RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

5.4. Belief in algebraic solubility shaken 83
remains very doubtful, and consequently that any proof whose entire strength
depends on this assumption in the current state of affairs has no weight.” 70
In 1799, GAUSS’S aim was to scrutinize EULER’S proof of the Fundamental Theo-
rem of Algebra. For this purpose, it was sufficient for GAUSS to express his suspicion
that the algebraic solution of general equations was not established with the necessary
rigor. Thus — at least as it stands — GAUSS’S criticism seems to confront the founda-
tions and not the validity of this hidden assumption in EULER’S proof. Although it
is doubtful whether or not GAUSS possessed a demonstration that the validity could
also be questioned, he certainly suggested the possibility. Two years later in his influ-
ential Disquisitiones 1801, GAUSS addressed the problem again in connection with the
cyclotomic equations (see quotation below). Possibly alluding to LAGRANGE’S “very
great computational work” GAUSS described the solution of higher degree equations
not merely beyond the existing tools of analysis but outright impossible.
“The preceding discussion had to do with the discovery of auxiliary equations.
Now we will explain a very remarkable property concerning their solution. Everyone
knows that the most eminent geometers have been unsuccessful in the search
for a general solution of equations higher than the fourth degree, or (to define the
search more accurately) for the reduction of mixed equations to pure equations.
And there is little doubt that this problem is not merely beyond the powers of contemporary
analysis but proposes the impossible (cf. what we said on this subject
in Demonstrationes nova, art. 9 [above]). Nevertheless it is certain that there are
innumerable mixed equations of every degree which admit a reduction to pure
equations, and we trust that geometers will find it gratifying if we show that our
equations are always of this kind.” 71
While GAUSS was voicing his opinion on the insolubility of higher degree equa-
tions in Latin from his position in Göttingen, the support for the solubility of the
quintic was shaken even more radically by an Italian. P. RUFFINI (1765–1822) had
published his first proof of the impossibility of solving the quintic in 1799, the same
year GAUSS had first uttered his doubts about its possibility. But while GAUSS had
70 “Seu, missis verbis, sine ratione sufficienti supponitur, cuiusvis aequationis solutionem ad solutionem
aequationum purarum reduci posse. 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.” (C. F. Gauss, 1799, 17–18); for a translation into German, see (C. F. Gauss,
1890, 20–21).
71 “Disquisitiones praecc. circa inventionem aequationum auxiliarium versabantur: iam de earum solutione
proprietatem magnopere insignem explicabimus. Constat, omnes summorum geometrarum
labores, aequationum ordinem quartum superantium resolutionem generalem, sive (ut accuratius
quid desideretur definiam) affectarum reductionem ad puras, inveniendi semper hactenus irritos
fuisse, et vix dubium manet, quin hocce problema non tam analyseos hodiernae vires superet, quam
potius aliquid impossibile proponat (Cf. quae de hoc argumento annotavimus in Demonstr. nova
etc. arg. 9). Nihilominus certum est, innumeras aequationes affectas cuiusque gradus dari, quae
talem reductionem ad puras admittant, geometrisque gratum fore speramus, si nostras aequationes
auxiliares semper huc referendas esse ostenderimus.” (C. F. Gauss, 1801, 449); English translation
from (C. F. Gauss, 1986, 445). Bold-face has been substituted for the original small-caps.