INVnUTlOX AXH 



INVOLUTION AND KVOUTTK'X. 



The old ntfen*. and especially Briggs's form of it, employ* thin prin- 

 ciple ; fa u calculated, and either fa or f(a + 1) - fa. Brigjr, 

 who proceed* by several figure* at a time, and vine* f'a, does really use 

 what wa* afterwards called Newton's method, and assist* it by opera- 

 tion* suggested by Vieta. 



When the exegesis wa* abandoned by Raphaon and others, in favour 

 of Newton's form of operation, no further improvement was made in 

 the direct numerical solution of equations, until the time ( Mr. 

 Homer; at least no further improvement was published. Mr. Unity 

 Atkinson, a young man of Newcastle, re-invented the whole method in 

 1801, applying Newton's divisor, and giving rules by which one divisor 

 was made to help in forming the next. This was read to the Philo- 

 sophical Society of Newcastle in 1809, and published posthumously, as 

 ' A new Method of extracting the Roots of Equations,' Newcastle, 

 1831, 4to. In our article in the ' Companion to the Almanac, already 

 cited, we have supposed that no one can be shown either to have used 

 +'a. or to have made each value of it help the next, before Mr. Atkin- 

 son : but we now find that Briggs was before him in both points. 



Lagrange's method of transforming the root into a continued fraction 

 [TIIKUBT or EQUATIONS] does not need notice here, because it belongs 

 to another mode of expression. But it ought to be noticed that 

 Homer's process very much abridges the labour of Lagrange's method, 

 a* much indeed as it does that of Vieta's exegesis, and for the same 

 reason. Mr. Exley, of Bristol, in the ' Imperial Encyclopaedia,' article 

 ARITHMETIC, improved (according to Horner himself) the common 

 method of extracting the cube root, so as to precede Horner in this 

 particular case. We believe more than one method had been given 

 for reducing the enormous labour of the ordinary extraction of the 

 cube root : we may mention one, which is ingenious and effective, and 

 almost exactly a particular case of Homer's method, given by Mr. A. 

 Ingram, in his edition of Mutton's ' Arithmetic,' Hawick, 1811, 8vo. : 

 and Horner himself refers to an edition of Melrose's ' Arithmetic,' by 

 Mr. Ingram (the same, we suppose) as containing such a method. 



Homer's paper was read to the Royal Society on the 1st of July, 

 1819, and was published in the current volume of the Transactions, 

 on the 1st of December. These dates are of importance : the publica- 

 tion of the above paper was the signal for more than one person to 

 make a nibbling claim to the invention. Horner was unfortunate in 

 two points. First, he had not sufficient knowledge of ancient algebra 

 to be nware that his method contained the process of Vieta, and that 

 his real claim consisted in the discovery of the beautiful process by 

 which the labour is immensely reduced, and completely systematise^ : 

 we suspect that he completely re-invented Vieta's part for himself. 

 Secondly, he appears to desire to be the analyst rather than the arith- 

 metician, and will not show anything except to those who can take all 

 It is true, beyond a doubt, that his method Is adapted to every sort of 

 equation, and that it is as great a help to the person who desires to 

 solve ton x ax = 0, or "** .r, as to the other who wants nothing 

 but a common algebraical equation. So far, then, it is more Hum 

 Vieta's method simplified ; it is the same also extended. Rut if the 

 inventor had proceeded from simple algebra to the more complicated 

 eases, his merits would have been more rapidly appreciated. He did 

 not well see that his mode of solution applies as well to the integer part 

 of tin' n*>t as to the fractional ; nor did he fully comprehend how 

 much of his own discovery consisted in the general mode of calculating 

 the value of <rr, as given at the beginning of this article. But that 

 we may not do him injustice, and still more that we may enable those 

 of our readers who have not access to the original paper to see how 

 completely he had got hold even of the most convenient arithmetical 

 process, we give his solution of the famous Newtonian instance 

 * 2ar = 6. After reducing the root by 2, the heads of his column* 

 are 1, 6, 10, and 1 (the first column, which is always vacant, he does 

 not set down). He then annexes either dots or ciphers, and proceeds 

 exactly as follows 



6.. 



SOB 

 M 







' Si 



10.... 



1 1 - i 



l-.M-l 



SMI . 

 UKNM 



11129890 

 2611 



11 1.'.7'; 4 



81 

 81 



111011 



111.11 i 



I 



1000000(-0946514815 

 9491 



60671000 



44617684 



81M4T3 



6878826 



671351 



668066 



,1,1,6,1)16686(14816 

 11161 



~:..: 7 :. 



4405 



910 



893 



When Mr. Homer's paper had been published six months appeared 



' A new Method of Solving Equations,' by Theophilu* HoMreil, 1. 

 1820 (preface dated .Inn.- 1 ). 4to. The method is taken from Harriot ; 

 and a supplement U added, which gives Homer's method. Both arc 

 claimed as independent inventions, ami H.-M!. rs name is not mentioned. 

 Mr. Holdred asserts that, after having had hU method fur forty yean 

 he was led to that in thu supplement * by a mistake he committed in 

 solving an equation sent him by one of his subscribers. We have 

 piven, in the article of the ' Companion to the Almanac,' already cited, 

 our reasons for coming t the conclusion that Mr. Holdred to >k hi* 

 first method from Harriot, and hU second from Homer. 



A claim was made by Mr Peter Nicholson in various places, which is 

 quite futile. We acquit Mr. Nicholson (a highly respectable man, 

 eminent in the application of mathematics to the arts) of all unfair 

 intention : and we must remind our readers of a point without the 

 knowledge of which the various controversial writings on this subject 

 will be full of confusion. Hardly any one knew of Vieta's Exegesis, 

 which there is little doubt that both Horner and Atkinson reinvented. 

 In fact, so completely had this exegesis dropped out of sight, that even 

 Dr. Peacock, in his short account of Homer's method (' Report on 

 Analysts to the British Association ') does not allude to it. Accordingly, 

 all the re-inventors of Vieta's method speak of quite new rules 

 discovered for the solution of equations, and treat Homer's process as 

 a constituent part of one of the new inventions. But a person 

 acquainted with the history of the subject finds nothing n<- . v .pi 

 Homer's process. Vieta had the main system, Briggs ha<! tin; 

 Newtonian divisor, Wallia had the method of contraction, Briggs bad 

 a method of making one divisor help the rest : Horner had the method 

 which must finally be adopted. Budan, as we shall see, had only a 

 particular case of that method, and did not apply it to any mechanical 

 process of numerical solution. 



Mr. Nicholson claims Horner's identical process, and fairly refers to 

 the very place in which he says it is to be found. But on looking 

 there (see the article already cited in the ' Companion to the Almanac '), 

 we find that he has been deceived by a distant resemblance, :iml that, 

 though he has given a new and useful process for a useful purpose, 

 neither the process nor the purpose is Homer's. At the same time it 

 is but justice to Mr. Nicholson to say, that in his ' Elements of 

 Algebra,' London, 1819, t 12mo, he made as near on approach to 

 Homer's method as could well be done, and applied it in the case of 

 equations of the second and third degrees. The succession of columns 

 is seen, each column helps the next, and each step in any one column 

 helps the next step. But the grand simplification, which the con- 

 troversialists called the " non-figurate method," is wanting : so that this 

 process of Nicholson's is perhaps hardly more than Briggs was in 

 possession of. Mr. Nicholson had received Mr. Holdred's method, 

 whose name he properly mentions in the preface. This method he had 

 greatly improved ; and it seems he wished that Holdred should 

 publish his own method as amended by him ; but he asserts (in the 

 preface to his work on Involtion and Evolution) that the latter refused, 

 alleging that his own credit would be diminished, unless he could pass 

 them as his own. 



Dr. Peacock hod never seen Holdred's tract, and his result, derived 

 from the assertions of Mr. Nicholson and from Homer's paper, is that 

 Nicholson, by a combination of the methods of Holdred and Horner, 

 reduced the method to its present practicable form. But any one who 

 will solve ^2^=6 in the systematic form we have given, will see 

 that Horner had that form. Nicholson was, we believe, the one who 

 first clearly saw that the method, in its simplest organisation, applies 

 as well to the integer as to the fractional portion of a root. All Mr. 

 Nicholson's simplifications, as given in his latest writings, con 

 doing in the head some of the things which Homer put down on 

 paper. The form we give carries this still further; and those who can 

 do what we have recommended all arithmeticians to practise in COM- 

 PUTATION con follow us : but there is no invention in this. 



Some have been disposed to give a good deal of the merit of this 

 system to Budan ; and his claim must be considered. Two editions of 

 the ' Nouvelle Mdthode pour la Resolution des Equations numeriques,' 

 Paris, 4to., were published in 1807 and 1822. The basis of M. Budan's 

 operations is the simple case of Homer's process in which the root of 

 an equation is diminished by unity. This is done exactly in the mode 

 1'v which Horner afterwards proceeded. Thus to lessen the root of 

 ar 2x 5=0 by unity, Budan proceeds thus : 



1+0-2-6 

 1+1-1-fi 



(A) 1 + 2 + 1 



1 + 3 



1 

 Answer a? + 8-t 5 + x 6 = 



But to lessen the root by 2, Budan is never able to arrive at the 



* Wo cannot but believe that Mr. Holdred did sec Mr. Homer's piper. Hid 

 he mentioned It, and the name of the nubscribcr, his equation, the mistake 

 made, <rc. 4c,, dUUnctly declaring when and where he flrsl aw Mr. Horner's 

 paper, he might hare possibly established a claim to be a second in- 



t The preface l dated May 17, 1819, and the publication took place early in 

 July, Sir. Horner's paper luring been publicly read st the Royal Society on tho 

 firnt of that month. 



