Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History O ...

who proved that several ancient

geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

problems were impossible to solve using only

compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

. In a paper from 1837, Wantzel proved that the problems of #

doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related prob ...

, and #

trisecting the angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...

are impossible to solve if one uses only

. In the same paper he also solved the problem of determining which regular polygons are constructible: # a regular polygon is constructible

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...

the number of its sides is the product of a

power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative ...

and any number of distinct

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 42949672 ...

s (i.e. that the sufficient conditions given by

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

are also necessary) The solution to these problems had been sought for thousands of years, particularly by the ancient Greeks. However, Wantzel's work was neglected by his contemporaries and essentially forgotten. Indeed, it was only 50 years after its publication that Wantzel's article was mentioned either in a journal article or in a textbook. Before that, it seems to have been mentioned only once, by

Julius Petersen Julius Peter Christian Petersen (16 June 1839, Sorø, West Zealand – 5 August 1910, Copenhagen) was a Danish mathematician. His contributions to the field of mathematics led to the birth of graph theory. Biography Petersen's interests ...

, in his doctoral thesis of 1871. It was probably due to an article published about Wantzel by

Florian Cajori Florian Cajori (February 28, 1859 – August 14 or 15, 1930) was a Swiss-American historian of mathematics. Biography Florian Cajori was born in Zillis, Switzerland, as the son of Georg Cajori and Catherine Camenisch. He attended schools first ...

more than 80 years after the publication of Wantzel's article that his name started to be well-known among mathematicians. Wantzel was also the first person who proved, in 1843, that when a cubic polynomial with rational coefficients has three real roots but it is irreducible in (the so-called ''

casus irreducibilis In algebra, ''casus irreducibilis'' (Latin for "the irreducible case") is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed to numerically), i.e., by obtaining roots th ...

''), then the roots cannot be expressed from the coefficients using real radicals alone, that is, complex non-real numbers must be involved if one expresses the roots from the coefficients using radicals. This theorem would be rediscovered decades later by (and sometimes attributed to)

Vincenzo Mollame Vincenzo Mollame (Naples, 4 July 1848 – Catania, 23 June 1912) was an Italian mathematician. Mollame was privately tutored by Achille Sanni and then studied Mathematics at the University of Naples Federico II. After obtaining his degree, he beca ...

and

Otto Hölder Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart. Early life and education Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Christ ...

References

External links

Profile from School of Mathematics and Statistics; University of St Andrews, Scotland
1814 births 1848 deaths 19th-century French mathematicians French geometers {{France-mathematician-stub