901 



INVOLUTION AND EVOLUTION. 



INVOLUTION AND EVOLUTION. 



9G2 



The existence of equal roots, or of nearly equal roots, might be 

 here suspected from the slow increase of the divisor column ; but the 

 method cannot verify the fact of their being two absolutely equal roots. 

 The column preceding the divisor column being large and negative, 

 requires us to make trial of figures above, not below, those which the 

 divisor column seems to indicate. But nearly equal roots may some- 

 times be detected, as in the following instance. Let 7-z 3 10-r 2 14.r + 

 20=0, of which it is known that one root lies between 1 and 2. The 

 ordinary process gives 



-7 10 14 



17 



1300 

 748 

 8400 



10 

 3 



-4 

 -110 

 -138 

 -166 

 -1940 

 -1947 

 -1954 

 -19610 

 -19638 

 -19666 

 -19694 



6453 



449900 



371848 



292684 



288745 



284806 



28460 



28440 



20 (1-41421356 

 3000 

 8000 

 1547000 

 61608 

 3859 



1013 



160 



18 



1 



This root may be carried on without difficulty. But at the end of 

 the second process, when the dividend is reduced to 8000, the divisor 

 only 8400, and the preceding column as much as 1940, it may be 

 worth while to try another figure. This state of things gives a sus- 

 picion that there is another root in the immediate vicinity of the one 

 in hand. If the three last columns be + a, b, and c, and if we find 

 that pa + b is nearly e-r-p, which is the trial test of p being a new 

 figure of the root, we are sure that (p + 1) a + b will not be near 

 e~(p + \): and moreover p (pa + b) = c has not two positive roota. 

 But if the three last columns be a, b, and c, it may very easily happen 

 that 6 pa may be nearly c -=-/>, and b qa nearly c-r-q ; for p (b pa) = c 

 has two positive roots. Perfect certainty, in the absence of an easy 

 algebraical criterion, may only be attainable by trying every figure. In 

 the instance before us, finding T41 succeed, with a presumption of a 

 larger root, we try 1'43, beginning with 



7 1940 8400 8000 



-1961 2517 449 



-1982 3429 

 -2003 



This figure will not do, for now a permanent difference of sign is 

 established between the dividend and divisor columns. We then try 

 2, as follows : 



7 1940 8400 8000 



-1951 4492 -984 



1968 556 

 1982 



There is now a difference of sign between the two last columns, but, 

 looking at the second column, we see that agreement may be restored 

 by the next figure. The figure 8 will do it, as follows : 



7 19820 55600 984000 



19876 10S408 156736 



-19932 262864 

 -19988 



and 26286400 is not contained 10 times in 156736000. All the signs 

 being now negative, we may change them all. If we had tried 7 

 instead of 8, we should have had 



19820 

 19869 

 19918 

 19967 



55600 

 83483 

 -S2290S 



-984000 



-399619 



But now 22290900 is contained more than 10 times in 399619000, 

 which shows that 7 is not high enough. If we try 9, we have 



7 19820 65600 984000 



19883 123347 +126123 



and a permanent difference of sign is established between the two last 

 columns, whence 9 is too high. Proceed then with 



.7 199880 26286400 156736000 (1-428 



and we find 1-42857142857 ... for a root. The reader may watch the 

 operation in the following equation : 



(9 x - 10) (9 x 11) (30 x - 37) = 0, 

 or 243<te* - 8667** + 10293*- 4070 = 

 the roota of which are 1-111 ..., 1-222 .... and V283 . . . 



Whatever common figures two roots of Qx = may have begun with, 

 there must be a root of $'x = which begins with these figures. And 

 whatever common figures three roots may begin with, there must be 

 two roota of f'x=0, and one root of <f>"x=0 which begin with those 

 figure* : and so on. If there were a difficult equation having three 

 roota nearly equal, no method of detecting them would be easier, of all 

 those known at present, than solving contemporaneously the three 

 equations <fuc = Q, f'x=0, <(>"x=Q, not making any step in one till all 



ARTS AND SCI. DFV. VOL. IV. 



had been brought up ; that is, one step of each first, then the second 

 of each, and so on. 



It may happen that a finite root is established, and yet that the 

 process must be continued to obtain another root beginning with the 

 same figures. For example, 



- 38 = 



It will be seen in the following process that 2 is a root, with a pre- 

 sumption, from the appearance of the divisor column and the one 

 before it, that there is another root beginning with 2. And by trial 

 2-1111 ... is found to succeed. 



9 



-46 



-28 



--10 



80 



89 



98 



1070 

 1079 

 1088 

 1097 



75 



19 



-100 

 11 



8700 



9779 



10867 



10977 



11087 



38 (2-11111 

 



11000 



1221 



23 



12 



1 



We shall now proceed to a short account of the history of this 

 problem, and of the controversies which have existed, and to some 

 extent still exist. For a fuller account of it up to the time of Mr. 

 Homer, see a paper by the writer of this article in the ' Companion to 

 the Almanac,' for 1839. 



Before the time of Vieta, evolution consisted in the rules for the 

 performance of division, and extraction of the square and cube roots, in 

 forms probably derived from the East. To him [ViETA, in Bioo. Div.] 

 we owe the first publication of a numerical method of finding the suc- 

 cessive figures of the root of an algebraical equation by means of the 

 value of the function equated to zero in the equation. This method 

 of Vieta is in fact that which Horner's process now makes so easily 

 practicable. If <(>x be the equation, and a a part of the root, it uses 

 0a, and <t>(a + l) <pa as a divisor. The process is so cumbersome, 

 that Vieta does not attempt to apply it to equations having more than 

 two figures in the root. 



This method attracted but little attention on the continent : but in 

 England, where everything relating to numerical calculation has been 

 always diligently studied, it was much noticed, and received extensions 

 of power. In the posthumous work (1631) of Harriot [HARRIOT, in 

 BIOG. Div.] examples of it are given with the improvement of forming 

 only so many -figures of the divisor as are wanted : and he ventures 

 upon roots of three places. In the second edition of Oughtred's 

 ' Clavis Mathematica' (1647) Vieta's method is given without Harriot's 

 improvements. But the first who used Vieta's method to any great 

 extent was Briggs, in the calculation of the sines, 4c., in the ' Trigono- 

 metria Britannica.' In the preface the method is applied to equations 

 of the third and fifth degrees, and partially described for the seventh 

 and higher degrees : with examples carried to fifteen and sixteen figures 

 of the root. It is for the facilitation of these solutions that the Abacus 

 foyxpijoros is given, which some have unreasonably inteipreted aa giving 

 Briggs a claim on the binomial theorem. Gellibrand tells us that 

 Briggs formed his tables of sines by algebraic equations and differences 

 about thirty years before his death. Now Briggs died in 1630, and 

 Vieta's tract appeared in 1600 : the former must then have received 

 the work soon, immediately seen the importance of the method, and 

 commenced operations by means of it. We cannot give Briggs any 

 independent title to the invention ; for it is likely enough that he was 

 in correspondence with Vieta, whose works he certainly knew. One 

 of his examples is the solution of what would now be written 



of Sx = 1-298896096660366 



for which he gets x = 1-917639469736386. He puts down the work 

 as far as ... 697, proceeding towards the end by several figures at a 

 time : and he has got what Vieta had not, the Newtonian divisor <f>'x 

 instead of <p(x + 1) $-. Of course it adds materially to .the his- 

 torical value of this method that it was thus used in an operation of so 

 much importance to the progress of mathematics in general. The 

 dates above given may even cause a suspicion that it was the power of 

 solving equations thus suddenly acquired, which first suggested the 

 calculation of the natural sines, &c., in the ' Trigonometria Bri- 

 tannica.' 



Wallia, in his Algebra (1684), gives the method of the " numerose 

 exegetw," as he calls it (Vieta had called it potestatum adfectarum ad 

 exegeiin resolutio) with an example of the fourth degree worked to 

 seventeen places of the root. He makes use of the method of con- 

 tracting the figures towards the end. In this same Algebra appeared, 

 for the first time, what is called Newton's method of approximation, 

 which soon superseded the exegesis, into which however it had been 

 virtually incorporated by Briggs. Newton's approximation, at least in 

 the general form which it took in the hands of Taylor, is as follows. 

 If a be a near value of x in <fx = 0, then, except when there are two 

 nearly equal roots, a nearer value is 



<t>a 





 <t> a, 



3q 



