Gauss represented a clear expression of a great mathematician of a small town called Gottingen. He is known in history for his remarkable geometrical discoveries. He is known for his discoveries in method of least squares, quadratic reciprocity, and non-Euclidean geometry. One of his greater works is also seen in astronomy. I totally agree with the works of Gauss on construction of polygons, least squares method, the fundamental theorem of algebra or the non-Euclidean's - differential geometry. Though he never published these discoveries anywhere but his work is highly remarkable.
Gauss started with these discoveries at a very early age. He proved the construction of regular 17 sided polygons called heptadecagon. He proved that this can be constructed simply with the help of a ruler and a compass and thinks this is one of his greatest achievements in the history of geometry. Because as opposed to Kepler, Gauss proved that not only a triangle, square, pentagon, hexagon are constructible but then he proved it right that 17 sided figures can also be constructed with the equal lengths. He further added that 17 gon can be constructed using four quadratic equations (Swetz, 1994).
One more important discovery of Gauss is the theory of least squares and normal distribution. He proved that every curve led to the least squares. He believed that the problems can be simplified by solving the errors evenly distributed. As a result, this gave the accurate estimates by solving the errors incurred in the equation. The construction was possible with trigonometric functions along with arithmetic and square roots. Gaussian distribution curve is a bell shaped curve used for normal distribution. In the Gaussian distribution, all the values combined give the value as 1.
Gauss gave the fundamental theorem of algebra where he proved that any algebraic equation to the degree n, where n is a positive integer will have n number of roots. I totally agree with Gauss in his work on Disquisitiones Arithmeticae where he investigated the number theory within mathematics. Also, he made it possible to draw a circle into equal arch's just with the help of a ruler and a compass. In the number theory, he came up with an idea of congruence in numbers with the help of which infinite series of whole numbers can be broken into smaller chunks of numbers. This can e explained by taking an example: 700 - 400 = 300 right. Here the remainder is 300. This remainder can further be divided into smaller chunks of numbers like 100, 50, and 30 and so on. Here 700 and 400 are congruent to each other by modulo 100. This concept was very much popular among the digital watches. The gauss theory of numbers has its relevance even today and many great mathematicians of today hold this opinion. It plays a crucial role in the Internet world today through security technologies (Struik 1987).
In is theory of geometry, he never agreed to Euclidean's indeed known for his non-Euclidean geometry. He found that parallel postulate fails in the Euclid's geometrical theory that through a point which is not on the line, in this case either there is none or more than one parallel line. The basic difference between the Euclid and Non Euclid's theory on geometry was the nature of parallel lines. Non Euclid theory discovered the geometry of space. The non Euclidean's geometry studied Elliptic geometry