots will 

 ohanKcd, and the results give limits for the ne: 

 of the original. 



84. A celebrated mode of examining the roots of equa- 

 tions, but too complicated for ordinary use, consists in 

 forming the equation whose roots are the squares of the 



ires of the roots of the original. Any quantity ! 

 found lew than the least positive root of this new eq'i 

 it* square root U lew than the difference of any two root* of 



which can always be done by division, when the dimen- 

 sion is even, and assuming y = .r+.- , an equation of 

 the 2wth degree can be reduced to one of the th and 

 n quadratics. But when the dimension is odd. either 1 

 or -4-1 must be a root, and the equation can be depressed 

 to an even degree by division by r-fl or x 1. 



The student who is acquainted with the precrdhu,' re- 

 sults, namely, such as are cither stated or referred to in this 

 article, will find no difficulty either in leading on the his- 

 tory of this subject, or in its'applicntion. It is peculiarly 

 a subject on which selection should be made for the 

 bi'iriniier. 



THKK.V (iWipnX an island in the Grecian Archipcl 

 and the chief of the group known by the name of Spo 

 although called by some antient 'writers one of tin 

 Its modern name is Santu Thiia, which i- 

 uouneed and iisuallv wiittcn Santorini. It : 

 Si mho i \.4s4.Casaub.') to be 200 stadia in eirci 

 but by modern tiavcllers thirty-six miies. and in t 



island of Dia, and distant from Cn-te TIKI 

 the island of los, which lay to the north of it, 25 R 

 miles. 'Pliny, Ilixf. Nit..\\. 23.1 When it first emerged 

 from the sea, it is said to have been railed ('allistc : Tile- 

 jasia, a small island to the west, and . ! by 



.us torn away from i'. :'iiny. 



'ion seems at one time to hsvc hccii actively 

 at work in this part of the sea. S 

 that on on .1 the sea 



between Tliera and Thera*ia. vdiich lasteil ; r four . 

 and that an inland was formed in coiise.jiuiice, tv 

 stadia in circumference. The same phenomenon has also 

 taken place in modem times, ami is |, iibed 



by .1. fhfvenot in his Travels in the Levanf'(part 1.1. 

 Pliny also speaks of an island which arose between Tliera 



