Sania Ali Carl Friedrich Gauss’ life

Carl Fridrich Gauss was born in Brunswick, to a working class family on the 30th of April 1777. His father made a living by laying bricks and gardening, hence it was not likely that Gauss would pursue an education. His father also hoped for his son to follow in his footsteps and work as a labourer. His father was also of the mindset that education was not very helpful and would not make a difference to a labourer. With his own father discouraging him to get an education and his family’s financial situation, Gauss did not look like he would end up in school anytime soon. However, his mother and uncle played a huge role in getting him there when they realized in his younger years how bright he was. When he was only five years old, he managed to find an error in his fatherr’s payroll. His potential was even discovered later on by his teachers while he was still in grade school. Gauss’ class was challenged by a teacher to find the sum of all the integers numbers from one to 101. Astonishingly, Gauss was the first to finish, while the rest of his class took understandably much longer. When his teacher checked the work, the vast majority of the children’s solutions were incorrect, but Gauss’s had the correct number, 5050, written as his answer. The child prodigy had recognized that if he paired off the integers,for example 1+100=101, 2+99=101, 3+98=101, etc he would have 50 pairs of integers, and because those 50 pairs all added up to 101, he multiplied 101 by 50 and got a total of 5050.

As if this was not amazing enough, Gauss only continued to excell in the field of mathematics. With the help of Bartels and Bttne Gauss began his education at Gymnasium (a senior secondary education institute in Germany as well as other portions of mainland Europe) at the age of twelve in 1788 where he further learned more mathematics as well High German and Latin and began to question concepts such as Euclid’s . After ingesting many nontrivial concepts so quickly, Gauss caught the attention of Karl Wilhelm Ferdinand, the Duke of Brunswick, whom agreed to provide a stipend to help the prodigy learn more so that he could himself contribute to the field. That was how Friedrich Gauss found himself at the Brunswick Collegium Carolinum in 1972 where he independently rediscovered the binomial theorem, Bode’s Law, the arithmetic- geometric mean, and the law of quadratic reciprocity.

Towards the end of his stay at the Collegium, Gauss managed to discover a method to create a regular polygon with 17 sides, known as a heptadecagon, with the use of a straight edge and a compass. Many mathematicians believed such a feat was impossible and Guass was so confident in the importance of this discovery that he not only insisted that it be engraved on his tombstone, but it also inspired him to continue studying in the field of mathematics. During his stay in college, Gauss was very keen to learn and study languages, but he now knew that he would rather study and discover more in mathematics instead.

Duke Ferdinand continued to support Gauss financially so that he may continue to discover more. Gauss then found himself attending the University of G¨ottingen in 1795. Here he befriended Farkas Bolyai whom he met in 1799 and continued to correspond with many years later.

Gauss’ next discovery was in an astronomical application. In 1801, some astronomers had discovered what they thought to be a planet and named it Ceres. Later on in the year, they lost track of where it was. However, Gauss had been in communication with them about the matter, and he then managed to pinpoint its exact location, therefore rediscovering it. He used this new means of calculations to create a method to determine the orbits of certain new asteroids.

In the year 1818, Gauss applied his practical uses of mathematics to help him conduct a geodesic survey of the Kingdom of Hanover and used previous Danish surveys for further data. For the purpose of helping him with this survey, Gauss invented the heliotrope which was an instrument that used a mirror to reflect sunlight over vast distances to measure position. Such a project piked Gauss’ interest in topology and diffrential geometry which are fields of mathematics that deal with curved surfaces. This led to the establishment of Gauss’ Theorema Egregium (which is Latin for The Remarkable Theorem) in 1828 which was an incredibly important document that dealt with the notion of curvature. The theorem states that the curvature of an object can be determined by measuring only angles and distances on the surface, and therefore curvature is independent of whether or not the surface is embedded in a 2-dimensional or 3-dimensional space. This coined the phrase of a Gaussian curvature or a Gauss curvature of a surface in differential geometry.

These such discoveries lead Gauss to his appointment as a professor of mathematics and the director of the observatory at the Gottingen University which is where he worked in this official position until his death on the 23rd of February 1855.

Carl Friedrich Gauss’ mathematical and other works

Using Gauss’ Theorema Egregium (translates from Latin into The Remarkable Theorem), the curvature of a surface such as Gaussian curvature seen in differential geometry can be calculated using K = k1 ∗ k2 where k1 and k2 are the principal curvatures. The following equation gives us the total curvature of a surface in an integral form:

3 X ZZ θi = π + KdA (1) i=1 T where the total curvature of a geodesic triangle is given by the deviation of the sum of its angles from π. The sum of the angles of a geodesic triangle’s curvature that is positive will be greater than π, while the sum of the angles of a geodesic triangle’s curvature that is negative will be less than π. This implies that for a surface with zero curvature, such as the Euclidian plane, the sum of angles will add up to exactly π radians, which is indeed the case.

Friedrich Gauss was not only a renown mathematician, but he also made some very useful and important contributions to the field of physics. One of the most well known laws discovered by Guass would be his equation of Gauss’s Law for electric fields, an equation that contributes greatly to the Maxwell’s equations of electricity and mangetism. These diffrentiable equations lay the foundation towards many electrodynamic and classical circuit problems.

The following is Gauss’ Law as expressed with the electric field E. Q ΦE = (2) εo

Where ΦE is the electric flux through an enclosed surface, Q is the total enclosed charge by the body, and εo is the electric constant.

This equation can also be written in its integral form as follows: I ΦE = EdA (3)

Where ΦE once again represents the electric flux through an enclosed surface, E is the electric field, and dA is the infinitesimal area that is thought of as a planar.

Besides this, Gauss also found the Fundemental Theorem of Algebra which was the proof for how every algebraic equation has at least one root or solution. This theorem shook the basis of algebra and impacted

2 mathematics greatly for centuries afterwards. Other mathematicians had also attempted to prove the same theory such as Jean le Rond d’Alembert who had produced a false proof before him. In Gauss’ dissertation he critiqued the mathematician’s work although his own proof was questionable nowadays due to the use of the Jordan curve method. However, by 1849 he had rigorously stayed at improving the proof and managed to produce three more, and in the process substantially cleared up the use of complex numbers in the proof.

In his book Disquisitiones Arithmeticae (Latin for Arithmetical Investigations), he also created the sign for congruency which he used to present clear modular arithmetics as well as contributed greatly towards number theory. Gauss’s symbol for congruency can be seen as =∼

Despite the seemingly random occurence of prime numbers, Gauss attempted to find some sort of relation or pattern to their order. He approached the problem from a different angle such that he graphed the incedence of primes as the numbers increased. He noticed something that resembled a trend which was that as the primes increased by ten, the the probability of a prime number appearing reduced approximately by a factor of two. Although this was progress, Gauss was aware that it was not significant enough proof as it only managed to give an approximation of whether a prime number would appear or not. Hence he kept this discovery secret until much later. Gauss was a person that only published the work he was confident in and the work he thought would be both important and have an impact on the world. Therefore, whenever he wrote proofs he was not satisfied with or had any ideas that he had difficulty expanding, he never published them and kept them secret in his own journals and diaries which were discovered often after another mathematician had proved a certain relationship that Gauss had theorized about before as is supposedly the case here.

In 1840, Guass published his paper titled Diotrische Untersuchungen which is German for Dioptric Investi- gations that explored the idea of Gaussian optics. The paper expanded upon this technique that would be applicable to geometrical optics that describe the behaviour of light rays in optical systems using Gauss’ parax- ial approximation. This approximation only took into consideration the rays that would make small angles with the optical axis of the system and trignometric functions can be represented as linear functions of the angles. Gaussian optics apply to any systems in which all the optical surfaces are completely flat or belong to portions of spheres. Gauss showed that an optical system can be characterized by a series of cardinal points which helps one calculate its optical properties.

Collaboration with other scholars

Carl F. Gauss rarely collaborated with any scholars, if at all, and was considered incredibly aloof by many. It was because of this that most scholars would not approach him to work with, especially when you take into account how he claimed to have explored the existence of non-Euclidean geometry in the 1820s, but never published any of his work on it.

However, in 1831 Guass collaborated with Wilhelm Weber who was a physics professor at the time to discover more information about magnetism including finding a means of representing the unit of magnetism with respect to mass, charge, and time and the discovery of Kirchoff’s circuit laws in electricity. It was after these discoveries that Gauss also established his namesake law, the Gauss’ flux theorem or better known as Gauss’ law (which is further elaborated upon above in Gauss’ Mathematical Work).

Kirchoff’s Circuit Laws: The first of Kirchoff’s laws which is also known as Kirchoff’s first law, Kirchoff’s current law, and Kirchoff’s nodal law states that for every junction in a circuit, the sum of the currents that enter equal the sum of the currents that exit the junction for a circuit or alternatively that the total algebric sum of the currents at a junction equal zero. This can be written as the following:

3 N X Ik = 0 (4) k=1

The second of Kirchoff’s laws which is also known as Kirchoff’s second law, Kirchoff’s voltage law, and Kirchoff’s loof (or mesh) rule states that for any closed circuit loop the sum of the electric motive forces(emfs) equals the potential drops across that loop or alternatively that the total algebric sum of the voltages for a closed circuit equals zero. This can be written as the following:

N |sumk=1Vk = 0 (5)

The two collabortaed to invent the first electromechanical telegraph in 1833 which was then used to connect the observatory with the institute of physics at G¨ottingen.Weber and Gauss also ordered a magnetic observatory to be built in the garden of the observatory and then established the ”Magnetischer Verein” which is German for the Magnetic Club which had the main objective of measuring the Earth’s magnetic field in different regions across the globe. Guass then created a method for measuring the horizontal intensity of the magnetic field as well as developed a mathematical theory to distinguish between the Earth’s inner and outer mangnetospheric sources.

Historical events that marked Carl Friedrich Gauss’ life.

Gauss was living in Germany during the time when Russia and Austria had declared war against Napolean Bonaparte in 1805, which lead Napoleon to find allies within Germany that wanted to see Austria’s power reduced. Although Prussia remained neutral on this dispute, other territories such as Bavaria and two others belonging to southwest Germany decided to come to France’s aid. It was this region that encountered first signs of the war. Following the Danube river, Napoleon cuts off the Austrians and their approaching Russian support. Then, in October 1805, he surrounds the Austrians at Ulm and takes them in as hostages, all the while maintaining minimal losses.

Meanwhile, the French arrive at Vienna and are able to enter the city with no opposition. They quickly moved on to pursue a combined army of Russian and Austrains to Moravia, a large region in the Czech Republic. They finally caught up to the armies on 2nd December at Austerlitz which is where the famous Battle of Austerlitz took place. This was a decisive battle in the Napoleonic Wars. Under the command of the General Kutuzov, the joint army did outnumber the French about 90,000 to 68,000 men, yet the war was still won by the French. This victory secures Napoleon’s authority and dismisses any immediate danger from the Third Coalition. Defeated, the Russians set on their journey to trudge on back home after agreeing to the terms of a truce. The Austrian Emporer, Francis I, also signs a peace treaty with Napoleon on the 26th of December at Pressburg. The terms of this treaty cause the emporer to hand over the entire Northern coast of the Adriatics, which includes regions of Venetia (which is Venice and some surrounding territories), Dalmatia, and Istria. Francis I is also forced to recognize a change in rank with the German Allies of Napoleon which leads the rulers of Bavaria, Baden, and Wrttemberg to recieve an increase in status. A while later, in July 1806, Napoleon merges the grand duchy, and the two kingdoms, along with other smaller cities, to form the Confederation of Rhine which became a vassal state under the protection of France.

Significant historical events around the world during Gauss’ life

Gauss lived through the beginning half of the 19th century and hence was alive during a very eventful time

4 in history. It had been during Gauss’ lifetime that the French Revolution had raged on from 1789 to 1799 which had placed the world into disarray. Germany, Gauss’ homeland, was greatly affected by such brutal and violent battles that occured in France due to this.

In the Brunswick Manifesto, the Imperial and Prussian armies threatened retaliation on the French population if it were to resist their advance or the reinstatement of the monarchy. This among other things made King Louis XVI appear to be conspiring with the enemies of France. On 17 January 1793 Louis was condemned to death for ”conspiracy against the public liberty and the general safety” by a close majority in Convention: 361 voted to execute the king, 288 voted against, and another 72 voted to execute him subject to a variety of delaying conditions. The former Louis XVI, now simply named Citoyen Louis Capet (Citizen Louis Capet) was executed by guillotine on 21 January 1793 on the Place de la Rvolution, former Place Louis XV, now called the Place de la Concorde. Conservatives across Europe were horrified and monarchies called for war against revolutionary France.

When war went badly, prices rose and the sans-culottes poor labourers and radical Jacobins rioted; counter- revolutionary activities began in some regions. This encouraged the Jacobins to seize power through a parlia- mentary coup, backed up by force effected by mobilising public support against the Girondist faction, and by utilising the mob power of the Parisian sans-culottes. An alliance of Jacobin and sans-culottes elements thus became the effective centre of the new government. Policy became considerably more radical, as ”The Law of the Maximum” set food prices and led to executions of offenders.

This policy of price control was coeval with the Committee of Public Safety’s rise to power and the Reign of Terror. The Committee first attempted to set the price for only a limited number of grain products but, by September 1793, it expanded the ”maximum” to cover all foodstuffs and a long list of other goods. Widespread shortages and famine ensued. The Committee reacted by sending dragoons into the countryside to arrest farmers and seize crops. This temporarily solved the problem in Paris, but the rest of the country suffered. By the spring of 1794, forced collection of food was not sufficient to feed even Paris and the days of the Committee were numbered. When Robespierre went to the guillotine in July of that year the crowd jeered, ”There goes the dirty maximum!”

Significant mathematical progress during Carl F. Gauss’ lifetime

Despite having much of Europe in a state of turmoil due to events such as the French Revolution, both Germany and France still managed to produce mathematical advances during the 19th century. Such as Joseph Flourier’s progress of understanding infinite sums that consisted of terms that were trignometric functions. These periodic sums that can be expressed as an infinite series of sine and cosine are nowadays known as Flourier series. Flourier followed in the steps of Leibniz, Euler, and La Grange to name a few by contributing his own definition of what is exactly meant by a function, although many of the modern day definitions found in textbooks are greatly attributed to the 19th century mathematician Peter Dirichlet.

In 1806, the French Jean-Robert Argand√ published his paper expanding on how he had found that complex numbers of the form a + bi, where i = −1 could be represented on geometric diagrams and manipulated through the use of trignometric functions and vectors. Even though Dane Caspar Wessel had published a similar paper and Gauss had popularized the concept, the diagrams to this day are referred to as Argand Diagrams.

The Frenchman variste Galois proved in the late 1820s that there is no general algebraic method for solving polynomial equations of any degree greater than four, which showed further progress than the Norwegian Niels Henrik Abel who had shown the impossibility of solving quintic equations just a few years earlier and breaching an impasse which had existed for centuries. Galois’ work also laid the groundwork for further developments

5 such as the beginnings of the field of abstract algebra, including areas like algebraic group theory, geometry, rings, fields, modules, vector spaces and non-commutative algebra.

The German Bernhard Riemann worked on a different kind of non-Euclidean geometry than Gauss called elliptic geometry, as well as on a generalized theory of all the different types of geometry. He had been taught mathematics under Gauss but Riemann soon took this even further, breaking away completely from all the limitations of 2 and 3 dimensional geometry, whether flat or curved, and began to think in higher dimensions. His exploration of the zeta function in multi-dimensional complex numbers revealed an unexpected link with the distribution of prime numbers, and his famous Riemann Hypothesis, still unproven after 150 years, remains one of the worlds great unsolved mathematical mysteries.

British mathematics also accomplished much in the early and mid-19th century. For example, the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th Century, but it was Charles Babbage in 19th Century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. His large machine of 1823 was able to calculate logarithms and trigonometric functions, and was the true forerunner of the modern electronic computer.

In the mid-19th century, a British mathematician George Boole crafted the idea of an algebra that worked off of the concept that the only operators were AND, OR, and NOT. This is now called Boolean algebra, or Boolean logic, in which there are only two objects, ”on” or ”off”, 1 or 0, ”true” or ”false”, etc. It is in this that famously 1 + 1 = 1 and Boolean algebra has been used as the foundation of what is now known as computer science.

Connections between history and the development of mathematics

Due to the increase in global trade and overseas economical interations, countries were now associating with each other which lead to encouraging mathematicians from different regions to also collaborate and share ideas as well as discuss their theories and already published papers.

Remarks

Carl Friedrich Gauss is nowadays known as a powerhouse of knowledge that expanded the fields of mathe- matics, physics, and astronomy to name but a few and his genius has helped pave the way for many other brilliant scientists, mathematicians, and other prodigious minds to further advance mankind’s knowledge and understanding of the world around us and its workings.

References

1. http://www.math.uic.edu/\someone/notes.pdf

2. Bak, Newman: Complex Analysis. Springer 1989

3. http://www.math.wichita.edu/history/men/gauss.html

4. http://www.famousscientists.org/carl-friedrich-gauss/

5. http://www.storyofmathematics.com/19th gauss.html

6. http://www.historyworld.net/wrldhis/PlainTextHistories.asp?groupid=2803

6 7. http://www.thefamouspeople.com/profiles/carl-f-gauss-442.php

8. http://www.storyofmathematics.com/19th.html

9. Doyle 2002, p. 196

10. White, E. ”The French Revolution and the Politics of Government Finance, 17701815.” The Journal of Economic History 1995, p 244

11. http://www.history.com/topics/french-revolution

12. http://www.britannica.com/event/French-Revolution

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