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PROFESSOR SYLVESTER ON THE REAL 
Part I.— ON NEWTON’S RULE FOR THE DISCOVERY OF IMAGINARY ROOTS. 
(2) In the ‘Arithmetica Universalis,’ in the first chapter on equations, Newton has given 
a rule for discovering an inferior limit to the number of imaginary roots in an equation 
of any degree, without proof or indication of the method by which he arrived at it, or the 
evidence upon which it rests ( 2 ). Maclaurin, in vol. xxxiv. p. 104, and vol. xxxvi. p. 59 
of the Philosophical Transactions, Campbell( 3 ) in vol. xxxviii. p. 515 of the same, and 
other authors of reputation have sought in vain for a demonstration of this marvellous 
and mysterious rule( 4 ). Unwilling to rest my belief in it on mere empirical evidence, I 
( 2 ) It appears to be the prevalent belief among mathematicians who have considered the question, that 
Newton was not in possession of other than empirical evidence in support of his rule. 
( 3 ) Campbell’s memoir is rather on an analogous rule to Newton’s than on the rule itself, to which he refers 
only by way of comparison with his own. In it the same singular error of reasoning is committed as in the 
notes of the French edition of the ‘ Arithmetica,’ viz. of assuming, without a shadow of proof, that if each of a 
set of criteria indicates the existence of some imaginary roots, a succession of sets of such criteria must indicate 
the existence of at least as many distinct imaginary pairs of roots as there are such sets (see par. at foot of 
p. 528, Phil. Trans., vol. xxxv.) — much as if, supposing a number of dogs to be making a point in the same 
field, the existence could be assumed of as many birds as pointers. 
( 4 ) Mr. Archibald Smith has obligingly called my attention to Waring’s treatment of the question of New- 
ton’s rule in the ‘ Meditationes Analyticse.’ On superficial examination the reader might be induced to suppose 
that in part 9, p. 68, ed. 1782, Waring had deduced a proof of the rule from the preceding propositions ; but on 
looking into the ease will find that there is not the slightest vestige of proof, the rule being stated, but without 
any demonstration whatever being either adduced or alleged. In fact, on turning to the preface of this (the 
last) edition of the 8 Meditationes,’ the reader will find at p. 11 an explicit avowal of the demonstration being 
wanting. After referring in order to Campbell’s, Maclaurin’s, and Newton’s rules, as well as his own, for 
discovering the existence of impossible roots, he adds these words : 
“ At omnes hse regulie prasdictae perraro invenerunt verum numerum impossibilium radicum in sequationibus 
multarum dimensionum et adhuc demonstratione egent ; vulgares enim demonstrationes solummodo probant impos- 
sibiles radices in data aequatione contineri, non vero quod saltern tot sunt quot invenit regula.” 
Vera resolutio problematis est perdifficilis et valde laboriosa; cognitum est radices ex possibilitate per 
eequalitatem transire ad impossibilitatem ; ergo in generali resolutione hujusce problematis necesse est invenire 
casum in quo radices datse aequationis evadunt aequales ; resolutio autem hujus casus valde laboriosa est ; et 
consequenter resolutio generalis praedieti problematis magis erit laboriosa.” 
Written in Latin, and when the proper language of algebra was yet unformed, it is frequently a work of 
much labour to follow Waring’s demonstrations and deductions, and to distinguish his assertions from his 
proofs. I find he agrees with the opinion expressed by myself, that Newton’s rule will not “ pene,” as stated 
by Newton, but only “ perraro,” give the true number of imaginary roots. Like myself, too, in the body of the 
memoir Waring has given theorems of probability in connexion with rules of this kind, but without any clue 
to his method of arriving at them. Their correctness may legitimately be doubted. 
[Since the above was sent to press, I have been enabled to ascertain that the great name of Euler is to be 
added to the long list of those who have fallen into error in their treatment of this question : see Institutiones 
Calculi Differentialis, vol. ii. cap. xiii. He says (p. 555, edition of Prony), “ videndum est utrum haec duo 
criteria (meaning Newton’s criteria of imaginariness) sint contigua necne ; priori casu numerus radicum 
imaginarium non augebitur; posteriori vero quia criteria litteras prorsus diversas involvunt, unumquodque 
binas radices imaginarias monstrabit.” 
The force of the supposed argument is contained in the words in italics. It is sufficiently met by the ques- 
tion, why or how the conclusion follows from them ? Moreover the letters of two non-contiguous criteria are not 
necessarily prorsus diversce ; for two criteria with but a single other intervening between them will contain 
one letter in common.] 
