Gauss

The great German mathematician Carl Friedrich Gauss was born 30 April 1777 in Brunswick and died 23 February 1855 in Göttingen.

Gauss was the only son of poor peasants living in miserable conditions. His father was an honest man, condemned to a life of hard physical labour, but had little understanding of/or sympathy with the intellectual needs of his unusual bright son. Fortunately for the mother of Gauss, her brilliant son was the centre of her life, her pride and her joy.

Gauss' precocity is legendary. He exhibited such early genius that his family and neighbours called him "wonder child". When he was two years old, he gradually got his parents to tell him how to pronounce all the letters of the alphabet. Then, by sounding out combinations of letters, he learned (on his own) to read aloud. At the age of 3 he was correcting his father's weekly wage calculations and by age of ten, the teachers at his school told his family they could teach him no more. Fortunately, word of the child prodigy reached the Duke of Brunswick, and he became Gauss' patron for the next twenty years. The Duke's support enabled Gauss to devote himself to his studies and research.

In 1788 Gauss began his education at the Gymnasium, where he learnt High German and Latin. In 1792 he entered Brunswick Collegium Carolium. The young Gauss was adept at the study of languages and by the age of nineteen, he had already mastered German, Greek, Latin, English, Danish and French and he added Russian and Swedish in later life. At the academy Gauss independently discovered Bode's law, the binomial theorem and the arithmetic- geometric mean, as well as the law of quadratic reciprocity and the prime number theorem.

In 1795 Gauss left Brunswick to study at Göttingen University. Gauss left Göttingen in 1798 without a diploma, but by this time he had made one of his most important discoveries - the construction of a regular 17-sided polygon with compasses and straight edge. The prove was published as Section VII of Gauss's famous work, Disquisitiones Arithmeticae.

This was the most major advance in this field since the time of Euclid and the discovery so pleased Gauss that he decided to pursue mathematics rather than study languages. In addition he left instructions for a regular 17-sided polygon to be put on his memorial stone. Today, there is a 17-point star (the stone cutter thought a 17-sided polygon would look too much like a circle) on Gauss's memorial stone.

Gauss returned to Brunswick where he received his doctoral degree in 1799. His dissertation was a brilliant proof of the fundamental theorem of algebra. In 1801, when he was 24, he completed his work Disquisitiones Arithmeticae which became the most significant contribution to number theory up to that time.

In 1800 he set up a formulae to calculate the Easter date. The number of the year J is divided by 19 ; the remainder of this division is denoted by a. Next J is divided by 4, this remainder is denoted by b, and also divided by 7 with remainder denoted by c. Afterwards the remainder of the division of 19a + M by 30 is called d and the remainder of 2b + 4c + 6d + N divided by 7 is called e.

For M and N we must take the following values :
for years 1582 until 1699 : M = 22 and N = 3 ;
for years 1700 until 1799 : M = 23 and N = 3 ;
for years 1800 until 1899 : M = 23 and N = 4 ;
for years 1900 until 2099 : M = 24 and N = 5 ;
for years 2100 until 2199 : M = 24 and N = 6 ;

Easter day will be (22 + d + e) March or (d + e - 9) April. Herewith holds that instead of 26 April always 19 April will be taken and instead of 25 April always 18 April if d = 28 and a > 10. Such an exceptional case occurred in 1981 and will occur again in 2049.

Immediately following his abstract work in pure maths, Gauss plunged into the realm of applied maths, in particular, astronomy. The newly discovered asteroid Ceres had been observed by many astronomers for 40 days, but none of them could get a correct computation for its orbit. Gauss was able to accurately compute the orbit after only three observations. When Ceres was rediscovered by Zach on 7 December 1801 it was almost exactly where Gauss had predicted. Although he did not disclose his methods at the time, Gauss had used his least squares approximation method.

In 1807 Gauss left Brunswick to take up the position of director of the Göttingen observatory. In 1809, Gauss published his second book, Theoria motus corporum coelestium in sectionibus conicis Solem ambientium, in 1809, a major two volume treatise on the motion of celestial bodies. In the first volume he discussed differential equations, conic sections and elliptic orbits, while in the second volume, the main part of the work, he showed how to estimate and then to refine the estimation of a planet's orbit. Gauss's contributions to theoretical astronomy stopped after 1817, although he went on making observations until the age of 70.

Much of Gauss's time was spent on a new observatory, completed in 1816, but he still found the time to work on other subjects. His publications during this time include Disquisitiones generales circa seriem infinitam , a rigorous treatment of series and an introduction of the hypergeometric function, Methodus nova integralium valores per approximationem inveniendi , a practical essay on approximate integration, Bestimmung der Genauigkeit der Beobachtungen , a discussion of statistical estimators, and Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata . The latter work was inspired by geodesic problems(shortest distance between two points on a surface such as the Earth) and was principally concerned with potential theory. In fact, Gauss found himself more and more interested in geodesy in the 1820's.

Gauss had been asked in 1818 to carry out a geodesic survey of the state of Hanover to link up with the existing Danish grid. Gauss was pleased to accept and took personal charge of the survey, making measurements during the day and reducing them at night, using his extraordinary mental capacity for calculations. Because of the survey, Gauss invented the heliotrope which worked by reflecting the Sun's rays using a design of mirrors and a small telescope. He also developed the mathematics of error analysis for measurement in general, giving rise to probability analysis and hypothesis testing. The normal probability curve is known as the Gaussian curve.

From the early 1800's Gauss had an interest in the question of the possible existence of a non-Euclidean geometry. In a book review in 1816 he discussed proofs which deduced the axiom of parallels from the other Euclidean axioms, suggesting that he believed in the existence of non-Euclidean geometry, although he was rather vague. Gauss had a major interest in differential geometry, and published many papers on the subject. Disquisitiones generales circa superficies curva (1828) was his most renowned work in this field.

Gauss had worked on physics already before 1831, publishing Uber ein neues allgemeines Grundgesetz der Mechanik , which contained the principle of least constraint, and Principia generalia theoriae figurae fluidorum in statu aequilibrii which discussed forces of attraction. These papers were based on Gauss's potential theory, which proved of great importance in his work on physics. He later came to believe his potential theory and his method of least squares provided vital links between science and nature.

In 1832, Gauss and Weber began investigating the theory of terrestrial magnetism. Gauss was excited by this prospect and by 1840 he had written three important papers on the subject: Intensitas vis magneticae terrestris ad mensuram absolutam revocata (1832), Allgemeine Theorie des Erdmagnetismus (1839) and Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungskräfte (1840). Allgemeine Theorie... showed that there can only be two poles in the globe and went on to prove an important theorem, which concerned the determination of the intensity of the horizontal component of the magnetic force along with the angle of inclination. Gauss used the Laplace equation to aid him with his calculations, and ended up specifying a location for the magnetic South pole.

Gauss and Weber achieved much in their six years together. They discovered Kirchoff's laws, as well as building a primitive telegraph device which could send messages over a distance of 5000 ft. However, this was just an enjoyable pastime for Gauss. He was more interested in the task of establishing a world-wide net of magnetic observation points. This occupation produced many concrete results. The Magnetischer Verein and its journal were founded, and the atlas of geomagnetism was published, while Gauss and Weber's own journal in which their results were published ran from 1836 to 1841.

Gauss spent the years from 1845 to 1851 updating the Göttingen University widow's fund. This work gave him practical experience in financial matters, and he went on to make his fortune through shrewd investments in bonds issued by private companies. He was also able to attend the opening of the new railway link between Hanover and Göttingen, but this proved to be his last outing.

Gauss made it a rule of life to record only those ideas which he had fully formed. There were certain ideas that Gauss had done work on but did not publish, since he felt that they were incomplete. He described all his life's work with a crest containing a fruit tree and the motto : "Pauca, sed matura" or "Few, but ripe". What Gauss revealed to others was complete, with no unfinished proofs or partial explanations and left nothing more to be said. Gauss's kept his diary(only nineteen pages long, with but 146 entries !) for nearly eighteen years and then simply recorded his findings in a morass of notes and papers, some of which still await deciphering.

Gauss's political opinions were conservative and nationalistic. In philosophy he advocated the empirical reasoning and he was an opponent of the idealism of Kant and Hegel. Like Newton and Archimedes, Gauss was famous during his lifetime, even among military men. When the French army under Napoleon moved near Gauss's town of Brunswick, the emperor himself gave the command to spare Brunswick because "the foremost mathematician of all time lives there". The range of his scientific work is very extensive and without any doubt he is ranked as the last of the three great all-rounders of mathematics : Archimedes, Newton and Gauss himself.

by Wendy Maes

Carl Friedrich Gauss