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, mathematical structure, structure, space, Mathematica ...

who proved that several ancient geometric 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 Idealiz ...

. In a paper from 1837, Wantzel proved that the problems of # doubling the cube, and # trisecting the angle are impossible to solve if one uses only a

. 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

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 2, two as the Base (exponentiation), base and integer as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^ ...

and any number of distinct

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have 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, ...

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

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

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, in his doctoral thesis of 1871. It was probably due to an article published about Wantzel by Florian Cajori 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 to prove, in 1843, that if a cubic polynomial with rational coefficients has three real roots but is irreducible in (the so-called '' casus irreducibilis''), 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 and Otto Hölder. Wantzel is often overlooked for his contributions to mathematics. In fact, for over a century there was great confusion as to who proved the impossibility theorems.

References

External links

Profile from School of Mathematics and Statistics; University of St Andrews, Scotland
{{DEFAULTSORT:Wantzel, Pierre 1814 births 1848 deaths 19th-century French mathematicians French geometers