957 



INVOLUTION AND EVOLUTION. 



INVOLUTION AND EVOLUTION. 



958 



44521433589 }>p 

 109639800 q 

 82892291 s 

 4452285890991 s, 

 32892372 1 

 292377 v 

 445232170713 m 

 292377 v 



2192 w 



44523248501~'tne 



But considering that the process is one which no person will very 

 often perform, we doubt whether to recommend even this abridgment. 

 All such simplifications tend to make the computer lose sight of the 

 uniformity of method which runs through the whole ; and we have 

 always found them, in rules which only occur now and then, afford 

 greater assistance in forgetting the method than in abbreviating it. 

 But one abbreviation, when duly learnt, is found to have very great 

 advantage. The addition of the products which are carried from each 

 column to the next and for the last column, the subtraction might 

 be made at once, figure by figure, as fast as the figures are formed. 



On evolution of algebraical quantities we do not think it necessary 

 to speak, since either the binomial theorem [BINOMIAL THEOREM], or 

 some other method of development, is employed with more advantage 

 than the usual modification of the arithmetical process. We have also 

 omitted the process of division, the most simple of all evolutions, since 

 its connection with the preceding is sufficiently obvious. 



There is, however, a process of an evolutionary character which we 

 take this opportunity of suggesting, and of which any one moderately 

 conversant with algebra will easily arrive at the demonstration. In 

 finding the highest common divisor of two algebraical integral expres- 

 sions, and also in the process of Sturm's theorem, it is . required to 

 divide one expression, p, by another, Q, not for the sake of finding the 

 quotient, but the remainder ; and this remainder, cleared of all 

 numerical factors and of fractions, is preferable. The following rule 

 will render the application of Sturm's theorem much more easy : 



Question : Two expressions, p and Q, being given, .of which p is 

 lower than Q, required the remainder of Q divided by p, cleared of frac- 

 tions and of positive numerical factors. To take an example with us, let 

 p=2.z 3 *-4 

 <J= 4a 3*' + 'Lf x + 1. 



1. Add 1 to the difference of the degrees (5 3 = 2, 2 + 1 = 3) : this 

 is the number of operations to be expected. In this case it is three. 



2. Write down in two lines the coefficients of the divisor and 

 dividend, including the coefficient for every missing term, but change 

 the sign of every coefficient in the divisor, execjtt the firit. Clear both 

 expressions of all whole factors ; and if the two leading coefficients (2 

 and 4) have a common factor, divide by this factor before writing them 

 down. Write ciphers in all blank places. 



(p) 1 1 4 



(<j) 2-3 2-1 1 



3. Take the first vertical pair, and every other in succession, and 

 make cross multiplication and addition : thus, j " " ' j gives ad + be. 



Put the first result in the first column, the second in the second, and 

 so on. 



(1) -3 2 10-1 1 



4. Repeat this process with the first line, and the result just 

 obtained, and again with the result, making each new result out of the 

 first line and the last result, and so on till the number of operations 

 ascertained in the first clause of the rule has been performed. But if 

 the leading term of the first line have been divided, multiply that 

 leading term again after the result, unless the first term of the result 

 be also divisible by the game factor. 



5. But if ever the first column of a result should turn out a cipher, 

 throw it away, and bring forward the next column, and so on, making 

 every step stand in the next higher place ; and if the two first columns 

 of any result should be ciphers, throw them away, and bring forward 

 the third, and so on. And for every cipher thus thrown away, diminish 

 by one the number of operations required in the first clause. 



6. If any horizontal line thus obtained have a factor in all its terms, 

 divide by that factor, taken positively, before proceeding further ; and 

 if the leading term of any new result have a factor in common with 

 the leading term of the first line, divide both before proceeding. 



The table of results now is as follows, in which the various changes 

 of the leading terms are shown by putting them down as they occur, 

 and putting a bar over them as they disappear and are replaced by 

 others. In practice, the pen may be drawn through the figure which 

 i dismissed. 



- 1 



17x'-l2x + 10 Answer. 



7. When the last result has been obtained, make an algebraical 

 expression one degree lower than the divisor, the coefficients of which 

 are the numbers in the last result, with their signs. 



The real remainder in the preceding example is SJa; 2 6x+5. 



Let the next example be 



p=4.r 3 6 



Here the number of operations should be four ; but it is reduced by 

 the circumstance mentioned in the fourth clause of the rule. 



(1) 

 (2) 

 (3) 



4, 2,4,2 



2, 1 

 _0,*3 

 22,11 



6-1-1 



- 3 



1 



27 



- 65 



-27 

 53 



1 



- 6 

 32 





 3 



8 



- 3 



12* 1 65* + 53 Answer. 



The method of verifying the several processes, as their results arise, 

 is as follows : Make an additional proof column, in which place the 

 sums of the numbers in each line, taken with their signs ; making 

 these stuns vary with the variation of the leading factors : thus 



r 



og+op 



6 



1 

 ar + cp 



rroof. 

 A 

 p 

 z 



Here A is o + i + c+ ...; P is p + q + r+ ...; and ziaaq + bp + 

 or + cp + . . . . If then the process be correctly done, an exten- 

 sion of it to the proof column gives ap + Ap, which ought to exceed 

 z by 2 ap. 



We shall conclude this article with the process which will be applied 

 hereafter. [STURM'S THEOREM.] The object is to proceed as in 

 finding the greatest common divisor of p and Q, changing the sign of 

 every remainder before using it. 



P= ix 3 9x> 4x+l 

 Q = x* 3^2^ + x- 3 



The remainders therefore, with the sigus changed as directed, are 

 43.i- + 45, 151 36.r 19264, and the last is a negative whole number. 

 The following is the first instance of the use of the proof column : 



lx!6 + 4x(-6)-(-16) = 8 = 2(4xl) 



In the preceding part of this -article we gave an account, with 

 instances, of the method of solving equations, which is commonly 

 known by the name of the late Mr. W. G. Horner, schoolmaster, of 

 Bath. We believe we may usefully give what precedes a considerable 

 extension : first, because the method [COMPUTATION] is one of the 

 best exercises in computation ; secondly, because neither its meaning 

 nor its history is very generally understood, and the latter is very 

 instructive. 



The process of involution as above defined, is the formation of the 

 value of a rational and integral algebraical expression, such as 

 O.T 8 + bx* + ex + d, by a succession of multiplications separated by 

 additions, as in 



\(ax + l)x + c} x + d. 



Homer's mode of doing this takes the figures from left to right, or 

 takes those of largest value first ; and exhibits a plan of performing 

 the operation which combines the result of each figure with the joint 

 result of all that come before. Thus iti finding the value of the pre- 

 ceding when .r= 123-456 the value is first found when x = 100; then, 

 by help of the preceding, when .i'=120; then when .=123; then 

 when x= 123'4 ; and so on. By this means we are enabled to proceed, 

 when the value of the succeeding figures depends upon the results of 

 those already found, as happens in all the cases of evolution, the 

 inverse process. To take, however, the direct process first, we shall 



* Then bring forward the next column. 



