Date: Jul 25, 2013 6:09 PM
Subject: Re: An independent integration test suite

Waldek Hebisch schrieb:
> Nasser M. Abbasi <> wrote:

> >
> > I remember reading that some math people, 100 or 200 years ago,
> > decided that sqrt (of non-negative, non-complex values) was single
> > valued function and its value is the non-negative root (ie.
> > principal square root).

> Beware that mathematicians have Humpty Dumpty attitude:
> : When I use a word. it means just what I choose it to
> : mean -- neither more nor less.
> Another mathematician may use the same word with different
> meaning (of course, in good math text meaning is explained).
> Before complex numbers were invented only nonnegative
> square root was in use.

Euler in his "Vollständige Anleitung zur Algebra" (Complete guide to
algebra) aimed at laypeople and published in 1770 in St. Petersburg in
two volumes,


defines the square root as double-valued (§120): "es können daher von
jedem Quadrat zwei Quadratwurzeln angegeben werden, deren eine positive,
die andere negativ ist." He admits negative radicands as well (§147):
SQRT(-4) = SQRT(4)*SQRT(-1) = 2*SQRT(-1), etc., and these square roots
are taken to be double-valued too (§150). On the other hand, he
introduces cube roots as single-valued: CBRT(8) = 2, CBRT(27) = 3 in
§164 and CBRT(-8) = -2, CBRT(-27) = -3 in §167, and then remarks (§167):
"Also werden wir hier nicht zu unmöglichen oder eingebildeten Zahlen
geführt, wie bei den Quadratwurzeln der Negativzahlen geschehen.".

Later on, however, in the context of solving cubic equations, he treats
all solutions (§151): any cubic equation x^3 = a has the three solutions
x = CBRT(a), x = (-1 + SQRT(-3))/2*CBRT(a), x = (-1 - SQRT(-3))/2*
CBRT(a), and he adds: "woraus erhellt, daß jede Cubikwurzel dreierlei
Werthe hat, von denen der erste möglich, die beiden anderen aber
unmöglich sind."

Euler always writes SQRT(-1) for the imaginary unit, and (§143) calls
square roots of negative numbers "unmögliche" (impossible) or
"eingebildete" (imagined) numbers. These designations are justified as
follows ($144): "Von diesen behauptet man also mit vollem Recht, daß sie
weder größer noch kleiner sind als Nichts; und auch nicht einmal Nichts
selbst, weswegen sie für unmöglich gehalten werden müssen." On the other
hand (§145): "Dennoch stellen Sie unserem Verstande sich vor, und finden
in unserer Einbildung Platz; daher sie auch blos eingebildete Zahlen
genannt werden." And even though they are "impossible" we know how to
calculate with them: SQRT(-4)*SQRT(-4) = -4, etc.

But why do it? Euler's reason sounds rather weak to me ($151): "Endlich
muß noch das Bedenken gehoben werden, daß die Lehre von den unmöglichen
Zahlen als nutzlose Grille angesehen werden könne. Allein dies Bedenken
ist unbegründet; diese Lehre ist in der That von der größten
Wichtigkeit, indem oft Aufgaben vorkommen, von welchen man nicht sofort
wissen kann, ob sie Mögliches oder Unmögliches verlangen. Wenn nun die
Auflösung derselben zu solchen unmöglichen Zahlen führt, so hat man ein
sicheres Zeichen, daß die Aufgabe Unmögliches verlangt." We do it
because the occurrence of "impossible" numbers safely indicates that a
given problem has no solution.

Obviously, the invention of complex numbers was a gradual process.


PS: I am quoting from my undated (around 1900) Reclam version of Euler's
"Algebra" in one volume, for which grammar and spelling were somewhat
modernized. Interestingly, the original German edition was preceded by
the publication in 1768/69 of a Russian edition prepared by two students
of Euler under the title "Universal'naya arifmetika" (Universal
arithmetic), volume I of which can also be found on the internet. The
original German edition sells for something like Euro 7500,- nowadays:


Euler, Leonhard. Vollständige Anleitung zur Algebra.
St. Petersburg: Kaiserliche Academie der Wissenschaften, 1770.
Bound in 2 volumes, 188 x 121 mm, polished calf with raised bands and
richly gilt spine.

Euler composed his famous 'Algebra' in German in 1765-6, soon after his
return to St. Petersburg from Berlin. He was by then partially blind,
and dictated the work to a young valet. Publication of the original
German version [...] was, however, delayed until 1770 and thus came to
be preceded by a Russian translation by his students Peter Inokhodtsev
and Ivan Yudin which was issued in 1768-9. [...]

"Euler's Vollständige Anleitung zur Algebra is not only the most popular
textbook on elementary algebra, with the exception of Euclid's Elements
it is the most widely printed book on mathematics." (Truesdell). It "was
translated into Russian (1768-9), Dutch (1773), French (1774), Latin
(1790), English (1797, 1822) and Greek (1800). The popular German
edition from Reclam Verlag sold no less than 108,000 copies between 1883
and 1943 (Reich, 1992). After 240 years this algebra textbook is still
in print today, available in several languages and editions" (Heeffer).

"In 1770 Euler published his 'Vollständige Anleitung zur Algebra' ...
The first volume treats of determinate algebra. This contains one of the
earliest attempts to place the fundamental processes on a scientific
basis: the same subject had attracted D'Alembert's attention. This work
also includes the proof of the binomial theorem for an unrestricted real
index which is still known by Euler's name ... The second volume of the
algebra treats of indeterminate or Diophantine algebra. This contains
the solutions of some of the problems proposed by Fermat, and which had
hitherto remained unsolved." (Ball).

"Euler proved the impossibility of solving x^3+y^3=z^3 in which x, y,
and z are integers, x*y*z /= 0 (a particular case of Fermat's last
theorem); his demonstration, based on the method of infinite descent and
using complex numbers of the form a+b*(-3)^(1/2), is thoroughly
described in his Vollständige Anleitung zur Algebra, the second volume
of which has a large section devoted to Diophantine analysis." (DSB).

Truesdell, Leonard Euler: Supreme Geometer, 1972;
Albrecht Heeffer, The Rhetoric of Problems in Algebra Textbooks from
Pacioli to Euler, pp.4-5;
Ball, A Short Account of the History of Mathematics, p.397;
DSB presumably refers to: Charles Coulston Gillespie (ed.), Dictionary
of Scientific Biography, 1970-80.