INVOLUTION AND EVOLUTION. 



INVOLUTION AND EVOLUTION. 



exhibit at length the finding of 9x-8-141x s ---009.r-1427-409, when 

 xa 121-23; or, to avoid utl in the expression, we may consider 

 this as the thousandth part of OOar-S141i-l-x-1427499. The 

 proce** U a* follow* : a great many figure* (about 115) being repeated 

 twice over, in a manner wholly unneceasary in computation, in order to 

 facilitate the explanation. 



- ,/.. | 

 26937,1809 



- :.:!- : 

 : _.. .... 



: -- mjK 



888046169 

 391292028 

 394546887 



C 



: I4M6M700 

 1981 1--" 



: 1080 tffll : i 



8967168,401 

 15506343,181 



B 



15505343181 

 IK IMUM9 



D 



10890686X091 



15975675212760 



15975675212760000 

 15987553760654100 



MOO, - 31,41 - 1,427499(121-23 



8968,59 ... . , 8967,163401 



1 7968,50 

 2698,50 



A 



., , - , 

 .-:- 

 305685.9 B- 

 WMM 



3245859 C D- 



8254859 

 8263859 



82638590 D- 

 :. 3958535106000 

 32674590 3959515964700 

 3960496904400 



D 



: HMI 



: H '.--'. D 

 BM ;:: 



KM WOO 



We fint put down the coefficients an usual, not changing the sign 

 of the last (which is only a convenience for n-olution, and doeg not 

 alter any figure). The value of x being 121*23, we begin with 100, 

 which, having two ciphers, we mark off by commas from the several 

 coefficients 0, '2, 4, 6 places. We then proceed by Homer's process 

 with the figure 1 (not 100), taking care to make commas fall under 

 commas, or to use the commas as if they were decimal points (which 

 they are in fact, though not unit-point*). As soon as we have done the 

 first process, containing all that comes before the lines A, we learn as 

 follows. Let 



\r = 9-r 1 - 3-14U' + -009x - 1427'499 



then, x being 100, \r, *>'*, #".r, and fr'"x are severally 8967163-401, 

 269371-809, 2696-859, and 9. We then write down the results again, 

 after the lines A (which is not necessary in calculation), merely to show 

 the new disposition of the commas. We are now to proceed with 20 

 (from the first 2 in 12T23), which, having one cipher, we mark off 

 0, 1, 2, 3 places in the several columns. Immediately before the lines 

 B we learn that when x 120, <t>.r. <f>'s. and \^fx are severally 

 15505343-181, 888046-169, and 3236-859. We then write down these 

 resulU without any commas, and proceed with the second 1 in 121, 

 from which we find that when j--121,the functions are 15890635-209, 

 894546 887, and 8263-859. We then begin to provide for the decimal 

 point, by annexing one, two, and three ciphers to the working column*, 

 and taking the second 2 in 121'23 to work with, and applying Uorner's 

 procesi, we find, when jr=- 121-2, that ^r, fx, and J? x, are severally 

 (remembering that all the annexed ciphers are so many additional 

 decimal place*) 15975675-212760, 895853-51060, and 8269-2590. 

 Finally, we annex the cipher* again, and with the 8 wo find 

 that x = 121-23 give* 15987553-760654100, 396049-6904400, and 

 3280-06900. 



Let u* now compare the trouble of this process with that of any 

 other method of doing the same. If we throw out all the figures 

 which we have written twice over merely for explanation, and also the 

 last two and one line* in the second and third columns, which are only 

 wanted to go on further with, we have written down about 280 figure*. 

 The ordinary verification cost* about 840 figures. It U true that every 

 tep is both a multiplication and an addition in one : but this can be 

 don* and ought to be done in the use of this method, and is not done 

 in the ordinary method. And we have not only the advantage of a purely 

 mechanical method, in which the fint arrangement causes the succeeding 

 tops to require nothing except a look at the successive figures of the 

 value, but the still greater advantage of being able, at the end of the 

 proems, to make any small alteration of value with ease. If, for 

 tneUiy-e, having discovered that 121-297 would do better than 121-23, 

 we wish to get additional accuracy, we have but to rub out the last 

 8- process, and proceed with 9 and 7. In the ordinary mode, we must 

 ithr repeat the whole procea* again, or correct approximately by 

 substituting 121-28- -008, which wttlrequire us to calculate f'x, and 

 perhaps Jf'V. 



We shall now exhibit a common multiplication, and the formation of 

 a aquare : not, of course, that we attach any particular value to those 

 simple cases, but that we may show the uniformity of the process. 

 Ke<|uired 14706x82814, or the value of 14796x + when .r=32316. 

 We repeat the lines as before, which is more than u necessary, and 

 make* this process look very long. 



14790 



0.0000(32316 

 44388,0000 



443880,000 



473472,000 



47347 



I77!OOMO 



477H1080,0 



47Mi.W7iill 

 Answer 478147686 



Required the square of 279'46, or the value of 

 x= 279-46. 



1 ,00 



2,00 

 4,00 

 40,0 

 47,0 

 54,0 

 540 

 549 



, when 



558 



5584 



(in !; 



I. 



oo,oo 



7 -.".'on 

 77-11 



,100 

 780T. , 

 78064 

 7809789M 



55880 



Answer 78007'8916 



00893 



The process here described ia one which, we venture to say positively! 

 has neither been put in its right place, nor received its due reward- 

 It is the natural extension of the common process of multiplication, 

 and its inversion is aa naturally and necessarily the proper mode of 

 solving equations, as the inversion of multiplication is the same for 

 the simple equation ox=b, or common division. The inventor of it 

 must rank, not with the analyst or the algebraist, commonly so called, but 

 with the discoverer of the process of multiplication and division, and 

 the extraction of the square root. 



The application of this method to the solution of $u- = consist* in 

 finding the first figure by trial, and making use of the Newtonian 

 approximation to find successive figures : namely, that if a be nearly 

 a value of x, a <j>a : <p'a is more nearly so. This method become* 

 dillicult when two roots are nearly equal ; but the difficulty lies in what 

 may be called Newton's part of the complete method, not in Homer's 

 part. When the difficulty of algebra shall be conquered, the process of 

 arithmetic may easily be amended in the trial part ; but to suppose 

 that a capital improvement in the manner of conducting computations 

 is little worth, because it is not accompanied by a victory over difficulties 

 of quite another kind, is unreasonable. With a little more trial, H < 

 method may be applied to the case of nearly equal roots ; and as it is, it 

 is more efficacious in discovering them than any other method. 



To what has been said upon the method, we may add the 

 following remarks : 1. When the last term is positive, and would in 

 the ordinary process be made negative, it is often better, instead of 

 changing the sign of the last coefficient only, to change the sign of all 

 but the last. Thus in solving 2* llx+1 = 0, the beads of the columns, 

 should be 1, 0, 11, and 1, instead of 1, 0, 11, and 1. Also, 

 that if at any period of the process the divisor and dividend columns 

 should become negative, the signs of all should be immediately changed. 



2. In making the contractions, it will be advisable to make the 

 igutv which comes next after the separating line correct, to continue 

 it, in fact, till the next contraction, and to use it to carry from. This 

 is not done in what precedes, but it is done in the instance in COM- 

 PUTATION. In that instance, the following figures, seen one over the 

 other in the last column but one, as follows, 3, 5, 7, 6, 9, 1, 2, 2, 2, 

 are figures cut off by the contraction, but made up from the second 

 column to carry from into the fourth. 



3. If, at the beginning of the process, all the heads of the columns 

 Mi multiplied by 9, the root will not be altered, and, until the con- 

 traction begin*, the verification by casting out nines is rendered easy, 

 since every result in every column is divisible by 9. 



We shall now show how the process works in some equations which 

 lave equal, and nearly equal, roots. 

 Let j* 6j? + 9 = 0, which has two roots, each equal to */'A. ' 



-\ 







1 

 -2 

 -8 

 -40 

 -47 

 -54 

 -61 

 -680 

 -688 

 -686 

 -689 

 -692 



6 



5 



3 



000 

 -829 

 -707 

 -113400 

 - -Hi 5449 

 -117507 

 -119574 

 -11971 

 -11985 

 -11999 







5 



> 



697 



748000 



401653 



49132 



80190 



1220 



9 (1-7320 

 40000 

 1210000 

 5041 



