Popular Science Monthly/Volume 33/September 1888/Sketch of Carl Friedrich Gauss
|←A Fossil Continent||Popular Science Monthly Volume 33 September 1888 (1888)
Sketch of Carl Friedrich Gauss
"IF we except the great name of Newton," says Prof. H. J. S. Smith, "it is probable that no mathematician of any age or country has ever surpassed Gauss in the combination of an abundant fertility in invention with an absolute rigorousness in demonstration which the ancient Greeks themselves might have envied." Wagener says, in the sketch of Gauss in the "Biographie Universelle," that each work of his is an event in the history of science, a revolution, which, overturning the old theories and methods, replaces them by new ones, and advances science to a height which no one had ever before dreamed of. The scientific estimate of Gauss's quality took another form in the expression of Laplace, who, when asked who was the greatest mathematician in Germany, replied "Pfaff." His interrogator remarking that he should have thought Gauss was, Laplace retorted, "Oh, yes, Pfaff is the greatest mathematician in Germany, but Gauss is the greatest mathematician in Europe."
Carl Friedrich Gauss was born in Brunswick, April 23, 1777, and died in Göttingen, February 23, 1855. His father was a brick-layer, and desired that the boy should be brought up to the same trade. But the lad had other tastes, and is said by some of his biographers to have displayed a greater precocity in his aptitude for mathematics than even Pascal. At three years old he could calculate and solve problems in numbers, and amuse himself by tracing geometrical lines and figures in the sand. He had, in fact, hardly reached that age when he ventured to tell his father concerning a certain account, "That is not right; it should be so much" — and was correct. At the age of ten he was acquainted with the binomial theorem and theory of the infinite series. Such gifts could not fail to attract marked attention from his teachers. The report of them reached Bartels, afterward Professor of Mathematics at Dorpat, and he brought the youth to the notice of Charles William, Duke of Brunswick, who undertook the charge of his education. Having, rather in opposition to his father's designs, learned all that the professors at the Collegium Carolinum could teach him, he went to Göttingen, in 1795, "as yet undecided whether to pursue philology or mathematics." The scale was probably turned by circumstances; one of them, perhaps, being the rare gifts of the mathematical professor, Kaestner, whom Gauss described as the "first of geometers among poets, and the first of poets among geometers"; and another, his success in solving the problem of the division of the circle into seventeen equal parts. Henceforth he made mathematics, which he styled "the queen of the sciences," the main study of his life, interesting himself particularly in arithmetic, " the queen of mathematics." After com- pleting his course at the university, Gauss spent a short time in 1798 at Helmstadt, consulting the library there, and enjoying the society of his fellow-mathematician Pfaff. Having obtained a full supply of notes, he returned to Brunswick, and employed himself in the elaboration of the studies which have placed his name high in the list of eminent mathematicians.
In 1807 he was offered by the Czar of Russia a professorship in the Academy of St. Petersburg ; but he declined the position, at the instance of Olbers, and because he felt that such a professor- ship would cramp his studies. His desire was to obtain the post of astronomer at an observatory, so that he could spend all his time on his observations and his studies for the advancement of science. This desire was gratified in the same year, when he was appointed Director of the Observatory and Professor of Astron- omy at Gottingen. In this service he spent the rest of his life, never sleeping away from under the roof of the building, except in 1828, when he accepted an invitation to attend a meeting of the natural philosophers in Berlin, and in 1854 — the year before his death — when, on the opening of the railway to Hamburg, he for the first time saw a locomotive. He consecrated all his time, says Larousse, " his genius, and his indefatigable activity, to the most abstract and profound researches in all branches of mathematics, astronomy, and physics. Endowed with most favorable health, possessing simple and modest tastes, so indifferent to display that he never wore any of the numerous decorations that the various governments decreed to him. Gauss had a gentle, up- right, and correct character. Applying the greatest care to the preparation of his briefest as well as of his most elaborate mem- oirs, he would offer nothing to the public till it had received the last finishing touches from the workman's hand. He had en- graved on his seal a tree loaded with fruit, encircled with the legend, ' Pauca, sed matura' (few, but ripe). And he left a large number of works, which he did not consider mature enough to publish," but which arrangements were made after his death for having edited. " The genius of Gauss," Larousse continues, " was essentially original. If he was treating of a subject which had already engaged the attention of other students, it seemed as if their works were wholly unknown to him. He had his own man- ner of approaching the propositions, and his own method of treat- ing them, and his solutions were absolutely new. These solutions had the merit of being general, complete, and applicable to all the cases that could be included under the question. Unfortu- nately, the very originality of the methods, a particular mode of notation, and the exaggerated, perhaps affected, laconicism of his demonstrations, make the reading of Gauss's works extremely laborious. Hence, envious minds have not let the opportunity pass to reproach him with having made himself unintelligible in order to appear profound. The reason of this is, that Gauss does not leave visible any trace of the analytical course by which he has been led to the final solution. He used to say that when a monument is exhibited to the public there should remain no traces of the scaffoldings that have been used in constructing it. He was wrong in this; for, although it may be true that the scaffoldings ought to be withdrawn from the eye of the public, they should be for a certain time accessible to those of architects; and, even if they are out of use, they are sometimes the object of special descriptions, which make their merit understood. . . . Although Gauss is hard to understand as a writer, he was very clear as a professor. He was not, however, one of those mathe- maticians who are represented as being so deeply buried in their science as to have become strangers to the outer world. He used to talk pertinently and agreeably on subjects of philosophy, poli- tics, and literature."
The charge of obscurity here brought against Gauss is re- viewed by Prof. H. J. S. Smith, who says: "It may seem para- doxical, but it is probably nevertheless true, that it is precisely the effort after a logical perfection of form which has rendered the writings of Gauss open to the charge of obscurity and un- necessary difficulty. The fact is, that there is neither obscurity nor difficulty in his writings, so long as we read them in the sub- missive spirit in which an intelligent school-boy is made to read his Euclid. Every assertion that is made is fully proved, and the assertions succeed one another in a joerfectly just analogical order; there is nothing, so far, of which we can complain. But, when we have finished the perusal, we soon begin to feel that our work is bui begun, that we are still standing on the threshold of the temple, and that there is a secret which lies behind the veil, and is as yet concealed from us. . . . No vestige appears of the process by which the result itself was obtained, perhaps not even a trace of the considerations which suggested the successive steps of the demon- stration."
According to M. Wagener, as summarized by Prof. Tucker, though Gauss looked upon mathematics as the principal means for developing human knowledge, he yet fully recognized the beneficial influence of an acquaintance with classical literature. He had, indeed, a wonderful faculty for the acquisition of lan- guages; he was acquainted with most of the European languages, and could speak many of them well. At the age of sixty -two he took up the study of the Russian language, and he mastered it in two years. He took a great interest in politics till within a few weeks of his death. " His lectures, in which he adopted the analytic metliod, were exceedingly clear expositions; in them lie liked to discuss the methods and the roads by which he had arrived at his great results. He required the closest attention, and objected to the taking of notes, lest his hearers should lose the thread of his argument. The students seated round the lecture - table listened with delight to the lucid and animated addresses of their master; addresses more resembling conversations than set lectures."
Gauss's writings are upon subjects of arithmetic, algebra, and astronomy. The fullest list, that given in the Royal Society's catalogue of scientific papers, contains one hundred and twenty- four titles, but does not include his largest works. The most im- portant papers are on arithmetic, while only a very few of the number are algebraic, and they all relating to a single theorem. Prof. Cayley remarks that of the memoirs in pure mathematics "it may be safely said that there is not one which has not signally contributed to the progress of the branch of mathematics to which it belongs, or which would not require to be carefully analyzed in a history of the subject." One of his earliest discoveries was "the method of least squares," which, though first published by Legen- dre, he applied as early as 1795. His first published paper—a thesis for the Doctorate of Philosophy, in 1799—was devoted to the demonstration that every equation has a root; and of this theorem he made two other distinct demonstrations in 1815 and 181G. But these works, though he was the first in the field on the subject, gave him no fame. Lagrange seems not to have heard of the first one; and Cauchy, whose subsequent demonstrations have been preferred, received in France all the praise due to a first discoverer. The "Disquisitiones Arithmetica?," which is perhaps his principal work, contains many important researches, one of which, known as the celebrated Fundamental Theorem of Gauss, or the law of Quadratic Reciprocity of Legendre, of itself alone. Prof. Tucker says, "would have placed Gauss in the first rank of mathemati- cians." The author discovered it by induction before he was eighteen years old, and worked out the first proof which he pub- lished of it in the following year. He was not satisfied with this, but published other demonstrations resting on different principles, till the number reached six. He had, however, been anticipated in enunciating the theorem, but in a more complex form, by Euler, and Legendre had unsuccessfully attempted to prove it. "The question of priority of enunciation or of demonstrating by induction," says Prof. Tucker, "in this case is a trifling one; any rigorous demonstration of it involved apparently insuperable difficulties." Another discussion involves the theory of describing within a circle the polygon of seventeen sides; another, the theory of the congruence of numbers, or the relation that exists between all numbers that give the same remainder when they are divided by the same number. In his "General Disquisitions on Curved Surfaces" he established the famous theorem that in whatever way a flexible and inextensible surface may be deformed, the sum of the principal curvatures at each point will always be the same. The calculations of the elements of the asteroids Pallas, Ceres, Juno, and Vesta, were made by Gauss, and attracted all the more attention because they furnished the first occasions on which all the elements of a planet had to be determined in a short time and by a small number of observations. The methods were not yet even fixed, because no one had taken up the subject. The process adopted by Gauss was simple and confessedly worthy of the attention of astronomers. Gauss has been called the godfather of the planet Vesta, from his having selected the name for it. In two papers on the comets of 1811, he gave a new and much more simple method than had been practiced before to determine the elements of a comet with the smallest number of observations. While actively engaged in the measurement of the degree in Hanover, Gauss devised the instrument known as the helio- trope, which has since come into general use in all geodesic observations.
Gauss engaged also in researches on magnetism, concerning which he published a paper in 1833 on the intensity of terrestrial magnetism. He and Prof. Wilhelm "Weber invented new mag- netic apparatus, including the declination instrument and the bifilar magnetometer. They erected at Gottingen an observatory, free from iron, where he made magnetic observations, and—antici- pating the electro-magnetic telegraph—sent telegraphic signals to a neighboring town.
His collected works, edited by E. J. Schering, have been recently published by the Poyal Society of Gottingen in seven volumes. With them are included notices by him of many of the memoirs, and of works of other authors in the "Gottingen gelehrte An- zeigen," and a considerable amount of previously unpublished matter. Gauss was a member of all the important learned so- cieties.
One of Gauss's last acts was, a little while before his death, to have engraved at the foot of his portrait, as giving the best sum- mary of his views and labors, the lines from Shakespeare's "King Lear":
"Thou, Nature, art my goddess; to thy laws My services are bound."