P65 



INVOLUTION AND EVOLUTION. 



INVOLUTION AND EVOLUTION. 



966 



process on the left, which is Homer's : he must repeat the process of 

 diminishing the last root by one, as on the right. 



1 + 0-25 

 1+2+2-1 

 (B) 1 + 4 + 10 

 1 + 6 

 1 



1+3+1-6 

 1+4+5-1 

 1 + 5 + 10 (c) 

 1 + 6 ' 

 1 



Accordingly Sudan has both (A) and (c) to do, where Homer has 

 only (B). To diminish a root by 3, Budan has 3 processes, and so on. 

 To diminish a root by 10, 20, &c., he divides the roots of the original 

 equation by 10, then diminishes by 1, by 1 more, Ac , and then 

 multiplies the resulting roots by 10, 20, &c. ; and similarly for 100, &c. 

 It is obviously possible, by a large amount of calculation, to obtain the 

 root of an equation in this manner ; but Budan is not only obliged to 

 call hi other methods, and even thus to spend very great labour, but 

 he ends by presenting the root in the form of a sum of common 

 fractions, each of which must be reduced to a decimal. Thus for 

 x 3 2.r= 5, he gets 



=2 + - + _ + 



= 2094551481364 



_ 

 11 275 1651)25 



Budan's method is not then even of the same species * a Homer's. 

 In an appendix added to the edition of 1822, two years after Horner'a 

 paper, there is the method extended to the process for diminishing the 

 root by n (Horner's process), but no use is made of it, and singularly 

 enough the only example given is one in which n is 1. 



Horner (' Leybourn's Repository,' page 38, of part ii., vol. v.) denies 

 ever having seen Budan's work until 1818, after his method was 

 finished. This, in one point of view, counts for nothing ; for every 

 discoverer has a right to have it supposed that those who come after 

 him have used his works : that is to say, the first discoverer would 

 have a right to the credit therefrom arising, even though it could be 

 ghown that subsequent discoveries were made without his aid. If a 

 partial or unfinished method turn out to have a value of quite a new 

 character when made complete, it is impossible to deny to its author 

 the credit of having been further than his contemporaries on the road 

 towards the complete method : consequently, Budan must have, in one 

 sense, the merit of having proposed a particular case of that which 

 Horner afterwards used. But, as it happens, a contemporary of Mr. 

 Homer, in trying to insinuate that Horner had taken his method from 

 Budan, has furnished independent evidence to the contrary. Mr. 

 Nicholson, in a note to the preface (page ix.) of his essay ' On Involu- 

 tion and Evolution ' states : " I am informed by Mr. Dickson that 

 about twelve months ago he (Horner) purchased at his shop, in St. 

 Martin's-le-Grand, an ' Essay on the Numerical Solution of Equations/ 

 by Budan ; at which time he mentioned that he was engaged expressly 

 on this subject." This called forth the preceding statement from Mr. 

 Horner, who, had he any unfair intention, and had he really been 

 indebted to Budan, would have argued from the date of Mr. Nichol- 

 son's preface that he must have bought Budan only just time enough 

 to insert the note about him in his paper before he sent it to the 

 Royal Society. Instead of this, he answers in the most straight- 

 forward manner, that he bought Budan about July, ISIS, nearly tin, 

 years before Mr. Nicholson wrote ; but avers that his method was then 

 finished. And this we entirely believe ; and also that it would have 

 been impossible for him, fully engaged as he was in teaching a school, 

 to have produced his method, so as to send it to the Royal Society in 

 the spring of 1819, if he had only seen the first hint in the summer of 

 1818. But had he seen Sudan's work, and had he thence derived the 

 hint which he improved, his merit would not have been the less : 

 Lagrange, the greatest writer on equations then existing, had seen it ; 

 Legendre had seen it : and both had closely examined it, and reported 

 to the Institute upon it. The members of the Institute had seen it. 

 Lagrange, too, knew of Vieta's Exegesis. But no one, except the Bath 

 schoolmaster, ever brought forward Budan's method, or any extension 

 of it, either from Budan, or independently, to the improvement of 

 Vieta. Fourier had seen Budan's book, and had invented a method of 

 his own of solving equations ; or rather had given his own mode of 

 conducting Newton's approximation; but this method is far below 

 that of Horner. 



We have written so much on the discovery of this method, because 

 unfair attempts were made by claimants who had no title whatever to 



lie recommends that when more than two or three decimals of a root arc 

 wanted, the work should be turned over to tcorkmrn (mnna-urrirr.i) who are to 

 be a distinct clans from the mathematicians. The best comment on this will be 

 to insert in this little foot note every figure of the work for rix places of the 

 equation on which this remark was made, with a pucss at the terenth. 







2 



4 



600 



609 



6)8 



627 



-2 



2 



100000 



111043 



1H2'J4 

 11194.; 



in.-,; 



11160 



5 (20945515 



50671 



6153 



574 



16 



5 







deprive the author, who was a man of real genius, of his rights over 

 his own discovery. We refer to MM. Holdred and Nicholson : though 

 we do not believe the second was knowingly unfair. Mr. Atkinson, 

 when he first saw the " non-figurate method " (as some called the sub- 

 ject of this paper), saw and said that it was a " capital improvement." 

 We have written also because it can hardly yet be said that mathe- 

 maticians are alive to the value of this grand completion of the system 

 of arithmetic. The continental writers show no knowledge of it ; the 

 Oxford and Cambridge elementary works do not yet recognise its 

 existence, except so far as this, that one very recent Cambridge edition 

 makes an imperfect introduction of it. The fact is, that mathema- 

 ticians dislike calculation, and are apt to form hasty opinions on nume- 

 rical methods before they have given them sufficient trial. The first 

 elementary writer who brought Horner's method into instruction was 

 Mr. (afterwards Professor) Young, in his ' Elements of Algebra,' pub- 

 lished in 1823. 



In 1S31, eleven years after this method was published, appeared 

 Fourier's posthumous work on equations, containing an extended use 

 of Newton's method. It amounts to employing <pa, Q'a./t, !^"a,h-, &e., 

 to calculate the value of <f> (a + A), and <j>'a, <t>"a.h, &c., scfxiiMtJ;/, t.> 

 calculate q/ (a + h) ; and so on. Fourier was an expert arithmetician, 

 and in this very work shows his power of suggesting new forms of 

 arithmetical process ; but he does not come near anything like making 

 the previous calculation of ^>("' (a + A) give assistance to that of <f><"-'i 

 (a + A). The equation a 3 2.r=5, which Wallis happened to take as 

 his instance of Newton's method, has always been the example on 

 which numerical solvers have shown their power. No one can be said 

 to have carried a method beyond those which preceded, unless he has 

 solved this equation to more places than they have done. Fourier 

 went to thirty-two decimal places, which we do not know that any 

 one had done before. Some students of University College, London 

 (and one of King's College), none exceeding eighteen years of age, 

 carried Horner's process further still, their independent calculations 

 giving root to 52 figures. Some years afterwards, another student of 

 University College, Mr. W. H. Johnston, of Dundalk, carried the solu- 

 tion to 101 decimal places, and verified it by the independent solution 

 of a related equation. In 1851, Mr. J. Power Hicks, of Lincoln College, 

 Oxford, then a student of University College, carried the solution 

 to 152 decimal places, never having seen Mr. Johnston's result, with 

 which, so far as it went, his own agreed. This last solution* is as 

 follows, and it took about 50 hours of calculation : 



2-09455,14815,42326,59148,23865,40579, 

 30296,38573,06105,62823,91803,04128, 

 52904,53121,89983,48366,71462,67281, 

 77715,77578,60839,52118,00629,63459, 

 84514.03984,20812,82370,08437,22349, 

 91 



We insert this conclusion as a challenge to any who still hold the 

 opinion, which as a matter of course was maintained by some when 

 Horner's method first appeared, that some older methods were superior 

 to it. There were those who thought that the method of trial and 

 error, or of false position, as it was called, was preferable. Mr. 

 Nicholson gives, as the work of a young computer, the following solu- 

 tion of 



ij* + Ix* + 9x + 6x* + 5* + 3x = 792 



x =2-05204,21768,79605,36521,40434,01281,20197,34602,75599,54554 

 17242,14 



An ablet calculator informs us, that he makes the figures after 107 to 

 be 34660, 87786,99113,74218,13787,467. 



We have left entirely out of sight all the irrelevant controversy 

 relating to the method of finding the limits of the roots, conducting 

 the process when two roots are nearly equal, and so on. The claims of 

 Budan, Fourier, Horner, 4c., are here mixed up in a manner which 

 requires a sifting investigation. Very frequently the value of Horner's 

 method is stated as depending upon points of this kind. When any of 

 the doubtful cases arise, which we noticed at the beginning of this 

 article, we find, for ourselves, that the ease with which repeated trials 

 are made by Horner's process gives us more command of these ques- 

 tions than anything else ; in fact Fourier's theorem [STURM'S THEOREM] 

 is very easily brought to bear by means of it. But it must be admitted 

 that all methods which in any way include the Newtonian approxi- 

 mation are imperfect, when roots are nearly equal, in not having a 

 better addition to the root a already obtained than (pa : q/a. Let a 

 better method come, and we have no doubt that Horner's process is' 

 more ready to make easy use of it than any other. A student who i.s 

 very slow at finding out the trial figures of common division, might as 

 reasonably depreciate the rule of division altogether, as quarrel with 

 Horner's method because there is now and then a difficulty in ascer- 

 taining whether or no more than one figure will do to proceed with. 



1 As there is always a liability to defacement of figures, wo give the sums of 

 the digits in the horizontal and vertical lines. The sums in the horizontal lines 

 should be 137, 115, 140, 157, 121, 10. The sums in the vertical lines should 

 be 2 ; 32, 23, 27, 16, 24 ; 16, 25, 28, 25, 21 ; 20, 17, 29, 14, 25 ; 28, 23, 16, 

 20, 25 J 27, 13, 30, 17, 26 ; 18, 16, 15, 26, 36. 



f Mr. A. Davis, assistant master in University College School. 



