Gauss: AVCTORE CAROLO FRIDERICO GAVSS HELMSTADII APVD C. G. FLECKEISEN. 1799

DEMONSTRATIO NOVA
THEOREMATIS
OMNEM FVNCTIONEM ALGEBRAICAM
RATIONALEM INTEGRAM
VNIVS VARIABILIS
IN FACTORES REALES PRIMI VEL SECUNDI GRADVS
RESOLVI POSSE

AVCTORE
CAROLO FRIDERICO GAVSS
HELMSTADII
APVD C. G. FLECKEISEN. 1799

NEW DEMONSTRATION

OF THE THEOREM

THAT EVERY RATIONAL INTEGRAL ALGEBRAIC FUNCTION
OF ONE VARIABLE
CAN BE RESOLVED INTO REAL FACTORS OF THE FIRST OR SECOND DEGREE

BY
CARL FRIEDRICH GAUSS
IN HELMSTEDT
AT C. G. FLECKEISEN. 1799

Quaelibet aequatio algebraica determinata reduci potest ad formam x^m+Ax^m-1+Bx^m-2+ etc. +M=0, ita vt m sit numerus integer positiuus. Si partem primam huius aequationis per X denotamus, aequationique X=0 per plures valores inaequales ipsius x satisfieri supponimus, puta ponendo x=α, x=β, x=γ etc.

Any determined algebraic equation can be reduced to the form x^m+Ax^m-1+Bx^m-2+ etc. +M=0, so that m is a positive integer number. If we denote the first part of this equation by X, and suppose that the equation X=0 is satisfied for several unequal values of x itself, for example by setting x=α, x=β, x=γ etc.

functio X per productum e factoribus x-α, x-β, x-γ etc. diuisibilis erit. Vice versa, si productum e pluribus factoribus simplicibus x-α, x-β, x-γ etc.

the function X will be divisible by the product of the factors x-α, x-β,
x-γ etc. Conversely, if the product from several simple factors x-α, x-β, x-γ etc.

functionem X metitur: aequationi X=0 satisfiet, aequando ipsam x cuicunque quantitatum α, β, γ etc. Denique si X producto ex m factoribus talibus simplicibus aequalis est (siue omnes diuersi sint, siue quidam ex ipsis identici): alii factores simplices praeter hos functionem X metiri non poterunt. Quamobrem aequatio m^ti gradus plures quam m radices habere nequit; simul vero patet, aequationem m^ti gradus pauciores radices habere posse, etsi X in m factores simplices resolubilis sit: si enim inter hos factores aliqui sunt identici, multitudo modorum diuersorum aequationi satisfaciendi necessario minor erit quam m. Attamen concinnitatis caussa geometrae dicere maluerunt, aequationem in hoc quoque casu m radices habere, et tantummodo quasdam ex ipsis aequales inter se euadere: quod vtique sibi permittere potuerunt.

measures the function X: the equation X=0 will be satisfied by making x equal to any of the quantities α, β, γ etc. Finally, if X is equal to a product of m such simple factors (whether all are different, or some of them identical), other simple factors besides these will not be able to measure the function X. Wherefore an equation of the m^th degree cannot have more than m roots; yet at the same time it is evident that an equation of the m^th degree can have fewer roots, even if X is resoluble into m simple factors: for if among these factors some are identical, the multitude of different modes of satisfying the equation will necessarily be less than m. Nevertheless, for the sake of concinnity, geometers have preferred to say that the equation in this case too has m roots, and that only certain of them turn out equal among themselves: which of course they could permit themselves.

Quae hucusque sunt enarrata, in libris algebraicis sufficienter demonstrantur neque rigorem geometricum vspiam offendunt. Sed nimis praepropere et sine praeuia demonstratione solida adoptauisse videntur analystae theorema cui tota fere doctrina aequationum superstructa est: Quamuis functionem talem vt X semper in m factores simplices resolui posse, siue hoc quod cum illo prorsus conspirat, quamuis aequationem m^ti gradus reuera habere m radices. Quum iam in aequationibus secundi gradus saepissime ad tales casus perueniatur, qui theoremati huic repugnant: algebraistae, vt hos illi subiicerent, coacti fuerunt, fingere quantitatem quandam imaginariam cuius quadratum sit -1, et tum agnouerunt, si quantitates formae a+b√-1 perinde concedantur vt reales, theorema non modo pro aequationibus secundi gradus verum esse, sed etiam pro cubicis et biquadraticis.

The things that have been set forth thus far are sufficiently demonstrated in algebraic books and nowhere offend geometric rigor. But the analysts seem to have adopted too hastily and without a prior solid demonstration the theorem upon which almost the whole doctrine of equations is superstructured: that any function such that X can always be resolved into m simple factors, or this, which entirely agrees with it, that any equation of the mth degree truly has m roots. Since even in equations of the second degree one very often comes upon cases that conflict with this theorem, the algebraists, in order to bring these under it, were compelled to feign some imaginary quantity whose square is -1; and then they acknowledged that, if quantities of the form a+b√-1 are conceded in the same way as real ones, the theorem is true not only for equations of the second degree, but also for cubics and biquadratics.

Hinc vero neutiquam inferre licuit, admissis quantitatibus formae a+b√-1 cuiuis aequationi quinti superiorisue gradus satisfieri posse, aut vti plerumque exprimitur (quamquam phrasim lubricam minus probarem) radices cuiusuis aequationis ad formam a+b√-1 reduci posse. Hoc theorema ab eo, quod in titulo huius scripti enunciatum est, nihil differt, si ad rem ipsam spectas, huiusque demonstrationem nouam rigorosam tradere, constituit propositum praesentis dissertationis.

Hence indeed it was by no means permissible to infer that, with quantities of the form a+b√-1 admitted, any equation of the fifth or of higher degree can be satisfied; or—as it is for the most part expressed (although I should less approve the slippery phrase)—that the roots of any equation can be reduced to the form a+b√-1. This theorem differs in nothing from that which is enunciated in the title of this writing, if one looks to the matter itself; and to deliver a new rigorous demonstration of this constitutes the purpose of the present dissertation.

Ceterum ex eo tempore, quo analystae comperti sunt, infinite multas aequationes esse, quae nullam omnino radicem haberent, nisi quantitates formae a+b√-1 admittantur, tales quantitates fictiae tamquam peculiare quantitatum genus, quas imaginarias dixerunt, vt a realibus distinguerentur, consideratae et in totam analysin introductae sunt; quonam iure? hoc loco non disputo. — Demonstrationem meam absque omni quantitatum imaginarium subsidio absoluam, etsi eadem libertate, qua omnes recentiores analystae vsi sunt, etiam mihi vti liceret.

Moreover, from the time when the analysts discovered that there are infinitely many equations which would have no root at all unless quantities of the form a+b√-1 be admitted, such fictitious quantities, as a peculiar kind of quantities, which they called imaginary, so that they might be distinguished from the real, have been considered and introduced into the whole analysis; by what right? I do not dispute in this place. — I shall complete my demonstration without any support of imaginary quantities, although it would be permitted for me also to use the same liberty which all the more recent analysts have employed.

Quamuis ea, quae in plerisque libris elementaribus tamquam demonstratio theorematis nostri afferuntur, tam leuia sint, tantumque a rigore geometrico abhorreant, vt vix mentione sint digna tamen, ne quid deesse videatur, paucis illa attingam. ''Vt demonstrent, quamuis aequationem x^m+Ax^m-1+Bx^m-2 + etc. +M=0, siue X=0, reuera habere m radices, suscipiunt probare, X in m factores simplices resolui posse.

Although those things which in most elementary books are put forward as a demonstration of our theorem are so slight, and so far do they abhor geometric rigor, that they are scarcely worthy of mention, nevertheless, lest anything seem to be lacking, I shall touch on them briefly. ''In order to demonstrate that any equation x^m+Ax^m-1+Bx^m-2 + etc. +M=0, or X=0, actually has m roots, they undertake to prove that X can be resolved into m simple factors.

Ad hunc finem assumunt m factores simplices x-α, x-β, x-γ etc. vbi α, β, γ etc. adhuc sunt incognitae, productumque ex illis aequale ponunt functioni X. Tum ex comparatione coëfficientium deducunt m aequationes, ex quibus incognitas α, β, γ etc.

To this end they assume m simple factors x-α, x-β, x-γ etc., where α, β, γ etc. are still unknown, and they set the product of these equal to the function X. Then from a comparison of the coefficients they deduce m equations, from which they determine the unknowns α, β, γ etc.

determinari posse aiunt, quippe quarum multitudo etiam sit m. Scilicet m-1 incognitas eliminari posse, vnde emergere aequationem, quae, quam placuerit, incognitam solam contineat. '' Vt de reliquis, quae in tali argumentatione reprehendi possent, taceam, quaeram tantummodo, vnde certi esse possimus, vltimam aequationem reuera vllam radicem habere? Quidni fieri posset, vt neque huic vltimae aequationi neque propositae, vlla magnitudo in toto quantitatum realium atque imaginariarum ambitu satisfaciat?

they say can be determined, since their multitude is also m. Namely that m-1 unknowns can be eliminated, whence there emerges an equation which contains only the single unknown that one may please. '' That I may be silent about the rest which could be reprehended in such an argumentation, I will merely ask: whence can we be certain that the last equation really has any root? Why could it not happen that neither this last equation nor the proposed one is satisfied by any magnitude in the whole ambit of real and imaginary quantities?

— Ceterum periti facile perspicient, hanc vltimam aequationem necessario cum proposita omnino identicam fore, siquidem calculus rite fuerit institutus; scilicet eliminatis incognitis β, γ etc. aequationem α^m+Aα^m-1+Bα^m-2+ etc. +M=0 prodire debere.

— Moreover
the skilled will easily perceive that this ultimate equation will necessarily
be entirely identical with the proposed one, provided that the calculus has been rightly
instituted; namely, with the unknowns β, γ, etc., eliminated, the equation
α^m+Aα^m-1+Bα^m-2+ etc. +M=0 ought to emerge.

Plura de isto ratiocinio exponere necesse non est.

It is not necessary to expound further on this ratiocination.

Quidam auctores, qui debilitatem huius methodi percepisse videntur, tamquam axioma assumunt, quamuis aequationem reuera habere radicos, si non possibiles, impossibiles. Quid sub quantitatibus possibilibus et impossibilibus intellegi velint, haud satis distincte exposuisse videntur. Si quantitates possibiles idem denotare debent vt reales, impossibiles idem vt imaginariae: axioma illud neutiquam admitti potest, sed necessario demonstratione opus habet.

Certain authors, who seem to have perceived the debility of this method, assume as an axiom that every equation truly has roots, if not possible, then impossible. What they wish to be understood under possible and impossible quantities they seem not to have explained with sufficient distinctness. If possible quantities are to denote the same as real ones, and impossible the same as imaginary ones, that axiom can by no means be admitted, but necessarily requires a demonstration.

Attamen in illo sensu expressiones accipiendae non videntur, sed axiomatis mens haec potius videtur esse: "Quamquam nondum sumus certi, necessario dari m quantitates reales vel imaginarias, quae alicui aequationi datae m^ti gradus satisfaciant, tamen aliquantisper hoc supponemus; nam si forte con- tingeret, vt tot quantitates reales et imaginariae inueniri ne- queant, certe effugium patebit, vt dicamus reliquas esse impos- sibiles. '' Si quis hac phrasi vti mauult quam simpliciter dicere, aequationem in hoc casu tot radices non habituram, a me nihil obstat: at si tum his radicibus impossibilibus ita vtitur tamquam aliquid veri sint, et e. g. dicit, summam omnium radicum aequationis x^m+Ax^m-1+ etc. =0, esse =-A, etiamsi impossibiles inter illas sint (quae expressio proprie significat, etiamsi aliquae deficiant): hoc neutiquam probare possum.

However, the expressions do not seem to be to be taken in that sense, but the mind of the axiom seems rather to be this: "Although we are not yet certain that there necessarily are m real or imaginary quantities which satisfy a given equation of the m^th degree, nevertheless we shall for a while suppose this; for if by chance it should happen that so many real and imaginary quantities cannot be found, surely a loophole will be open, namely to say that the remaining ones are impossible.'' If someone prefers to use this phrase rather than simply to say that in this case the equation will not have so many roots, nothing stands in the way on my part: but if then he uses these impossible roots as though they were something true, and, e.g., says that the sum of all the roots of the equation x^m+Ax^m-1+ etc. =0 is =-A, even if impossible ones are among them (which expression properly signifies, even if some are lacking): this I can by no means approve.

Nam radices impossibiles, in tali sensu acceptae, tamen sunt radices, et tum axioma illud nullo modo sine demonstratione admitti potest, neque inepte dubitares, annon aequationes exstare possint, quae ne impossibiles quidem radices habeant? *1)

For impossible roots, accepted in such a sense, are nevertheless roots, and then that axiom can in no way be admitted without demonstration, nor would you ineptly doubt whether equations could exist that have not even impossible roots? *1)

Antequam aliorum geometrarum demonstrationes theorematis nostri recenseam, et quae in singulis reprehenda mihi videantur, exponam: obseruo sufficere si tantummodo ostendatur, omni aequationi quantiuis gradus x^m+Ax^m-1+Bx^m-2+ etc. +M=0 siue X=0 (vbi coëfficientes A, B etc. reales esse supponuntur) ad minimum vno modo satisfieri posse per valorem ipsius x sub forma a+b√-1 contentum.

Before I review the demonstrations of our theorem by other geometers, and set forth what in each seems to me blameworthy, I observe: it suffices if it be shown only that every equation of whatever degree x^m+Ax^m-1+Bx^m-2+ etc. +M=0 or X=0 (where the coefficients A, B etc. are supposed to be real) can be satisfied at least in one way by a value of x contained under the form a+b√-1.

Constat enim, X tunc diuisibilem fore per factorem realem secundi gradus xx-2ax+aa+bb, si b non fuerit =0, et per factorem realem simplicem x-a, si b=0. In vtroque casu quotiens erit realis, et inferioris gradus quam X; et quum hic eadem ratione factorem realem primi secundiue gradus habere debeat, patet, per continuationem huius operationis functionem X tandem in factores reales simplices vel duplices resolutum iri, aut, si pro singulis factoribus realibus duplicibus binos imaginarios simplices adhibere mauis, in m factores simplices.

For it is evident that X will then be divisible by the real factor of the second degree xx-2ax+aa+bb, if b is not =0, and by the simple real factor x-a, if b=0. In either case the quotient will be real, and of a lower degree than X; and since this in the same way ought to have a real factor of the first or of the second degree, it is clear that, by the continuation of this operation, the function X will at length be resolved into real factors, simple or double; or, if for each real double factor you prefer to employ two simple imaginaries, into m simple factors.

Prima theorematis demonstratio illustri geometrae d'Alembert debetur, Recherches sur le calcul intégral, Histoire de l'Acad. de Berlin, Année 1746, p. 182. sqq.

The first demonstration of the theorem is owed to the illustrious geometer d'Alembert, Recherches sur le calcul intégral, Histoire de l'Acad. de Berlin, Year 1746, p. 182 ff.

Eadem extat in Bongainville, Traité du calcul intégral, à Paris 1754. p. 47. sqq. Methodi huius praecipua momenta haec sunt.

The same is extant in Bongainville, Traité du calcul intégral, à Paris 1754. p. 47. sqq. The principal points of this method are these.

Primo ostendit, si functio quaecunque X quantitatis variabilis x fiat =0 aut pro x=0 aut pro x=∞, atque valorem infinite paruum realem positiuum nancisci possit tribuendo ipsi x valorem realem: hanc functionem etiam valorem infinite paruum realem negatiuum obtinere posse per valorem ipsius x vel realem vel sub forma imaginaria p+q√-1 contentum. Scilicet designante Ω valorem infinite paruum ipsius X, et ω valorem respondentem ipsius x, asserit ω per seriem valde conuergentem aΩ^α+bΩ^β+cΩ^γ etc. exprimi posse, vbi exponentes α, β, γ etc.

First he shows that, if any function X of the variable quantity
x becomes = 0 either for x = 0 or for x = ∞, and can obtain an infinitely
small positive real value by assigning to x a real value: then this function can also obtain an infinitely small
negative real value through a value of x either real or contained under the imaginary form p + q√-1.
Namely, letting Ω designate the infinitely small value of X, and ω the corresponding value
of x, he asserts that ω can be expressed by a very convergent series
aΩ^α + bΩ^β + cΩ^γ etc., where the exponents are α, β, γ
etc.

sint quantitates rationales continuo crescentes, et quae adeo ad minimum in distantia certa ab initio positiuae euadant, terminosque, in quibus adsint, infinite paruos reddant. Iam si inter omnes hos exponentes nullus occurat, qui sit fractio denominatoris paris, omnes terminos seriei reales fieri tum pro positiuo tum pro negatiuo valore ipsius Ω; si vero quaedam fractiones denominatoris paris inter illos exponentes repariantur, constare, pro valore negatiuo ipsius Ω terminos respondentes in forma p+q√-1 contentos esse. Sed propter infinitam seriei conuergentiam in casu priori sufficere, si terminus primus (i. e. maximus) solus retineatur, in posteriori vltra eum terminum, qui partem imaginariam primus producat, progredi opus non esse.

let the exponents be rational quantities continuously increasing, and which at least, at a certain distance from the beginning, turn out positive, and render the terms in which they occur infinitely small. Now, if among all these exponents none occurs that is a fraction with an even denominator, all the terms of the series become real both for the positive and for the negative value of Ω; but if certain fractions with an even denominator are found among those exponents, it is established that, for the negative value of Ω, the corresponding terms are contained in the form p+q√-1. But on account of the infinite convergence of the series, in the former case it suffices if the first term (i.e., the maximum) alone is retained; in the latter, there is no need to proceed beyond that term which first produces the imaginary part.

Per simila ratiocinia ostendi posse, si X valorem realem negatiuum infinite paruum ex valore reali ipsius x assequi possit: functionem illam valorem realem positiuum infinite paruum ex valore reali ipsius x vel ex imaginario sub forma p+q√-1 contento adipisci posse.

By similar ratiocinations it can be shown, if X can attain an infinitely small negative real value from the real value of x itself: that function can acquire an infinitely small positive real value from the real value of x itself, or from an imaginary contained under the form p+q√-1.

Hinc secundo concludit, etiam valorem aliquem realem finitum ipsius X dari, in casu priori negatiuum, in posteriori positiuum, qui ex valore imaginario ipsius x sub forma p+q√-1 contento produci possit.

Hence he secondly concludes that there is also some finite real value
of X, negative in the former case, positive in the latter,
which can be produced from the imaginary value of x in the form p+q√-1
contained.

Hinc sequitur, si X sit talis functio ipsius x, quae valorem realem V ex valore ipsius x reali v obtineat, atque etiam valorem realem quantitate infinite parua vel maiorem vel minorem ex valore reali ipsius x assequatur, eandem etiam valorem realem quantitate infinite parua atque adeo finite vel minorem vel maiorem quam V (resp.)recipere posse, tribuendo ipsi x valorem sub forma p+q√-1 contentum. Hoc nullo negotio ex praecc. deriuatur, si pro X substitui concipitur V+Y, et pro x, v+y.

Hence it follows, if X is such a function of x as obtains the real value V from the real value v of x, and also from a real value of x attains a real value greater or lesser by an infinitely small quantity, that the same function can also receive a real value greater or lesser by an infinitely small, and indeed by a finite, quantity than V (respectively)receive, by assigning to x a value contained under the form p+q√-1. This is derived with no trouble from the foregoing, if for X one conceives V+Y to be substituted, and for x, v+y.

Tandem affirmat ill. d'Alembert, si X totum interuallum aliquod inter duos valores reales R, S percurrere posse supponatur ( i. e. tum ipsi R, tum ipsi S, tum omnibus valoribus realibus intermediis aequalis fieri), tribuendo ipsi x valores semper in forma p+q√-1 contentos; functionem X quauis quantitate finita reali adhuc augeri vel diminui posse (prout S>R vel S<R), manente x semper sub forma p+q√-1. Si enim quantitas realis U daretur (inter quam et R supponitur S iacere), cui X per talem valorem ipsius x aequalis fieri non posset, necessario valorem maximum ipsius X dari (scilicet quando S>R; minimum vero, quando S<R), puta T, quem ex valore ipsius x, p+q√-1, consequeretur, ita vt ipsi x nullus valor sub simili forma contentus tribui posset, qui functionem X vel minimo excessu propius versus U promoueret. Iam si in aequatione inter X et x pro x vbique substituatur p+q√-1, atque tum pars realis, tum pars, quae factorem √-1 implicet, hoc omisso, cifrae aequentur: ex duabus aequationibus hinc prodeuntibus (in quibus p, q et X cum constantibus permixtae occurrent) per eliminationem duas alias elici posse, in quarum altera p, X et constantes reperiantur altera a p libera solas q, X et constantes inuoluat.

At length the illustrious d’Alembert affirms, if it be supposed that X can traverse some whole interval between two real values R, S (i. e. be made equal both to R and to S and to all the intermediate real values), by assigning to x values always contained in the form p+q√-1; that the function X can still be increased or diminished by any finite real quantity (as S>R or S<R), with x remaining always under the form p+q√-1. For if there were given a real quantity U (between which and R S is supposed to lie), to which X could not be made equal by such a value of x, then necessarily a maximum value of X would be given (namely when S>R; a minimum, truly, when S<R), to wit T, which it would attain from the value of x, p+q√-1, so that no value contained under a similar form could be assigned to x which would advance the function X toward U by even the least excess. Now if in the equation between X and x p+q√-1 be everywhere substituted for x, and then the real part and the part that involves the factor √-1, this being omitted, the coefficients be set equal: from the two equations hence proceeding (in which p, q and X occur commingled with constants) two others can be elicited by elimination, in one of which p, X and constants are found, the other, free from p, involves only q, X and constants.

Quamobrem quum X per valores reales ipsarum p,q omnes valores ab R vsque ad T percurrerit, per praecc. X versus valorem U adhuc propius accedere posse tribuendo ipsius p, q valores tales α+γ√-1, β+δ√-1 resp. Hinc vero fieri x=α-δ+(γ+β)√-1, i. e. adhuc sub forma p+q√-1 esse, contra hyp.

Wherefore
when X, by the real values of p, q, has run through all the values from R up to T, by the foregoing X can still approach more closely to the value U by assigning to p, q such values α+γ√-1, β+δ√-1 respectively. Hence indeed it comes to be x=α-δ+(γ+β)√-1, i. e.
still to be under the form p+q√-1, contrary to the hypothesis.

Iam si X functionem talem vt x^m+Ax^m-1+Bx^m-2 + etc. +M denotare supponitur, nullo negotio perspicitur, ipsi x tales valores reales tribui posse, vt X totum aliquod interuallum inter duos valores reales percurrat. Quare x valorem aliquem sub forma p+q√-1 contentum talem etiam nancisci poterit, vnde X fiat =0. Q. E. D. *2)

Now if X is supposed to denote a function such as x^m+Ax^m-1+Bx^m-2 + etc. +M, it is seen without difficulty that such real values can be assigned to x that X traverses some entire interval between two real values. Wherefore x will also be able to obtain some value contained under the form p+q√-1, whence X becomes =0. Q. E. D. *2)

Quae contra demonstrationem d'Alembertianam obiici posse videntur, ad haec fere redeunt.

The things which seem able to be objected against the d'Alembertian demonstration reduce for the most part to these.

1. Ill. d'A. nullum dubium mouet de existentia valorum ipsius x quibus valores dati ipsius X respondeant, sed illam supponit, solamque formam istorum valorum inuestigat.

1. The illustrious d'A. raises no doubt about theexistence of the values of x
to which the given values of X correspond, but presupposes that, and investigates only the form of those values.

Quamuis vero haec obiecto per se grauissima sit, tamen hic ad solam dictionis formam pertinet, quae facile ita corrigi potest, vt illa penitus destruatur.

Although indeed this objection is in itself most grave, nevertheless here it pertains only to the form of diction, which can easily be corrected thus, so that it is utterly destroyed.

2. Assertio,ω per talem seriem qualem ponit semper exprimi posse, certo est falsa, si X etiam funcionem quamlibet transscendentem designare debet (vti d'A. pluribus locis innuit). Hoc e. g. manifestum est, si ponitur X=e^1/x, siue x=1/log X. Attamen si demonstrationem ad eum casum restringimus, vbi X est functio algebraica ipsius x (quod in praesenti negotio sufficit), propositio vtique est vera. — Ceterum d'A. nihil pro confirmatione suppositionis suae attulit; cel. Bougainville supponit X esse functionem algebraicam ipsius x, et ad inuentionem seriei parallelogrammum Newtonianum commendat.

2. The assertion, thatω can always be expressed by such a series as he posits, is certainly false, if X is also to designate an arbitrary transcendental function (as d'A. hints in several places). This, e. g. is manifest, if one sets X=e^1/x, or x=1/log X. However, if we restrict the demonstration to that case where X is an algebraic function of x (which in the present matter suffices), the proposition is indeed true. — Besides, d'A. brought forward nothing for the confirmation of his supposition; the celebrated Bougainville supposes X to be an algebraic function of x, and for the invention of the series recommends the Newtonian parallelogram.

3. Quantitatibus infinite paruis liberius vtitur, quam cum geometrico rigore consistere potest aut saltem nostra aetate (vbi illae merito male audiunt) ab analysta scrupuloso concederetur, neque etiam saltum a valore infinite paruo ipsiusΩ ad finitum satis luculenter explicauit. Propositionem suam, Ω etiam valorem aliquem finitum consequi posse, non tam ex possibilitate valoris infinite parui ipsius Ω, concludere videtur quam inde potius, quod denotante Ω quantitatem valde paruam, propter magnam seriei conuergentiam, quo plures termini seriei accipiantur, eo propius ad valorem verum ipsius ω accedatur, aut, quo plurium partium summa pro ω accipiatur, eo exactius aequtioni, quae relationem inter ω et Ω siue x et X exhibeat, satisfactum iri. Praeterea quod tota haec argumentatio nimis vaga videtur, quam vt ulla conclusio rigorosa inde colligi possit: obseruo, vtique dari series, quae quantumuis paruus valor quantitati, secundum cuius potestates progrediuntur, tribuatur, nihilominus semper diuergant, ita vt si modo satis longe continuentur, ad terminos quauis quantitate data maiores peruenire possis.

3. He employs infinitesimal quantities more freely than can consist with geometric rigor, or at least than in our age (when they deservedly have a bad reputation) would be conceded by a scrupulous analyst, nor has he explained with sufficient clarity the leap from an infinitely small value ofΩ to a finite one. His proposition, that Ω also can attain some finite value, seems to be concluded not so much from the possibility of an infinitely small value of Ω, as rather from this: that, Ω denoting a very small quantity, on account of the great convergence of the series, the more terms of the series are taken, the nearer one approaches to the true value of ω; or, the more the sum of several parts is taken for ω, the more exactly the equation which exhibits the relation between ω and Ω, or x and X, will be satisfied. Moreover, since all this argumentation seems too vague for any rigorous conclusion to be gathered from it, I observe that there indeed are series which, however small a value may be assigned to the quantity according to whose powers they proceed, nevertheless always diverge, so that, if only they be continued far enough, you can arrive at terms greater than any given quantity.

*3) Hoc euenit, quando coëf- ficientes seriei progressionem hypergeometricam constituunt. Quamobrem necessario demonstrari debuisset, talem seriem hypergeometricam in casu praesenti prouenire non posse.

*3) This happens when the coëf-
ficients of the series constitute a hypergeometric progression. Wherefore it ought to have been necessarily demonstrated that such a hypergeometric series cannot arise in the present case.

Ceterum mihi videtur, ill. d'A. hic non recte ad series infinitas confugisse, hasque ad stabiliendum theorema hoc fundamentale doctrinae aequationum haud idoneas esse.

Moreover it seems to me that the illustrious d’A. has here not rightly resorted to infinite series,
and that these are by no means suitable for establishing this fundamental theorem of the doctrine of equations.

4. Ex suppositione,X obtinere posse valorem S neque vero valorem U, nondum sequitur, inter S et U necessario valorem T iacere, quem X attingere sed non superare possit. Superest adhuc alius casus: scilicet fieri posset, vt inter S et U limes situs sit, ad quem accedere quidem quam prope velis possit X, ipsum vero nihilominus numquam attingere. Ex argumentis ab ill.

4. From the supposition, thatX can obtain the value S but not
the value U, it does not yet follow that between S and U there necessarily lies the value T
which X could attain but not surpass. There still remains
another case: namely, it could happen that between S and U a limit is
situated, to which X can indeed approach as near as you please, but
nevertheless never attain it. From the arguments by ill.

d'A. allatis tantummodo sequitur, X omnem valorem, quem attigerit, adhuc quantitate finita superare posse, puta quando enaserit =S, adhuc quantitate aliqua finita Ω augeri posse; quo facto, nouum incrementum Ω′ accedere, tunc iterum augmentum Ω′′ etc., ita vt quotcunque incrementa iam adiecta sint, nullum pro vltimo haberi debeat, sed semper aliquod nouum accedere possit. At quamuis multitudo incrementorum possibilium nullis limitibus sit circumscripta: tamen vtique fieri posset, vt si incrementa Ω, Ω′, Ω′′ etc. continuo decrescerent, nihilominus summa S+Ω+Ω′+Ω′′ etc.

d'A. from those adduced it only follows that X can still exceed by a finite quantity any value it has reached,
for instance when it has turned out = S, it can still be increased by some finite quantity Ω; this being done, a new
increment Ω′ is added, then again an augmentation Ω′′, etc., so that, however many increments have already been
added, none ought to be held as the last, but some new one can always be added. But although the multitude of
possible increments is circumscribed by no limits: nevertheless it could indeed happen that, if the increments
Ω, Ω′, Ω′′, etc. continually decreased, nonetheless the sum S+Ω+Ω′+Ω′′, etc.

limitem aliquem numquam attingeret, quotcunque termini considerentur.

would never reach some limit, however many terms were considered.

Quamquam hic casus occurrere non potest, quando X designat functionem algebraicam integram ipsius x: tamen sine demonstratione, hoc fieri non posse, methodus necessario pro incompleta habenda est. Quando vero X est functio transscendens, siue etiam algebraica fracta, casus ille vtique locum habere potest, e. g. semper quando valori cuidam ipsius X valor infinite magnus ipsius x respondet. Tum methodus d'Alembertiana non sine multis ambagibus, et in quibusdam casibus nullo forsan modo, ad principia indubitata reduci posse videtur.

Although this case cannot occur, when X designates an entire algebraic function of x itself: nevertheless, without a demonstration that this cannot be done, the method must necessarily be held as incomplete. When indeed X is a transcendental function, or even a fractional algebraic one, that case indeed can have place, e. g. always whenever to some value of X there corresponds an infinitely great value of x. Then the d’Alembertian method seems, not without many circumlocutions, and in some cases perhaps in no way, to be reducible to indubitable principles.

Propter has rationes demonstrationem d'Alembertianam pro satisfaciente habere nequeo. Attamen hoc non obstante verus demonstrationis neruus probandi per omnes obiectiones neutiquam infringi mihi videtur, credoque eidem fundamento (quamuis longe diuersa ratione, et saltem maiori circumspicienta) non solum demonstrationem rigorosam theorematis nostri superstrui, sed ibinde omnia peti posse, quae circa aequationum transscendentium theoriam desiderari queant. De qua re grauissima alia occasione fusius agam; conf.

For these reasons I cannot hold the d'Alembertian demonstration as satisfactory. Nevertheless, this notwithstanding, the true nerve of the demonstration of proving seems to me by no means to be broken by all objections, and I believe that upon the same foundation (although by a far different method, and at least with greater circumspection) not only can a rigorous demonstration of our theorem be superstructured, but from there everything can be sought which might be desiderated concerning the theory of transcendental equations. On which most weighty matter I shall speak more fully on another occasion; conf.

interim infra art. 23.

meanwhile below, in article 23.

Post d'Alembertum ill. Euler disquisitiones suas de eodem argumento promulgauit, Recherches sur les racines imaginaires des équations, Hist. de l'Acad.

After d'Alembert, the illustrious Euler published his disquisitions on the same subject, Researches on the imaginary roots of
equations, Hist. of the Acad.

de Berlin A. 1749. p. 223 sqq. Methodum duplicem hic tradidit: prioris summa continetur in sequentibus.

of Berlin, in the year 1749, p. 223 and following. Here he set forth a twofold method: the substance of the former is contained in the following.

Primo ill. E. suscipit demonstrare, si m denotet quamcunque dignitatem numeri 2, functionem x^2m+Bx^2x-2+Cx^2m-3 + etc. +M=X (in qua coëfficiens termini secundi est =0) semper in duos factores reales resolui posse, in quibus x vsque ad m dimensiones ascendat.

First, ill. E. undertakes to demonstrate that, if m denote any dignity (degree/power) of the number 2, the function x^2m+Bx^2x-2+Cx^2m-3 + etc. +M=X (in which the coefficient of the second term is =0) can always be resolved into two real factors, in which x ascends up to m dimensions.

Ad hunc finem duos factores assumit, x^m-ux^m-1+α x^m-2+β x^m-3+ etc., et x^m+ ux^m-1+λ x^m-2+μ x^m-3 etc. vbi coëfficientes u,α, β etc. λ, μ etc.

To this end he assumes two factors, x^m-ux^m-1+α x^m-2+β x^m-3+ etc., and x^m+ ux^m-1+λ x^m-2+μ x^m-3 etc. where the coefficients u,α, β etc. λ, μ etc.

adhuc incogniti sunt, horumque productum aequale ponit functioni X. Tum coëfficientium comparatio suppeditat 2m-1 aequationes, manifestoque demonstrari tantummodo debet, incognitis u,α,β etc. λ, μ etc. (quarum multitudo etiam est 2-1) tales valores reales tribui posse, qui aequationibus illis satisfaciant.

are as yet unknown, and he sets their product equal to the function X. Then the comparison of coefficients supplies 2m-1 equations, and it need only be manifestly demonstrated that to the unknowns u, α, β etc., λ, μ etc. (whose multitude likewise is 2-1) real values of such a kind can be assigned as will satisfy those equations.

Iam E. affirmat, si primo u tamquam cognita consideretur, ita vt multitudo incognitarum vnitate minor sit quam multitudo aequationum, his secundum methodos algebraicas notas rite combinatis omnes α, β etc. λ, μ etc. rationaliter et sine vlla radicum extractione per u et coëfficientes B, C etc.

Now E. affirms that, if at the outset u be regarded as known, in such a way that the multitude of unknowns is less by one than the multitude of equations, then, these being duly combined according to the well-known algebraic methods, all α, β etc., λ, μ etc., can be obtained rationally and without any extraction of roots, by means of u and the coefficients B, C etc.

determinari posse, adeoque valores reales nancisci, simulac u realis fiat. Praeterea vero omnes α, β etc. λ, μ etc.

can be determined, and thus obtain real values, as soon as u becomes real. Moreover, all α, β etc. λ, μ etc.

eliminari poterunt, ita vt prodeat aequatio U=0, vbi U erit functio integra solius u et coëfficientium cognitorum. Hanc aequationem ipsam per methodum eliminationis vulgarem euoluere, opus immensum fore, quando aequatio proposita X=0 est gradus aliquantum alti; et pro gradu indeterminato, plane impossibile (iudice ipso E. p. 239.). Attamen sufficit, vnam illius aequationis proprietatem nouisse, scilicet quod terminus vltimus in U (qui incognitam u non implicat) necessario est negatiuus, vnde sequi constat, aequationem ad minimum vnum radicem realem habere, siue u et proin etiam α, β etc. λ, μ etc.

can be eliminated, so
that there comes forth the equation U=0, where U will be an integral function of u alone and of
the known coefficients. To work out this very equation by the common method
of elimination would be an immense labor,
when the proposed equation X=0 is of somewhat high degree; and
for an indeterminate degree, plainly impossible (in the judgment of E. himself, p. 239).
Nevertheless it suffices to have known one property of that equation,
namely that the last term in U (which does not involve the unknown u)
is necessarily negative, whence it is evident to follow that the equation
has at least one real root, namely u, and therefore also
α, β etc., λ, μ etc.

ad minimum vno modo realiter determinari posse: illam vero proprietatem per sequentes reflexiones confirmare licet. Quum x^m-ux^m-1+α x^m-2+ etc. supponatur esse factor functionis X: necessario u erit summa m radicum aequationis X=0, adeoque totidem valores habere debebit, quot modis diuersis ex 2m radicibus m excerpi possunt, siue per principia calculi combinationum (2m . 2m-1 . ... . m+1) / (1 . 2 . 3 ... m) valores.

to be able at a minimum in one way to be determined in reality; and indeed one may confirm that property by the following reflections. Since x^m-u x^m-1+α x^m-2+ etc. is supposed to be a factor of the function X: necessarily u will be the sum of m roots of the equation X=0, and so it must have just as many values as there are different ways in which out of 2m roots m can be selected, that is, by the principles of the calculus of combinations, (2m . 2m-1 . ... . m+1) / (1 . 2 . 3 ... m) values.

Hic numerus semper erit impariter par (demonstrationem haud difficilem supprimo): si itaque ponitur =2k, ipsius semissis k impar erit; aequatio U=0 vero erit gradus 2k^ti. Iam quoniam in aequatione X=0 terminus secundus deest: summa omnium 2m radicum erit 0; vnde patet, si summa quarumcunque m radicum fuerit +p, reliquarum summam fore -p, i. e. si +p est inter valores ipsius u, etiam -p inter eosdem erit. Hinc E. concludit, U esse productum ex k factoribus duplicibus talibus uu-pp, uu-qq, vu-rr etc., denotantibus +p, -p, +q, -q etc. omnes 2k radices aequationis U=0, vnde, propter multitudinem imparem horum factorum, terminus vltimus in U erit quadratum producti pqr etc.

This number will always be oddly even (I suppress a not-difficult demonstration): if therefore it is set =2k, its half k will be odd; the equation U=0 indeed will be of degree 2k^ti. Now, since in the equation X=0 the second term is lacking, the sum of all the 2m roots will be 0; whence it is clear that, if the sum of any m roots is +p, the sum of the remaining will be -p, i. e. if +p is among the values of u, then -p will be among the same. Hence E. concludes that U is the product of k double factors such as uu-pp, uu-qq, vu-rr etc., denoting +p, -p, +q, -q etc., all the 2k roots of the equation U=0; whence, because of the odd multitude of these factors, the last term in U will be the square of the product pqr etc.

signo negatiuo affectum. Productum autem pqr etc. semper ex coëfficientibus B, C etc.

affected with a negative sign. But the product pqr, etc., is always derived from the coefficients B, C, etc.

rationaliter determinari potest, adeoque necessario erit quantitas realis. Huius itaque quadratum signo negatiuo affectum certo erit quantitas negatiua. Q. E. D.

can be determined rationally, and so will necessarily be a real quantity. Therefore the square of this, affected with a negative sign, will certainly be a negative quantity. Q. E. D.

Quum hi duo factores reales ipsius X sint gradus m^ti atque m potestas numeri 2: eadem ratione vterque rursus in duos factores reales 1/2m dimensionum resolui poterit. Quoniam vero per repetitam dimidiationem numeri m necessario tandem ad binarium peruenitur, manifestum est, per continuationem operationis functionem X tandem in factores reales secundi gradus resolutam haberi.

Since these two real factors of X are of the m^th degree, and m is a power of the number 2: by the same reasoning each in turn can be resolved into two real factors of 1/2m dimensions. But since by repeated halving of the number m one necessarily at length arrives at the number two, it is manifest that, by continuation of the operation, the function X is at length found resolved into real factors of the second degree.

Quodsi vero functio talis proponitur, in qua terminus secundus non deest, puta x^2m+Ax^2m-1+Bx^2m-2+etc. +M, designante etiamnum 2m potestatem binariam, haec per substitutionem x=y-A/(2m) transibit in similem functionem termino secundo carentem. Vnde facile concluditur, etiam illam functionem in factores reales secundi gradus resolubilem esse.

But if indeed a function is proposed of such a sort in which the second term is not lacking, for instance x^2m+Ax^2m-1+Bx^2m-2+etc. +M, with 2m still designating a binary power, this, by the substitution x=y-A/(2m), will pass into a similar function lacking the second term. Whence it is easily concluded that that function too is resolvable into real factors of the second degree.

Denique proposita functione gradus n^ti, designante n numerum, qui non est potestas binaria: ponatur potestas binaria proxime maior quam n, =2m, multiplicetur functio proposita per 2m-n factores simplices reales quoscunque. Ex resolubilitate producti in factores reales secundi gradus, nullo negotio deriuatur, etiam functionem propositam in factores reales secundi vel primi gradus resolubilem esse debere.

Finally, with a function of degree n^th proposed, n designating a number
which is not a binary power: let the binary power next greater than n, =2m, be posited; let the proposed function be multiplied by
2m-n arbitrary simple real factors whatsoever. From the resolvability
of the product into real factors of the second degree, it is derived with no trouble
that the proposed function also must be resoluble into real factors of the second or
first degree.

Contra hanc demonstrationem obiici potest.

Against this demonstration it can be objected.

1. Regulam, secundum quam E. concludit, ex2m-1 aequationibus 2m-2 incognitas α,β etc. λ, μ etc. omnes rationaliter determinari posse, neutiquam esse generalem, sed saepissime exceptionem pati.

1. The rule, according to which E. concludes that from2m-1 equations the 2m-2 unknowns α, β etc. λ, μ etc. can all be rationally determined, is by no means general, but very often admits an exception.

Si quis e. g. in art. 3, aliaque incognitarum tamquam cognita spectata, reliquas per hanc et coefficientes datos rationaliter exprimere tentat, facile inueniet, hoc esse impossibile, nullamque quantitatum incognitarum aliter quam per aequationem m-1^ti gradus determinari posse. Quamquam vero hic statim a priori perspici potest, illud necessario ita euenire debuisse: tamen merito dubitari posset, annon etiam in casu praesenti pro quibusdam valoribus ipsius m res eodem modo se habebat; vt incognitae α,β etc.

If someone, e. g. in art. 3, with other unknowns regarded as known, attempts to express the remaining ones rationally by means of this and the given coefficients, he will easily find that this is impossible, and that none of the unknown quantities can be determined otherwise than by an equation of the m-1^th degree. Although indeed here one can at once perceive a priori that it must necessarily have turned out so; nevertheless one might rightly doubt whether also in the present case, for certain values of m itself, the matter stood in the same way; so that the unknowns α,β etc.

λ, μ etc. ex u,B,C etc. aliter quam per aequationem gradus forsan maioris quam 2m determinari nequeant.

λ, μ etc. from u,B,C etc. cannot be determined otherwise than by an equation of degree perhaps greater than 2m.

Pro eo casu, vbi aequatio X=0 est quarti gradus, E. valores rationales coëfficientium per u et coëfficientes datos eruit; idem vero etiam in omnibus aequationibus altioribus fieri posse, vtique explicatione ampliori egebat. — Ceterum operae pretium esse videtur, in formulas illas, quae α, β etc. rationaliter per u, B, C etc.

For that case where the equation X=0 is of the fourth degree, E. extracts the rational values of the coefficients by means of u and the given coefficients; but that the same also can be done in all higher equations certainly required a fuller explication. — Moreover, it seems worth the work, to go into those formulas which express α, β etc. rationally through u, B, C etc.

exprimant, profundius et generalissime inquirere; de qua re aliisque ad eliminationis theoriam (argumentum haudquaquam exhaustum) pertinentibus alia occasione fusius agere suscipiam.

express, to inquire more profoundly and in the most general way; concerning which matter and others pertaining to the theory of elimination (a topic by no means exhausted) I will on another occasion undertake to treat more at length.

2. Etiamsi autem demonstratum fuerit, cuiusuis gradus sit aequatioX=0, semper formulas inueniri posse, quae ipsas α, β etc. λ, μ etc. rationaliter per u, B, C etc.

2. Even if, however, it should be demonstrated, of whatever degree the equationX=0 may be, that formulas can always be found which express the very α, β etc., λ, μ etc., rationally through u, B, C etc.

exhibeant: tamen certum est, pro valoribus quibusdam determinatis coëfficientium B, C etc. formulas illas indeterminatas euadere posse, ita vt non solum impossibile sit, incognitas illas rationaliter ex u, B, C etc. definire, sed adeo reuera quibusdam in casibus valori alicui reali ipsius u nulli valores reales ipsarum α, β etc.

they may exhibit: nevertheless
it is certain that, for certain determinate values of the coefficients
B, C etc., those formulas can turn out indeterminate, so that not
only is it impossible to define those unknowns rationally from u, B, C
etc., but indeed, in some cases, to some real value of the
u itself there correspond no real values of α, β etc.

λ, μ etc. respondeant. Ad confirmationem huius rei breuitatis gratia ablego lectorem ad diss.

λ, μ etc. correspond. For the confirmation of this matter, for brevity’s sake, I refer the reader to the dissertations.

ipsam E. vbi p. 236. aequatio quarti gradus fusius explicata est. Statim quisque videbit, formulas pro coëfficientibus α, β indeterminatas fieri, si C=0 et pro u assumatur valor 0, illorumque valores non solum sine extractione radicum assignari non posse, sed adeo ne reales quidem esse, si fuerit BB-4D quantitas negatiua.

that same E., where on p. 236 the equation of the fourth degree is more fully explained. At once anyone will see that the formulas for the coefficients α, β become indeterminate if C=0 and the value 0 is assumed for u, and that their values not only cannot be assigned without extraction of roots, but indeed are not even real, if BB-4D is a negative quantity.

Quamquam vero in hoc casu u adhuc alios valores reales habere, quibus valores reales ipsarum α, β respondeant, facile perspici potest: tamen vereri aliquis posset, ne huius difficultatis enodatio (quam E. omnino non attigit) in aequationibus altioribus multo maiorem operam facessat. Certe haec res in demonstratione exacta neutiquam silentio praeteriri debet.

Although indeed in this case u can still have other real values, to which real values of α, β correspond, this can easily be perceived: nevertheless someone might fear, lest the resolution of this difficulty (which E. did not touch at all) in higher equations would occasion much greater labor. Certainly this matter ought by no means to be passed over in silence in an exact demonstration.

3. Ill. E. supponit tacite, aequationemX=0 habere 2m radices, harumque summam statuit =0 ideo quod terminus secundus in X abest. Quomodo de hac licentia (qua omnes auctores de hoc argumento vtuntur) sentiam, iam supra art.

3. The Ill. E. supposes tacitly that the equationX=0 has 2m roots, and he posits their sum =0 for the reason that the second term in X is absent. How I judge about this license (which all authors on this argument use), already above art.

3. declaraui. Propositio, summam omnium radicum aequationis alicuius coëfficienti primo, mutato signo, aequalem esse, ad alias aequationes applicanda non videtur, nisi quae radices habent: iam quum per hanc ipsam demonstrationem euinci debeat, aequationemX=0 reuera radices habere, haud permissum videtur, harum existentiam supponere. Sine dubio ii, qui huius paralogismi fallaciam nondum penetrauerunt, respondebunt, hic non demonstrari, aequationi X=0 satisfieri posse (nam hoc dicere vult expressio, eam habere radices), sed tantummodo, ipsi per valores ipsios x sub forma a+b√-1 contentos satisfieri posse; illud vero tamquam axioma supponi.

3. I have declared. The proposition, that the sum of all the roots of some equation is equal to the first coefficient, with the sign changed, does not seem applicable to other equations, except those which have roots: now, since by this very demonstration it ought to be proved that the equationX=0 really has roots, it does not seem permitted to assume their existence. Without doubt those who have not yet penetrated the fallacy of this paralogism will reply that what is not being demonstrated here is that the equation X=0 can be satisfied (for this is what the expression means, that it has roots), but only that it can itself be satisfied by the values of x contained under the form a+b√-1; whereas that is being assumed as an axiom.

At quum aliae quantitatum formae, praeter realem et imaginariam a+b√-1 concipi nequeant, non satis luculentum videtur, quomodo id quod demonstrari debet ab eo, quod tamquam axioma supponitur, differat; quin adeo si possibile esset adhuc alias formas quantitatum excogitare, puta formam F, F′, F′′ etc. tamen sine demonstratione admitti non deberet, cuius aequationi per aliquem valorem ipsius x aut realem, aut sub forma a+b√-1, aut sub forma F, aut sub F′ etc. contentum satisfieri posse.

But since other forms of quantities, besides the real and the imaginary a+b√-1, cannot be conceived, it does not seem sufficiently lucid how that which must be demonstrated differs from that which is assumed as an axiom; indeed, even if it were possible still to devise other forms of quantities, for example the form F, F′, F′′ etc., nevertheless it ought not to be admitted without demonstration that there is an equation for which it is possible to be satisfied by some value of x either real, or contained under the form a+b√-1, or under the form F, or under F′ etc.

Quamobrem axioma illud alium sensum habere nequit quam hunc: Cuius aequationi satisfieri potest aut per valorem realem incognitae, aut per valorem imaginariam sub forma a+b√-1 contentum, aut forsan per valorem sub forma alia hucusque ignota contentum, aut per valorem, qui sub nulla omnino forma continetur. Sed quomodo huiusmodi quantitates de quibus ne ideam quidem fingere potes — vera vmbrae vmbra — summari aut multiplicari possint; hoc ea perspicuitate, quae in mathesi semper postulatur, certo non intelligitur.*4)

Wherefore that axiom can have no other sense than this: An equation can be satisfied either by a real value of the unknown, or by an imaginary value contained under the form a+b√-1, or perhaps by a value contained under some other form hitherto unknown, or by a value that is contained under no form at all. But how quantities of this sort — of which you cannot even fashion an idea, a true shadow of a shadow — could be summed or multiplied; this, with that perspicuity which is always demanded in mathematics, is certainly not understood.*4)

Ceterum conclusiones, quas E. ex suppositione sua elicuit, per has obiectiones haudquaquam suspectas reddere volo; quin potius certus sum, illas per methodum neque difficilem neque ab Euleriana multum diuersam ita comprobari posse, vt nemini vel minimus scrupulus superesse debeat. Solam formam reprehendo, quae quamuis in inueniendis nouis veritatibus magnae vtilitatis esse possit, tamen in demonstrando, coram publico, minime probanda videtur.

Moreover, the conclusions which E. elicited from his supposition,
I by no means wish to render suspect by these objections; rather
I am certain that they can be confirmed by a method neither difficult nor
much different from the Eulerian, so that not even the least scruple
ought to remain to anyone. I censure only the form,
which, although in finding new truths it can be of great utility,
yet in demonstrating, before the public, seems by no means
to be approved.

4. Pro demonstratione assertionis, productumpqr etc. ex coëfficientibus in X rationaliter determinari posse, ill. E. nihil omnino attulit.

4. For a demonstration of the assertion that the productpqr etc., from the coefficients in X, can be determined rationally, the illustrious E. adduced nothing at all.

Omnia, quae hac de re in aequationibus quarti gradus explicat, haec sunt (vbi a, b, c, d sunt radices aequationes propositae x⁴+Bxx+Cx+D=0):

All that he explains on this matter in equations of the fourth degree are these (where a, b, c, d are the roots of the proposed equation x⁴+Bxx+Cx+D=0):

''On m'objectera sans doute, que j'al supposé ici, que la quantité pqr étoit une quantité réelle, et que son quarré ppqqrr étoit affirmatif; ce qui étoit encor douteux, vu que les racines a, b, c, d étant imaginaire, il pourroit bien arriver, que le quarré de la quantité pqr, qui en es composée, fut negatif. Or je reponds à cela que ce cas ne sauroit jamais avoir lieu; car quelque imaginaires que soient les racines a,b,c,d, on sait pourtant, qu'il doit y avoir a+b+c+d=0; ab+ac+ad+bc +bb+cd=B; abc+abd+acd+bcd=-C *5); abcd =D, ces quantités B,C,D étant réelles. Mais puisque p= a+b, q=a+c, r=a+b, leur produit pqr=(a+b)(a+c)a+d) est determinable comme on sait, par les quantités B,C,D, et sera par conseéquent réel, tout comme nous avons vu, qu'il est effectivement pqr=-C, et ppqqrr=CC. On reconnoitra aisément de même, que dans les plus hautes équations cette même circonstance doit avoir lieu, et qu'on ne sauroit me faire des objections de ce coté.'' Conditionem, productum pqr etc.

''One will no doubt object to me that I have supposed here that the quantity pqr was a real quantity, and that its square ppqqrr was affirmative; which was still doubtful, since the roots a, b, c, d being imaginary, it might well happen that the square of the quantity pqr, which is composed from them, was negative. Now I reply to this that this case can never take place; for however imaginary the roots a, b, c, d, may be, one nevertheless knows that there must be a+b+c+d=0; ab+ac+ad+bc +bb+cd=B; abc+abd+acd+bcd=-C *5); abcd =D, these quantities B, C, D being real. But since p= a+b, q=a+c, r=a+b, their product pqr=(a+b)(a+c)a+d) is determinable, as is known, by the quantities B, C, D, and will consequently be real, just as we have seen that in fact pqr=-C, and ppqqrr=CC. One will easily recognize likewise that in the higher equations this same circumstance must obtain, and that one cannot make objections to me on this side.'' The condition, the product pqr etc.

rationaliter per B,C etc. determinari posse, E. nullibi adiecit, attamen semper subintellexisse videtur, quum absque illa demonstratio nullam vim habere possit. Iam verum quidem est in aequationibus quarti gradus, si productum (a+b)(a+c)(a+d) euoluatur obtineri aa(a+b+c+d)+abc+abd+acd+bcd=-C, attamen non satis perspicuum videtur, quomodo in omnibus aequationibus superioribus productum rationaliter per coëfficientes determinari possit.

rationally to be able to be determined through B, C, etc., E. added nowhere; nevertheless he seems always to have tacitly understood it, since without that the demonstration can have no force. Now it is indeed true in equations of the fourth degree, if the product (a+b)(a+c)(a+d) is expanded, that one obtains aa(a+b+c+d)+abc+abd+acd+bcd=-C; nevertheless it does not seem sufficiently perspicuous how in all higher equations the product can be determined rationally through the coëfficients.

Clar. de Foncenex, qui primus hoc obseruauit (Miscell. phil.

The illustrious de Foncenex, who first observed this
(Miscell. phil.

math. soc. Taurin.

math. soc. Turin.

T. I. p. 117.), recte contendit, sine demonstratione rigorosa huius propositionis methodum omnem vim perdere, illam vero satis difficilem sibi videri confitetur, et quam viam frustra tentauerit, enarrat. *6) Attamen haec res haud difficulter per methodum sequentem (cuius summam addigitare tantummodo hic possum) obsoluitur: Quamquam in aequationibus quarti gradus non satis clarum est, productum (a+b)(a+c)(a+d) per coëfficientes B,C,D determinabile esse, tamen facile perspici potest, idem productum etiam esse = (b+a)(b+c)(b+d), nec non =(c+a)(c+b)(c+d), denique etiam =(d+a)(d+b)(d+c). Quare productum pqr erit quadrans summae (a+b)(a+c)(a+d)+(b+a)(b+c)(b+d)+ (c+a)(c+b)(c+d)+(b+a)(b+d)(d+c), quam, si evoluatur, fore functionem rationalem integram radicum a,b,c,d talem, in quam omnes eadem ratione ingrediantur, nullo negotio a priori praeuideri potest. Tales vero functiones semper rationaliter per coëfficientes aequationis, cuius radices sunt a, b, c, d, exprimi possunt.

T. 1. p. 117.), rightly contends that, without a rigorous demonstration of this proposition, the method loses all force; he confesses, moreover, that that seems to him rather difficult, and he relates which path he tried in vain. *6) Nevertheless, this matter is disposed of without much difficulty by the following method (the sum of which I can only append here): Although in equations of the fourth degree it is not sufficiently clear that the product (a+b)(a+c)(a+d) is determinable through the coefficients B, C, D, yet it is easy to perceive that the same product is also = (b+a)(b+c)(b+d), and likewise =(c+a)(c+b)(c+d), and finally also =(d+a)(d+b)(d+c). Wherefore the product pqr will be a quarter of the sum (a+b)(a+c)(a+d)+(b+a)(b+c)(b+d)+ (c+a)(c+b)(c+d)+(b+a)(b+d)(d+c), which, if it be expanded, will be a rational entire function of the roots a, b, c, d, such that all enter in the same manner—something that can be foreseen a priori without any trouble. Such functions, moreover, can always be expressed rationally through the coefficients of the equation whose roots are a, b, c, d.

— Idem etiam manifestum est, si productum pqr sub hanc formam redigatur:

— The same is also manifest, if the product pqr be reduced under this form:

1/2 (a+b-c-d) x 1/2 (a+c-b-d) x 1/2 (a+d-b-c),

quod productum euolutum omnes a, b, c, d eodem modo implicaturum esse facile praeuideri potest. Simul periti facile hinc colligent, quomodo hoc ad altiores aequationes applicare debeat. — Completam demonstrationis expositionem, quam hic apponere breuitas non permittit, vna cum vberiori disquisitione de functionibus plures variabiles eodem modo inuoluentibus ad aliam occasionem mihi reseruo.

that the expanded product will easily be foreseen to involve all a, b, c, d in the same way. At the same time the skilled will easily gather from this how one ought to apply this to higher equations. — A complete exposition of the demonstration, which brevity does not permit to append here, together with a more copious disquisition concerning functions involving several variables in the same way, I reserve for another occasion.

Ceterum obseruo, praeter has quatuor obiectiones, adhuc quaedam alia in demonstratione E. reprehendi posse, quae tamen silentio praetereo, ne forte censor nimis seuerus esse videar, praesertim quum praecedentia satis ostendere videatur, demonstrationem in ea quidem forma, in qua ab E. proposita est, pro completa neutiquam haberi posse.

Moreover I observe, besides these four objections, still
that certain other things in E.’s demonstration can be reprehended, which, however,
I pass over in silence, lest perhaps I should seem a too severe censor,
especially since what precedes seems sufficiently to show that the demonstration,
in the very form in which it has been proposed by E., can by no means be
held as complete.

Post hanc demonstrationem, E. adhuc aliam viam theorema pro aequationibus, quarum gradus non est potestas binaria, ad talium aequationum resolutionem reducendi ostendit: attamen quum methodus haec pro aequationibus quarum gradus est potestas binaria, nihil doceat, insuperque omnibus obiectionibus praecc. (praeter quartam) aeque obnoxia sit vt demonstratio prima generalis: haud necesse est illam hic fusius explicare.

After this demonstration, E. further showed another way of reducing the theorem for equations whose degree is not a binary power to the resolution of such equations: however, since this method, for equations whose degree is a binary power, teaches nothing, and moreover is equally liable to all the preceding objections (except the fourth) as the first general demonstration, it is by no means necessary to explain it here more fully.

In eadem commentatione ill. E. theorema nostrum adhuc alia via confirmare annixus est p. 263, cuius summa continetur in his: Proposita aequatione xⁿ+Ax^n-1+Bx^n-2 etc. =0, hucusque quidem expressio analytica, quae ipsius radices exprimat, inueniri non potuit, si exponens n>4; attamen certum esse videtur (vti asserit E.), illam nihil aliud continere posse, quam operationes arithmeticas et extractiones radicum eo magis complicatas, quo maior sit n. Si hoc conceditur, E. optime ostendit, quantumuis inter se complicata sint signa radicalia, tamen formulae valorem semper per formam M+N√-1 repraesentabilem fore, ita vt M, N sint quantitates reales.

In the same commentary the ill. E. has endeavored still by another way to confirm our theorem on p. 263, the sum of which is contained in these points: the equation xⁿ+Ax^n-1+Bx^n-2 etc. =0 being proposed, up to this point indeed an analytic expression that would express its roots has not been able to be found, if the exponent n>4; yet it seems certain (as E. asserts) that it can contain nothing else than arithmetical operations and extractions of roots, the more complicated the greater n is. If this is conceded, E. shows very well that, however complicated the radical signs may be among themselves, nevertheless the value of the formula will always be representable by the form M+N√-1, so that M, N are real quantities.

Contra hoc ratiocinium obiici potest, post tot tantorum geometrarum labores perexiguam spem superesse, ad resolutionem generalem aequationum algebraicarum vmquam perueniendi, ita vt magis magisque verisimile fiat, talem resolutionem omnino esse impossibilem et contradictoriam. Hoc eo minus paradoxum videri debet, quum id quod vulgo resolutio aequationis dicitur proprie nihil aliud sit quam ipsius reductio ad aequationes puras. Nam aequationum purarum solutio hinc non docetur sed supponitur, et si radicem aequationis x^m=H per ^m√H exprimis illam neutiquam soluisti, neque plus fecisti, quam si ad denotandam radicem aequationis xⁿ+Ax^n-1+ etc.

Against this reasoning it can be objected that, after so many labors of so many geometers, a very slight hope remains of ever attaining to the general resolution of algebraic equations, so that it becomes more and more likely that such a resolution is altogether impossible and contradictory. This ought to seem the less paradoxical, since that which is commonly called the resolution of an equation is properly nothing else than its reduction to pure equations. For the solution of pure equations is here not taught but presupposed, and if you express the root of the equation x^m=H by ^m√H you have by no means solved it, nor have you done more than if, for denoting the root of the equation xⁿ+Ax^n-1+ etc.

=0 signum aliquod excogitares, radicemque huic aequalem poneres. Verum est, aequationes puras propter facilitatem ipsarum radices per approximationem inueniendi, et propter nexum elegantem, quem omnes radices inter se habent, prae omnibus reliquis multum praestare, adeoque neutiquam vituperandum esse, quod analystae harum radices per signum peculiare denotauerunt: attamen ex eo, quod hoc signum perinde vt signa arithmetica additionis, subtractionis, multiplicationis, diuisionis et euectionis ad dignitatem sub nomine expressionum analyticarum complexi sunt, minime sequitur cuiusuis aequationis radicem per illas exhibere posse. Seu, missis verbis, sine ratione sufficienti supponitur, cuiusuis aequationis solutionem ad solutionem aequationum purarum reduci posse.

=0 you might devise some sign, and set the root equal to this. It is true that pure equations,
on account of the facility of finding their roots by approximation,
and because of the elegant nexus which all the roots have among
themselves, excel greatly above all the rest; and so it is by no means to be blamed
that analysts have denoted the roots of these by a peculiar sign: nevertheless, from the fact that this sign,
just like the arithmetic signs of addition, subtraction, multiplication,
division, and elevation to a power, has been embraced into dignity under the name analytical
expressions, it by no means follows that one can exhibit the root of any equation by them. Or, in short, it is
assumed without sufficient reason that the solution of any equation can be reduced to the solution
of pure equations.

Forsan non ita difficile foret, impossibilitatem iam pro quinto gradu omni rigore demonstrare, de qua re alio loco disquisitiones meas fusius proponam. Hic sufficit, resolubilitatem generalem aequationum in illo sensu acceptam, adhuc valde dubiam esse, adeoque demonstrationem, cuius tota vis ab illa suppositione pendet, in praesenti rei statu nihil ponderis habere.

Perhaps it would not be so difficult to demonstrate, already with full rigor, the impossibility for the fifth degree, on which matter elsewhere I shall more fully propose my disquisitions. Here it suffices that the general resolvability of equations, taken in that sense, is still very doubtful, and thus the demonstration whose whole force depends on that supposition, in the present state of the matter, has no weight.

10.

Postea etiam clar. de Foncenex, quum in demonstratione prima Euleri defectum animaduertisset (supra art. 8. obiect.

Afterwards also the distinguished de Foncenex, when he had observed a defect in Euler’s first demonstration (above, art. 8, objected.

4.), quem tollere non poterat, adhuc aliam viam tentauit et in comment. laudata p. 120. in medium protulit*7). Quae consistit in sequentibus.

4.), which
he could not remove, he tried still another way and, in the lauded commentary,
p. 120, brought it forward *7). Which consists in the following.

Proposita sit aequatio Z=0, designante Z functionem m^ti gradus incognitae z. Si m est numerus impar, iam constat, aequationem hanc habere radicem realem; si vero m est par, clar. F. sequenti modo probare conatur, aequationem ad minimum vnam radicem formae p+q√-1 habere. Sit m=2ⁿi, designante i numerum imparem, supponaturque zz+uz+M esse diuisor functionis Z. Tunc singuli valores ipsius u erunt summae binarum radicum aequationis Z=0 (mutato signo), quamobrem u habebit (m . m-t) / (1 . 2)=m′ valores, et si u per aequationem U=0 determinari supponitur (designante U functionem integram ipsius u et coëfficientium cognitorum in Z), haec erit gradus m′^ti. Facile vero perspicitur m′ fore numerum formae 2^n-1i′, designante i′ numerum imparem.

Let the equation Z=0 be proposed, with Z designating a function of the m^ti degree of the unknown z. If m is an odd number, it is already established that this equation has a real root; but if m is even, the renowned F. tries to prove in the following way that the equation has at least one root of the form p+q√-1. Let m=2ⁿi, with i denoting an odd number, and let it be supposed that zz+uz+M is a divisor of the function Z. Then the individual values of u will be the sums of pairs of roots of the equation Z=0 (with the sign changed), wherefore u will have (m . m-t) / (1 . 2)=m′ values; and if u is supposed to be determined by the equation U=0 (with U designating an integral function of u itself and of the coefficients known in Z), this will be of the m′^ti degree. It is, indeed, easily seen that m′ will be a number of the form 2^n-1i′, with i′ denoting an odd number.

Iam nisi m′ est impar, supponatur iterum, uu+uu′+M′ esse diuisorem ipsius U, patetque per similia ratiocinia u′ determinari per aequationem U′=0, vbi U′ sit functio (m′ . m′-1) / (1 . 2)ti gradus ipsius u′. Posito vero (m′ . m′-1) / (1 . 2)=m′′, erit m′′ numerus formae 2^n-2i′′, designante i′′ numerum imparem. Iam nisi m′′ est impar, statuatur u′ u′+u′′u′+M′′ esse diuisorem functionis U′, determinabiturque u′′ per aequationem U′′=0, quae si supponitur esse gradus m′′′^ti, m′′′ erit numerus formae s^n-3i′′′. Manifestum est, in serie aequationum U=0, U′=0, U′′=0 etc. n^tam fore gradus imparis adeoque radicem realem habere.

Now unless m′ is odd, let it again be supposed that uu+uu′+M′ is a divisor of U itself, and it is clear by similar reasonings that u′ is determined by the equation U′=0, where U′ is a function of the (m′ . m′-1) / (1 . 2)th degree of u′. And if one sets (m′ . m′-1) / (1 . 2)=m′′, then m′′ will be a number of the form 2^n-2i′′, with i′′ denoting an odd number. Now unless m′′ is odd, let it be established that u′ u′+u′′u′+M′′ is a divisor of the function U′, and then u′′ will be determined by the equation U′′=0, which, if it is supposed to be of the m′′′th degree, m′′′ will be a number of the form s^n-3i′′′. It is manifest that, in the series of equations U=0, U′=0, U′′=0, etc., the n^th will be of odd degree and so will have a real root.

Statuemus breuitatis gratia n=3, ita vt aequatio U′′=0 radicem realem u′′ habeat, nullo enim negotio perspicitur pro quouis alio valore ipsius n idem ratiocinium valere. Tunc coëfficientem M′′ per u′′ et coëfficientes in U′ (quos fore functiones integras coëfficientium in Z facile intelligitur), siue per u′′ et coëfficientes in Z rationaliter determinabilem fore asserit clar. de F., et proin realem.

We will set, for brevity’s sake, n=3, so that the equation U′′=0 has a real root u′′; for with no trouble it is evident that for any other value of n the same ratiocination holds. Then the celebrated de F. asserts that the coefficient M′′ is rationally determinable through u′′ and the coefficients in U′ (which are easily understood to be integral functions of the coefficients in Z), or through u′′ and the coefficients in Z, and therefore real.

Hinc sequitur, radices aequationis u′ u′+u′′u′+M′′=0 sub forma p+q√-1 contentas fore; eadem vero manifesto aequationi U′=0 satisfacient: quare dabitur valor aliquis ipsius u′ sub forma p+q√-1 contentus. Iam coëfficiens M′ (eodem modo vt ante) rationaliter per u′ et coëfficientes in Z determinari potest, adeoque etiam sub forma p+q√-1 contentus erit; quare aequationis uu+u′ u+M′ radices sub eadem forma contentae erunt, simul vero aequationi U=0 satisfacient, i. e. aequatio haec habebit radicem sub forma p+q√-1 contentam. Denique hinc simili ratione sequitur, etiam M sub eadem forma contineri, nec non radicem aequationis zz+uz+M=0, quae manifesto etiam aequationi propositae Z=0 satisfaciet.

Hence it follows that the roots of the equation u′ u′+u′′u′+M′′=0 will be contained under the form p+q√-1; and these same will plainly
satisfy the equation U′=0: wherefore some value of u′ will be given, contained under the form p+q√-1. Now the coefficient M′ (in the same
way as above) can be determined rationally through u′ and the coefficients in Z, and thus will likewise be contained under the form p+q√-1;
wherefore the roots of the equation uu+u′ u+M′ will be contained under the same form, and at the same time will satisfy the equation U=0, i. e. the equation
will have a root contained under the form p+q√-1. Finally,
hence by a like reasoning it follows that M also is contained under the same form, and likewise the root of the equation zz+uz+M=0, which
will plainly also satisfy the proposed equation Z=0.

Quamobrem quaeuis aequatio ad minimum vnam radicem formae p+q√-1 habebit.

Wherefore every equation will have at least one root of the form p+q√-1.

11.

Obiectiones 1, 2, 3, quas contra Euleri demonstrationem primam feci (art. 8.), eandem vim contra hanc methodum habent, ea tamen differentia, vt obiectio secunda, cui Euleri demonstratio tantummodo in quibusdam casibus specialibus obnoxia erat, praesentem in omnibus casibus attingere debeat. Scilicet a priori demonstrari potest, etiamsi formula detur, quae coëfficientem M′ rationaliter per u′ et coëfficientes in Z exprimat, hanc pro pluribus valoribus ipsius u′ necessario indeterminatam fieri debere; similiterque formulam, quae coëfficientem M′′ per u′′ exhibeat, indeterminatam fieri pro quibusdam valoribus ipsius u′′ etc.

Objections 1, 2, 3, which I made against Euler’s first demonstration (art. 8.), have the same force against this method,
with this difference, however, that the second objection, to which Euler’s demonstration
was liable only in certain special cases,
ought to apply to the present method in all cases. Namely, it can be demonstrated a priori
that, even if a formula is given which expresses the coefficient M′
rationally in terms of u′ and the coefficients in Z, this must necessarily become indeterminate for several
values of u′ itself; and similarly that the formula which exhibits the coefficient M′′ by means of u′′
becomes indeterminate for certain values of u′′ itself, etc.

Hoc luculentissime perspicietur, si aequationem quarti gradus pro exemplo assumimus. Ponamus itaque m=4, sintque radices aequationis Z=0, hae α, β, γ, δ. Tum patet aequationem U=0 fore sexti gradus ipsiusque radices -(α+β), -(α+γ), -(α+δ), -(β+γ), -(β+δ), -(γ+δ). Aequatio U′=0 autem erit decimi quinti gradus, et valores ipsius u′ hi 2α+β+γ, 2α+β+δ, 2α+γ+δ, 2β+α+γ, 2β+α+δ, 2β+γ+δ, 2γ+α+β, 2γ+α+δ, 2γ+β+δ, 2δ+α+β, 2δ+α+γ, 2δ+β+γ, α+β+γ+δ, α+β+γ+δ, α+β+γ+δ. Iam in hac aequatione, quippe cuius gradus est impar, subsistendum erit, habebitque ea reuera radicem realem α+β+γ+δ (quae primo coëfficienti in Z mutato signo aequalis adeoque non modo realis sed etiam rationalis erit, si coëfficientes in Z sunt rationales). Sed nullo negotio perspici potest, si formula detur, quae valorem ipsius M′ per valorem respondentem ipsius u′ rationaliter exhibeat, hanc necessario pro u′=α+β+γ+δ indeterminatam fieri. Hic enim valor ter erit radix aequationis U′=0, respondebuntque ipsi tres valores ipsius M′, puta (α+β)(γ+δ), (α+γ)(β+δ) et (α+δ)(β+γ), qui omnes irrationales esse possunt.

This will be seen most lucidly, if we assume for an example an equation of the fourth degree. Let us therefore set m=4, and let the roots of the equation Z=0 be these α, β, γ, δ. Then it is clear that the equation U=0 will be of the sixth degree and will have as its roots -(α+β), -(α+γ), -(α+δ), -(β+γ), -(β+δ), -(γ+δ). But the equation U′=0 will be of the fifteenth degree, and the values of u′ are these 2α+β+γ, 2α+β+δ, 2α+γ+δ, 2β+α+γ, 2β+α+δ, 2β+γ+δ, 2γ+α+β, 2γ+α+δ, 2γ+β+δ, 2δ+α+β, 2δ+α+γ, 2δ+β+γ, α+β+γ+δ, α+β+γ+δ, α+β+γ+δ. Now in this equation, since its degree is odd, one must stop, and in fact it will truly have the real root α+β+γ+δ (which, with the first coefficient in Z having its sign changed, is equal thereto, and thus not only real but also rational, if the coefficients in Z are rational). But it can be seen without any trouble that, if a formula is given which exhibits the value of M′ rationally by the corresponding value of u′, this must necessarily become indeterminate for u′=α+β+γ+δ. For this value will be a root of the equation U′=0 ter, and there will correspond to it three values of M′, namely (α+β)(γ+δ), (α+γ)(β+δ) and (α+δ)(β+γ), which can all be irrational.

Manifesto autem formula rationalis neque valorem irrationalem ipsius M′ in hoc casu producere posset, neque tres valores diuersos. Ex hoc specimine satis colligi potest, methodum clar. de Foncenexii neutiquam esse satisfacientem, sed si ab omni parte completa reddi debeat, multo profundius in theoriam eliminationis inquiri oportere.

Manifestly, however, a rational formula could in this case produce neither an irrational value of M′, nor three distinct values. From this specimen it can be sufficiently gathered that the method of the illustrious de Foncenex is by no means satisfactory, but that, if it is to be rendered complete in every respect, one must inquire much more profoundly into the theory of elimination.

12.

Denique ill. LaGrange de theoremate nostro egit in comm. Sur la forme des racines imaginaires des equations, Nouv.

Finally, the illustrious LaGrange treated our theorem in the commentary On the form of the imaginary roots of equations, Nouv.

Mem. de l' Acad. de Berlin 1772.

Memoirs of the Academy of Berlin 1772.

p. 222 sqq. Magnus hic geometra imprimis operam dedit, defectus in Euleri demonstratione prima supplere et reuera praesertim ea, quae supra (art. 8.) obiectionem tertiam et quartam constituunt, tam profunde perscrutatus est, vt nihil amplius desiderandum restet, nisi forsan in disquisitione anteriori supra theoria eliminationis (cui inuestigatio haec tota innititur) quaedam dubia superesse videantur.

p. 222 sqq. This great geometer in the first place devoted his efforts to supplement the defects in Euler’s first demonstration, and indeed he investigated especially those things which above (art. 8) constitute the third and fourth objection so profoundly that nothing further remains to be desired, unless perhaps in the prior disquisition on the theory of elimination (on which this whole investigation rests) certain doubts seem to remain.

— Attamen obiectionem primam omnino non attigit, quin etiam tota disquisitio superstructa est suppositioni, quamuis aequationem m^ti gradus reuera m radices habere.

— Yet the first objection
he did not touch at all; nay even the whole disquisition
is superstructed upon the supposition that an equation of the m^th degree
really has m roots.

Probe itaque iis, quae hucusque exposita sunt, perpensis, demonstrationem nouam theorematis grauissimi ex principiis omnino diuersis petitam peritis haud ingratam fore spero, quam exponere statim aggredior.

Accordingly, with those things that have been set forth thus far duly weighed, I hope that a new demonstration of a most weighty theorem, derived from principles altogether diverse, will be by no means unwelcome to the learned, which I proceed at once to expound.

13.

LEMMA. Denotante m numerum integrum positiuum quemcunque, functio sin φ. x^m-sin mφ. r^m-1x+sin(m-1)φ. r^m diuisibilis erit per xx-2cos φ. rx+rr.

LEMMA. With m denoting any positive integer, the function sin φ. x^m-sin mφ. r^m-1x+sin(m-1)φ. r^m will be divisible by xx-2cos φ. rx+rr.

Demonstr. Pro m=1 functio illa sit =0 adeoque per quemcunque factorem diuisibilis; pro m=2 quotiens sit sin φ, et pro quouis valore maiori quotiens erit sin φ. x^m-2+sin 2φ. rx^m-3 +sin3φ. rrx^m-4+ etc. +sin(m-1)φ. r^m-2. Facile enim confirmatur multiplicata hac functione per xx-2cosφ. rx+rr, productum functioni propositae aequale fieri.

Demonstr. For m=1 let that function be =0 and so divisible by any factor; for m=2 let the quotient be sin φ, and for any greater value the quotient will be sin φ. x^m-2+sin 2φ. rx^m-3 +sin3φ. rrx^m-4+ etc. +sin(m-1)φ. r^m-2. For it is easily confirmed that, this function having been multiplied by xx-2cosφ. rx+rr, the product becomes equal to the proposed function.

14.

LEMMA. Si quantitas r angulusque φ ita sunt determinati, vt habeantur aequationes
r^m cos mφ+Ar^m-1 cos(m-1)φ+Br^m-2 cos(m-2)φ+ etc. +Krrcos2φ+Lrcosφ+M=0

LEMMA. If the quantity r and the angle φ are so determined that there are equations
r^m cos mφ+Ar^m-1 cos (m-1)φ+Br^m-2 cos (m-2)φ+ etc. +Krrcos2φ+Lrcosφ+M=0

[1]
r^m sin mφ+Ar^m-1 sin(m-1)φ+Br^m-2 sin(m-2)φ+ etc. +Krrsin2φ+Lrsinφ=0

[2]
functio x^m+Ax^m-1+Bx^m-2+ etc. +Kxx+Lx+M=X diuisibilis erit per factorem duplicem xx-2cosφ. rx+rr, si modo r. sinφ non =0; si vero r. sinφ=0, eadem functio diuisibilis erit per factorem simplicem x-rcosφ.

[2]
the function x^m+Ax^m-1+Bx^m-2+ etc. +Kxx+Lx+M=X will be divisible by the double factor xx-2cosφ. rx+rr, provided that r. sinφ is not =0; if, however, r. sinφ=0, the same function will be divisible by the simple factor x-rcosφ.

Demonstr. I. Ex art. praec.

Proof. 1. From the preceding article.

omnes sequentes quantitates diuisibiles erunt per xx-2cosφ. rx+rr:

all the following quantities will be divisible by xx-2cosφ. rx+rr:

sinφ. rx^m	-	sin mφ. r^mx	+	sin(m-1)φ. r_m⁺¹
Asinφ. rx^m-1	-	Asin(m-1)φ. r^m-1	+	Asin(m-2)φ. r^m
Bsinφ. rx^m-2	-	Bsin(m-2)φ. r^m-2	+	Bsin(m-2)φ. r^m-1
	etc.			etc.
Ksinφ. rxx	-	Ksin2φ. rxx	+	Ksinφ. r³
Lsinφ. rx	-	Lsinφ. rx		*
Msinφ. r		*	+	Msin(-φ)r.

sinφ. rx^m	-	sin mφ. r^mx	+	sin(m-1)φ. r_m⁺¹
Asinφ. rx^m-1	-	Asin(m-1)φ. r^m-1	+	Asin(m-2)φ. r^m
Bsinφ. rx^m-2	-	Bsin(m-2)φ. r^m-2	+	Bsin(m-2)φ. r^m-1
	etc.			etc.
Ksinφ. rxx	-	Ksin2φ. rxx	+	Ksinφ. r³
Lsinφ. rx	-	Lsinφ. rx		*
Msinφ. r		*	+	Msin(-φ)r.

Quamobrem etiam summa harum quantitatum per xx-2cosφ. rx+rr diuisibilis erit. At singularum partes primae constituunt summam sinφ. rX; secundae additae dant 0, propter

Wherefore also the sum of these quantities will be divisible by xx-2cosφ. rx+rr. But the first parts of each constitute the sum sinφ. rX; the second, when added, give 0, on account of

[2]; tertiarum vero aggregatum quoque euanescere, facile perspicitur, si

[2]; moreover, that the aggregate of the third parts likewise disappears is easily perceived, if

[1] multiplicatur persinφ,

[1] is multiplied bysinφ,

[2] percosφ, productumque illud ab hoc subducitur. Vnde sequitur, functionem sinφ. rX diuisibilem esse per xx-2cosφ. rx+rr, adeoque, nisi fuerit rsinφ=0, etiam functionem X. Q. E. P.

[2] bycosφ, and that product is subtracted from this. Whence it follows that the function sinφ. rX is divisible by xx-2cosφ. rx+rr, and hence, unless rsinφ=0, the function X as well. Which was to be proved.

II. Si verorsinφ=0, erit aut r=0 aut sinφ=0. In casu priori erit M=0, propter

2. But ifrsinφ=0, there will be either r=0 or sinφ=0. In the former case there will be M=0, because

[1], adeoqueX per x siue per x-rcosφ diuisibilis; in posteriori erit cosφ=±1, cos2φ= +1, cos3φ=±1 et generaliter cos nφ=cosφⁿ. Quare propter

[1], and thusX is divisible by x or by x-rcosφ; in the latter case there will be cosφ=±1, cos2φ= +1, cos3φ=±1 and in general cos nφ=cosφⁿ. Wherefore, on account of

[1] fietX=0, statuendo x=cosφ, et proin functio X per x-cosφ erit diuisibilis. Q. E. S.

[1] it will beX=0, by setting x=cosφ, and hence the function X by
x-cosφ will be divisible. Q. E. S.

15.

Theorema praecedens plerumque adiumento quantitatum imaginariarum demonstratur, vid. Euler Introd. in Anal.

The preceding theorem is for the most part demonstrated with the aid of imaginary quantities; see Euler, Introd. in Anal.

Inf. T. I. p. 110; operae pretium esse duxi, ostendere, quomodo aeque facile absque illarum auxilio erui possit. Manifestum iam est, ad demonstrationem theorematis nostri nihil aliud requiriri quam vt ostendatur: Proposita functione quacunque X formae x^m+Ax^m-1+ Bx^m-2+ etc.+Lx+M, r et φ ita determinari posse, vt aequationes

Inf. T. I.
p. 110; I judged it worth the effort to show how just as easily it can be derived without the aid of those. It is now manifest that for the demonstration of our theorem nothing else is required than that it be shown: Given any function X of the form x^m+Ax^m-1+Bx^m-2+ etc.+Lx+M, r and φ can be so determined, that the equations

[1] et

[1] and

[2]locum habeant. Hinc enim sequetur, X habere factorem realem primi vel secundi gradus; diuisio autem necessario producet quotientem realem inferioris gradus, qui ex eadem ratione quoque factorem primi vel secundi gradus habebit. Per continuationem huius operationis X tandem in factores reales simplices vel duplices resoluetur. Illud itaque theorema demonstrare, propositum est sequentium disquisitionum.

[2]hold good. Hence indeed it will follow that X has a real factor of the first or second degree; but division will necessarily produce a real quotient of lower degree, which for the same reason will also have a factor of the first or second degree. By the continuation of this operation X will at length be resolved into real factors, simple or double. To demonstrate that theorem, therefore, is the aim of the following disquisitions.

16.

Concipiatur planum fixum infinitum (planum tabulae, fig. 1.), et in hoc recta fixa infinita GC per punctum fixum C transiens. Assumta aliqua longitudine pro vnitate vt omnes rectae per numeros exprimi possint, erigatur in quouis puncto plani P, cuius distantia a centro C est r angulusque GCP=φ, perpendiculum aequale valori expressionis r^msin mφ+Ar^m-1sin(m-1)φ+ etc.+Lrsinφ, quem breuitatis gratia in sequentibus semper per T designabo.

Let a fixed infinite plane be conceived (the plane of the board, fig. 1.), and in it a fixed infinite straight line GC passing through the fixed point C. Some length having been assumed for the unit so that all straight lines can be expressed by numbers, let there be erected at any point of the plane P, whose distance
from the center C is r and whose angle GCP=φ, a perpendicular
equal to the value of the expression r^msin mφ+Ar^m-1sin(m-1)φ+
etc.+Lrsinφ, which for the sake of brevity I shall always designate in what follows by T.

Distantiam r semper tamquam positiuam considero, et pro punctis, quae axi ab altera parte iacent, angulus φ aut tamquam duobus rectis maior, aut tamquam negatiuus (quod hic eodem redit) spectari debet. Extremitates horum perpendiculorum (quae pro valore positiuo ipsius T supra planum accipiendae sunt, pro negatiuo infra, pro euanescente in plano ipsio) erunt ad superficiem curuam continuam quaquauersum infinitam, quam breuitatis gratia in sequentibus superficiem primam vocabo. Prorsus simili modo ad idem planum et centrum eundemque axem referatur alia superficies, cuius altitudo supra quoduis plani punctum sit r^mcos mφ+Ar^m-1cos(m-1)φ+ etc.

I always consider the distance r as positive, and for the points which lie on the other side of the axis, the angle φ must be regarded either as greater than two right angles, or as negative (which here comes to the same thing). The extremities of these perpendiculars (which, for a positive value of T, are to be taken above the plane, for a negative below, for a vanishing one in the plane itself) will be on a continuous curved surface, infinite in all directions, which for brevity in what follows I shall call the first surface. In entirely similar manner, with reference to the same plane and center and the same axis, let another surface be referred, whose height above any point of the plane is r^mcos mφ+Ar^m-1cos(m-1)φ+ etc.

+Lrcosφ+M, quam expressionem breuitatis gratia semper per U denotabo. Superficiem vero hanc, quae etiam continua et quaquauersum infinita erit, per denominationem superficiei secundae a priori distinguam. Tunc manifestum est, totum negotium in eo versari, vt demonstretur, ad minimum vnum punctum dari, quod simul in plano, in superficie prima et in superficie secunda iaceat.

+Lrcosφ+M, which expression for brevity I will always denote by U. This surface, moreover, which also will be continuous and in every direction infinite, I shall distinguish from the former by the denomination of the second surface. Then it is manifest that the whole business turns on this: that it be demonstrated that at least one point is given which at the same time lies in the plane, in the first surface, and in the second surface.

17.

Facile perspici potest, superficiem primam partim supra planum partim infra planum iacere; patet enim distantiam a centro r tam magnam accipi posse, vt reliqui termini in T prae primo r^msin mφ euanescant; hic vero, angulo φ rite determinato, tam positiuus quam negatiuus fieri potest. Quare planum fixum necessario a superficie prima secabitur; hanc plani cum superficie prima intersectionem vocabo lineam primam; quae itaque determinabitur per aequationem T=0. Ex eadem ratione planum a superficie secunda secabitur; intersectio constituet curuam per aequationem U=0 determinatam, quam lineam secundam appellabo. Proprie vtraque curua ex pluribus ramis constabit, qui omnino seiuncti esse possunt, singuli vero erunt lineae continuae.

It can easily be perceived that the first surface lies partly above the plane partly below the plane; for it is evident that the distance from the center r can be taken so great that the remaining terms in T, in comparison with the first r^msin mφ, vanish; but this, the angle φ being properly determined, can be made both positive and negative. Wherefore the fixed plane will necessarily be cut by the first surface; this intersection of the plane with the first surface I shall call the first line; which therefore will be determined by the equation T=0. By the same reasoning the plane will be cut by the second surface; the intersection will constitute a curve determined by the equation U=0, which I shall call the second line. Properly, each curve will consist of several branches, which can be altogether disjoined, but each will be a continuous line.

Quin adeo linea prima semper erit talis, quam complexam vocant, axisque GC tamquam pars huius curuae spectanda; quicunque enim valor ipsi r tribuatur, U semper fiet =0, quando φ aut =0 aut =180°. Sed praestat complexum cunctorum ramorum per omnia puncta, vbi T=0, transeuntium tamquam vnam curuam considerare (secundum vsum in geometria sublimiori generaliter receptum), similiterque cunctos ramos per omnia puncta transeuntes, vbi U=0. Patet iam, rem eo reductam esse, vt demonstretur, ad minimum vnum punctum in plano dari, vbi ramus aliquis lineae primae a ramo lineae secundae secetur. Ad hunc finem indolem harum linearum propius contemplari oportebit.

Nay even, the first line will always be of such a sort as they call a complex, and the axis GC is to be regarded as a part of this curve; for whatever value is assigned to r, U will always become =0 when φ is either =0 or =180°. But it is better to consider as one curve the complex of all the branches passing through all the points where T=0 (according to the usage generally received in higher geometry), and similarly all the branches passing through all the points where U=0. It is now clear that the matter has been reduced to this: that it be demonstrated that there exists in the plane at least one point where some branch of the first line is cut by a branch of the second line. To this end, it will be necessary to contemplate more closely the character of these lines.

18.

Ante omnia obseruo, vtramque curuam esse algebraicam, et quidem, si ad coordinates orthogonales reuocetur, ordinis m^ti. Sumto enim initio abscissarum in C, abscissisque x versus G, applicatis y versus P, erit x=rcosφ, y=rsinφ, adeoque generaliter, quidquid sit n, rⁿsin nφ=nx^n-1y-(n. n-1. n-2) / (1. 2. 3) x^n-3y³ +(n. .. n-4) / (1. .. .5) x^n-5y⁵- etc., rⁿcos nφ=xⁿ-(n. n-1) / (1. 2) x^n-2yy +(n. n-1. n-2. n-3) / (1. 2. 3. 4) x^n-4y⁴- etc. Quamobrem tum T tum U constabunt ex pluribus huiusmodi terminis ax^α y^β, denotantibus α, β numeros integros positiuos, quorum summa, vbi maxima est, sit =m. Ceterum facile praeuideri potest, cunctos terminos ipsius T factorem y inuoluere, adeoque lineam primam proprie ex recta (cuius aequatio y=0) et curua ordinis m-1^ti compositam esse; sed necesse non est ad hanc distinctionem hic respicere.

Before all things I observe that each curve is algebraic, and indeed, if it be referred to orthogonal coordinates, of order m^th. For, with the origin of abscissas taken at C, the abscissas x directed toward G, the ordinates y toward P, it will be x=rcosφ, y=rsinφ, and thus in general, whatever n may be, rⁿsin nφ=nx^n-1y-(n. n-1. n-2) / (1. 2. 3) x^n-3y³ +(n. .. n-4) / (1. .. .5) x^n-5y⁵- etc., rⁿcos nφ=xⁿ-(n. n-1) / (1. 2) x^n-2yy +(n. n-1. n-2. n-3) / (1. 2. 3. 4) x^n-4y⁴- etc. Wherefore both T and U will consist of several terms of this kind ax^α y^β, α, β denoting positive integers, whose sum, where it is maximal, is =m. Moreover, it can easily be foreseen that all the terms of T involve the factor y, and so the first line is properly composed of a straight line (whose equation is y=0) and of a curve of order m-1^th; but it is not necessary here to have regard to this distinction.

Maioris momenti erit inuestigatio, an linea prima et secunda crura infinita habeant, et quot qualiaque. In distantia infinita a puncto C linea prima, cuius aequatio sin mφ+1/r sin(m-1)φ +B/rr sin(m-2)φ etc. =0, confundetur cum linea, cuius aequatio sin mφ=0. Haec vero exhibet m lineas rectas in puncto A se secantes, quarum prima est axis GCG′, reliquae contra hanc sub angulis 1/m 180, 2/m 180, 3/m 180 etc.

Of greater moment will be the investigation, whether the first and second line have infinite branches, and how many and of what sort. At infinite distance from the point C the first line, whose equation is sin mφ+1/r sin(m-1)φ +B/rr sin(m-2)φ etc. =0, will be confounded with the line whose equation is sin mφ=0. This indeed exhibits m straight lines intersecting one another at the point A, of which the first is the axis GCG′, the remaining against this under angles 1/m 180, 2/m 180, 3/m 180 etc.

graduum inclinatae. Quare linea prima 2m ramos infinitos habet, qui peripheram circuli radio infinito descripti in 2m partes aequales dispertiuntur, ita vt peripheria a ramo primo secetur in concursu circuli et axis, a secundo in distantia 1/m 180°, a tertio in distantia 2/m 180° etc. Eodem modo linea secunda in distantia infinita a centro habebit asymptotam per aequationem cos mφ=0 expressam, quae est complexus m rectarum in puncto C sub aequalibus angulis itidem se secantium, ita tamen, vt prima cum axi CG constituat angulum 1/m 90°, secunda angulum 3/m 90°, tertia angulum 5/m 90° etc.

inclined by degrees. Wherefore the first line has 2m infinite branches, which are distributed over the periphery of a circle described with infinite radius into 2m equal parts, so that the periphery is cut by the first branch at the concurrence of the circle and the axis, by the second at a distance 1/m 180°, by the third at a distance 2/m 180°, etc. In the same way the second line, at infinite distance from the center, will have an asymptote expressed by the equation cos mφ=0, which is a complex of m straight lines intersecting one another at equal angles at point C, yet in such a way that the first with the axis CG forms an angle 1/m 90°, the second an angle 3/m 90°, the third an angle 5/m 90°, etc.

Quare linea secunda etiam 2m ramos infinitos habebit, quorum singuli medium locum inter binos ramos proximos linea primae occupabunt, ita vt peripheriam circuli radio infinite magno descripti in punctis, quae 1/m 90°, 3/m 90°, 5/m 90° etc. ab axe distant, secent. Ceterum palam est, axem ipsum semper duos ramos infinitos lineae primae constituere, puta primum et m+1^tum. Luculentissime hic ramorum situs exhibetur in fig.

Therefore the second line will likewise have 2m infinite branches, each of which will occupy the middle place between the two nearest branches of the first line, so that they cut the periphery of the circle described with an infinitely great radius at the points which are distant from the axis by 1/m 90°, 3/m 90°, 5/m 90°, etc. Moreover, it is plain that the axis itself always constitutes two infinite branches of the first line, to wit the first and the m+1^th. Most lucidly the disposition of the branches is exhibited here in the fig.

2, pro casu m=4 constructa, vbi rami lineae secundae, vt a ramis lineae primae distinguantur, punctati exprimuntur, quod etiam de figura quarta est tenendum. *8) — Quum vero hae conclusiones maximi momenti sint, quantitatesque infinite magnae quosdam lectores offendere possint: illas etiam absque infinitorum subsidio in art. sequ.

2, constructed for the case m=4, where the branches of the second line, so that they may be distinguished from the branches of the first line, are expressed as dotted, which likewise is to be held for the fourth figure. *8) — Since indeed these conclusions are of the greatest moment, and infinitely great quantities may offend some readers: these also, without the assistance of infinities, will be set forth in the following article.

eruere docebo.

I will teach how to extract them.

19.

THEOREMA. Manentibus cunctis vt supra, ex centro C describi poterit circulos, in cuius peripheria sint 2m puncta, in quibus T=0, totidemque, in quibus U=0, et quidem ita, vt singula posteriora inter bina priorum iaceant.

THEOREM. With all things remaining as above, from the center C it will be possible to describe circles, on whose periphery there are 2m points at which T=0, and just as many at which U=0, and indeed in such a way that each of the latter lies between a pair of the former.

Sit summa omnium coëfficientium A,B etc. K,L,M, positiue acceptorum =S, accipiaturque R simul >S√2 et >1 *9): tum dico in circulo radio R descripto ea, quae in theoremate enunciata sunt, necessario locum habere. Scilicet designato breuitatis gratia eo puncto huius circumferentiae, quod 1/m 45 gradibus ab ipsius concursu cum laeua parte axis distat, siue pro quo φ= 1/m 45°, per (1); similiter eo puncto, quod φ=3/m 45°, ab hoc concursu distat, siue pro quo φ=3/m 45°, per (3); porro eo, vbi φ=5/m 45°, per (5) etc.

Let the sum of all the coefficients A, B etc. K, L, M, taken as positive, be = S, and let R at the same time be taken > S√2 and > 1 *9): then I say that in the circle described with radius R those things that were stated in the theorem necessarily hold. Namely, for brevity’s sake, having designated that point of this circumference which is distant by 1/m 45 degrees from its intersection with the left-hand part of the axis, that is, for which φ= 1/m 45°, by (1); similarly that point which, being φ=3/m 45°, is distant from this intersection, that is, for which φ=3/m 45°, by (3); further, that where φ=5/m 45°, by (5) etc.

vsque ad (8m-1), quod (8m-1)/m 45 gradibus ab illo concursu distat, si semper versus eandem partem progrederis, (aut 1/m 45° a parte opposita), ita vt omnino 4m puncta in peripheria habeantur, aequalibus interuallis dissita: iacebit inter (8m-1) et (1) vnum punctum, pro quo T=0; nec non sita erunt similia puncta singula inter (3) et (5); inter (7) et (9); inter (11) et (13) etc. quorum itaque multitudo 2m; eodemque modo singula puncta, pro quibus U=0, iacebunt inter (1) et (3); inter (5) et (7); inter (9) et (11), quorum multitudo igitur etiam =2m; denique praeter haec 4m puncta alia, in tota peripheria non dabuntur, pro quibus vel T vel U sit =0.

up to (8m-1), which is distant by (8m-1)/m 45 degrees from that concurrence, if you always advance toward the same side (or by 1/m 45° from the opposite side), so that in all 4m points are had on the periphery, separated by equal intervals: there will lie between (8m-1) and (1) one point, for which T=0; and likewise similar single points will be situated between (3) and (5); between (7) and (9); between (11) and (13), etc., whose multitude therefore is 2m; and in the same way single points, for which U=0, will lie between (1) and (3); between (5) and (7); between (9) and (11), whose multitude therefore likewise =2m; finally, besides these 4m points, there will not be given any others on the whole periphery, for which either T or U is =0.

Demonstr. I. In puncto (1) eritmφ=45° adeoque T= R^m-1(R√1/2 +Asin(m-1)φ+B/R sin(m-2)φ+ etc. + L/(R^m-2) sinφ); summa vero Asin(m-1)φ+B/R sin(m-2)φ etc. certo non poterit esse maior quam S, adeoque necessario erit minor quam R√1/2: vnde sequitur in hoc puncto valorem ipsius T certo esse positiuum.

Proof. 1. At the point (1) there will bemφ=45°, and so T= R^m-1(R√1/2 +Asin(m-1)φ+B/R sin(m-2)φ+ etc. + L/(R^m-2) sinφ); but indeed the sum Asin(m-1)φ+B/R sin(m-2)φ etc. certainly cannot be greater than S, and so necessarily will be less than R√1/2: whence it follows that at this point the value of T is certainly positive.

A potiori itaque T valorem positiuum habebit, quando mφ inter 45° et 135° iacet, i. e. a puncto (1) vsque ad (3) valor ipsius T semper positiuus erit. Ex eadem ratione T a puncto (9) vsque ad (11) positiuum valorem vbique habebit, et generaliter a quouis puncto (8k+1) vsque ad (8k+3), denotante k integrum quemcunque. Simili modo T vbique inter (5) et (7), inter (13) et (15) etc.

By the stronger reason, therefore, T will have a positive value when mφ lies between 45° and 135°, i. e. from point (1) up to (3) the value of T itself will always be positive. By the same reasoning, T will have a positive value everywhere from point (9) up to (11), and in general from any point (8k+1) up to (8k+3), with k denoting any integer whatsoever. In a similar manner, T is everywhere between (5) and (7), between (13) and (15), etc.

et generaliter inter (8k+5) et (8k+7a) valorem negatiuum habebit, adeoque in omnibus his interuallis nullibi poterit esse =0. Sed quoniam in (3) hic valor est positiuus, in (5) negatiuus: necessario alicubi inter (3) et (5) erit =0; nec non alicubi inter (7) et (9); inter (11) et (13) etc. vsque ad interuallum inter (8m-1) et (1) incl., ita vt omnino in 2m punctis habeatur T=0. Q. E. P.

and in general between (8k+5) and (8k+7a) it will have a negative value, and so in all these intervals it can nowhere be =0. But since at (3) this value is positive, at (5) negative: necessarily somewhere between (3) and (5) it will be =0; and likewise somewhere between (7) and (9); between (11) and (13), etc., up to the interval between (8m-1) and (1), incl., so that altogether at 2m points one has T=0. Q. E. D.

II. Quod vero praeter haec2m puncta, alia, hac proprietate praedita, non dantur, ita cognoscitur. Quum inter (1) et (3); inter (5) et (7) etc. nulla sint, aliter fieri non posset, vt plura talia puncta exstent, quam si in aliquo interuallo inter (3) et (5), vel inter (7) et (9) etc.

2. But that, indeed, besides these2m points, others endowed with this property
are not given, is recognized thus. Since between (1) and
(3); between (5) and (7) etc. there are none, it could not otherwise happen that
more such points should exist than if in some interval between (3)
and (5), or between (7) and (9) etc.

ad minimum duo iacerent. Tum vero nesessario in eodem interuallo T alicubi esset maximum, vel minimum, adeoque (dT)/(dφ)=0. Sed (dT)/(dφ)=mR^m-2(Rcos mφ+(m-1)/m Acos(m-1)φ+ etc.) et cos mφ inter (3) et (5) semper est negatiuus et >√1/2. Vnde facile perspicitur in toto hoc interuallo (dT)/(dφ) esse quantitatem negatiuam; eodemque modo inter (7) et (9) vbique positiuam; inter (11) et (13) negatiuam etc. ita vt in nullo horum interuallorum esse possit 0, adeoque suppositio consistere nequeat.

at a minimum two would lie. Then indeed necessarily in the same interval T would somewhere be a maximum, or a minimum, and so (dT)/(dφ)=0. But (dT)/(dφ)=mR^m-2(Rcos mφ+(m-1)/m Acos(m-1)φ+ etc.) and cos mφ between (3) and (5) is always negative and >√1/2. Whence it is easily perceived that in this whole interval (dT)/(dφ) is a negative quantity; and in the same way between (7) and (9) everywhere positive; between (11) and (13) negative, etc., so that in none of these intervals can it be 0, and therefore the supposition cannot stand.

Quare etc. Q. E. S.

Wherefore etc. Which was to be shown.

III. Prorsus simili modo demonstratur,U habere valorem negatiuum vbique inter (3) et (5), inter (11) et (13) etc. et generaliter inter (8k+3) et (8k+5); positiuum vero inter (7) et (9), in- ter (15) et (17) etc.

3. In entirely similar fashion it is demonstrated thatU has a negative value everywhere between (3) and (5), between (11) and (13), etc., and generally between (8k+3) and (8k+5); but a positive one between (7) and (9), in- ter (15) and (17), etc.

et generaliter inter (8k+7) et (8k+9). Hinc statim sequitur, U=0 fieri debere alicubi inter (1) et (3), inter (5) et (7) etc., i. e. in 2m punctis. In nullo vero horum interuallorum fieri poterit (dT)/(dφ)=0 (quod facile simili modo vt supra probatur): quamobrem plura quam illa 2m puncta in circuli peripheria non dabuntur, in quibus fiat U=0. Q. E. T. et Q.

and generally between (8k+7) and (8k+9). Hence it follows at once that U=0 must occur somewhere between (1) and (3), between (5) and (7), etc., i. e. in 2m points. But in none of these intervals will it be possible that (dT)/(dφ)=0 (which is easily proved in a similar way as above): wherefore more than those 2m points on the periphery of the circle will not be given, at which U=0 occurs. Q. E. T. et Q.

Ceterum ea theorematis pars, secundum quam plura quam 2m puncta non dantur, in quibus T=0, neque plura quam 2m, in quibus U=0, etiam inde demonstrari potest, quod per aequationes T=0, U=0 exhibentur curuae m^ti ordinis, quales a circulo tamquam curuae secundi ordinis in pluribus quam 2m punctis secari non posse, ex geometria sublimiori constat.

Moreover, that part of the theorem, according to which more than 2m points are not given in which T=0, nor more than 2m in which U=0, can also be demonstrated from this: that by the equations T=0, U=0 there are exhibited curves of the m^th order, which by a circle, as a curve of the second order, cannot be cut in more than 2m points, as is established by higher geometry.

20.

Si circulus alius radio maiori quam R ex eodem centro describitur, eodemque modo diuiditur: etiam in hoc inter puncta (3) et (5) iacebit punctum vnum, in quo T=0, itemque inter (7) et (9) etc., perspicieturque facile, quo minus radius huius circuli a radio R differat, eo propius huiusmodi puncta inter (3) et (5) in vtriusque circumferentia sita esse debere. Idem etiam locum habebit, si circulus radio aliquantum minori quam R, attamen maiori quam S√2 et 1 describitur. Ex his nullo negotio intelligitur, circuli radio R descripti circumferentiam in eo puncto inter (3) et (5), vbi T=0, reuera secari ab aliquo ramo lineae primae; idemque valet de reliquis punctis, vbi T=0. Eodem modo patet, circumferentiam circuli huius in omnibus 2m punctis, vbi U=0, ab aliquo ramo lineae secundae secari.

If another circle with a radius greater than R is described from the same center, and is divided in the same manner: likewise in this one between the points (3) and (5) there will lie one point at which T=0, and likewise between (7) and (9), etc.; and it will be easily perceived that, the less the radius of this circle differs from the radius R, by so much the nearer such points between (3) and (5) must lie on the circumference of each. The same will also hold if a circle with a radius somewhat less than R, yet greater than S√2 and 1, is described. From these facts it is understood without any trouble that the circumference of the circle described with radius R, at that point between (3) and (5) where T=0, is indeed cut by some branch of the first line; and the same holds for the remaining points where T=0. In the same way it is clear that the circumference of this circle, at all 2m points where U=0, is cut by some branch of the second line.

Hae conclusiones etiam sequenti modo exprimi possunt: Descripto circulo debitae magnitudinis e centro C, in hunc intrabunt 2m rami lineae primae totidemque rami lineae secundae, et quidem ita, vt bini rami proximi lineae primae per aliquem ramum lineae secundae ab inuicem separentur. Vid. fig.

These conclusions also can be expressed in the following way: With a circle of appropriate magnitude described from center C, into it there will enter 2m branches of the first line and just as many branches of the second line, and indeed so that pairs of neighboring branches of the first line are separated from one another by some branch of the second line. See fig.

2, vbi circulus iam non infinitae sed finitae magnitudinis erit, numerique singulis ramis adscripti cum numeris, per quos in art. praec. et hoc limites certos in peripheria breuitatis caussa designaui, non sunt confundendi.

2, where the circle will now be not of infinite but of finite magnitude, and the numbers assigned to the individual branches are not to be confused with the numbers by which, in the preceding article and in this one, for the sake of brevity, I have designated fixed limits on the periphery.

21.

Iam ex hoc situ relatiuo ramorum in circulum intrantium tot modis diuersis deduci potest, intersectionem alicuius rami lineae primae cum ramo lineae secundae intra circulum necessario dari, vt quaenam potissimum methodus prae reliquis eligenda sit, propemodum nesciam. Luculentissima videtur esse haec: Designemus (fig. 2.) punctum peripheriae circuli, vbi a laeua axis parte (quae ipsa est vnus ex 2m ramis lineae primae) secatur, per 0; punctum proximum, vbi ramus lineae secundae intrat, per 1; punctum huic proximum, vbi secundus lineae primae ramus intrat, per 2, et sic porro vsque ad 4m-1, ita vt in quouis puncto numero pari signato ramus lineae secundae in circulum intret, contra ramus lineae secundae in omnibus punctis per numerum imparem expressis.

Now from this relative situation of the branches entering into the circle in so many diverse ways it can be deduced that the intersection of some branch of the first line with a branch of the second line is necessarily given within the circle, so that which method ought preferably to be chosen before the rest I almost do not know. This seems most lucid: Let us designate (fig. 2.) the point of the periphery of the circle, where on the left part of the axis (which itself is one of the 2m branches of the first line) it is cut, by 0; the nearest point, where a branch of the second line enters, by 1; the point next to this, where the second branch of the first line enters, by 2, and so on up to 4m-1, in such a way that at every point marked by an even number a branch of the second line enters into the circle, conversely a branch of the second line in all the points expressed by an odd number.

Iam ex geometria sublimori constat, quamuis curuam algebraicam, (siue singulas cuiusuis curuae algebraicae partes, si forte e pluribus composita sit) aut in se redientem aut vtrimque in infinitum excurrentem esse, adeoque si ramus aliquis curuae algebraicae in spatium definitum intret, eundem necessario ex hoc spatio rursus alicubi exire debere. *10) Hinc concluditur facile, quoduis punctum numero pari signatum (seu, breuitatis caussa, quoduis punctum par) per ramum lineae primae cum alio puncto pari intra circulum iunctum esse debere, similiterque quoduis punctum numero impari notatum cum alio simili puncto per ramum lineae secundae. Quamquam vero haec binorum punctorum connexio secundum indolem functionis X perquam diuersa esse potest, ita vt in genere determinari nequeat, tamen facile demonstrari potest, quaecunque demum illa sit, semper intersectionem lineae primae cum linea secunda oriri.

Now from higher geometry it is established that any algebraic curve (or the individual parts of any algebraic curve, if perchance it is composed of several) is either returning into itself or running out on both sides into infinity; and so, if some branch of an algebraic curve enters a definite region, the same must necessarily somewhere again exit from this region. *10) Hence it is easily concluded that any point marked with an even number (or, for the sake of brevity, any even point) must be joined within the circle by a branch of the first line with another even point; and similarly any point marked with an odd number with another like point by a branch of the second line. Although indeed this connection of pairs of points, according to the character of the function X, can be very diverse, so that in general it cannot be determined, nevertheless it can easily be shown that, whatever it may be, the intersection of the first line with the second line always arises.

22.

Demonstratio huius necessitatis commodissime apagogice repraesentari posse videtur. Scilicet supponamus, iunctionem binorum quorumque punctorum parium, et binorum quorumque punctorum imparium ita adornari posse, vt nulla intersectio rami lineae primae cum ramo lineae secundae inde oriatur. Quoniam axis est pars lineae primae, manifesto punctum 0 cum puncto 2m iunctum erit.

The demonstration of this necessity seems most conveniently able to be represented apagogically. Namely, let us suppose that the joining of each and every pair of even points, and of each and every pair of odd points, can be arranged in such a way that no intersection of a branch of the first line with a branch of the second line arises therefrom. Since the axis is a part of the first line, manifestly the point 0 will be joined with the point 2m.

Punctum 1 itaque cum nullo puncto vltra axem sito, i. e. cum nullo puncto per numerum maiorem quam 2m expresso iunctum esse potest, alioquin enim linea iungens necessario axem secaret. Si itaque 1 cum puncto n iunctum esse supponitur, erit n<2m. Ex simili ratione si 2 cum n′ iunctum esse statuitur, erit n′<n, quia alioquin ramus 2 ... n′ ramum 1 ... n necessario secaret. Ex eadem caussa punctum 3 cum aliquo punctorum inter 4 et n′ iacentium iunctum erit, patetque si 3,4,5 etc.

The point 1 therefore cannot be joined with any point situated beyond the axis, i. e. with any point expressed by a number greater than 2m, otherwise the joining line would necessarily cut the axis. If therefore it is supposed that 1 is joined with the point n, there will be n<2m. By a similar reasoning, if it is posited that 2 is joined with n′, there will be n′<n, because otherwise the branch 2 ... n′ would necessarily cut the branch 1 ... n. From the same cause the point 3 will be joined with some one of the points lying between 4 and n′, and it is clear likewise for 3,4,5 etc.

iuncta esse supponantur cum n′′, n′′′, n′′′′ etc., n′′′ iacere inter 5 et n′′, n′′′′ inter 6 et n′′′ etc. Vnde perspicuum est, tandem ad aliquod punctum h peruentum iri, quod cum puncto h+2 iunctum sit, et tum ramus, qui in puncto h+1 in circulum intrat, necessario ramum puncta h et h+2 iungentem secabit. Quia autem alter horum duorum ramorum ad lineam primam, alter ad secundam pertinebit, manifestum iam est, suppositionem esse contradictoriam, adeoque necessario alicubi intersectionem lineae primae cum linea secunda fieri.

let [them] be supposed to be joined with n′′, n′′′, n′′′′ etc., n′′′ to lie between 5 and n′′, n′′′′ between 6 and n′′′, etc. Whence it is perspicuous that at length one will be brought to some point h, which is joined with the point h+2, and then the branch which at the point h+1 enters into the circle will necessarily cut the branch joining the points h and h+2. But since one of these two branches will pertain to the first line, the other to the second, it is now manifest that the supposition is contradictory, and so of necessity somewhere an intersection of the first line with the second takes place.

Si haec cum praecedentibus iunguntur, ex omnibus disquisitionibus explicatis colligetur, theorema, quamuis functionem algebraicam rationalem integram vnius indeterminatae in factores reales primi vel secundi gradus resolui posse, omni rigore esse demonstratum.

If these are joined with the preceding, from all the disquisitions
set forth it will be gathered, the theorem, that any entire rational algebraic
function of one indeterminate can be resolved into real factors
of the first or second degree, has been demonstrated with full rigor.

23.

Ceterum haud difficile ex iisdem principiis deduci potest, non solum vnam sed ad minimum m intersectiones lineae primae cum secunda dari, quamquam etiam fieri potest, vt linea prima a pluribus ramis lineae secundae in eodem puncto secetur, in quo casu functio X plures factores aequales habebit. Attamen quum hic sufficiat, vnius intersectionis necessitatem demonstrauisse, fusius huic rei breuitatis caussa non immoror. Ex eadem ratione etiam alias harum linearum proprietates hic vberius non persequor, e. g. intersectionem semper fieri sub angulis rectis; aut si plura crura vtriusque curuae in eodem puncto conueniant, totidem crura lineae primae affore, quot crura lineae secundae, haecque alternatim posita esse, et sub aequalibus angulis se secare etc.

Moreover, it can be deduced not at all with difficulty from the same principles that not only one but at least m intersections of the first line with the second are given, although it can also happen that the first line is cut by several branches of the second line at the same point, in which case the function X will have several equal factors. However, since here it suffices to have demonstrated the necessity of a single intersection, I do not dwell more fully on this matter for the sake of brevity. From the same reasoning I likewise do not pursue here more copiously other properties of these lines, e. g. that the intersection always occurs under right angles; or if more branches of each curve meet at the same point, that there will be as many branches of the first line present as branches of the second line, and that these are placed alternately and cut each other under equal angles, etc.

Denique obseruo, minime impossibile esse, vt demonstratio praecedens, quam hic principiis geometricis superstruxi, etiam in forma mere analytica exhibeatur: sed eam repraesentationem, quam hic explicaui, minus abstractam euadere credidi, verumque neruum probandi hic multo clarius ob oculos poni, quam a demonstratione analytica exspectari possit.

Finally I observe that it is by no means impossible that the preceding demonstration, which I have here superstructured upon geometric principles, also be exhibited in a purely analytic form: but I have judged that that representation, which I have here explained, turns out less abstract, and that the true nerve of proving is here set much more clearly before the eyes, than can be expected from an analytic demonstration.

Coronidis loco adhuc aliam methodum theorema nostrum demonstrandi addigitabo, quae primo aspectu non modo a demonstratione praecedente, sed etiam ab omnibus demonstrationibus reliquis supra enarratis maxime diuersa esse videbitur, et quae nihilominus cum d'Alembertiana, si ad essentiam spectas, proprie eadem est. Cum qua illam comparare, parallelismumque inter vtramque explorare peritis committo, in quorum gratiam vnice subiuncta est.

In place of a coronis I will still append another method of demonstrating our theorem, which at first sight will seem to be not only most different from the preceding demonstration, but also from all the remaining demonstrations narrated above; and which nonetheless, with the d’Alembertian, if you look to the essence, is properly the same. To compare this with that one, and to explore the parallelism between the two, I commit to the experts, for whose sake alone it has been subjoined.

24.

Supra planum figurae 4. relatiue ad axem CG punctumque fixum C descriptas suppono superficiem primam et secundam eodem modo vt supra. Accipe punctum quodcunque in aliquo ramo lineae primae situm siue vbi T=0, ( e. g. quodlibet punctum M in axe iacens), et nisi in hoc etiam U=0, progredere ex hoc puncto in linea prima versus eam partem, versus quam magnitudo absoluta ipsius U decrescit. Si forte in puncto M valor absolutus ipsius U versus vtramque partem decrescit, arbitrarium est, quorsum progrediaris; quid vero faciendum sit, si U versus vtramque partem crescat, statim docebo.

Above the plane of figure 4, relative to the axis CG and the fixed point C, I suppose the first and second surface to be described in the same manner as above. Take any point situated on some branch of the first line, that is where T=0, ( e. g. any point M lying on the axis), and, unless here also U=0, proceed from this point along the first line toward that side toward which the absolute magnitude of U decreases. If by chance at the point M the absolute value of U decreases toward either side, it is arbitrary which way you proceed; but what is to be done if U increases toward either side, I will teach immediately.

Manifestum est itaque, dum semper in linea prima progrediaris, necessario tandem te ad punctum peruenturum, vbi U=0, aut ad tale, vbi valor ipsius U fiat minimum, e. g. punctum N. In priori casu quod quaerebatur inuentum est; in posteriori vero demonstrari potest, in hoc puncto plures ramos lineae primae sese intersecare (et quidem multitudinem parem ramorum), quorum semissis ita comparati sint, vt si in aliquem eorum deflectas (siue huc siue illuc) valor ipsius U adhucdum decrescere pergat. (Demonstrationem huius theorematis, prolixiorem quam difficiliorem breuitatis gratia supprimere debeo.) In hoc itaque ramo iterum progredi poteris, donec U aut fiat =0 (vti in fig. 4. euenit in P), aut denuo minimum.

It is manifest, therefore, that while you always proceed along the first line, you will necessarily at length come to a point where U=0, or to such a point where the value of U becomes minimal, e. g. the point N. In the former case what was being sought has been found; in the latter, however, it can be demonstrated that at this point several branches of the first line intersect (indeed an even multitude of branches), whose half-branches are so arranged that, if you deflect into any one of them (either this way or that), the value of U still continues to decrease. (The demonstration of this theorem, more prolix than difficult, I ought to suppress for the sake of brevity.) In this branch, therefore, you can proceed again, until U either becomes =0 (as in fig. 4. it happens at P), or again becomes minimal.

Tum rursus deflectes, necessarioque tandem ad punctum peruenies, vbi sit U=0.

Then you will deflect again, and necessarily at length you will arrive at a point where U=0.

Contra hanc demonstrationem obiici posset dubium, annon possibile sit, vt quantumuis longe progrediaris, et quamuis valor ipsius U semper decrescat, tamen haec decrementa continuo tardiora fiant, et nihilominus ille valor limitem aliquem nusquam attingat; quae obiectio responderet quartae in art. 6. Sed haud difficile foret, terminum aliquem assignare, quem simulac transieris, valor ipsius U necessario non modo semper rapidius mutari debeat, sed etiam decrescere non amplius possit, ita vt antequam ad hunc terminum perueneris, necessario valor 0 etiam affuisse debeat. Hoc vero et reliqua, quae in hac demonstratione addigitare tantummodo potui, alia occasione fusius exsequi mihi reseruo.

Against this demonstration a doubt could be raised, whether it is not possible that, however far you proceed, and although the value of U always decreases, nevertheless these decrements become continually slower, and nonetheless that value never reaches any limit; which objection would correspond to the fourth objection in Art. 6. But it would not be difficult to assign some bound such that, as soon as you have passed it, the value of U must of necessity not only change ever more rapidly, but also can no longer decrease, so that before you arrive at this bound, of necessity the value 0 must also already have been present. This, however, and the other points which in this demonstration I have only been able to touch upon, I reserve to set forth more fully on another occasion.

E r r a t a.

Pag. 7. annot. lin.

Page 7. note. line.

5. pro imaginarium l. imaginariam. p. 13. l. vlt. pro 23. l. 24. p. 14. l. 17. pro 2-1, l. 2m-1. p. 15. l. 4. por vnum l. vnam.

5. for imaginariumread imaginariam. p. 13. l. last. for 23. read24. p. 14. l. 17. for2-1, read 2m-1. p. 15. l. 4. for por vnum read vnam.

p. 16. l. 12. pro multiplicetur, l. multipliceturque. ibid. l. 21. pro aliaque, l. aliqua.

p. 16. l. 12. for multiplicetur, l. multipliceturque. ibid. l. 21. for aliaque, l. aliqua.

p. 28. l. 7 et 8. pro cosφ, l. rcosφ.

p. 28. l. 7 and 8. for cosφ, read. rcosφ.

*1) Sub quantitate imaginaria hic semper intelligo quantitatem in forma a+b√-1 contentam, quamdiu b non est =0. In hoc sensu expressio illa semper ab omnibus geometris primae notae accepta est, neque audiendos censeo, qui quantitatem a+b√-1 in eo solo casu imaginarium vocare voluerunt vbi a=0, impossibilem vero quando non sit a=0, quum haec distinctio neque necessaria sit neque vllius vtilitatis. — Si quantitates imaginariae omnino in analysi retineri debent (quod pluribus rationibus consultius videtur, quam ipsas abolere, modo satis solide stabiliantur): necessario tamquam aeque possibiles ac reales spectandae sunt; quamobrem reales et imaginarias sub denominatione communi quantitatum possibilium complecti mallem: contra, impossibilem dicerem quantitatem, quae conditionibus satisfacere debeat, quibus ne imaginariis quidem concessis satisfieri potest, attamen ita, vt phrasis haec idem significet ac si dicas, talem quantitatem in toto magnitudinum ambitu non dari. Hinc vero genus peculiare quantitatum formare, neutiquam concederem.

*1) By an imaginary quantity I here always understand a quantity contained in the form a+b√-1, so long as b is not =0. In this sense that expression has always been accepted by all geometers of the first rank, nor do I judge those to be listened to who wished to call the quantity a+b√-1 imaginary only in the case where a=0, but impossible when a is not =0, since this distinction is neither necessary nor of any utility. — If imaginary quantities ought altogether to be retained in analysis (which for several reasons seems more advisable than to abolish them, provided they are established on a sufficiently solid basis): they must necessarily be regarded as equally possible as real ones; wherefore I would prefer to embrace real and imaginary under the common denomination of possible quantities: on the contrary, I would call a quantity impossible which would have to satisfy conditions to which, even with imaginary quantities admitted, it cannot be satisfied—yet in such a way that this phrase means the same as if you should say that such a quantity is not given anywhere in the whole compass of magnitudes. From this, however, I would by no means allow a peculiar genus of quantities to be formed.

Quodsi quis dicat, triangulum rectilineum aequilaterum rectangulum impossibile esse, nemo erit qui neget. At si tale triangulum impossibile tamquam nouum triangulorum genus contemplari, aliasque triangulorum proprietates ad illud applicare voluerit, ecquis risum teneat? Hoc esset verbis ludere seu potius abuti.

But if someone says that a rectilinear, equilateral, right-angled triangle is impossible, there will be no one to deny it. But if someone should wish to contemplate such an impossible triangle as a new genus of triangles, and to apply other properties of triangles to it, who could hold back laughter? This would be to play with words, or rather to abuse them.

— Quamuis vero etiam summi mathematici saepius veritates, quae quantitatum ad quas spectant possibilitatem manifesto supponunt, ad tales quoque applicauerint quarum possibilitas adhuc dubia erat; neque abnuerim, huiusmodi licentias plerumque ad solam formam et quasi velamen rationiorum pertinere, quod veri geometrae acies mox penetrare possit: tamen consultius, scientiaeque, quae tamquam perfectissimum claritatis et certitudinis exemplar merito celebratur, sublimitate magis dignum videtur, tales libertates aut omnino proscribere, aut saltem parcius neque alias ipsis vti, nisi vbi etiam minus exercitati perspicere valeant, rem etiam absque illarum subsidio etsi forsan minus breuiter tamen aeque rigorose absolui potuisse. — Ceterum haud negauerim, ea quae hic contra impossibilium abusum dixi, quodam respectu etiam contra imaginarias obiici posse: sed harum vindicationem nec non totius huius rei expositionem vberiorem ad aliam occasionem mihi reseruo.

— Although indeed even the foremost mathematicians have often applied truths which manifestly presuppose the possibility of the quantities to which they pertain, also to such as whose possibility was still doubtful; nor would I deny that such licenses for the most part belong to mere form and, as it were, a veil of reasonings, which the keenness of the true geometer can presently penetrate: nevertheless it seems more advisable, and more worthy of the loftiness of that science which is rightly celebrated as the most perfect exemplar of clarity and certainty, either wholly to proscribe such liberties, or at least to employ them more sparingly, nor to use them otherwise, unless where even the less practiced are able to perceive that the matter could have been brought to completion equally rigorously even without their aid, though perhaps less briefly. — Moreover I would by no means deny that the things which I have said here against the abuse of impossibles can in a certain respect be objected also against imaginaries: but the vindication of these, and likewise a fuller exposition of this whole affair, I reserve for another occasion.

*2) Obseruare conuenit, ill. d'Alembert in sua huius demonstrationis expositione considerationes geometricas adhibuisse, atque X tamquam abscissam, x tamquam ordinatam curuae spectauisse (secundum morem omnium geometrarum primae huius saeculi partis, apud quos notio functionum minus usitata erat). Quia vero omnia ipsius ratiocinia, si ad ipsorum essentiam solam respicis, nullis principiis geometricis, sed pure analyticis innituntur, et curua imaginaria, ordinataeque imaginariae expressiones duriores esse lectoremque hodiernum facilius offendere posse videntur, formam repraesentationis mere analyticam hic adhibere malui. Hanc annotationem ideo adieci, ne quis demonstrationem d'Alembertianam ipsam cum hac succinta expositione comparans aliquid essentiale immutatum esse suspicetur.

*2) It is appropriate to observe that the illustrious d’Alembert, in his exposition of this demonstration, employed geometric considerations, and regarded X as the abscissa, x as the ordinate of the curve (according to the custom of all the geometers of the first part of this century, among whom the notion of functions was less usual). But since all his ratiocinations, if you regard their essence alone, rest on no geometric principles but purely analytic ones, and since an imaginary curve and imaginary expressions of the ordinates seem harsher and may more easily offend the modern reader, I have preferred here to employ a purely analytic form of representation. I have added this annotation for this reason, lest anyone, comparing the d’Alembertian demonstration itself with this succinct exposition, suspect that something essential has been altered.

*3) Hacce occasione obiter adnoto, ex harum serierum numero plurimas esse, quae primo aspectu maxime conuergentes videantur, e. g. ad maximam partem eas, quibus ill. Euler in parte poster. Inst.

*3) On this occasion I note in passing that, out of the number of these series, there are very many which at first aspect seem most convergent, e. g. for the most part those with which the illustrious Euler, in the posterior part of the Institutions,

Calc. Diff. Cap.

Differential Calculus, Chapter.

VI. ad summam aliarum serierum quam proxime assignandam vtitur p. 441--474. (reliquae enim series p. 475-- 478 reuera conuergere possunt), quod, quantum scio, a nemine hucusque obseruatum est. Quocirca magnopere optandum esset, vt dilucide et rigorose ostenderetur, cur huiusmodi series, quae primo citissime, dein paullatim lentius lentiusque conuergunt, tandemque magis magisque diuergunt, nihilominus summam proxime veram suppeditent, si modo non mimis multi termini capiantur, et quousque talis summa pro exacta tuto haberi possit?

6. for assigning as nearly as possible the sum of other series he employs pp. 441--474. (for the remaining series on pp. 475--478 can truly converge), which, so far as I know, has hitherto been observed by no one. Wherefore it would be greatly to be desired that it be shown lucidly and rigorously why series of this kind, which at first converge very swiftly, then little by little more and more slowly, and finally diverge more and more, nonetheless supply a sum approximately true, provided only that not too many terms be taken, and how far such a sum can safely be regarded as exact?

*4) Tota haec res multum illustrabitur per aliam disquisitionem sub prelo iam sudantem vbi in argumento longe quidem diuerso, nihilominus tamen analogo, liceutiam similem prorsus eodem iure vsurpare potuissem, vt hic in aequationibus ab omnibus analystis factum est. Quamquam vero plurium veritatum demonstrationes adiumento talium fictionum paucis verbis absoluere licuisset, quae absque his perquam difficiles euadunt et subtilissima artificia requirunt, tamen illis omnino abstinere malui, speroque, paucis me satisfacturum fuisse, si analystarum methodum imitatus essem.

*4) This whole matter will be much illuminated by another disquisition
now sweating under the press, where in an argument indeed far different, nonetheless
analogous, I could, by the same right, have usurped a similar license,
as here in equations has been done by all analysts.
Although in truth it would have been permitted to dispatch the demonstrations of more truths with the aid of such
fictions in a few words, which without these
turn out exceedingly difficult and require the most subtle artifices, nevertheless
I preferred to abstain from them altogether, and I hope that I would have satisfied a few,
had I imitated the analysts’ method.

*5) E. per errorem habet C, vnde etiam postea perperam statuit pqr=C.

*5) E., by mistake, reads C, whence also later he wrongly set pqr=C.

*6) In hanc expositionem error irrepsisse videtur, scilicet p. 118. l. 5. loco characteris v (on choissoit seulement celles où entroit p etc. ), necessario legere oportet, une même racine quelconque de l'équation proposée, aut simile quid, quum illud nullum sensum habeat.

*6) Into this exposition an error seems to have crept, namely p. 118, l. 5.
in place of the character v (one chose only those in which p entered, etc.),
one must necessarily read, one and the same arbitrary root of the proposed equation, or something similar, since that has no sense.

*7) In tomo secundo eorundem Miscellaneorum p. 337. dilucidationes ad hanc commentationem continentur: attamen hae ad disquisitionem praesentam non pertinent, sed ad logarithmos quantitatum negatiuarum, de quibus in eadem comm. sermo fuerat.

*7) In the second tome of the same Miscellanies, p. 337, elucidations concerning this commentation are contained: however, these do not pertain to the present disquisition, but to the logarithms of negative quantities, about which in the same comm. there had been discourse.

*8) Figura quarta constructa est supponendo X=x⁴-2xx+3x+10, in qua itaque lectores disquisitionibus generalibus et abstractis minus assueti situm respectiuum vtriusque curuae in concreto intueri poterunt. Longitudo lineae CG assumta est =10.

*8) The fourth figure is constructed by supposing X=x⁴-2xx+3x+10, in which accordingly readers less accustomed to general and abstract disquisitions will be able to view the relative position of each curve in concrete. The length of the line CG is assumed =10.

*9) Quando S>√1/2, conditio prima secundam; quando vero S<√1/2 secunda primam implicabit.

*9) When S>√1/2, the first condition will imply the second; whereas when S<√1/2 the second will imply the first.

*10) Satis bene certe demonstratum esse videtur, curuam algebraicam neque alicubi subito abrumpi posse (vti e. g. euenit in curua transscendente, cuius aequatio y=1/log x), neque post spiras infinitas in aliquo puncto se quasi perdere (vt spiralis logarithmica), quantumque scio nemo dubium contra rem mouit. Attamen si quis postulat, demonstrationem nullis dubiis obnoxiam alia occasione tradere suscipiam. In casu praesenti vero manifestum est, si aliquis ramus e. g. 2, ex circulo nullibi exiret (fig.

*10) It seems certainly to have been demonstrated quite well that an algebraic curve can neither anywhere be suddenly broken off (as, e. g., happens in a transcendental curve whose equation is y=1/log x), nor, after infinite spirals, as it were lose itself at some point (as the logarithmic spiral), and so far as I know no one has raised a doubt against the matter. Yet if anyone demands it, I will undertake to deliver on another occasion a demonstration subject to no doubts. In the present case, however, it is manifest that, if some branch, e. g. 2, were nowhere to exit from the circle (fig.

3.), te in circulum inter 0 et 2 intrare, postea circa totum hunc ramum (qui in circuli spatio se perdere deberet) circummeare, et tandem inter 2 et 4 rursus ex circulo egredi posse, ita vt nullibi in tota via in lineam primam incideris. Hoc vero absurdum esse inde patet, quod in puncto, vbi in circulum ingressus es, superficiem primam supra te habuisti, in egressu, infra; quare necessario alicubi in superficiem primam ipsam incidere debuisti, siue in punctum lineae primae. — Ceterum ex hoc ratiocinio principiis geometriae situs innixo, quae haud minus valida sunt, quam principia geometriae magnitudinis, sequitur tantummodo, si in aliquo ramo lineae primae in circulum intres, te alio loco ex circulo rursus egredi posse, semper in linea prima manando, neque vero, viam tuam esse lineam continuam in eo sensu, quo in geometria sublimori accipitur.

3.), that you enter the circle
between 0 and 2, afterwards go around the whole of this branch (which in
the circle’s space ought to lose itself), and at last between 2 and
4 go out of the circle again, so that nowhere on the whole way you fall upon the
first line. But that this is absurd is evident from this, that at the
point where you entered the circle you had the first surface above you, at the egress,
below; wherefore of necessity somewhere you must have fallen upon the first
surface itself, that is, upon a point of the first line. —
Moreover, from this ratiocination resting on the principles of the geometry of position, which
are no less valid than the principles of the geometry of magnitude,
there follows only this: if on some branch of the first line you enter the
circle, you may at another place go out of the circle again, always remaining on the first
line, yet not that your way is a continuous line in the sense in which it is taken in
higher geometry.

Sed hic sufficit, viam esse lineam continuam in sensu communi, i. e. nullibi interruptam sed vbique cohaerentem.

But here it suffices that the path be a continuous line in the common sense, i. e. nowhere interrupted but everywhere coherent.