Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work
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In his publication of 1805 Legendre proposed the principle of least squares as an advantageous and convenient method of adjusting observations. He called it méthode des moindres quarrés, showed that the rule of the arithmetical mean is a particular case of the general principle, deduced the method of normal equations, and gave examples of its application to the determination of the orbit of a comet and to the form of a meridian section of the earth. Although Legendre gave no demonstration that the results thus determined were the most probable or best, his remarks indicated that he recognized the advantages of the method in equilibrating the errors.
Adrain showed from the law of probability of error that the arithmetical mean followed, and that the most probable position of an observation point in space is the center of gravity of all the given points. He also applied it to the discussion of two practical problems in surveying and navigation.
In 1795 Gauss deduced the law of probability of error and from it gave a full development of the method. To Gauss is due the algorism of the method, the determination of weights from normal equations, the investigation of the precision of results, the method of correlatives for conditional observations, and numerous practical applications. Few branches of science owe so large a proportion of subject matter to the labors of one man.
The method thus thoroughly established spread rapidly among astronomers. The theory was subjected during the following fifty years to rigid analysis by Encke, Gauss, Hagen, Ivory, and Laplace, while the labor of Bessel, Gerling, Hansen, and Puissant developed its practical applications to astronomical and geodetic observations. During the period since 1850, the literature on the subject has been greatly extended. The writings of Airy and De Morgan in England, of Liagre and Quetelet in Belgium, of Bienaymé in France, of Schiaparelli in Italy, of Andrä in Denmark, of Helmert and Jordan in Germany, and of Chauvenet and Schott in the United States have brought the method to a high degree of perfection in all its branches and have caused it to be universally adopted by scientific men as the only proper method for the discussion of observations.
There falls to the part of Gauss the service of having rendered the method reliable against all exceptions and of having equipped it so comfortably for use that its utility could not be unfolded until the twenties when he published his “polished” supplementary essays on the subject. From that time on no further application of observations was to be thought of.
Gauss’ fundamental paper on this subject is “Theoria combinationis observationum erroribus minimis obnoxiae,” 1821, and is given in Volume IV of his Works. Woodward wrote:
No single adjunct has done so much as this to perfect plans of reduction, and to give definiteness to computed results. The effect of the general adoption of this method has been somewhat like the effect of the general adoption by scientific men of the metric system; it has furnished common modes of procedure, common measures of precision, and common terminology, thus increasing to an untold extent the availability of the priceless treasures which have been recorded in the century’s annals of astronomy and geodesy.12
At school in 1792 or 1793 Gauss also investigated the law of primes, i.e., of their rarer and rarer occurrence in the series of natural numbers. Euler he studied for the rich content; Newton and Lagrange for the form presented. Newton’s rigor of proof influenced him, and Lagrange’s work on the theory of numbers interested him.
Gauss stayed at the Collegium Carolinum well into the fourth year; on August 21, 1795, the order was issued from the ducal office to arrange “that 158 thalers yearly shall be paid to the student named Gauss, going to Göttingen, for assistance and that he be informed of this, as well as that the ‘free table’ is open to him in Göttingen.” His stipend was raised to 400 thalers in 1801 and in 1803 to 600 thalers, besides free apartments.
On October 11 the young scientist left Brunswick. He indicated later on the occasion of his receiving the doctorate that Göttingen University (the Georgia Augusta) was chosen because of the great wealth of mathematical literature in its library. The Duke made no objection, thus furnishing proof of his lively interest in the development of the young man’s talent. On this occasion high social circles in Brunswick spoke a great deal of Gauss. It was believed that the Duke added a considerable, specified amount to the stipend from his own pocket. Even then the augurs of the court were setting up a horoscope for the young man, which was well suited to recommend him to the attention of those who listened to such things.
III
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Student Days
The same age saw the founding of three German universities: Halle, Göttingen, and Erlangen. Göttingen, founded in 1737 by King George II of England, soon took the lead among all German universities and held it until the end of the nineteenth century. George II, from whom comes the name Georgia Augusta, endowed the institution in a princely manner.
Göttingen was equipped, with Halle as a model, by its first curator, Gerlach Adolph von Münchhausen, though it differed from its model in several respects. The legal and political faculty stood forth prominently rather than the theological, as in Halle. Lehrfreiheit ruled from the very first. Its theologians followed historical and critical studies rather than controversy. In the first decades there were men like Albrecht von Haller, to whom Göttingen owes the establishment of its medical and scientific school, and the great philologist Johann Matthias Gesner, the former colleague of Johann Sebastian Bach in Leipzig at the St. Thomas School, who founded in Göttingen the first philological seminar in Germany. Great jurists like Georg Ludwig Böhmer labored there. Johann Stephan Putter made Göttingen an outstanding school for the study of civil law.
German youth from every corner of the land, especially the West and South, people of rank, princes and counts, in fact almost everybody who had an interest in general culture soon streamed into Göttingen. Foreign countries, England above all, took notice of Göttingen. Students came from northern Europe. The faculty included such men as Gatterer, Achenwall, Schlözer, Spittler, and later Heeren, so that at the end of the eighteenth century one could almost speak of a Göttingen historical school. These men exerted a very strong influence.
That was the foundation for the activity of the man to whom Göttingen owes the most, except perhaps Münchhausen: Christian Gottlob Heyne, the successor of Gesner.13 Heyne was a pure classicist; the ancient humanities and classical literature, he thought, were the means of every nobler “training of the mind” for the true, the good, and the beautiful. His lectures were attended not only by philologists but by students of all faculties, especially lawyers. Besides, he was a splendid administrative officer. The fate of the university was tied up with him until the close of the eighteenth century. Just as Hollman kept the university from the deeper shocks of the impact of the French occupation at the time of the Seven Years’ War, so Heyne, during the Westphalian period, knew how to sweep away the danger of Göttingen’s being transformed according to a French model.
Into this atmosphere came young Gauss. When he reached Göttingen, he secured a room at Gothmarstrasse 11. Worries about the future had not left him. He was still undecided whether to follow philology or mathematics. The former offered a surer and speedier prospect of livelihood. For a time, therefore, he was among the auditors of Heyne, who attracted him personally more than did Kästner,14 whom he dubbed “the leading mathematician among poets and leading poet among mathematicians.” N. Fuss once remarked that Euler’s Tentamen novae theoriae musicae contained “too much geometry for musicians and too much music for geometers.”
At the same time there was another young man from Brunswick striving toward the same goal as Gauss. Johann Joseph Anton Ide, born on January 26, 1775, was the son of a mill inspector who died young and left his family in reduced circumstances. Even in the lifetime of his father the boy had attended the orphanage school; on recommendation of the inspector Jenner he was received in 1788 into the instruction department of the Freemasons’ lodge, which at the time was under the dire
ction of Hellwig, who discovered in him, as he himself reported, “an eminent genius for the mathematical sciences” and urged him to devote himself to the study of them. But, remarked Hellwig further, “as it is becoming easy for many a young man in this splendid world, even lacking natural abilities, industry, and good rearing, to acquire assistance through various channels, so it was hard for Ide.” Indeed, he found a ready reception in the Martineum, and, after Hellwig had become teacher of mathematics and natural history at the reorganized Katharineum, at this school also. In 1794 he got a “free place” at the Carolineum, where he was attracted to Zimmermann and Lueder. But mathematics and its related areas belonged to those sciences “whose culture unfortunately deserved no special assistance, in the eyes of high patrons.” For a while the efforts of those two men were in vain. The twenty-five thalers yearly from Count Veltheim were worthy of thanks, but of course did not go very far. There was nothing left except to approach the Duke again. Almost contrary to expectation, this step was of consequence. The Duke promised the necessary means; a half year after Gauss, Ide was able to enter the University of Göttingen at Easter, 1796, where he continued his studies for five years.
Ide belonged to the small circle to which Gauss limited his social intercourse during his student days. He was the only one of the older acquaintances at Göttingen until Arnold Wilhelm Eschenburg (born on September 15, 1778) arrived in 1797. The latter was a son of the Carolineum professor already mentioned; he and Gauss had been close friends there. Eschenburg matriculated to study law and finance. In 1800 after the completion of his Studies Eschenburg became a lawyer in Brunswick, and a year later secretary of the lower court. He was appointed cabinet secretary to the Duke in 1805. It was said to be almost incredible how much he influenced the Duke in favor of Gauss while in this position. Eschenburg died in 1861 as governmental councilor and treasurer at Detmold.
The lectures of the brilliant physicist Georg Christoph Lichtenberg (1744–1799) seem to have offered Gauss considerable stimulation, for he called him “the leading adornment of Göttingen.” Also Carl Felix Seyffer (1762–1822), after 1789 the assistant professor of astronomy, should be mentioned; Gauss was in friendly association with him, and they carried on a correspondence after Seyffer had moved to Munich.
Ide was called to Moscow in 1803 as professor of mathematics and died there in 1806 as a result of the Russian climate. To the circle of Gauss’ student friends belong also Heinrich Wilhelm Brandes (1777–1834), later professor of mathematics at Breslau and Leipzig, and Johann Albrecht Friedrich Eichhorn of Wertheim (1779–1856), a lawyer who was Prussian minister of religion from 1840 to 1848. Gauss frequently came in touch with Johann Friedrich Benzenberg (1777–1846) and later assisted him with his work Experiments on the Laws of Gravity, the Resistance of the Air, and the Rotation of the Earth (Hamburg, 1804).
By far the most intimate friend of Gauss at this time was Wolfgang Bolyai von Bolya, descendant of an old Hungarian noble family whose records reach back into the thirteenth or fourteenth century of Magyar history. He was born on the Bolya estate, which lies about fifteen miles north of Hermannstadt; his mother was Christine Vajna of Pava. His father, Caspar, took him in 1781 to the Evangelical Reformed College at Nagy-Enyed. The quiet, introspective boy seldom took part in the games of his schoolmates; indeed, he had to be forced to do so. On the other hand, he had special aptitude for languages, poetry, and mental arithmetic. At the various festivities of the college he was presented as the child prodigy.
At the college Bolyai became a close friend of the cultured Baron Simon Kemény. About 1790 the two went to Klausenburg and lived in the house of the famous professor of theology Michael Szathmáry, who managed for a while to interest Bolyai exclusively in theology. George Méhes was the instructor in mathematics there. In Klausenburg, Bolyai’s eyes were injured by the explosion of some gunpowder, which he had himself prepared, so that for a long while he could read only with the greatest difficulty.
It was at that time the fashion for sons of the Hungarian nobility to attend German universities for their higher education. An older, talented student acted as “mentor.” Thus Bolyai accompanied Baron Simon Kemény, who went to Jena in the summer of 1796. In Vienna, Bolyai got sick and had to remain behind. Here he saw the Artillery School and was so carried away that he wanted to enter a military career. Such enthusiasm was a notable characteristic of his. A letter from Kemény caused him to drop this plan and go to Jena and from there to Göttingen, where he arrived in September, 1796.
Bolyai wrote thus:
We [he and his fellow countryman Baron Simon Kemény] went to Göttingen, where Kästner and Lichtenberg were able to stand us, and I became acquainted with Gauss, at that time a student there. Even today I am a friend of his, although I am far from being able to compare myself with him. He was very modest and didn’t make much showing; not three days as in the case of Plato, but for years one could be with him without recognizing his greatness. What a shame, that I didn’t understand how to open up this silent “book without a title” and read it! I didn’t know how much he knew, and after he saw my temperament, he regarded me highly without knowing how insignificant I am. The passion for mathematics (not externally manifested) and our moral agreement bound us together so that while often out walking we were silent for hours at a time, each occupied with his own thoughts.
Gauss once said that Bolyai was the only one who understood how to penetrate into his views on the foundations of mathematics.
In discussing his studies of Euler and Lagrange at this time, Gauss later wrote: “I became animated with fresh ardor, and by treading in their footsteps, I felt fortified in my resolution to push forward the boundaries of this wide department of science,” Then on March 30, 1796, the nineteen-year-old student made a discovery which determined above all else his future career. That evening remained vividly impressed on his memory. In the Allgemeine Literaturzeitung of April, 1796, appeared this notice:
It is known to every beginner in geometry that various regular polygons, viz., the triangle, tetragon, pentagon, 15-gon, and those which arise by the continued doubling of the number of sides of one of them, are geometrically constructible.
One was already that far in the time of Euclid, and, it seems, it has generally been said since then that the field of elementary geometry extends no farther: at least I know of no successful attempt to extend its limits on this side.
So much the more, methinks, does the discovery deserve attention . . . that besides those regular polygons a number of others, e.g., the 17-gon, allow of a geometrical construction. This discovery is really only a special supplement to a theory of greater inclusiveness, not yet completed, and is to be presented to the public as soon as it has received its completion.
carl friedrich gauss
Student of Mathematics at Göttingen
It deserves mentioning, that Mr. Gauss is now in his 18th year, and devoted himself here in Brunswick with equal success to philosophy and classical literature as well as higher mathematics.
18 April, 1796 e. a. w. zimmermann, Prof.
This was his first publication, and Gauss always considered the discovery one of his greatest. No mathematician in two thousand years had thought of it. It proved to be a rich field for the discoverer. He told Bolyai that this polygon of seventeen sides should adorn his tombstone. That could not be carried out, but there was a design on the side of the base of the Brunswick monument to him.15
In his youth the lad was less interested in geometry than in algebra, according to Sartorius. What equation could be closer to him than xp = 1, whose roots were so closely connected with the problem of circle division? He opened his scientific diary with the discovery of March 30. The realization that the division of a circle into p equal parts depends on the solution of the equation xp = 1 is due to Cotes and Demoivre; it was first explained and firmly established by Euler in 1748. Gauss was acquainted with the later work of Vandermonde, as shown by his letter of Octobe
r 12, 1802, to Olbers. The roots of the equation
if we set
are represented by the powers r, r2, r3 . . . rp–1, where k has the values 0, 1, 2 . . . n−1; in this manner the points of the regular n-gon are determined by the complex quantities just given. This figure is inscribed in a circle of radius unity. The quantities cos 2kπ/p and sin 2kπ/p are the rectangular Cartesian coordinates of the points concerned, if one chooses the center of the circle as the starting point and if the axis of abscissae is laid through the point for which x = 0.
The above values show that, if p is presupposed to be an uneven prime, the powers 1, g, g2 . . . gp–1 of a primitive root g (mod p), without respect to order, are congruent to the residues 1, 2, 3 . . . p–1, which is the peculiar relation that in the cyclic order, r, rg, rg² . . .