953 



INVOLUTION AND EVOLUTION. 



INVOLUTION AND EVOLUTION. 



954 



631064798(158 

 101989900 

 529074898 : 

 410987450 

 118087448 ; 

 118087448 



2 : -1 -1 



100 10200 1019900 



102 10199 1019889 



100 20200 3039900 



202 30399 4059799 : 



100 30200 4159950 



302 60599 : 8219749 



100 22600 5414950 



402 : 83199 13634699 ; 



50 25100 1126232 



452 108299 14760931 



_50 27600 



502 135899 ; 



50 4880 



552 140779 

 _5p 



602; 

 _8 

 610 



Assuming 100 as a first approximation, we find that 

 605990- 1 + 4059799* 529074898 = is an equation having roots less 

 by 100 than those of the given equation. And 529074898 contains 

 4059799 upwards of 130 times; but if any number of tens greater 

 than 50 be taken, the accumulations of the next involution will give 

 more than 5290, &c., as must be found by trial. Repeating the process, 

 we find that a* + 602.r> + 135899^ + 13634699 x 118087448 = is an 

 equation all whose roots are less by 50 than those of the last. We can 

 now depend upon 118087448 divided by 13634699 giving one figure of 

 the root, and the qxiotient is between 8 and 9. Assuming 8, the first 

 step of the third process shows that 8 is a root of the last equation, 

 and 58 of the preceding, and 158 of the given equation. 



We now give an example in which approximation is carried on. Let 

 the equation be *> 6a*+7a;+ 4 = 0, of which one root lies between 

 2 and 3. 



1-6 7 -4(2-414213562 



2 -8 -2 



-2: 

 -1-936 

 -0-064 ; 

 -0-045079 

 -0-P18921t 

 -0-017963056 

 -0-000957944J 

 -0-000897113 

 -0 : 000060831|| 

 44853 

 15978 

 13456 

 "2522 

 2243 

 279 

 269 

 10 

 8 

 "2 



The root of this equation is found to be 2-414213562, as follows. 

 Beginning with the multiplier 2, one set of involutions brings us to the 

 figure* followed by colons, andx' + Ox* 5x + 2 = is an equation on which 

 the process is to be repeated. Dividing 2 by 5 we find that "4 is 

 most probably the next figure, which is verified in the next trial, since 

 the result of involution, 1-938, is less than 2. We proceed in this way 

 until 2"4142, containing half the number of figures wanted, is found, 

 and this being a, we have found 0-000060831 for <pa, and 4-485316 

 for <p'a. The first divided by the second may be depended upon for 

 doubling the number of figures, as commonly practised in the extraction 

 of the square root. [APPROXIMATION.] The figures 13562 are found 

 by a contracted division shown in the example. 



But it is more convenient to avoid decimals in the process, which 

 may be done as follows. 1. If there be decimals in the coefficients of 

 the equation, annex ciphers to every place in such manner that the 

 number of decimals in the several places may be in increasing arith- 

 metical progression. Then strike out the decimal points entirely, and 

 proceed as with whole numbers, remembering that the root thus 

 obtained will be 10 times too great if the progression increase by 

 units, 100 times too great if it increase by twos, and so on. Thus 

 l-81z -6x'+ 33x + 18'4 should be changed into I'Slx 8 -600.r + 

 33-0000^+ 18-40000, and 181x> 6002* + 330000* + 1840000 will give 

 ten times the required root. 2. When all the whole figures of the root 

 have been obtained, and the decimal part in about to enter the calcu- 

 lation, before attempting to obtain the first decimal figure annex a 



W438 



1-2424 



-1 

 4 



-5: 

 0-16 



-4-84 



MH 

 -4-52 ; 



0-0121 

 -4-5079 



0^0122 

 -4'4957t 



0-004936 

 -4-490764 



0-004952 

 -4-485812? 



0-000248 

 4-485564 



0-000248 



cipher to the first working column on the left, two ciphers to the 

 second, and so on to the end. Then proceed with the new figure as if 

 it were a whole number, and make a new involution. When this is 

 finished annex ciphers again as before. One additional advantage will 

 be that the ciphers will serve to mark the places of completion of the 

 individual involutions. If in any case <pa should not contain <t>'a, place 

 a cipher in the root, annex ciphers again, and then proceed. In some 

 t>f the older algebraists, Oughtred for instance, the several vertical 

 lines of figures are kept in their places by a set of ruled columns, the 

 use of which is difficult. Mr. Horner has a similar contrivance ; but 

 the employment of ciphers removes all the difficulty, as in common 

 division and the extraction of the square root. See the last example in 

 this article. The method might easily be extended to the whole part 

 of the root. The following is an instance of, the method : 



y* + aa* 2*- 2 = 



1 



2 



3 



4 



50 



54 



58 



62 



660 



661 



662 



663 



6640 



6644 



6648 



6652 



6656 



1 



2 



1 



3 



4 



4 



800 



216 

 1016 



232 

 1248 



248 

 149600 

 661 



150261 

 662 



150923 

 _663_ 



15158600 



-2 



1 



1 



4 



3000 

 4064 

 7064 

 4992 

 12056000 

 150261 

 12206261 

 150923 

 12357184000 

 60740704 

 12417924704 

 60847072 

 12478771770 



2(1-414,2136 

 -1 



"30000 



28256 

 17440000 

 12206261 



52337390000 



49671698816 



2t)656yil84 



2495754355 



169936829 



124787718 



45149111 



37436315 



7712796 



7487263 



15185176 



15211768 

 26608 



15238376 



Many of the preceding figures are useless, but we have judged it 

 best to present the whole process. The best method of abbreviation is 

 to fix a point of the process from and after which the number of figures 

 in the last column is not to increase, striking off at every step one figure 

 from the last column but one, two from the last but two, and so on. 

 The consequence will be, that the several columns on the right will 

 disappear one after the other ; the process will be legitimately reduced 

 to termination with a contracted division, independently of the theo- 

 rem cited ; and the result will be true to the last place. The effect of 

 this will be, that as soon as the remaining part of the root is too small 

 for its highest power to show itself in the process, an equation of the 

 ( l)th degree takes the place of the nth, and so on, until there 

 remains only an equation of the first degree, and the approximation 

 then proceeds by the Newtonian method. All this was pointed out by 

 Mr. Homer, whose view of his own method was very complete, in 

 everything but historical information. Had he given in his paper an 

 example from Oughtred, also worked by his own method, pointing out 

 the difference of the two, we feel sure that the question about the 

 right to the invention never would have been discussed. 



Taking up the preceding example at the point with which we left 

 off (neglecting the division), and following the process, we have 



(Root obtained 1-414) 213562373 



15 23 83|76 124787 7 1 7 7)6 

 30 4 7 9 4 



6 | 656 

 Disappears 



at 



next 

 step. 



13 



2665691184 

 2496303944 



15 24|23 

 15 1 24 



Disappears, leaving 

 1 for carriage. 



124818 1972 

 30 4 8 2 

 124848 6 7 9|2 

 1524 



124850 2 0~3 

 1524 



169327240 

 124850203 



44477037 



37455657 



'^7021380 



6242640 



124851 



2 1 7 

 6 



Dividend * 



124852 1|9 



4^6 



124852 6|5 

 1 



124852 8 

 1 



Divisor 12485219 



