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On the dividing
line between Eighteenth and Nineteenth Century mathematics towers the majestic figure of
Carl Friedrich Gauss. He was born in the German city of Brunswick, the son of a day laborer. The duke of Brunswick gracefully recognized in young Gauss an infant prodigy
and took charge of his education. The young genius studied from 1795-98 at Gottingen and in 1799 obtained his doctor's degree at Helmstadt.
From 1807 until his death in 1855 he worked quietly and undisturbed as the director of the
astronomical observatory and professor at the university of his alma mater. His
comparative isolation, his grasp of "applied" as well as "pure"
mathematics, his preoccupation with astronomy and his frequent use of Latin have the touch
of the Eighteenth Century, but his work breathes the spirit of a new period.
Kant, Goethe, Beethoven, and
Hegel, on the side lines of a great political struggle raging in other countries, but
expressed in his own field the new ideas of his age in a most powerful way.
Gauss diaries show that already
in his seventeenth year he began to make startling discoveries. In 1795, for instance, he
discovered independently of Euler the law of quadratic reciprocity in number theory. Some
of his early discoveries were published in his Helmstadt dissertation of 1799 and in the
impressive "Disquisitiones arithmeticae" of 1801. The dissertation gave the
first rigorous proof of the so-called fundamental theorem of algebra, the theorem that
every algebraic equation with real coefficients has at least one root and hence has n
roots. The theorem itself goes back to Albert Girard, the editor of Stevin's works
("Invention nouvelle en algebre", 1629), and D’Alembert had tried to give a
proof in 1746. Gauss loved this theorem and later gave two more demonstrations, returning
in 1846 to his first proof. The third demonstration (1816) used complex integrals and
showed Gauss' early mastery of the theory of complex numbers.
The "Disquisitiones
arithmeticae" collected all the masterful work in number theory of Gauss'
predecessors and enriched it to such an extent that the beginning of modern number theory
is sometimes dated from the publication of this book. Its core is the theory of quadratic
congruencies, forms, and residues; it culminates in the law of quadratic residues, that
"theorema aureum" of which Gauss gave the first complete proof. Gauss was as
fascinated by this theorem as by the fundamental theorem of algebra and later published
five more demonstrations; one more was found after his death among his papers. The
"Disquisitiones" also contain Gauss' studies on the division of the circle, in
other words, on the roots of the equation xn = 1. They led up to the remarkable theorem that the
sides of the regular polygon of 17 sides (more general, of n sides, n = 2v + 1, p = 2x, n prime, and k = 0, 1, 2, 3, ....n prime, k = 0, 1, 2, 3 can be constructed
with compass and ruler alone, a striking extension of the Greek type of geometry.
Gauss interest in astronomy was aroused when, on the first day of the new century,
on Jan. 1, 1801, Piazzj in Palermo discovered the first planetoid, which was given the name
of Ceres. Since only a few observations of the new planetoid could be made the problem
arose to compute the orbit of a planet from a smaller number of observations. Gauss solved
the problem completely; it leads to an equation of degree eight. When in 1802, Pallas,
another planetoid, was discovered, Gauss began to interest himself in the secular
perturbations of planets. This led to the "Theoria motus corporum coelestium"
(1809), to his paper on the attraction of general ellipsoids (1813), to his work on
mechanical quadrature (1814), and to his study of secular perturbations (1818). To this
period belongs also Gauss' paper on the hypergeometric series (1812), which allows a
discussion of a large number of functions from a single point of view. It is the first
systematic investigation of the convergence of a series.
After 1820 Gauss began to be actively interested in geodesy. Here he combined
extensive applied work in triangulation with his theoretical research. One of the results
was his exposition of the method of least squares (1821, 1823), already the subject of
investigation by Legendre (1806) and Laplace. Perhaps the most important contribution of this period
in Gauss' life was the surface theory of the "Disquisitiones generales circa
superficies curvas" (1827), which approached its subject in a way strikingly
different from that of Mongc. Here again practical considerations, now in the field of
higher geodesy, were intimately connected with subtle theoretical analysis. In this
publication appeared the so-called intrinsic geometry of a surface, in which curvilinear
coordinates are used to express the linear element ds
in a quadratic differential form ds2 = Edu2 +
Fdu dv + Gdv2. There was also a climax, the "theorema
egregium", which states that the total curvature of the surface depends only on E, F,
and G and their derivatives, and thus is a bending invariant. But Gauss did not neglect
his first love, the "queen of mathematics," even in this period of concentrated
activity on problems of geodesy, for in 1825 and 1831 appeared his work on biquadratic
residues. It was a continuation of his theory of quadratic residues in the
"Disquisitiones arithmeticae," but a continuation with the aid of a new method,
the theory of complex numbers. The treatise of 1831 did not only give an algebra of
complex numbers, but also arithmetic. A new prime number theory appeared, in which 3
remains prime but 5 = (1 + 2i) (1 — 2i) is no longer a prime. This new complex number
theory clarified many dark points in arithmetic so that the quadratic law of reciprocity
became simpler than in real numbers. In this paper Gauss dispelled forever the mystery
which still surrounded complex numbers by his representation of them by points in a plane.
A statue in Gottingen represents Gauss and his younger colleague
Wilhelm Weber in the process of inventing the electric telegraph. This happened in 1833-34
at a time when Gauss' attention began to be drawn toward physics. In this period he did
much experimental work on terrestrial magnetism. But he also found time for a theoretical
contribution of the first importance— his "Allgemeine Lehrsatze" on the theory
of forces acting inversely proportional to the square of the distance (1839, 1840). This
was the beginning of potential theory as a separate branch of mathematics, (Green's paper
in 1828 was practically unknown at that time) and it led to certain minimal principles
concerning space integrals, in which we recognize "Dirichlet's" principle. For
Gauss the existence of a minimum was evident; this later became a much debated question
which was finally solved by Hilbert.
Gauss remained active until his death in 1855. In his later years he concentrated
more and more on applied mathematics. His publications, however, do not give an adequate
picture of his full greatness. The appearance of his diaries and of some of his letters
has shown that he kept some of his most penetrating thoughts to himself. We now know that
Gauss, as early as 1800, had discovered elliptic functions and around 1816 was in
possession of non-Euclidean geometry. He never published anything on these subjects;
indeed, only in some letters to friends did he disclose his critical position toward
attempts to prove Euclid's parallel axiom. Gauss seems to have been
unwilling to venture publicly into any controversial subject. In letters he wrote about
the wasps who would then fly around his ears and of the "shouts of the
Boeotians" that would be heard if his secrets were not kept. For himself Gauss
doubted the validity of the accepted Kantian doctrine that space conception is Euclidean a
priori; for him the real geometry of space was a physical fact to be discovered by
experiment.
In his history of mathematics of the Nineteenth Century Felix Klein has invited
comparison between Gauss and the twenty-five year older French mathematician Adrien Marie
Legendre. It is perhaps not entirely fair to compare Gauss with any mathematician except
the very greatest; but this particular comparison shows how Gauss' ideas were " in
the air," since Legendre in his own independent way worked on most subjects which
occupied Gauss. Legendre taught from 1775 to 1780 at the military school in Paris and later filled different government positions such as
professor at the Ecole Normale, examiner at the Ecole Polytechnique, and geodetic
surveyor.
Like Gauss he did fundamental work on number theory ("Essai sur les
nombres," 1798 "Theorie des nombres," 1830) where he gave a formulation of
the law of quadratic reciprocity. He also did important work on geodesy and on theoretical
astronomy, was as assiduous a computer of tables as Gauss, formulated in
1806 the method of least squares, and studied the attraction of ellipsoids—even
those which are not surfaces of revolution. Here he introduced the "Legendre"
functions. He also shared Gauss' interest in elliptic and Eulerian integrals as well as in
the foundations and methods of Euclidean geometry.
Although Gauss penetrated deeper into the nature of all these different fields of
mathematics, Legeudre produced' works of outstanding importance. His comprehensive
textbooks were for a long time authoritative, especially his "Exercises du calcul
integral" (3 vols, 1811-19) and his "Trait des functions elliptiques et des
integrates euleriennes" (1827-32), still a standard work. In his "Elements de
geometric" (1794) he broke with the Platonic ideals of Euclid and presented a textbook of elementary geometry based on
the requirements of modern education. This book passed through many editions and was
translated into several languages; it has had a lasting influence.
Euler made signal contributions
in every field of mathematics which existed in his day. He published his results not only
in articles of varied length but also in an impressive number of large textbooks which
ordered and codified the material assembled during the ages. In several fields Euler's
presentation has been almost final. An example is our present trigonometry with its
conception of trigonometric values as ratios and its usual notation, which dates from
Euler's "Introductio in analysin infinitorum" (1748). The tremendous prestige of
his textbooks settled for ever many moot questions of notation in algebra and calculus;
Lagrange, Laplace, and Gauss knew and followed Euler in all their works.
There are books by Euler on hydraulics, on ship construction, on artillery. In
1769-71 there appeared three tomes of a "Dioptrica" with a theory of the passage
of rays through a system of lenses. In 1739 appeared his new theory of music, of which it
has been said that it was too musical for mathematicians and too mathematical for
musicians. Euler's philosophical exposition of the most important problems of natural
science in his "Letters to a German Princess" (written 1760-61) remained a model
of popularization.
The enormous fertility of Euler has been a continuous source of surprise and
admiration for everyone who has attempted to study his work, a task not so difficult as it
seems, since Euler's Latin is very simple and his notation is almost modern—or perhaps
we should better say that our notation is almost Euler's! A long list can be made of the
known discoveries of which Euler possesses priority, and another a list of ideas which are
still worth elaborating. Great mathematicians have always appreciated their indebtedness
to Euler. "Lisez Euler," Laplace used to say to younger mathematicians, "lisez Euler,
c'est notre maitre a tous." And Gauss, more ponderously, expressed himself: "The
study of Euler's works will remain the best school for the different fields of mathematics
and nothing else can replace it." Riemann knew Euler's works well and some of his
most profound works have an Eulerian touch.
Publishers might do worse than offer translations of some of Euler's works together
with modern commentaries. From the times of ancient Babylon until those of Euler and
Laplace astronomy had guided and inspired the most sublime discoveries in mathematics; now
this development seemed to have reached its climax. However, a new generation, inspired by
the new perspectives opened by the French Revolution and the flowering of the natural
sciences, was to show how unfounded this pessimism was. This great new impulse came only
in part from France; it also came, as often in the history of
civilization, from the periphery of the political and economical centers, in this case
from Gauss in Gottingen.
Cauchy, along with his contemporaries Gauss, Abel, and Bolzano, belongs to the pioneers of the new insistence on rigor
in mathematics. The Eighteenth Century had been essentially a period of experimentation,
in which results came pouring in with luxurious abundance. The mathematicians of this age
had not bothered too much about the foundation of their work—"allez en avant, et la
foi vous viendra," D'Alembert is supposed to have said. When they worried about
rigor, as Euler and Lagrange occasionally did, their arguments were not always convincing.
The time had now arrived for a close concentration on the meaning of the results. What was
a "function" of a real variable, which showed such different behavior in the
case of a Fourier series and in the case of a power series? What was its relation to the
entirely different "function" of a complex variable? These questions brought all
the unsolved problems about the foundation of the calculus and the existence of the
potentially and the actually infinite again into the foreground of mathematical thinking.
What Eudoxos had done in the period after the fall of Athenian democracy, Cauchy and his
meticulous contemporaries began to accomplish in the period of expanding industrialism.
This difference of social setting produced different results, and where Eudoxos' success
had the tendency to stifle productivity, the success of the modern reformers stimulated
mathematical productivity to a high degree. Cauchy and Gauss were followed by Weierstrass
and Cantor.
Peter Lejeune Dirichlet was closely associated with Gauss and Jacobi, as well as
with the French mathematicians. He lived from 1822-27 as a private tutor and met Fourier,
whose book he studied; he also became familiar with Gauss' "Disquisitions
arithmeticae." He later taught at the University of Breslau and in 1855 succeeded Gauss at Gottingen. His personal acquaintance with French as well as German
mathematics and mathematicians made him the appropriate man to serve as the interpreter of
Gauss and to subject Fourier series to a penetrating analysis. Dirichlet's beautiful
"Vorlesungen iiber Zahlentheorie" (publ. 1863) still form one of the best
introductions into Gauss' investigations in number theory. They also contain many new
results.
During these years of almost feverish 'productivity in the new projective and
algebraic geometries another novel and even more revolutionary type of geometry lay hidden
in a few obscure publications discarded by most leading mathematicians. The question
whether Euclid's parallel postulate is an independent axiom
or can be derived from other axioms had puzzled mathematicians for two thousand years.
Ptolemy had tried to find an answer in Antiquity, Nasir al-din in the middle Ages, Lambert
and Legendre in the Eighteenth Century. All these men had tried to prove the axiom and had
failed, even if they reached some very interesting results in the course of their
investigation. Gauss was the first man who believed in the independent nature of the
parallel postulate, which implied that other geometries, based on another choice of axiom,
were logically possible. Gauss never published his thoughts on this subject. The first to
challenge openly the authority of two millennia and to construct a non-Euclidean geometry
were a Russian, Nikolai Ivanovitch Lobachevsky, and a Hungarian, Janos Bolyai. The first
in time to publish his idea was Lobachevsky, who was a professor in Kazan and lectured on the subject of Euclid's parallel axiom in 1826. His first book appeared in
1829-30 and was written in Russian. Few people took notice of it. Even a later German
edition with the title "Geometrische Untersuchungen zur Theorie der
Parallellinien" received little attention, even though Gauss showed interest. By that
time Bolyai had already published his ideas on the subject.
Janos (Johann) Bolyai was the son of a mathematics teacher in a provincial town of Hungary. This teacher, Farkas (Wolfgang) Bolyai, had studied at Gottingen when Gauss was also a student there. Both men kept up an
occasional correspondence. Farkas spent much time in trying to prove Euclid's fifth postulate (p. 60), but could not come to a
definite conclusion. His son inherited his passion and also began to work on a proof
despite his father's plea to do something else:
Two products of the algebra of the United Kingdom deserve our special attention: Hamilton's quaternions and Clifford's biquaternions. Hamilton, the
Royal Astronomer of Ireland, having completed his work on mechanics and optics, turned in
1835 to algebra. His "Theory of Algebraic Couples" (1835) denned algebra as the
science of pure tune and constructed a rigorous algebra of complex numbers on the
conception of a complex number as a number pair. This was probably independent of Gauss,
who in his theory of biquadratic residues (1831) had also constructed a rigorous algebra
of complex numbers, but based on the geometry of the complex plane. Both conceptions are
now equally accepted. Hamilton subsequently tried to penetrate
The theory of groups made possible a synthesis of the geometrical and algebraic
work of Monge, Poncelet, Gauss, Cayley, Clebsch, Grassmann, and Riemann. Riemann's theory
of space, which had offered so many suggestions of the Erlangen program, inspired not only Klein but also Helmholtz and
Lie. Helmholtz in 1868 and 1884 studied Riemann's conception of space, partly by looking
for a geometrical image of his theory of colors, partly by inquiring into the origin of
our ocular measure. This led him to investigate the nature of geometrical axioms and
especially Riemann's quadratic measurement. Lie improved on Helmholtz' speculations
concerning the nature of Riemann's measurement by analyzing the nature of the underlying
groups of transformations (1890). This "Lie-Helmholtz" space problem has been of
importance not only to relativity and group theory, but also to physiology.
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