What Is Carl Friedrich Gauss Famous For? The Prince of Mathematics Explained

Carl Friedrich Gauss (1777–1855) is famous for an astonishing range of discoveries: he proved that a regular 17-sided polygon can be drawn with compass and ruler, wrote the book that founded modern number theory, predicted where the lost dwarf planet Ceres would reappear, developed the method of least squares that underlies modern data analysis, and, with Wilhelm Weber, built the first electromagnetic telegraph. The bell-shaped "normal distribution" in statistics is often called the Gaussian distribution after him. Soon after his death, a medal called him "the Prince of Mathematicians", a title that has stuck.

A bricklayer's son in Brunswick

Gauss was born on 30 April 1777 in Brunswick, in what is now Germany. His family was poor and of low social standing. His father worked at various times as a butcher, bricklayer and gardener, and his mother could barely read.

His talent was noticed early by his schoolteachers, who brought him to the attention of the Duke of Brunswick. The Duke paid for him to attend the Collegium Carolinum from 1792 to 1795 and then the University of Göttingen until 1798. Without that patronage, one of the greatest minds in history might have spent his life laying bricks.

The famous sum: 1 + 2 + 3 + ... + 100

You may have heard this story. A teacher told the class to add up all the numbers from 1 to 100, expecting it to keep them busy for an hour. Young Gauss wrote the answer almost immediately: 5,050.

It is best treated as a popular anecdote rather than a documented event. But the trick is real and worth knowing. Pair the first and last numbers: 1 plus 100 is 101. Then 2 plus 99 is 101, 3 plus 98 is 101, and so on. There are fifty such pairs, so the total is 50 times 101, or 5,050. The lesson is the heart of Gauss's genius: look for a pattern instead of grinding through the work.

A 17-sided shape that chose his career

Since ancient Greece, mathematicians had known how to draw regular triangles, squares, pentagons and some other shapes using only a compass and an unmarked ruler. Then progress stopped for two thousand years.

In 1796, aged just 19, Gauss showed that a regular heptadecagon, a polygon with 17 equal sides, can also be constructed this way. It was the first progress in constructing regular polygons in over 2,000 years. The reason it works is that 17 belongs to a rare family of numbers called Fermat primes; the only known ones are 3, 5, 17, 257 and 65,537. Gauss had been torn between languages and mathematics, and this result helped him decide to choose mathematics. Lovers of philology might regret it; the rest of us should not.

The book that founded number theory

Gauss earned his doctorate in 1799 from the University of Helmstedt with a thesis proving the fundamental theorem of algebra, which says, roughly, that every polynomial equation has a solution if we allow complex numbers.

Two years later, in 1801, he published Disquisitiones Arithmeticae ("Arithmetical Investigations"), written when he was about 21. It gathered scattered results about whole numbers, added many new ones, and turned number theory into an organised discipline. Much of today's cryptography, which keeps your bank card and messages safe, rests on the branch of mathematics this book shaped.

Finding a lost world: Ceres

On 1 January 1801 the Italian astronomer Giuseppe Piazzi discovered a small object orbiting the Sun and named it Ceres. He tracked it for a short time before it vanished into the Sun's glare. With so few observations, nobody knew where to look for it again.

Gauss took up the challenge. Using new mathematical methods, he calculated its orbit and predicted where it would reappear. His prediction was accurate to within about half a degree, and in December 1801 the astronomer Franz Xaver von Zach found Ceres again, with Heinrich Olbers confirming it independently. The young mathematician became famous across Europe overnight.

Least squares: making sense of messy data

Behind the Ceres success was an idea that is everywhere today. Measurements always contain small errors. How do you find the "best" answer from many imperfect observations? Gauss's method of least squares chooses the answer that makes the total of the squared errors as small as possible. He discovered it before Adrien-Marie Legendre published it. He also explained his methods for calculating orbits in a major work of 1809.

Every time a phone estimates your location, or a scientist draws a "line of best fit" through scattered points, a descendant of this idea is at work.

Measuring the Earth and its magnetism

In November 1807 Gauss became professor at Göttingen and director of its observatory, a post he kept until his death. From 1820 to 1844 he led the geodetic survey of the Kingdom of Hanover, measuring the land with triangles. For this he invented the heliotrope, an instrument with mirrors and a small telescope that reflects sunlight to distant survey points.

The survey led him to think deeply about curved surfaces. His "remarkable theorem", the Theorema Egregium, says that the curvature of a surface can be found purely from measurements made on the surface itself. An ant that never leaves a ball could, in principle, discover that it lives on a curved world.

He then turned to magnetism. In 1832 he gave the first absolute measurement of the Earth's magnetic field, and in 1833 he and the physicist Wilhelm Weber built the first electromagnetic telegraph.

Few, but ripe

Gauss published less than he discovered. His seal carried the Latin motto Pauca sed matura, "few, but ripe", and he refused to release work he considered unfinished. He is credited with being the first to study non-Euclidean geometry, the geometry in which parallel lines behave differently, but in an 1824 letter he strongly forbade the recipient from using his ideas publicly. When the young János Bolyai published similar ideas, Gauss wrote that they matched thoughts he had had decades earlier.

His private life had real sorrow. His first wife, Johanna, died in 1809, and his second wife, Minna, died in 1831. Gauss died in Göttingen on 23 February 1855, aged 77.

Learn with Gauss: English phrases for problem-solving

Phrase	Meaning	Example
to spot a pattern	to notice something that repeats	Once I spotted the pattern, the verbs were easy.
a shortcut	a quicker way to do something	Pairing numbers is a clever shortcut.
to work something out	to solve or calculate	I worked out the meaning from the context.
quality over quantity	better to do less but do it well	Gauss believed in quality over quantity.
to make sense of	to understand something confusing	Grammar charts help me make sense of tenses.

Try this: Gauss looked for patterns instead of doing everything the long way. Write three English sentences about a pattern you have noticed in the language you are learning, using "spot a pattern", "work out" and "make sense of".