“ It is characteristic of higher arithmetic that many of the most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand and are often found only after many fruitless investigations with the aid of deep analysis and lucky combinations. This significant phenomenon arises from the wonderful concatenation of different teachings of this branch of mathematics, and from this it often happens that many proofs for years were sought in vain, are later proved in many different ways. As soon as a new result is discovered by induction, one must consider as the first requirement the finding of a proof by any possible means. But after such good fortune, one must not in higher arithmetic consider the investigation closed or view the search for other proofs as a superfluous luxury. For sometimes one does not at first come upon the most beautiful and simplest proof, and then it is just the insight into the wonderful concatenation of truth in higher arithmetic that is the chief attraction for study and often leads to the discovery of new truths. For these reasons the finding of new proofs for known truths is often at least as important as the discovery itself.”- Gauss.
Johann Carl Friedrich Gauss was born on April 30, 1777 in Brunswick, Germany. Gauss started school at the age of seven and is known to possess phenomenal memory from that time onwards and had extraordinary capability of counting and arithmetic. It is said he could calculate before he could read. He learned Latin along with official High German, philology and mathematics. Later Gauss enrolled at Brunswick Collegium Carolinum where he studied Newton’s Principia. Soon he started having his own ideas though most of them had already been discovered. After that he studied at the University of Gottingen.
Gauss’ professional success came in the form of his construction of the regular seventeen-sided polygon using ruler and compass only which he always regarded as his one of the major achievements. After came his first major publication, the monumental ‘Disquisitiones arithmeticae’. This was one of the first attempts to organize the study of the theory of numbers. This was a masterpiece and made him famous worldwide. At the same time, Gauss became interested in astronomy and created a sensation by calculating the orbit of the asteroid Ceres from the extremely limited observational data.
Gauss was the recipient of many academic and other honors. In later years he was addressed as Geheimrat Gauss. He was elected a foreign member of the Royal Society of London in 1804 and was also awarded the Copley Medal in 1838.
Gauss had a rich list of scientific work which included both mathematics and science. Just the way Newton is regarded a physicist first, mathematician later, Gauss is regarded as mathematician first, physicist later. In mathematics the theory of number, algebra, analysis, geometry, mechanics, celestial mechanics, probability, the theory of errors, and actuarial science were prominent fields of his interest. In pure and applied science his interests included observational astronomy, surveying, geodesy, capillarity, geomagnetism, electromagnetism, optics, and the design of scientific equipment. In mathematics, it is the penetration of his work which impresses the most; in science it is more the virtuosity.
The collected works of Gauss include over three hundred papers, many written in Latin. His unpublished notebooks have been worked over by historians but many problems of interpretation remain. His voluminous professional and personal correspondence has also been carefully studied. Gauss was a perfectionist, he published only after thorough study. Gauss disliked formal teaching but he enjoyed good rapport with young mathematicians. The mathematicians in his circle included his gifted protégé Eisenstein, Richard Dedekind, Johann Benedikt Listing, August Ferdinand Mobius, Bernhard Riemann and Karl von Staudt.
British Mathematician Henry Smith writes about Gauss as follows-
“If we except the great name of Newton (and the exception is one which Gauss himself would have delighted to make), it is probable that no mathematician of any age or country has ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute righteousness in demonstration, which the ancient Greeks themselves might have envied. It may be admitted, without disparagement to the eminence of such great mathematicians as Euler and Cauchy, that they were so overwhelmed with the exuberant wealth of their own creations, and so fascinated by the interest attaching to the results at which they had arrived, that they did not greatly care to expend their time in arranging their ideas in a strictly logical order, or even in establishing by irrefragable proof results which they instinctively felt, and could almost see, to be true. With Gauss the case was otherwise. It may seem paradoxical, but it is probably nevertheless true, that it is precisely the effort after logical perfection of form which has rendered the writings of Gauss open to the charge of obscurity and unnecessary difficulty. Every assertion that is made is fully proved, and the assertions succeed one another in a perfectly just and logical 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 but 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 yet concealed from us. No vestige appears of the process by which the result itself is obtained, perhaps not even a trace of the considerations which suggested the successive steps of the demonstration. If on the other hand we turn to a memoir of Euler’s there is a sort of free and luxuriant gracefulness about the whole performance, which tells of the quiet pleasure which Euler must have taken in each step of his work; but we are conscious nevertheless that we are at an immense distance from the severe grandeur of design which is the characteristic of all Gauss’ greatest efforts . it is not the greatest, but it is perhaps not the least, of Gauss’ claims to the vastness of the science, he exacted the utmost rigorousness in every part of it, never passé over difficulty as if it did not exist, and never accepted a theorem as true beyond the limits within which it could actually be demonstration.”