# Algebraic Symmetries

III. Neue Entdeckungen.

Es ist jedem der Geometrie bekannt, dass

verschiedene ordentliche Vielecke, namentlich das

Dreyeck, Viereck, Funfzehneck, und die,welche durch

wiederholte Verdoppelung der Seitenzahl eines derselben

entstehen, sich geometrisch construiren lassen. So weit

war man schon zu Euklids Zeit, und es scheint, man habe

sich seitdem allgemein , dass das Gebiet der

Elementargeometrie sich nicht weiter erstrecke: wenigstens

kenne ich keinen Versuch, ihre Grenzen

auf dieser Seite zu erweiten.

Desto mehr, mich, verdient die Entdeckung

Aufmerksamkeit, dass ausser jenen ordentlichen Vielecken

noch eine Menge anderer, z. B,. das Siebenzehneck, einer geometrischen

Construction ist. Diese Entdeckung ist

eigentlich nur ein Corollarium einer noch nicht ganz vollendeten

Theorie von Umfange, und sie soll,

sobald diese ihre Vollendung erhalten hat, dem Publicum

vorgelegt werden.

C. F. Gauss, a. Braunschweig,

Stud. der Mathematik zu .

Es verdient angemerkt zu werden, dass Hr. Gauss

jetzt in seinem 18ten Jahr steht, und sich hier in Braunschweig

mit eben so gl¨ucklichem Erfolg der Philosophie

und der classischen Litteratur als der
Mathematik

gewidmet hat.

Den 18 April 96.

E. A. W. Zimmermann, Prof.

**Die Allgemeine Literatur-Zeitung
**

From Meyers Lexikon:

This review appeared in Jena from 1785 to 1803 and in Halle from 1804 to 1849 and

was a leading organ of German classical and romantic literature. Goethe, Schiller and

Kant were among the editors and authors. It moved to Halle as a result of an e ort of

the romantics to increase their control over the review and was replaced in Jena, on the

initiative of Goethe by the Jenaische Allgemeine Literatur-Zeitung.

**Gauss**

Gauss, who was born in 1777, wrote the Disquisitones between 1796 and 1798, thus

between his nineteenth and twenty- first years. It did not appear until 1801. It contains a

great deal in the way both of theorems and theories, the most important being:

**The statement was already known at the**

1. Proof of law of quadratic reciprocity.

1. Proof of law of quadratic reciprocity.

time, but even the best of the eighteenth century mathematicians were unable to find a

proof. It remains a central mathematical theorem.

**2. Developed the theory of binary quadratic forms.**In particular, he introduced

the notion of composition of quadratic forms and established its properties . Although

in some respects, namely in the context of the notion of ideal number, composition has

become a common working tool of all algebraists and number -theorists, Gauss's theory

itself is still difficult and little known. His form of the theory would appear to be that best

suited to computation.

**3. Cyclotomic fields.**The construction of the regular heptadecagon, and, more generally,

the analysis of the numbers formed from roots of unity . This is thus one of the

earliest manifestations of Galois theory. Gauss presumably knew more than he included

in the book, but he, apparently, published very little more on the subject.

His thesis of 1799 established, in effect and for the first time, that every polynomial

equation has a root.

As the root may be complex and the thesis did not refer to
complex numbers, the formulation

of the thesis was necessarily somewhat different . I have already emphasized that
this

is a basic mathematical fact.

These are all theories and results to which the contributions of Gauss are
clear. In

addition , he appears to have occupied himself as a very young man, even as an
adolescent,

with other important problems and observations, for example, with the nature of
geometries

in which the parallel axiom of Euclid is not satisfied and with the
arithmetic -geometric

mean, which is both an elementary and an advanced topic. Here, however, the
evidence is

different. It consists of Gauss's recollections as a somewhat saturnine older
man, so that

it appears to be difficult to disentangle what he himself discovered later, what
he learned

from other authors as an adolescent { he had an early and extensive acquaintance
with the

work of various leading eighteenth century mathematicians { as well as what he
learned

later, from what he discovered early. One could certainly spend a lot of time
with Gauss's

Collected Works and with his correspondence, reflecting on these matters.

One extremely useful reference is a diary that Gauss kept between 1796 and 1814,

with 146 entries that record his principal discoveries. The lemniscate function
to which, as

I noticed, Gauss alludes in the first lines of chapter 7 of the Disquisitiones
appears several

times . I was asked what Gauss what may have meant with his allusion to the
lemniscate.

A brief examination of the diary and of the collected works, which contain ample
editorial

comments, makes perfectly clear what Gauss knew. Although it is a digression
from my

main purpose, it is worthwhile to tarry a little on the matter.

The goal of this year's lectures is after all to communicate, starting at the
beginning,

some genuine mathematical understanding, beyond the gee-whiz or what I refer to
as the

Jack Horner manner, of recent achievements in number theory, and it would be a
shame,

since we are now in a position to appreciate a more detailed explanation of
Gauss's allusion,

to let slip the occasion of acquiring more concrete information.

Gauss was an overwhelming presence in nineteenth century mathematics, Even
though

he exerted little active influence. The German pre-eminence in mathematics as a
whole,

and in certain domains such as number theory in particular, that lasted
throughout the

nineteenth century and until the early thirties is due in good part to him,
although the

Prussian university system was probably also a significant factor. I have not
studied these

matters. Oddly enough it appears to have been
Weil who was most thoroughly

imbued with the aspirations of German number theory, both through direct,
personal

experience as a young man during the twenties and through his studies of variou
nineteenth

century authors. Simplifying, for the purposes of brevity, an elaborate
development in

which a large number of mathematicians took part, one might say that he not only
brought

it intact through the war at its highest level but also was the principal source
of its

transformation into the theories that were finally and successfully exploited in
the proof

of Fermat's theorem.

His major contribution was, oddly enough, a set of conjectures, the Weil
conjectures,

now demonstrated, but by others. He is, in various comments to his papers, quite
explicit

about the relation of these conjectures to Gauss and the lemniscate.

I quote from Weil's Two lectures on number theory, past and present, an essay I

recommend to your attention.

"In 1947, in Chicago, I felt bored and depressed and, not knowing what to do, I
started reading

Gauss's two memoirs on biquadratic residues, which I had never read
before....This

led me in turn to some conjectures."

A few lines later Weil draws attention to the very last
entry in Gauss's diary, an entry to

which we shall come in a moment.

**Digression**

The digression at first sight seems to demand some knowledge of the

calculus, but it does not. I first write down a formula that may be familiar

to some, but not to all. No matter! Do not puzzle over the left side. It is

no more than the mathematician's usual fastidious way of writing down the

length of the arc from the point (1, 0) on the circle to the point (x, y) and

that is what we mean by θ, which of course has to be measured in radians,

thus in units in which the radius is 1, but that is the unit chosen. If

then y = sin( θ). If y = 1 then θ= π/2.

We now do something similar for the lemniscate, a curve
defined by the

equation

The lemniscate is the curve in the form of a bow. The
point (x, y) is the

point where the line through the points (0, 0) and (1, u) cut the curve. The

length of the curve from (1, 0) to the point (x, y) is expressed mathematically

as

Once again, there is no need to be troubled by the
integral. It is again just a

way of expressing the length of an arc of a curve. Observe that θ plays
here

the role of the angle in a circle measured in radians. Gauss wrote

If u = 1, then θ is some number that I call, following Gauss,
. Thus

is the length of the upper loop on the right
running from (1, 0) to (0, 0).

Constructing a regular triangle, a regular pentagon, or a
regular heptadecagon

is the problem of dividing the total circumference of the circle into

three, five or seventeen arcs of equal length. We could consider the same

problem for the lemniscate, taking the initial point, which is now important

as it was not, because of symmetry, for the circle, to be the point (0, 1). In

an entry for March 19, 1797 Gauss notes that this leads to an equation for

u of degree m^{2}, whereas for the circle it was an equation of degree
m. In

the cases already considered, m was 3 or 5. We remove one easy root, u = 0

corresponding to the first point of division. This leads to equations of degree

m − 1 or for a lemniscate m^{2} − 1. In a later entry, apparently for
April 15,

he observes there is a problem of separating the real roots of this equation

from the complex. He is seeking the real roots. The corresponding equation

for the circle, thus for y, has only real roots. It is the numbers x + iy that

are complex. The real roots of the equation of degree m^{2}−1 give the
division

points and there are m− 1 of them.

In an entry dated March 21, he observes implicitly (all the entries are

cryptic) that these m − 1 roots are numbers that can be constructed with a

ruler and compass .

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